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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..d7b82bc --- /dev/null +++ b/.gitattributes @@ -0,0 +1,4 @@ +*.txt text eol=lf +*.htm text eol=lf +*.html text eol=lf +*.md text eol=lf diff --git a/41654-h/41654-h.htm b/41654-h/41654-h.htm new file mode 100644 index 0000000..72dd61a --- /dev/null +++ b/41654-h/41654-h.htm @@ -0,0 +1,9534 @@ +<!DOCTYPE html> +<html lang="en"> +<head> +<meta charset="UTF-8"> +<title>Introduction to Mathematical Philosophy | Project Gutenberg</title> + +<link href="images/cover.jpg" rel="icon" type="image/x-cover"> + +<style> + +body { + font-family: "Times New Roman", Times, serif; + margin-left: 10%; + margin-right: 10%; +} + +/* General headers */ + +h1 { + text-align: center; + clear: both; +} + +h2 { + text-align: center; + font-weight: bold; + margin-top: 1em; + margin-bottom: 1em; + } + +p { + margin-top: .51em; + text-align: justify; + margin-bottom: .49em; + text-indent:4%; +} + +.nind {text-indent:0%;} + +.hanging2 {padding-left: 2em; + text-indent: -1em; + } + +div.chapter { + page-break-before: always; + margin-top: 4em + } + +.center {text-align: center;text-indent:0%;} + +.footnote {margin-left: 10%; margin-right: 10%; font-size: 0.9em;} + +.footnote .label {position: absolute; right: 84%; text-align: right;} + +.fnanchor { + vertical-align: super; + margin-top: 1em; + font-size: .8em; + text-decoration: + none; +} + +/* Images */ + +img { + max-width: 100%; + width: 100%; + height: auto + } + +.width500 { + max-width: 500px + } + +.x-ebookmaker img { + width: 80% + } + +.x-ebookmaker .width500 { + width: 100% + } + +.figcenter { + margin: 3% auto 3% auto; + clear: both; + text-align: center; + text-indent: 0% +} + +img.floatleft { + float: left; + clear: left; + margin-left: 0; + margin-bottom: 1em; + margin-top: 1em; + margin-right: 1em; + padding: 0; + text-align: center; + page-break-inside: avoid; + max-width: 100%; + } + +.indx { + font-size: 85%; + margin-left: 20%; + margin-right: 20%; + text-align: left; + text-indent: 0; + line-height: 100% + } + +.pagenum { + position: absolute; + left: 92%; + font-size: small; + text-align: right; + font-style: normal; + font-weight: normal; + font-variant: normal; + text-indent: 0; +} + +.transnote {background-color: #E6E6FA; + color: black; + font-size:smaller; + padding:0.5em; + margin-bottom:5em; + font-family:sans-serif, serif; } + +/* css needed in m2svg output: displayed equations and prevention of bad breaks*/ + .align-center { + display: block; + text-align: center; + text-indent: 0; + margin-top: 1em; + margin-bottom: 1em; + } + .nowrap { + white-space: nowrap; + } + + </style> +</head> + +<body> +<div style='text-align:center'>*** START OF THE PROJECT GUTENBERG EBOOK 41654 ***</div> + +<div class="figcenter width500" style="width: 1524px;"> +<img src="images/cover.jpg" width="1524" height="2560" alt="Title page of the book Introduction to Mathematical Philosophy."> +</div> + +<div class='chapter'> +<p class="center">Library of Philosophy</p> + +<p class="center"><i>EDITED BY J. H. MUIRHEAD, LL.D.</i></p> + +<p><br><br><br></p> + +<p class="center"><b>INTRODUCTION TO MATHEMATICAL<br> +PHILOSOPHY</b></p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<p class="center"><i>By the same Author.</i></p> + +<p class="hanging2"> +PRINCIPLES OF SOCIAL RECONSTRUCTION. +<i>3rd Impression.</i> Demy 8vo. 7s. 6d. +net. +</p> +<p class="hanging2"> +"Mr Russell has written a big and living book."—<i>The +Nation</i>. +</p> +<p class="hanging2"> +ROADS TO FREEDOM: SOCIALISM, +ANARCHISM, AND SYNDICALISM. Demy 8vo. +7s. 6d. net. +</p> +<p class="hanging2"> +An attempt to extract the essence of these three doctrines, +first historically, then as guidance for the coming reconstruction. +</p> + +<p><br><br></p> + +<p class="center"><i>London: George Allen & Unwin, Ltd.</i> +<span class="pagenum" id="Page_iii">[Pg iii]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h1>INTRODUCTION TO +MATHEMATICAL PHILOSOPHY</h1> + +<p><br><br></p> + +<p class="center"><b>BY</b></p> + +<div style='text-align:center; font-size:1.2em;'>BERTRAND RUSSELL</div> + +<p><br><br></p> + +<p class="center"><b>LONDON: GEORGE ALLEN & UNWIN, LTD.</b></p> + +<p class="center"><b>NEW YORK: THE MACMILLAN CO.</b> +<span class="pagenum" id="Page_iv">[Pg iv]</span> +</p> +</div> + +<p><br><br></p> + +<p class="center"><i>First published May</i> 1919</p> + +<p class="center"><i>Second Edition April</i> 1920</p> + +<p><br><br></p> + +<p class="center">[All rights reserved] +<span class="pagenum" id="Page_v">[Pg v]</span> +</p> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='PREFACE'><a id="PREFACE">PREFACE</a></h2> + +<p class="nind"> +THIS book is intended essentially as an "Introduction," and +does not aim at giving an exhaustive discussion of the problems +with which it deals. It seemed desirable to set forth certain +results, hitherto only available to those who have mastered +logical symbolism, in a form offering the minimum of difficulty +to the beginner. The utmost endeavour has been made to +avoid dogmatism on such questions as are still open to serious +doubt, and this endeavour has to some extent dominated the +choice of topics considered. The beginnings of mathematical +logic are less definitely known than its later portions, but are of +at least equal philosophical interest. Much of what is set forth +in the following chapters is not properly to be called "philosophy," +though the matters concerned were included in philosophy so +long as no satisfactory science of them existed. The nature of +infinity and continuity, for example, belonged in former days +to philosophy, but belongs now to mathematics. Mathematical +<i>philosophy</i>, in the strict sense, cannot, perhaps, be held to include +such definite scientific results as have been obtained in this +region; the philosophy of mathematics will naturally be expected +to deal with questions on the frontier of knowledge, as +to which comparative certainty is not yet attained. But +speculation on such questions is hardly likely to be fruitful +unless the more scientific parts of the principles of mathematics +are known. A book dealing with those parts may, therefore, +claim to be an <i>introduction</i> to mathematical philosophy, though +it can hardly claim, except where it steps outside its province, +<span class="pagenum" id="Page_vi">[Pg vi]</span> +to be actually dealing with a part of philosophy. It does deal, +however, with a body of knowledge which, to those who accept +it, appears to invalidate much traditional philosophy, and even +a good deal of what is current in the present day. In this way, +as well as by its bearing on still unsolved problems, mathematical +logic is relevant to philosophy. For this reason, as well as on +account of the intrinsic importance of the subject, some purpose +may be served by a succinct account of the main results of +mathematical logic in a form requiring neither a knowledge of +mathematics nor an aptitude for mathematical symbolism. +Here, however, as elsewhere, the method is more important than +the results, from the point of view of further research; and the +method cannot well be explained within the framework of such +a book as the following. It is to be hoped that some readers +may be sufficiently interested to advance to a study of the +method by which mathematical logic can be made helpful in +investigating the traditional problems of philosophy. But that +is a topic with which the following pages have not attempted +to deal. +</p> + +<p style="text-align:right">BERTRAND RUSSELL.</p> +<p><span class="pagenum" id="Page_vii">[Pg vii]</span></p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='EDITORS NOTE'><a id="EDITORS_NOTE">EDITOR'S NOTE</a></h2> + +<p class="nind"> +THOSE who, relying on the distinction between Mathematical +Philosophy and the Philosophy of Mathematics, think that this +book is out of place in the present Library, may be referred to +what the author himself says on this head in the Preface. It is +not necessary to agree with what he there suggests as to the +readjustment of the field of philosophy by the transference from +it to mathematics of such problems as those of class, continuity, +infinity, in order to perceive the bearing of the definitions and +discussions that follow on the work of "traditional philosophy." +If philosophers cannot consent to relegate the criticism of these +categories to any of the special sciences, it is essential, at any +rate, that they should know the precise meaning that the science +of mathematics, in which these concepts play so large a part, +assigns to them. If, on the other hand, there be mathematicians +to whom these definitions and discussions seem to be an elaboration +and complication of the simple, it may be well to remind +them from the side of philosophy that here, as elsewhere, apparent +simplicity may conceal a complexity which it is the business of +somebody, whether philosopher or mathematician, or, like the +author of this volume, both in one, to unravel. +<span class="pagenum" id="Page_viii">[Pg viii]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2>CONTENTS</h2> +<p class="nind"> +CHAP.<br> +<a href="#PREFACE">PREFACE</a><br> + +<a href="#EDITORS_NOTE">EDITOR'S NOTE</a><br> + +1. <a href="#chap01">THE SERIES OF NATURAL NUMBERS</a><br> + +2. <a href="#chap02">DEFINITION OF NUMBER</a><br> + +3. <a href="#chap03">FINITUDE AND MATHEMATICAL INDUCTION</a><br> + +4. <a href="#chap04">THE DEFINITION OF ORDER</a><br> + +5. <a href="#chap05">KINDS OF RELATIONS</a><br> + +6. <a href="#chap06">SIMILARITY OF RELATIONS</a><br> + +7. <a href="#chap07">RATIONAL, REAL, AND COMPLEX NUMBERS</a><br> + +8. <a href="#chap08">INFINITE CARDINAL NUMBERS</a><br> + +9. <a href="#chap09">INFINITE SERIES AND ORDINALS</a><br> + +10. <a href="#chap10">LIMITS AND CONTINUITY</a><br> + +11. <a href="#chap11">LIMITS AND CONTINUITY OF FUNCTIONS</a><br> + +12. <a href="#chap12">SELECTIONS AND THE MULTIPLICATIVE AXIOM</a><br> + +13. <a href="#chap13">THE AXIOM OF INFINITY AND LOGICAL TYPES</a><br> + +14. <a href="#chap14">INCOMPATIBILITY AND THE THEORY OF DEDUCTION</a><br> + +15. <a href="#chap15">PROPOSITIONAL FUNCTIONS</a><br> + +16. <a href="#chap16">DESCRIPTIONS</a><br> + +17. <a href="#chap17">CLASSES</a><br> + +18. <a href="#chap18">MATHEMATICS AND LOGIC</a><br> + +<a href="#INDEX">INDEX</a></p> + +<p><span class="pagenum" id="Page_ix">[Pg ix]</span></p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2>INTRODUCTION TO MATHEMATICAL PHILOSOPHY</h2> +</div> + +<div class='chapter'> +<h2 title='I: THE SERIES OF NATURAL NUMBERS'><a id="chap01"></a>CHAPTER I +<br><br> +THE SERIES OF NATURAL NUMBERS</h2> + +<p class="nind"> +MATHEMATICS is a study which, when we start from its most +familiar portions, may be pursued in either of two opposite +directions. The more familiar direction is constructive, towards +gradually increasing complexity: from integers to fractions, +real numbers, complex numbers; from addition and multiplication +to differentiation and integration, and on to higher +mathematics. The other direction, which is less familiar, +proceeds, by analysing, to greater and greater abstractness +and logical simplicity; instead of asking what can be defined +and deduced from what is assumed to begin with, we ask instead +what more general ideas and principles can be found, in terms +of which what was our starting-point can be defined or deduced. +It is the fact of pursuing this opposite direction that characterises +mathematical philosophy as opposed to ordinary mathematics. +But it should be understood that the distinction is one, not in +the subject matter, but in the state of mind of the investigator. +Early Greek geometers, passing from the empirical rules of +Egyptian land-surveying to the general propositions by which +those rules were found to be justifiable, and thence to Euclid's +axioms and postulates, were engaged in mathematical philosophy, +according to the above definition; but when once the +axioms and postulates had been reached, their deductive employment, +as we find it in Euclid, belonged to mathematics in the +<span class="pagenum" id="Page_1">[Pg 1]</span> +ordinary sense. The distinction between mathematics and +mathematical philosophy is one which depends upon the interest +inspiring the research, and upon the stage which the research +has reached; not upon the propositions with which the research +is concerned. +</p> +<p> +We may state the same distinction in another way. The +most obvious and easy things in mathematics are not those that +come logically at the beginning; they are things that, from +the point of view of logical deduction, come somewhere in the +middle. Just as the easiest bodies to see are those that are +neither very near nor very far, neither very small nor very +great, so the easiest conceptions to grasp are those that are +neither very complex nor very simple (using "simple" in a +<i>logical</i> sense). And as we need two sorts of instruments, the +telescope and the microscope, for the enlargement of our visual +powers, so we need two sorts of instruments for the enlargement +of our logical powers, one to take us forward to the higher +mathematics, the other to take us backward to the logical +foundations of the things that we are inclined to take for granted +in mathematics. We shall find that by analysing our ordinary +mathematical notions we acquire fresh insight, new powers, +and the means of reaching whole new mathematical subjects +by adopting fresh lines of advance after our backward journey. +It is the purpose of this book to explain mathematical philosophy +simply and untechnically, without enlarging upon those +portions which are so doubtful or difficult that an elementary +treatment is scarcely possible. A full treatment will be found +in <i>Principia Mathematica</i>;<a id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a> +the treatment in the present volume is intended merely as an +introduction. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a>Cambridge University Press, vol. I., 1910; vol. II., 1911; vol. III., 1913. +By Whitehead and Russell.</p></div> + +<p> +To the average educated person of the present day, the +obvious starting-point of mathematics would be the series of +whole numbers, +<span class="align-center"><img style="vertical-align: -0.439ex; width: 18.301ex; height: 1.971ex;" src="images/1.svg" alt="" data-tex=" +1,\ 2,\ 3,\ 4,\ \dots\ \text{etc.} +"></span> +<span class="pagenum" id="Page_2">[Pg 2]</span> +Probably only a person with some mathematical knowledge +would think of beginning with 0 instead of with 1, but we will +presume this degree of knowledge; we will take as our starting-point +the series: +<span class="align-center"><img style="vertical-align: -0.439ex; width: 26.624ex; height: 1.946ex;" src="images/2.svg" alt="" data-tex=" +0,\ 1,\ 2,\ 3,\ \dots\ n,\ n + 1,\dots +"></span> +and it is this series that we shall mean when we speak of the +"series of natural numbers." +</p> +<p> +It is only at a high stage of civilisation that we could take +this series as our starting-point. It must have required many +ages to discover that a brace of pheasants and a couple of days +were both instances of the number 2: the degree of abstraction +involved is far from easy. And the discovery that 1 is a number +must have been difficult. As for 0, it is a very recent addition; +the Greeks and Romans had no such digit. If we had been +embarking upon mathematical philosophy in earlier days, we +should have had to start with something less abstract than the +series of natural numbers, which we should reach as a stage on +our backward journey. When the logical foundations of mathematics +have grown more familiar, we shall be able to start further +back, at what is now a late stage in our analysis. But for the +moment the natural numbers seem to represent what is easiest +and most familiar in mathematics. +</p> +<p> +But though familiar, they are not understood. Very few +people are prepared with a definition of what is meant by +"number," or "0," or "1." It is not very difficult to see that, +starting from 0, any other of the natural numbers can be reached +by repeated additions of 1, but we shall have to define what +we mean by "adding 1," and what we mean by "repeated." +These questions are by no means easy. It was believed until +recently that some, at least, of these first notions of arithmetic +must be accepted as too simple and primitive to be defined. +Since all terms that are defined are defined by means of other +terms, it is clear that human knowledge must always be content +to accept some terms as intelligible without definition, in order +<span class="pagenum" id="Page_3">[Pg 3]</span> +to have a starting-point for its definitions. It is not clear that +there must be terms which are <i>incapable</i> of definition: it is +possible that, however far back we go in defining, we always +<i>might</i> go further still. On the other hand, it is also possible +that, when analysis has been pushed far enough, we can reach +terms that really are simple, and therefore logically incapable +of the sort of definition that consists in analysing. This is a +question which it is not necessary for us to decide; for our +purposes it is sufficient to observe that, since human powers +are finite, the definitions known to us must always begin somewhere, +with terms undefined for the moment, though perhaps +not permanently. +</p> +<p> +All traditional pure mathematics, including analytical geometry, +may be regarded as consisting wholly of propositions +about the natural numbers. That is to say, the terms which +occur can be defined by means of the natural numbers, and +the propositions can be deduced from the properties of the +natural numbers—with the addition, in each case, of the ideas +and propositions of pure logic. +</p> +<p> +That all traditional pure mathematics can be derived from +the natural numbers is a fairly recent discovery, though it had +long been suspected. Pythagoras, who believed that not only +mathematics, but everything else could be deduced from +numbers, was the discoverer of the most serious obstacle in +the way of what is called the "arithmetising" of mathematics. +It was Pythagoras who discovered the existence of incommensurables, +and, in particular, the incommensurability of the +side of a square and the diagonal. If the length of the side is +1 inch, the number of inches in the diagonal is the square root +of 2, which appeared not to be a number at all. The problem +thus raised was solved only in our own day, and was only solved +<i>completely</i> by the help of the reduction of arithmetic to logic, +which will be explained in following chapters. For the present, +we shall take for granted the arithmetisation of mathematics, +though this was a feat of the very greatest importance. +<span class="pagenum" id="Page_4">[Pg 4]</span> +</p> +<p> +Having reduced all traditional pure mathematics to the +theory of the natural numbers, the next step in logical analysis +was to reduce this theory itself to the smallest set of premisses +and undefined terms from which it could be derived. This work +was accomplished by Peano. He showed that the entire theory +of the natural numbers could be derived from three primitive +ideas and five primitive propositions in addition to those of +pure logic. These three ideas and five propositions thus became, +as it were, hostages for the whole of traditional pure mathematics. +If they could be defined and proved in terms of others, +so could all pure mathematics. Their logical "weight," if one +may use such an expression, is equal to that of the whole series +of sciences that have been deduced from the theory of the natural +numbers; the truth of this whole series is assured if the truth +of the five primitive propositions is guaranteed, provided, of +course, that there is nothing erroneous in the purely logical +apparatus which is also involved. The work of analysing mathematics +is extraordinarily facilitated by this work of Peano's. +</p> +<p> +The three primitive ideas in Peano's arithmetic are: +<span class="align-center"><img style="vertical-align: -0.439ex; width: 20.663ex; height: 2.009ex;" src="images/3.svg" alt="" data-tex=" +\text{0, number, successor.} +"></span> +By "successor" he means the next number in the natural +order. That is to say, the successor of 0 is 1, the successor of +1 is 2, and so on. By "number" he means, in this connection, +the class of the natural numbers.<a id="FNanchor_2_1"></a><a href="#Footnote_2_1" class="fnanchor">[2]</a> +He is not assuming that +we know all the members of this class, but only that we know +what we mean when we say that this or that is a number, just +as we know what we mean when we say "Jones is a man," +though we do not know all men individually. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_2_1"></a><a href="#FNanchor_2_1"><span class="label">[2]</span></a>We shall use "number" in this sense in the present chapter. Afterwards +the word will be used in a more general sense.</p></div> + +<p> +The five primitive propositions which Peano assumes are: +</p> +<p class="hanging2"> +(1) 0 is a number. +</p> +<p class="hanging2"> +(2) The successor of any number is a number. +</p> +<p class="hanging2"> +(3) No two numbers have the same successor. +<span class="pagenum" id="Page_5">[Pg 5]</span> +</p> +<p class="hanging2"> +(4) 0 is not the successor of any number. +</p> +<p class="hanging2"> +(5) Any property which belongs to 0, and also to the successor +of every number which has the property, belongs to all +numbers. +</p> +<p class="nind"> +The last of these is the principle of mathematical induction. +We shall have much to say concerning mathematical induction +in the sequel; for the present, we are concerned with it only +as it occurs in Peano's analysis of arithmetic. +</p> +<p> +Let us consider briefly the kind of way in which the theory +of the natural numbers results from these three ideas and five +propositions. To begin with, we define 1 as "the successor of 0," +2 as "the successor of 1," and so on. We can obviously go +on as long as we like with these definitions, since, in virtue of (2), +every number that we reach will have a successor, and, in +virtue of (3), this cannot be any of the numbers already defined, +because, if it were, two different numbers would have the same +successor; and in virtue of (4) none of the numbers we reach +in the series of successors can be 0. Thus the series of successors +gives us an endless series of continually new numbers. In virtue +of (5) all numbers come in this series, which begins with 0 and +travels on through successive successors: for (<i>a</i>) 0 belongs to +this series, and (<i>b</i>) if a number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> belongs to it, so does its successor, +whence, by mathematical induction, every number belongs to +the series. +</p> +<p> +Suppose we wish to define the sum of two numbers. Taking +any number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m">,</span> we define <img style="vertical-align: -0.186ex; width: 5.883ex; height: 1.692ex;" src="images/48.svg" alt="" data-tex="m + 0"> as <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m">,</span> and +<img style="vertical-align: -0.566ex; width: 11.767ex; height: 2.262ex;" src="images/49.svg" alt="" data-tex="m + (n + 1)"> as the +successor of <span class="nowrap"><img style="vertical-align: -0.186ex; width: 6.11ex; height: 1.505ex;" src="images/50.svg" alt="" data-tex="m + n">.</span> In virtue of (5) this gives a definition of +the sum of <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> whatever number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> may be. Similarly +we can define the product of any two numbers. The reader can +easily convince himself that any ordinary elementary proposition +of arithmetic can be proved by means of our five premisses, +and if he has any difficulty he can find the proof in Peano. +</p> +<p> +It is time now to turn to the considerations which make it +necessary to advance beyond the standpoint of Peano, who +<span class="pagenum" id="Page_6">[Pg 6]</span> +represents the last perfection of the "arithmetisation" of +mathematics, to that of Frege, who first succeeded in "logicising" +mathematics, <i>i.e.</i> in reducing to logic the arithmetical notions +which his predecessors had shown to be sufficient for mathematics. +We shall not, in this chapter, actually give Frege's definition of +number and of particular numbers, but we shall give some of the +reasons why Peano's treatment is less final than it appears to be. +</p> +<p> +In the first place, Peano's three primitive ideas—namely, "0," +"number," and "successor"—are capable of an infinite number +of different interpretations, all of which will satisfy the five +primitive propositions. We will give some examples. +</p> +<p> +(1) Let "0" be taken to mean 100, and let "number" be +taken to mean the numbers from 100 onward in the series of +natural numbers. Then all our primitive propositions are +satisfied, even the fourth, for, though 100 is the successor of 99, +99 is not a "number" in the sense which we are now giving +to the word "number." It is obvious that any number may be +substituted for 100 in this example. +</p> +<p> +(2) Let "0" have its usual meaning, but let "number" +mean what we usually call "even numbers," and let the +"successor" of a number be what results from adding two to +it. Then "1" will stand for the number two, "2" will stand +for the number four, and so on; the series of "numbers" now +will be +<span class="align-center"><img style="vertical-align: -0.466ex; width: 25.175ex; height: 2.061ex;" src="images/4.svg" alt="" data-tex=" +\text{0, two, four, six, eight} \dots. +"></span> +All Peano's five premisses are satisfied still. +</p> +<p> +(3) Let "0" mean the number one, let "number" mean +the set +<span class="align-center"><img style="vertical-align: -1.602ex; width: 21.656ex; height: 4.638ex;" src="images/5.svg" alt="" data-tex=" +1,\ \dfrac{1}{2},\ \dfrac{1}{4},\ \dfrac{1}{8},\ \dfrac{1}{16},\ \dots +"></span> +and let "successor" mean "half." Then all Peano's five +axioms will be true of this set. +</p> +<p> +It is clear that such examples might be multiplied indefinitely. +In fact, given any series +<span class="align-center"><img style="vertical-align: -0.439ex; width: 26.427ex; height: 1.439ex;" src="images/6.svg" alt="" data-tex=" +x_{0},\ x_{1},\ x_{2},\ x_{3},\ \dots\ x_{n},\ \dots +"></span> +<span class="pagenum" id="Page_7">[Pg 7]</span> +which is endless, contains no repetitions, has a beginning, and +has no terms that cannot be reached from the beginning in a +finite number of steps, we have a set of terms verifying Peano's +axioms. This is easily seen, though the formal proof is somewhat +long. Let "0" mean <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/51.svg" alt="" data-tex="x_{0}">,</span> let "number" mean the whole +set of terms, and let the "successor" of <img style="vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;" src="images/52.svg" alt="" data-tex="x_{n}"> mean <span class="nowrap"><img style="vertical-align: -0.471ex; width: 4.486ex; height: 1.471ex;" src="images/53.svg" alt="" data-tex="x_{n+1}">.</span> Then +</p> +<p> +(1) "0 is a number," <i>i.e.</i> <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/51.svg" alt="" data-tex="x_{0}"> is a member of the set. +</p> +<p> +(2) "The successor of any number is a number," <i>i.e.</i> taking +any term <img style="vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;" src="images/52.svg" alt="" data-tex="x_{n}"> in the set, <img style="vertical-align: -0.471ex; width: 4.486ex; height: 1.471ex;" src="images/53.svg" alt="" data-tex="x_{n+1}"> is also in the set. +</p> +<p> +(3) "No two numbers have the same successor," <i>i.e.</i> if <img style="vertical-align: -0.357ex; width: 2.887ex; height: 1.357ex;" src="images/54.svg" alt="" data-tex="x_{m}"> +and <img style="vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;" src="images/52.svg" alt="" data-tex="x_{n}"> are two different members of the set, <img style="vertical-align: -0.471ex; width: 4.931ex; height: 1.471ex;" src="images/55.svg" alt="" data-tex="x_{m+1}"> and +<img style="vertical-align: -0.471ex; width: 4.486ex; height: 1.471ex;" src="images/53.svg" alt="" data-tex="x_{n+1}"> are +different; this results from the fact that (by hypothesis) there +are no repetitions in the set. +</p> +<p> +(4) "0 is not the successor of any number," <i>i.e.</i> no term in +the set comes before <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/51.svg" alt="" data-tex="x_{0}">.</span> +</p> +<p> +(5) This becomes: Any property which belongs to <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/51.svg" alt="" data-tex="x_{0}">,</span> and +belongs to <img style="vertical-align: -0.471ex; width: 4.486ex; height: 1.471ex;" src="images/53.svg" alt="" data-tex="x_{n+1}"> provided it belongs to <span class="nowrap"><img style="vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;" src="images/52.svg" alt="" data-tex="x_{n}">,</span> belongs +to all the <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">'</span>s. +</p> +<p> +This follows from the corresponding property for numbers. +</p> +<p> +A series of the form +<span class="align-center"><img style="vertical-align: -0.439ex; width: 22.574ex; height: 1.439ex;" src="images/7.svg" alt="" data-tex=" +x_{0},\ x_{1},\ x_{2},\ \dots\ x_{n},\ \dots +"></span> +in which there is a first term, a successor to each term (so that +there is no last term), no repetitions, and every term can be +reached from the start in a finite number of steps, is called a +<i>progression</i>. Progressions are of great importance in the principles +of mathematics. As we have just seen, every progression +verifies Peano's five axioms. It can be proved, conversely, +that every series which verifies Peano's five axioms is a progression. +Hence these five axioms may be used to define the +class of progressions: "progressions" are "those series which +verify these five axioms." Any progression may be taken as +the basis of pure mathematics: we may give the name "0" +to its first term, the name "number" to the whole set of its +terms, and the name "successor" to the next in the progression. +The progression need not be composed of numbers: it may be +<span class="pagenum" id="Page_8">[Pg 8]</span> +composed of points in space, or moments of time, or any other +terms of which there is an infinite supply. Each different +progression will give rise to a different interpretation of all the +propositions of traditional pure mathematics; all these possible +interpretations will be equally true. +</p> +<p> +In Peano's system there is nothing to enable us to distinguish +between these different interpretations of his primitive ideas. +It is assumed that we know what is meant by "0," and that +we shall not suppose that this symbol means 100 or Cleopatra's +Needle or any of the other things that it might mean. +</p> +<p> +This point, that "0" and "number" and "successor" +cannot be defined by means of Peano's five axioms, but must +be independently understood, is important. We want our +numbers not merely to verify mathematical formulæ, but to +apply in the right way to common objects. We want to have +ten fingers and two eyes and one nose. A system in which "1" +meant 100, and "2" meant 101, and so on, might be all right +for pure mathematics, but would not suit daily life. We want +"0" and "number" and "successor" to have meanings which +will give us the right allowance of fingers and eyes and noses. +We have already some knowledge (though not sufficiently +articulate or analytic) of what we mean by "1" and "2" and +so on, and our use of numbers in arithmetic must conform to +this knowledge. We cannot secure that this shall be the case +by Peano's method; all that we can do, if we adopt his method, +is to say "we know what we mean by '0' and 'number' and +'successor,' though we cannot explain what we mean in terms +of other simpler concepts." It is quite legitimate to say this +when we must, and at <i>some</i> point we all must; but it is the +object of mathematical philosophy to put off saying it as long +as possible. By the logical theory of arithmetic we are able to +put it off for a very long time. +</p> +<p> +It might be suggested that, instead of setting up "0" and +"number" and "successor" as terms of which we know the +meaning although we cannot define them, we might let them +<span class="pagenum" id="Page_9">[Pg 9]</span> +stand for <i>any</i> three terms that verify Peano's five axioms. They +will then no longer be terms which have a meaning that is definite +though undefined: they will be "variables," terms concerning +which we make certain hypotheses, namely, those stated in the +five axioms, but which are otherwise undetermined. If we adopt +this plan, our theorems will not be proved concerning an ascertained +set of terms called "the natural numbers," but concerning +all sets of terms having certain properties. Such a procedure +is not fallacious; indeed for certain purposes it represents a +valuable generalisation. But from two points of view it fails +to give an adequate basis for arithmetic. In the first place, it +does not enable us to know whether there are any sets of terms +verifying Peano's axioms; it does not even give the faintest +suggestion of any way of discovering whether there are such sets. +In the second place, as already observed, we want our numbers +to be such as can be used for counting common objects, and this +requires that our numbers should have a <i>definite</i> meaning, not +merely that they should have certain formal properties. This +definite meaning is defined by the logical theory of arithmetic. +<span class="pagenum" id="Page_10">[Pg 10]</span> +</p></div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='II: DEFINITION OF NUMBER'><a id="chap02"></a>CHAPTER II +<br><br> +DEFINITION OF NUMBER</h2> + +<p class="nind"> +THE question "What is a number?" is one which has been +often asked, but has only been correctly answered in our own +time. The answer was given by Frege in 1884, in his <i>Grundlagen +der Arithmetik</i>.<a id="FNanchor_3_1"></a><a href="#Footnote_3_1" class="fnanchor">[3]</a> +Although this book is quite short, not difficult, +and of the very highest importance, it attracted almost no +attention, and the definition of number which it contains remained +practically unknown until it was rediscovered by the +present author in 1901. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_3_1"></a><a href="#FNanchor_3_1"><span class="label">[3]</span></a>The same answer is given more fully and with more development in +his <i>Grundgesetze der Arithmetik</i>, vol. I., 1893.</p></div> + +<p> +In seeking a definition of number, the first thing to be clear +about is what we may call the grammar of our inquiry. Many +philosophers, when attempting to define number, are really +setting to work to define plurality, which is quite a different +thing. <i>Number</i> is what is characteristic of numbers, as <i>man</i> +is what is characteristic of men. A plurality is not an instance +of number, but of some particular number. A trio of men, +for example, is an instance of the number 3, and the number 3 +is an instance of number; but the trio is not an instance of +number. This point may seem elementary and scarcely worth +mentioning; yet it has proved too subtle for the philosophers, +with few exceptions. +</p> +<p> +A particular number is not identical with any collection of +terms having that number: the number 3 is not identical with +<span class="pagenum" id="Page_11">[Pg 11]</span> +the trio consisting of Brown, Jones, and Robinson. The number 3 +is something which all trios have in common, and which distinguishes +them from other collections. A number is something +that characterises certain collections, namely, those that have +that number. +</p> +<p> +Instead of speaking of a "collection," we shall as a rule speak +of a "class," or sometimes a "set." Other words used in +mathematics for the same thing are "aggregate" and "manifold." +We shall have much to say later on about classes. For +the present, we will say as little as possible. But there are +some remarks that must be made immediately. +</p> +<p> +A class or collection may be defined in two ways that at first +sight seem quite distinct. We may enumerate its members, as +when we say, "The collection I mean is Brown, Jones, and +Robinson." Or we may mention a defining property, as when +we speak of "mankind" or "the inhabitants of London." The +definition which enumerates is called a definition by "extension," +and the one which mentions a defining property is called +a definition by "intension." Of these two kinds of definition, +the one by intension is logically more fundamental. This is +shown by two considerations: (1) that the extensional definition +can always be reduced to an intensional one; (2) that the +intensional one often cannot even theoretically be reduced to +the extensional one. Each of these points needs a word of +explanation. +</p> +<p> +(1) Brown, Jones, and Robinson all of them possess a certain +property which is possessed by nothing else in the whole universe, +namely, the property of being either Brown or Jones or Robinson. +This property can be used to give a definition by intension of +the class consisting of Brown and Jones and Robinson. Consider +such a formula as "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is Brown or <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is Jones or <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is Robinson." +This formula will be true for just three <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">'</span>s, namely, Brown and +Jones and Robinson. In this respect it resembles a cubic equation +with its three roots. It may be taken as assigning a property +common to the members of the class consisting of these three +<span class="pagenum" id="Page_12">[Pg 12]</span> +men, and peculiar to them. A similar treatment can obviously +be applied to any other class given in extension. +</p> +<p> +(2) It is obvious that in practice we can often know a great +deal about a class without being able to enumerate its members. +No one man could actually enumerate all men, or even all the +inhabitants of London, yet a great deal is known about each of +these classes. This is enough to show that definition by extension +is not <i>necessary</i> to knowledge about a class. But when we come +to consider infinite classes, we find that enumeration is not even +theoretically possible for beings who only live for a finite time. +We cannot enumerate all the natural numbers: they are 0, 1, 2, +3, <i>and so on</i>. At some point we must content ourselves with +"and so on." We cannot enumerate all fractions or all irrational +numbers, or all of any other infinite collection. Thus our knowledge +in regard to all such collections can only be derived from a +definition by intension. +</p> +<p> +These remarks are relevant, when we are seeking the definition +of number, in three different ways. In the first place, numbers +themselves form an infinite collection, and cannot therefore +be defined by enumeration. In the second place, the collections +having a given number of terms themselves presumably form an +infinite collection: it is to be presumed, for example, that there +are an infinite collection of trios in the world, for if this were +not the case the total number of things in the world would be +finite, which, though possible, seems unlikely. In the third +place, we wish to define "number" in such a way that infinite +numbers may be possible; thus we must be able to speak of +the number of terms in an infinite collection, and such a collection +must be defined by intension, <i>i.e.</i> by a property common to all +its members and peculiar to them. +</p> +<p> +For many purposes, a class and a defining characteristic of +it are practically interchangeable. The vital difference between +the two consists in the fact that there is only one class having a +given set of members, whereas there are always many different +characteristics by which a given class may be defined. Men +<span class="pagenum" id="Page_13">[Pg 13]</span> +may be defined as featherless bipeds, or as rational animals, +or (more correctly) by the traits by which Swift delineates the +Yahoos. It is this fact that a defining characteristic is never +unique which makes classes useful; otherwise we could be +content with the properties common and peculiar to their +members.<a id="FNanchor_4_1"></a><a href="#Footnote_4_1" class="fnanchor">[4]</a> +Any one of these properties can be used in place +of the class whenever uniqueness is not important. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_4_1"></a><a href="#FNanchor_4_1"><span class="label">[4]</span></a>As will be explained later, classes may be regarded as logical fictions, +manufactured out of defining characteristics. But for the present it will +simplify our exposition to treat classes as if they were real.</p></div> + +<p> +Returning now to the definition of number, it is clear that +number is a way of bringing together certain collections, namely, +those that have a given number of terms. We can suppose +all couples in one bundle, all trios in another, and so on. In +this way we obtain various bundles of collections, each bundle +consisting of all the collections that have a certain number of +terms. Each bundle is a class whose members are collections, +<i>i.e.</i> classes; thus each is a class of classes. The bundle consisting +of all couples, for example, is a class of classes: each +couple is a class with two members, and the whole bundle of +couples is a class with an infinite number of members, each of +which is a class of two members. +</p> +<p> +How shall we decide whether two collections are to belong +to the same bundle? The answer that suggests itself is: "Find +out how many members each has, and put them in the same +bundle if they have the same number of members." But this +presupposes that we have defined numbers, and that we know +how to discover how many terms a collection has. We are so +used to the operation of counting that such a presupposition +might easily pass unnoticed. In fact, however, counting, +though familiar, is logically a very complex operation; moreover +it is only available, as a means of discovering how many +terms a collection has, when the collection is finite. Our definition +of number must not assume in advance that all numbers +are finite; and we cannot in any case, without a vicious circle, +<span class="pagenum" id="Page_14">[Pg 14]</span> +use counting to define numbers, because numbers are used in +counting. We need, therefore, some other method of deciding +when two collections have the same number of terms. +</p> +<p> +In actual fact, it is simpler logically to find out whether two +collections have the same number of terms than it is to define +what that number is. An illustration will make this clear. +If there were no polygamy or polyandry anywhere in the world, +it is clear that the number of husbands living at any moment +would be exactly the same as the number of wives. We do +not need a census to assure us of this, nor do we need to know +what is the actual number of husbands and of wives. We know +the number must be the same in both collections, because each +husband has one wife and each wife has one husband. The +relation of husband and wife is what is called "one-one." +</p> +<p> +A relation is said to be "one-one" when, if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation +in question to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> no other term <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">'</span> has the same relation to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> +and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> does not have the same relation to any term <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">'</span> other +than <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">.</span> When only the first of these two conditions is fulfilled, +the relation is called "one-many"; when only the second is +fulfilled, it is called "many-one." It should be observed that +the number 1 is not used in these definitions. +</p> +<p> +In Christian countries, the relation of husband to wife is +one-one; in Mahometan countries it is one-many; in Tibet +it is many-one. The relation of father to son is one-many; +that of son to father is many-one, but that of eldest son to father +is one-one. If <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is any number, the relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1"> is +one-one; so is the relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <img style="vertical-align: -0.025ex; width: 2.489ex; height: 1.532ex;" src="images/59.svg" alt="" data-tex="2n"> or to <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.489ex; height: 1.554ex;" src="images/60.svg" alt="" data-tex="3n">.</span> When we are +considering only positive numbers, the relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <img style="vertical-align: -0.025ex; width: 2.345ex; height: 1.912ex;" src="images/61.svg" alt="" data-tex="n^{2}"> is +one-one; but when negative numbers are admitted, it becomes +two-one, since <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> and <img style="vertical-align: -0.186ex; width: 3.118ex; height: 1.505ex;" src="images/62.svg" alt="" data-tex="-n"> have the same square. These instances +should suffice to make clear the notions of one-one, one-many, +and many-one relations, which play a great part in the principles +of mathematics, not only in relation to the definition of +numbers, but in many other connections. +</p> +<p> +Two classes are said to be "similar" when there is a one-one +<span class="pagenum" id="Page_15">[Pg 15]</span> +relation which correlates the terms of the one class each with +one term of the other class, in the same manner in which the +relation of marriage correlates husbands with wives. A few +preliminary definitions will help us to state this definition more +precisely. The class of those terms that have a given relation +to something or other is called the <i>domain</i> of that relation: +thus fathers are the domain of the relation of father to child, +husbands are the domain of the relation of husband to wife, +wives are the domain of the relation of wife to husband, and +husbands and wives together are the domain of the relation of +marriage. The relation of wife to husband is called the <i>converse</i> +of the relation of husband to wife. Similarly <i>less</i> is the converse +of <i>greater</i>, <i>later</i> is the converse of <i>earlier</i>, and so on. Generally, +the converse of a given relation is that relation which holds +between <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> whenever the given relation holds between +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">.</span> The <i>converse domain</i> of a relation is the domain of +its converse: thus the class of wives is the converse domain +of the relation of husband to wife. We may now state our +definition of similarity as follows:— +</p> +<p> +<i>One class is said to be "similar" to another when there is a +one-one relation of which the one class is the domain, while the +other is the converse domain.</i> +</p> +<p> +It is easy to prove (1) that every class is similar to itself, (2) that +if a class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is similar to a class <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">,</span> then <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> is +similar to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> (3) that +if <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is similar to <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> to <span class="nowrap"><img style="vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;" src="images/65.svg" alt="" data-tex="\gamma">,</span> +then <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is similar to <span class="nowrap"><img style="vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;" src="images/65.svg" alt="" data-tex="\gamma">.</span> A +relation is said to be <i>reflexive</i> when it possesses the first of these +properties, <i>symmetrical</i> when it possesses the second, and <i>transitive</i> +when it possesses the third. It is obvious that a relation +which is symmetrical and transitive must be reflexive throughout +its domain. Relations which possess these properties are an +important kind, and it is worth while to note that similarity is +one of this kind of relations. +</p> +<p> +It is obvious to common sense that two finite classes have +the same number of terms if they are similar, but not otherwise. +The act of counting consists in establishing a one-one correlation +<span class="pagenum" id="Page_16">[Pg 16]</span> +between the set of objects counted and the natural numbers +(excluding 0) that are used up in the process. Accordingly +common sense concludes that there are as many objects in the +set to be counted as there are numbers up to the last number +used in the counting. And we also know that, so long as we +confine ourselves to finite numbers, there are just <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> numbers +from 1 up to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">.</span> Hence it follows that the last number used in +counting a collection is the number of terms in the collection, +provided the collection is finite. But this result, besides being +only applicable to finite collections, depends upon and assumes +the fact that two classes which are similar have the same number +of terms; for what we do when we count (say) 10 objects is to +show that the set of these objects is similar to the set of numbers +1 to 10. The notion of similarity is logically presupposed in +the operation of counting, and is logically simpler though less +familiar. In counting, it is necessary to take the objects counted +in a certain order, as first, second, third, etc., but order is not +of the essence of number: it is an irrelevant addition, an unnecessary +complication from the logical point of view. The +notion of similarity does not demand an order: for example, +we saw that the number of husbands is the same as the number +of wives, without having to establish an order of precedence +among them. The notion of similarity also does not require +that the classes which are similar should be finite. Take, for +example, the natural numbers (excluding 0) on the one hand, +and the fractions which have 1 for their numerator on the other +hand: it is obvious that we can correlate 2 with <span class="nowrap"><img style="vertical-align: -1.552ex; width: 2.127ex; height: 4.588ex;" src="images/66.svg" alt="" data-tex="\dfrac{1}{2}">,</span> 3 +with <span class="nowrap"><img style="vertical-align: -1.602ex; width: 2.127ex; height: 4.638ex;" src="images/67.svg" alt="" data-tex="\dfrac{1}{3}">,</span> and +so on, thus proving that the two classes are similar. +</p> +<p> +We may thus use the notion of "similarity" to decide when +two collections are to belong to the same bundle, in the sense +in which we were asking this question earlier in this chapter. +We want to make one bundle containing the class that has no +members: this will be for the number 0. Then we want a bundle +of all the classes that have one member: this will be for the +number 1. Then, for the number 2, we want a bundle consisting +<span class="pagenum" id="Page_17">[Pg 17]</span> +of all couples; then one of all trios; and so on. Given any collection, +we can define the bundle it is to belong to as being the class +of all those collections that are "similar" to it. It is very easy +to see that if (for example) a collection has three members, the +class of all those collections that are similar to it will be the +class of trios. And whatever number of terms a collection may +have, those collections that are "similar" to it will have the same +number of terms. We may take this as a <i>definition</i> of "having +the same number of terms." It is obvious that it gives results +conformable to usage so long as we confine ourselves to finite +collections. +</p> +<p> +So far we have not suggested anything in the slightest degree +paradoxical. But when we come to the actual definition of +numbers we cannot avoid what must at first sight seem a paradox, +though this impression will soon wear off. We naturally think +that the class of couples (for example) is something different +from the number 2. But there is no doubt about the class of +couples: it is indubitable and not difficult to define, whereas +the number 2, in any other sense, is a metaphysical entity about +which we can never feel sure that it exists or that we have tracked +it down. It is therefore more prudent to content ourselves with +the class of couples, which we are sure of, than to hunt for a +problematical number 2 which must always remain elusive. +Accordingly we set up the following definition:— +</p> +<p> +<i>The number of a class is the class of all those classes that are +similar to it.</i> +</p> +<p> +Thus the number of a couple will be the class of all couples. +In fact, the class of all couples will <i>be</i> the number 2, according +to our definition. At the expense of a little oddity, this definition +secures definiteness and indubitableness; and it is not difficult +to prove that numbers so defined have all the properties that we +expect numbers to have. +</p> +<p> +We may now go on to define numbers in general as any one of +the bundles into which similarity collects classes. A number +will be a set of classes such as that any two are similar to each +<span class="pagenum" id="Page_18">[Pg 18]</span> +other, and none outside the set are similar to any inside the set. +In other words, a number (in general) is any collection which is +the number of one of its members; or, more simply still: +</p> +<p> +<i>A number is anything which is the number of some class.</i> +</p> +<p> +Such a definition has a verbal appearance of being circular, +but in fact it is not. We define "the number of a given class" +without using the notion of number in general; therefore we may +define number in general in terms of "the number of a given +class" without committing any logical error. +</p> +<p> +Definitions of this sort are in fact very common. The class +of fathers, for example, would have to be defined by first defining +what it is to be the father of somebody; then the class of fathers +will be all those who are somebody's father. Similarly if we want +to define square numbers (say), we must first define what we +mean by saying that one number is the square of another, and +then define square numbers as those that are the squares of +other numbers. This kind of procedure is very common, and +it is important to realise that it is legitimate and even often +necessary. +</p> +<p> +We have now given a definition of numbers which will serve +for finite collections. It remains to be seen how it will serve +for infinite collections. But first we must decide what we mean +by "finite" and "infinite," which cannot be done within the +limits of the present chapter. +<span class="pagenum" id="Page_19">[Pg 19]</span> +</p></div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='III: FINITUDE AND MATHEMATICAL INDUCTION'><a id="chap03"></a>CHAPTER III +<br><br> +FINITUDE AND MATHEMATICAL INDUCTION</h2> + +<p class="nind"> +THE series of natural numbers, as we saw in Chapter I., can all +be defined if we know what we mean by the three terms "0," +"number," and "successor." But we may go a step farther: +we can define all the natural numbers if we know what we mean +by "0" and "successor." It will help us to understand the +difference between finite and infinite to see how this can be done, +and why the method by which it is done cannot be extended +beyond the finite. We will not yet consider how "0" and "successor" +are to be defined: we will for the moment assume that +we know what these terms mean, and show how thence all other +natural numbers can be obtained. +</p> +<p> +It is easy to see that we can reach any assigned number, say +30,000. We first define "1" as "the successor of 0," then we +define "2" as "the successor of 1," and so on. In the case of +an assigned number, such as 30,000, the proof that we can reach +it by proceeding step by step in this fashion may be made, if we +have the patience, by actual experiment: we can go on until +we actually arrive at 30,000. But although the method of +experiment is available for each particular natural number, it +is not available for proving the general proposition that <i>all</i> such +numbers can be reached in this way, <i>i.e.</i> by proceeding from 0 +step by step from each number to its successor. Is there any +other way by which this can be proved? +</p> +<p> +Let us consider the question the other way round. What are +the numbers that can be reached, given the terms "0" and +<span class="pagenum" id="Page_20">[Pg 20]</span> +"successor"? Is there any way by which we can define the +whole class of such numbers? We reach 1, as the successor of 0; +2, as the successor of 1; 3, as the successor of 2; and so on. It +is this "and so on" that we wish to replace by something less +vague and indefinite. We might be tempted to say that "and +so on" means that the process of proceeding to the successor +may be repeated <i>any finite number</i> of times; but the problem +upon which we are engaged is the problem of defining "finite +number," and therefore we must not use this notion in our definition. +Our definition must not assume that we know what a +finite number is. +</p> +<p> +The key to our problem lies in <i>mathematical induction</i>. It will +be remembered that, in Chapter I., this was the fifth of the five +primitive propositions which we laid down about the natural +numbers. It stated that any property which belongs to 0, and +to the successor of any number which has the property, belongs +to all the natural numbers. This was then presented as a principle, +but we shall now adopt it as a definition. It is not difficult +to see that the terms obeying it are the same as the numbers +that can be reached from 0 by successive steps from next to +next, but as the point is important we will set forth the matter +in some detail. +</p> +<p> +We shall do well to begin with some definitions, which will be +useful in other connections also. +</p> +<p> +A property is said to be "hereditary" in the natural-number +series if, whenever it belongs to a number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> it also belongs to +<span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">,</span> +the successor of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">.</span> Similarly a class is said to be "hereditary" +if, whenever <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is a member of the class, so is <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">.</span> It is +easy to <i>see</i>, though we are not yet supposed to know, that to say +a property is hereditary is equivalent to saying that it belongs +to all the natural numbers not less than some one of them, <i>e.g.</i> +it must belong to all that are not less than 100, or all that are +less than 1000, or it may be that it belongs to all that are not +less than 0, <i>i.e.</i> to all without exception. +</p> +<p> +A property is said to be "inductive" when it is a hereditary +<span class="pagenum" id="Page_21">[Pg 21]</span> +property which belongs to 0. Similarly a class is "inductive" +when it is a hereditary class of which 0 is a member. +</p> +<p> +Given a hereditary class of which 0 is a member, it follows +that 1 is a member of it, because a hereditary class contains the +successors of its members, and 1 is the successor of 0. Similarly, +given a hereditary class of which 1 is a member, it follows that +2 is a member of it; and so on. Thus we can prove by a step-by-step +procedure that any assigned natural number, say 30,000, +is a member of every inductive class. +</p> +<p> +We will define the "posterity" of a given natural number +with respect to the relation "immediate predecessor" (which +is the converse of "successor") as all those terms that belong +to every hereditary class to which the given number belongs. It +is again easy to <i>see</i> that the posterity of a natural number consists +of itself and all greater natural numbers; but this also we +do not yet officially know. +</p> +<p> +By the above definitions, the posterity of 0 will consist of those +terms which belong to every inductive class. +</p> +<p> +It is now not difficult to make it obvious that the posterity of 0 +is the same set as those terms that can be reached from 0 by +successive steps from next to next. For, in the first place, 0 belongs +to both these sets (in the sense in which we have defined +our terms); in the second place, if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> belongs to both sets, +so does <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">.</span> +It is to be observed that we are dealing here with the +kind of matter that does not admit of precise proof, namely, the +comparison of a relatively vague idea with a relatively precise +one. The notion of "those terms that can be reached from 0 +by successive steps from next to next" is vague, though it <i>seems</i> +as if it conveyed a definite meaning; on the other hand, "the +posterity of 0" is precise and explicit just where the other idea +is hazy. It may be taken as giving what we <i>meant</i> to mean +when we spoke of the terms that can be reached from 0 by +successive steps. +</p> +<p> +We now lay down the following definition:— +</p> +<p> +<i>The "natural numbers" are the posterity of 0 with respect to the +<span class="pagenum" id="Page_22">[Pg 22]</span> +relation "immediate predecessor"</i> (which is the converse of +"successor"). +</p> +<p> +We have thus arrived at a definition of one of Peano's three +primitive ideas in terms of the other two. As a result of this +definition, two of his primitive propositions—namely, the one +asserting that 0 is a number and the one asserting mathematical +induction—become unnecessary, since they result from the definition. +The one asserting that the successor of a natural number +is a natural number is only needed in the weakened form "every +natural number has a successor." +</p> +<p> +We can, of course, easily define "0" and "successor" by means +of the definition of number in general which we arrived at in +Chapter II. The number 0 is the number of terms in a class +which has no members, <i>i.e.</i> in the class which is called the "null-class." +By the general definition of number, the number of terms +in the null-class is the set of all classes similar to the null-class, +<i>i.e.</i> (as is easily proved) the set consisting of the null-class all +alone, <i>i.e.</i> the class whose only member is the null-class. (This +is not identical with the null-class: it has one member, namely, +the null-class, whereas the null-class itself has no members. A +class which has one member is never identical with that one +member, as we shall explain when we come to the theory of +classes.) Thus we have the following purely logical definition:— +</p> +<p> +<i>0 is the class whose only member is the null-class.</i> +</p> +<p> +It remains to define "successor." Given any number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> let +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> be a class which has <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> members, and let <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> be a term which +is not a member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> Then the class consisting of +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> with <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> +added on will have <img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1"> members. Thus we have the following +definition:— +</p> +<p> +<i>The successor of the number of terms in the class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is the number +of terms in the class consisting of a together with <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> +where <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is any +term not belonging to the class.</i> +</p> +<p> +Certain niceties are required to make this definition perfect, +but they need not concern us.<a id="FNanchor_5_1"></a><a href="#Footnote_5_1" class="fnanchor">[5]</a> +It will be remembered that we +<span class="pagenum" id="Page_23">[Pg 23]</span> +have already given (in Chapter II.) a logical definition of the +number of terms in a class, namely, we defined it as the set of all +classes that are similar to the given class. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_5_1"></a><a href="#FNanchor_5_1"><span class="label">[5]</span></a>See <i>Principia Mathematica</i>, vol. II. * 110.</p></div> + +<p> +We have thus reduced Peano's three primitive ideas to ideas +of logic: we have given definitions of them which make them +definite, no longer capable of an infinity of different meanings, +as they were when they were only determinate to the extent of +obeying Peano's five axioms. We have removed them from the +fundamental apparatus of terms that must be merely apprehended, +and have thus increased the deductive articulation of +mathematics. +</p> +<p> +As regards the five primitive propositions, we have already +succeeded in making two of them demonstrable by our definition +of "natural number." How stands it with the remaining three? +It is very easy to prove that 0 is not the successor of any number, +and that the successor of any number is a number. But there +is a difficulty about the remaining primitive proposition, namely, +"no two numbers have the same successor." The difficulty +does not arise unless the total number of individuals in the +universe is finite; for given two numbers <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> neither of +which is the total number of individuals in the universe, it is +easy to prove that we cannot have <img style="vertical-align: -0.186ex; width: 14.155ex; height: 1.692ex;" src="images/68.svg" alt="" data-tex="m + 1 = n + 1"> unless we have +<span class="nowrap"><img style="vertical-align: -0.186ex; width: 6.361ex; height: 1.505ex;" src="images/69.svg" alt="" data-tex="m = n">.</span> But let us suppose that the total number of individuals +in the universe were (say) 10; then there would be no class of +11 individuals, and the number 11 would be the null-class. So +would the number 12. Thus we should have 11 = 12; therefore +the successor of 10 would be the same as the successor of 11, +although 10 would not be the same as 11. Thus we should have +two different numbers with the same successor. This failure of +the third axiom cannot arise, however, if the number of individuals +in the world is not finite. We shall return to this topic +at a later stage.<a id="FNanchor_6_1"></a><a href="#Footnote_6_1" class="fnanchor">[6]</a> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_6_1"></a><a href="#FNanchor_6_1"><span class="label">[6]</span></a>See Chapter XIII.</p></div> + +<p> +Assuming that the number of individuals in the universe is +not finite, we have now succeeded not only in defining Peano's +<span class="pagenum" id="Page_24">[Pg 24]</span> +three primitive ideas, but in seeing how to prove his five primitive +propositions, by means of primitive ideas and propositions belonging +to logic. It follows that all pure mathematics, in so far +as it is deducible from the theory of the natural numbers, is only +a prolongation of logic. The extension of this result to those +modern branches of mathematics which are not deducible from +the theory of the natural numbers offers no difficulty of principle, +as we have shown elsewhere.<a id="FNanchor_7_1"></a><a href="#Footnote_7_1" class="fnanchor">[7]</a> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_7_1"></a><a href="#FNanchor_7_1"><span class="label">[7]</span></a>For geometry, in so far as it is not purely analytical, see <i>Principles of +Mathematics</i>, part VI.; for rational dynamics, <i>ibid.</i>, part VII.</p></div> + +<p> +The process of mathematical induction, by means of which +we defined the natural numbers, is capable of generalisation. +We defined the natural numbers as the "posterity" of 0 with +respect to the relation of a number to its immediate successor. +If we call this relation <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.545ex;" src="images/70.svg" alt="" data-tex="\mathrm N">,</span> any number <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> will have this relation +to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.883ex; height: 1.692ex;" src="images/71.svg" alt="" data-tex="m + 1">.</span> A property is "hereditary with respect to <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.545ex;" src="images/70.svg" alt="" data-tex="\mathrm N">,</span>" or +simply "<img style="vertical-align: 0; width: 1.697ex; height: 1.545ex;" src="images/70.svg" alt="" data-tex="\mathrm N">-hereditary," if, whenever the property belongs to a +number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m">,</span> it also belongs to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.883ex; height: 1.692ex;" src="images/71.svg" alt="" data-tex="m + 1">,</span> <i>i.e.</i> to the number to which +<img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> has the relation <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.545ex;" src="images/70.svg" alt="" data-tex="\mathrm N">.</span> And a number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> will be said to belong to +the "posterity" of <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> with respect to the relation <img style="vertical-align: 0; width: 1.697ex; height: 1.545ex;" src="images/70.svg" alt="" data-tex="\mathrm N"> if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> has +every <img style="vertical-align: 0; width: 1.697ex; height: 1.545ex;" src="images/70.svg" alt="" data-tex="\mathrm N">-hereditary property belonging to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m">.</span> These definitions +can all be applied to any other relation just as well as to <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.545ex;" src="images/70.svg" alt="" data-tex="\mathrm N">.</span> Thus +if <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is any relation whatever, we can lay down the following +definitions:<a id="FNanchor_8_1"></a><a href="#Footnote_8_1" class="fnanchor">[8]</a>— +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_8_1"></a><a href="#FNanchor_8_1"><span class="label">[8]</span></a>These definitions, and the generalised theory of induction, are due to +Frege, and were published so long ago as 1879 in his <i>Begriffsschrift</i>. In +spite of the great value of this work, I was, I believe, the first person who +ever read it—more than twenty years after its publication.</p></div> + +<p> +A property is called "<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-hereditary" when, if it belongs to +a term <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> +then it belongs to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">.</span> +</p> +<p> +A class is <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-hereditary when its defining property +is <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-hereditary. +</p> +<p> +A term <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is said to be an "<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-ancestor" of the +term <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> if <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> has +every <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-hereditary property that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has, +provided <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a term +which has the relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to something or to which something +has the relation <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">.</span> (This is only to exclude trivial cases.) +<span class="pagenum" id="Page_25">[Pg 25]</span> +</p> +<p> +The "<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-posterity" of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is all the terms of +which <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is an <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-ancestor. +</p> +<p> +We have framed the above definitions so that if a term is the +ancestor of anything it is its own ancestor and belongs to its own +posterity. This is merely for convenience. +</p> +<p> +It will be observed that if we take for <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> the relation "parent," +"ancestor" and "posterity" will have the usual meanings, +except that a person will be included among his own ancestors +and posterity. It is, of course, obvious at once that "ancestor" +must be capable of definition in terms of "parent," but until +Frege developed his generalised theory of induction, no one could +have defined "ancestor" precisely in terms of "parent." A +brief consideration of this point will serve to show the importance +of the theory. A person confronted for the first time with the +problem of defining "ancestor" in terms of "parent" would +naturally say that <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A"> is an ancestor of <img style="vertical-align: 0; width: 1.382ex; height: 1.545ex;" src="images/74.svg" alt="" data-tex="\mathrm Z"> if, +between <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A"> and <span class="nowrap"><img style="vertical-align: 0; width: 1.382ex; height: 1.545ex;" src="images/74.svg" alt="" data-tex="\mathrm Z">,</span> +there are a certain number of people, <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">,</span> <span class="nowrap"><img style="vertical-align: -0.048ex; width: 1.633ex; height: 1.643ex;" src="images/76.svg" alt="" data-tex="\mathrm C">,</span> ..., of whom +<img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B"> is a child of <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A">,</span> each is a parent of the next, until the last, who +is a parent of <span class="nowrap"><img style="vertical-align: 0; width: 1.382ex; height: 1.545ex;" src="images/74.svg" alt="" data-tex="\mathrm Z">.</span> But this definition is not adequate unless we +add that the number of intermediate terms is to be finite. Take, +for example, such a series as the following:— +<span class="align-center"><img style="vertical-align: -0.816ex; width: 36.934ex; height: 2.773ex;" src="images/8.svg" alt="" data-tex=" +1,\ -\tfrac{1}{2},\ -\tfrac{1}{4},\ -\tfrac{1}{8},\ \dots\ +\tfrac{1}{8},\ \tfrac{1}{4},\ \tfrac{1}{2},\ 1. +"></span> +Here we have first a series of negative fractions with no end, +and then a series of positive fractions with no beginning. Shall +we say that, in this series, <img style="vertical-align: -0.816ex; width: 3.556ex; height: 2.773ex;" src="images/77.svg" alt="" data-tex="-\frac{1}{8}"> is an ancestor of <span class="nowrap"><img style="vertical-align: -0.816ex; width: 1.795ex; height: 2.773ex;" src="images/78.svg" alt="" data-tex="\frac{1}{8}">?</span> It will be +so according to the beginner's definition suggested above, but +it will not be so according to any definition which will give the +kind of idea that we wish to define. For this purpose, it is +essential that the number of intermediaries should be finite. +But, as we saw, "finite" is to be defined by means of mathematical +induction, and it is simpler to define the ancestral relation +generally at once than to define it first only for the case of the +relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">,</span> and then extend it to other cases. Here, +as constantly elsewhere, generality from the first, though it may +<span class="pagenum" id="Page_26">[Pg 26]</span> +require more thought at the start, will be found in the long run +to economise thought and increase logical power. +</p> +<p> +The use of mathematical induction in demonstrations was, +in the past, something of a mystery. There seemed no reasonable +doubt that it was a valid method of proof, but no one quite +knew why it was valid. Some believed it to be really a case +of induction, in the sense in which that word is used in logic. +Poincaré<a id="FNanchor_9_1"></a><a href="#Footnote_9_1" class="fnanchor">[9]</a> +considered it to be a principle of the utmost importance, +by means of which an infinite number of syllogisms could be +condensed into one argument. We now know that all such views +are mistaken, and that mathematical induction is a definition, +not a principle. There are some numbers to which it can be +applied, and there are others (as we shall see in Chapter VIII.) +to which it cannot be applied. We <i>define</i> the "natural numbers" +as those to which proofs by mathematical induction can be +applied, <i>i.e.</i> as those that possess all inductive properties. It +follows that such proofs can be applied to the natural numbers, +not in virtue of any mysterious intuition or axiom or principle, +but as a purely verbal proposition. If "quadrupeds" are +defined as animals having four legs, it will follow that animals +that have four legs are quadrupeds; and the case of numbers +that obey mathematical induction is exactly similar. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_9_1"></a><a href="#FNanchor_9_1"><span class="label">[9]</span></a><i>Science and Method</i>, chap. IV.</p></div> + +<p> +We shall use the phrase "inductive numbers" to mean the +same set as we have hitherto spoken of as the "natural numbers." +The phrase "inductive numbers" is preferable as affording a +reminder that the definition of this set of numbers is obtained +from mathematical induction. +</p> +<p> +Mathematical induction affords, more than anything else, +the essential characteristic by which the finite is distinguished +from the infinite. The principle of mathematical induction +might be stated popularly in some such form as "what can be +inferred from next to next can be inferred from first to last." +This is true when the number of intermediate steps between +first and last is finite, not otherwise. Anyone who has ever +<span class="pagenum" id="Page_27">[Pg 27]</span> +watched a goods train beginning to move will have noticed how +the impulse is communicated with a jerk from each truck to +the next, until at last even the hindmost truck is in motion. +When the train is very long, it is a very long time before the last +truck moves. If the train were infinitely long, there would be +an infinite succession of jerks, and the time would never come +when the whole train would be in motion. Nevertheless, if +there were a series of trucks no longer than the series of inductive +numbers (which, as we shall see, is an instance of the smallest +of infinites), every truck would begin to move sooner or later +if the engine persevered, though there would always be other +trucks further back which had not yet begun to move. This +image will help to elucidate the argument from next to next, +and its connection with finitude. When we come to infinite +numbers, where arguments from mathematical induction will +be no longer valid, the properties of such numbers will help to +make clear, by contrast, the almost unconscious use that is made +of mathematical induction where finite numbers are concerned. +<span class="pagenum" id="Page_28">[Pg 28]</span> +</p></div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='IV: THE DEFINITION OF ORDER'><a id="chap04"></a>CHAPTER IV +<br><br> +THE DEFINITION OF ORDER</h2> + +<p class="nind"> +WE have now carried our analysis of the series of natural numbers +to the point where we have obtained logical definitions of the +members of this series, of the whole class of its members, and +of the relation of a number to its immediate successor. We +must now consider the <i>serial</i> character of the natural numbers +in the order 0, 1, 2, 3,.... We ordinarily think of the numbers +as in this <i>order</i>, and it is an essential part of the work +of analysing our data to seek a definition of "order" or "series" +in logical terms. +</p> +<p> +The notion of order is one which has enormous importance +in mathematics. Not only the integers, but also rational fractions +and all real numbers have an order of magnitude, and +this is essential to most of their mathematical properties. The +order of points on a line is essential to geometry; so is the +slightly more complicated order of lines through a point in a +plane, or of planes through a line. Dimensions, in geometry, +are a development of order. The conception of a <i>limit</i>, which +underlies all higher mathematics, is a serial conception. There +are parts of mathematics which do not depend upon the notion +of order, but they are very few in comparison with the parts +in which this notion is involved. +</p> +<p> +In seeking a definition of order, the first thing to realise is +that no set of terms has just <i>one</i> order to the exclusion of others. +A set of terms has all the orders of which it is capable. Sometimes +one order is so much more familiar and natural to our +<span class="pagenum" id="Page_29">[Pg 29]</span> +thoughts that we are inclined to regard it as <i>the</i> order of that +set of terms; but this is a mistake. The natural numbers—or +the "inductive" numbers, as we shall also call them—occur +to us most readily in order of magnitude; but they are capable +of an infinite number of other arrangements. We might, for +example, consider first all the odd numbers and then all the +even numbers; or first 1, then all the even numbers, then all +the odd multiples of 3, then all the multiples of 5 but not of +2 or 3, then all the multiples of 7 but not of 2 or 3 or 5, and so +on through the whole series of primes. When we say that we +"arrange" the numbers in these various orders, that is an +inaccurate expression: what we really do is to turn our attention +to certain relations between the natural numbers, which themselves +generate such-and-such an arrangement. We can no +more "arrange" the natural numbers than we can the starry +heavens; but just as we may notice among the fixed stars +either their order of brightness or their distribution in the sky, +so there are various relations among numbers which may be +observed, and which give rise to various different orders among +numbers, all equally legitimate. And what is true of numbers +is equally true of points on a line or of the moments of time: +one order is more familiar, but others are equally valid. We +might, for example, take first, on a line, all the points that have +integral co-ordinates, then all those that have non-integral +rational co-ordinates, then all those that have algebraic non-rational +co-ordinates, and so on, through any set of complications +we please. The resulting order will be one which the +points of the line certainly have, whether we choose to notice +it or not; the only thing that is arbitrary about the various +orders of a set of terms is our attention, for the terms themselves +have always all the orders of which they are capable. +</p> +<p> +One important result of this consideration is that we must +not look for the definition of order in the nature of the set of +terms to be ordered, since one set of terms has many orders. +The order lies, not in the <i>class</i> of terms, but in a relation among +<span class="pagenum" id="Page_30">[Pg 30]</span> +the members of the class, in respect of which some appear as +earlier and some as later. The fact that a class may have many +orders is due to the fact that there can be many relations holding +among the members of one single class. What properties must +a relation have in order to give rise to an order? +</p> +<p> +The essential characteristics of a relation which is to give rise +to order may be discovered by considering that in respect of +such a relation we must be able to say, of any two terms in +the class which is to be ordered, that one "precedes" and the +other "follows." Now, in order that we may be able to use +these words in the way in which we should naturally understand +them, we require that the ordering relation should have three +properties:— +</p> +<p> +(1) If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> precedes <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> must not also precede <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> This is an +obvious characteristic of the kind of relations that lead to series. +If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is less than <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is not also less than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> +If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is earlier in +time than <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is not also earlier than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is +to the left of <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> +<img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is not to the left of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> On the other hand, relations which +do not give rise to series often do not have this property. If +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a brother or sister of <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is a brother or +sister of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is +of the same height as <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is of the same height as <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is of a +different height from <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is of a different height from <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> In +all these cases, when the relation holds between <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> it also +holds between <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> But with serial relations such a thing +cannot happen. A relation having this first property is called +<i>asymmetrical</i>. +</p> +<p> +(2) If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> precedes <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> precedes <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">,</span> <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> must precede <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">.</span> This +may be illustrated by the same instances as before: <i>less</i>, <i>earlier</i>, +<i>left of</i>. But as instances of relations which do <i>not</i> have this +property only two of our previous three instances will serve. +If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is brother or sister of <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">,</span> <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> may not be brother +or sister of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">,</span> since <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> may be the same person. The same +applies to difference of height, but not to sameness of height, +which has our second property but not our first. The relation +"father," on the other hand, has our first property but not +<span class="pagenum" id="Page_31">[Pg 31]</span> +our second. A relation having our second property is called +<i>transitive</i>. +</p> +<p> +(3) Given any two terms of the class which is to be ordered, +there must be one which precedes and the other which follows. +For example, of any two integers, or fractions, or real numbers, +one is smaller and the other greater; but of any two complex +numbers this is not true. Of any two moments in time, one +must be earlier than the other; but of events, which may be +simultaneous, this cannot be said. Of two points on a line, +one must be to the left of the other. A relation having this +third property is called <i>connected</i>. +</p> +<p> +When a relation possesses these three properties, it is of the +sort to give rise to an order among the terms between which it +holds; and wherever an order exists, some relation having these +three properties can be found generating it. +</p> +<p> +Before illustrating this thesis, we will introduce a few +definitions. +</p> +<p> +(1) A relation is said to be an aliorelative,<a id="FNanchor_10_1"></a><a href="#Footnote_10_1" class="fnanchor">[10]</a> +or to <i>be contained +in</i> or <i>imply diversity</i>, if no term has this relation to itself. +Thus, for example, "greater," "different in size," "brother," +"husband," "father" are aliorelatives; but "equal," "born +of the same parents," "dear friend" are not. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_10_1"></a><a href="#FNanchor_10_1"><span class="label">[10]</span></a>This term is due to C. S. Peirce.</p></div> + +<p> +(2) The <i>square</i> of a relation is that relation which holds between +two terms <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> when there is an intermediate term <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> such +that the given relation holds between <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and between +<img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">.</span> Thus "paternal grandfather" is the square of "father," +"greater by 2" is the square of "greater by 1," and so on. +</p> +<p> +(3) The <i>domain</i> of a relation consists of all those terms that +have the relation to something or other, and the <i>converse domain</i> +consists of all those terms to which something or other has the +relation. These words have been already defined, but are +recalled here for the sake of the following definition:— +</p> +<p> +(4) The <i>field</i> of a relation consists of its domain and converse +domain together. +<span class="pagenum" id="Page_32">[Pg 32]</span> +</p> +<p> +(5) One relation is said to <i>contain</i> or <i>be implied by</i> another if +it holds whenever the other holds. +</p> +<p> +It will be seen that an <i>asymmetrical</i> relation is the same thing +as a relation whose square is an aliorelative. It often happens +that a relation is an aliorelative without being asymmetrical, +though an asymmetrical relation is always an aliorelative. For +example, "spouse" is an aliorelative, but is symmetrical, +since if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is the spouse of <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is the spouse of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> But among +<i>transitive</i> relations, all aliorelatives are asymmetrical as well +as <i>vice versa</i>. +</p> +<p> +From the definitions it will be seen that a <i>transitive</i> relation +is one which is implied by its square, or, as we also say, "contains" +its square. Thus "ancestor" is transitive, because +an ancestor's ancestor is an ancestor; but "father" is not +transitive, because a father's father is not a father. A transitive +aliorelative is one which contains its square and is contained +in diversity; or, what comes to the same thing, one whose +square implies both it and diversity—because, when a relation +is transitive, asymmetry is equivalent to being an aliorelative. +</p> +<p> +A relation is <i>connected</i> when, given any two different terms +of its field, the relation holds between the first and the second +or between the second and the first (not excluding the possibility +that both may happen, though both cannot happen if the relation +is asymmetrical). +</p> +<p> +It will be seen that the relation "ancestor," for example, +is an aliorelative and transitive, but not connected; it is because +it is not connected that it does not suffice to arrange the human +race in a series. +</p> +<p> +The relation "less than or equal to," among numbers, is +transitive and connected, but not asymmetrical or an aliorelative. +</p> +<p> +The relation "greater or less" among numbers is an aliorelative +and is connected, but is not transitive, for if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is greater +or less than <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is greater or less than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">,</span> it may happen +that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> are the same number. +</p> +<p> +Thus the three properties of being (1) an aliorelative, (2) transitive, +<span class="pagenum" id="Page_33">[Pg 33]</span> +and (3) connected, are mutually independent, since +a relation may have any two without having the third. +</p> +<p> +We now lay down the following definition:— +</p> +<p> +A relation is <i>serial</i> when it is an aliorelative, transitive, and +connected; or, what is equivalent, when it is asymmetrical, +transitive, and connected. +</p> +<p> +A <i>series</i> is the same thing as a serial relation. +</p> +<p> +It might have been thought that a series should be the <i>field</i> +of a serial relation, not the serial relation itself. But this would +be an error. For example, +<span class="align-center"><img style="vertical-align: -0.439ex; width: 55.564ex; height: 1.946ex;" src="images/9.svg" alt="" data-tex=" +1,\ 2,\ 3;\quad +1,\ 3,\ 2;\quad +2,\ 3,\ 1;\quad +2,\ 1,\ 3;\quad +3,\ 1,\ 2;\quad +3,\ 2,\ 1 +"></span> +are six different series which all have the same field. If the +field <i>were</i> the series, there could only be one series with a given +field. What distinguishes the above six series is simply the +different ordering relations in the six cases. Given the ordering +relation, the field and the order are both determinate. Thus +the ordering relation may be taken to <i>be</i> the series, but the field +cannot be so taken. +</p> +<p> +Given any serial relation, say <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span> we shall say that, in respect +of this relation, <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> "precedes" <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation +<img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> +which we shall write "<img style="vertical-align: -0.464ex; width: 3.943ex; height: 2.009ex;" src="images/81.svg" alt="" data-tex="x\mathrm Py">" for short. The three characteristics +which <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> must have in order to be serial are: +</p> +<p class="hanging2"> +(1) We must never have <span class="nowrap"><img style="vertical-align: -0.025ex; width: 4.129ex; height: 1.57ex;" src="images/82.svg" alt="" data-tex="x\mathrm Px">,</span> <i>i.e.</i> no term must precede +itself. +</p> +<p class="hanging2"> +(2) <img style="vertical-align: 0; width: 2.528ex; height: 1.887ex;" src="images/83.svg" alt="" data-tex="\mathrm P^{2}"> must imply <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span> <i>i.e.</i> +if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> precedes <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> precedes <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">,</span> <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> must +precede <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">.</span> +</p> +<p class="hanging2"> +(3) If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are two different terms in the field of <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span> we shall +have <img style="vertical-align: -0.464ex; width: 3.943ex; height: 2.009ex;" src="images/81.svg" alt="" data-tex="x\mathrm Py"> or <span class="nowrap"><img style="vertical-align: -0.464ex; width: 3.943ex; height: 2.009ex;" src="images/84.svg" alt="" data-tex="y\mathrm Px">,</span> <i>i.e.</i> one of the two must precede the +other. +</p> +<p class="nind"> +The reader can easily convince himself that, where these three +properties are found in an ordering relation, the characteristics +we expect of series will also be found, and <i>vice versa</i>. We are +therefore justified in taking the above as a definition of order +<span class="pagenum" id="Page_34">[Pg 34]</span> +or series. And it will be observed that the definition is effected +in purely logical terms. +</p> +<p> +Although a transitive asymmetrical connected relation always +exists wherever there is a series, it is not always the relation +which would most naturally be regarded as generating the series. +The natural-number series may serve as an illustration. The +relation we assumed in considering the natural numbers was +the relation of immediate succession, <i>i.e.</i> the relation between +consecutive integers. This relation is asymmetrical, but not +transitive or connected. We can, however, derive from it, +by the method of mathematical induction, the "ancestral" +relation which we considered in the preceding chapter. This +relation will be the same as "less than or equal to" among +inductive integers. For purposes of generating the series of +natural numbers, we want the relation "less than," excluding +"equal to." This is the relation of <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> to <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> when <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> is an ancestor +of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> but not identical with <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> or (what comes to the same thing) +when the successor of <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> is an ancestor of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> in the sense in which +a number is its own ancestor. That is to say, we shall lay down +the following definition:— +</p> +<p> +An inductive number <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> is said to be <i>less than</i> another number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> +when <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> possesses every hereditary property possessed by the +successor of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m">.</span> +</p> +<p> +It is easy to see, and not difficult to prove, that the relation +"less than," so defined, is asymmetrical, transitive, and connected, +and has the inductive numbers for its field. Thus by +means of this relation the inductive numbers acquire an order +in the sense in which we defined the term "order," and this order +is the so-called "natural" order, or order of magnitude. +</p> +<p> +The generation of series by means of relations more or less +resembling that of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1"> is very common. The series of the +Kings of England, for example, is generated by relations of each +to his successor. This is probably the easiest way, where it is +applicable, of conceiving the generation of a series. In this +method we pass on from each term to the next, as long as there +<span class="pagenum" id="Page_35">[Pg 35]</span> +is a next, or back to the one before, as long as there is one before. +This method always requires the generalised form of mathematical +induction in order to enable us to define "earlier" and +"later" in a series so generated. On the analogy of "proper +fractions," let us give the name "proper posterity of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> with respect +to <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">" to the class of those terms that belong to the <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-posterity +of some term to which <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">,</span> in the sense which +we gave before to "posterity," which includes a term in its own +posterity. Reverting to the fundamental definitions, we find that +the "proper posterity" may be defined as follows:— +</p> +<p> +The "proper posterity" of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> with respect to <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> consists of +all terms that possess every <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-hereditary property possessed by +every term to which <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">.</span> +</p> +<p> +It is to be observed that this definition has to be so framed +as to be applicable not only when there is only one term to which +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">,</span> but also in cases (as <i>e.g.</i> that of father and +child) where there may be many terms to which <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">.</span> +We define further: +</p> +<p> +A term <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a "proper ancestor" of <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> with respect to +<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> if <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> belongs +to the proper posterity of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> with respect to <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">.</span> +</p> +<p> +We shall speak for short of "<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-posterity" and "<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-ancestors" +when these terms seem more convenient. +</p> +<p> +Reverting now to the generation of series by the relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> +between consecutive terms, we see that, if this method is to be +possible, the relation "proper <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-ancestor" must be an aliorelative, +transitive, and connected. Under what circumstances will +this occur? It will always be transitive: no matter what sort +of relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> may be, "<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-ancestor" and "proper <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-ancestor" +are always both transitive. But it is only under certain circumstances +that it will be an aliorelative or connected. Consider, +for example, the relation to one's left-hand neighbour at a round +dinner-table at which there are twelve people. If we call this +relation <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">,</span> the proper <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-posterity of a person consists of all who +can be reached by going round the table from right to left. This +includes everybody at the table, including the person himself, since +<span class="pagenum" id="Page_36">[Pg 36]</span> +twelve steps bring us back to our starting-point. Thus in such +a case, though the relation "proper <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-ancestor" is connected, +and though <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> itself is an aliorelative, we do not get a series +because "proper <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-ancestor" is not an aliorelative. It is for +this reason that we cannot say that one person comes before +another with respect to the relation "right of" or to its ancestral +derivative. +</p> +<p> +The above was an instance in which the ancestral relation was +connected but not contained in diversity. An instance where +it is contained in diversity but not connected is derived from the +ordinary sense of the word "ancestor." If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a proper ancestor +of <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> cannot be the same person; but it is not true that +of any two persons one must be an ancestor of the other. +</p> +<p> +The question of the circumstances under which series can be +generated by ancestral relations derived from relations of consecutiveness +is often important. Some of the most important +cases are the following: Let <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> be a many-one relation, and let +us confine our attention to the posterity of some term <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> When +so confined, the relation "proper <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-ancestor" must be connected; +therefore all that remains to ensure its being serial is that it shall +be contained in diversity. This is a generalisation of the instance +of the dinner-table. Another generalisation consists in taking +<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to be a one-one relation, and including the ancestry of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> as +well as the posterity. Here again, the one condition required +to secure the generation of a series is that the relation "proper +<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-ancestor" shall be contained in diversity. +</p> +<p> +The generation of order by means of relations of consecutiveness, +though important in its own sphere, is less general than the +method which uses a transitive relation to define the order. It +often happens in a series that there are an infinite number of intermediate +terms between any two that may be selected, however +near together these may be. Take, for instance, fractions in order +of magnitude. Between any two fractions there are others—for +example, the arithmetic mean of the two. Consequently there is +no such thing as a pair of consecutive fractions. If we depended +<span class="pagenum" id="Page_37">[Pg 37]</span> +upon consecutiveness for defining order, we should not be able +to define the order of magnitude among fractions. But in fact +the relations of greater and less among fractions do not demand +generation from relations of consecutiveness, and the relations +of greater and less among fractions have the three characteristics +which we need for defining serial relations. In all such cases +the order must be defined by means of a <i>transitive</i> relation, since +only such a relation is able to leap over an infinite number of +intermediate terms. The method of consecutiveness, like that +of counting for discovering the number of a collection, is appropriate +to the finite; it may even be extended to certain infinite +series, namely, those in which, though the total number of terms is +infinite, the number of terms between any two is always finite; +but it must not be regarded as general. Not only so, but care +must be taken to eradicate from the imagination all habits of +thought resulting from supposing it general. If this is not done, +series in which there are no consecutive terms will remain difficult +and puzzling. And such series are of vital importance for the +understanding of continuity, space, time, and motion. +</p> +<p> +There are many ways in which series may be generated, but +all depend upon the finding or construction of an asymmetrical +transitive connected relation. Some of these ways have considerable +importance. We may take as illustrative the generation +of series by means of a three-term relation which we may +call "between." This method is very useful in geometry, and +may serve as an introduction to relations having more than two +terms; it is best introduced in connection with elementary +geometry. +</p> +<p> +Given any three points on a straight line in ordinary space, +there must be one of them which is <i>between</i> the other two. This +will not be the case with the points on a circle or any other closed +curve, because, given any three points on a circle, we can travel +from any one to any other without passing through the third. +In fact, the notion "between" is characteristic of open series—or +series in the strict sense—as opposed to what may be called +<span class="pagenum" id="Page_38">[Pg 38]</span> +"cyclic" series, where, as with people at the dinner-table, a +sufficient journey brings us back to our starting-point. This +notion of "between" may be chosen as the fundamental notion +of ordinary geometry; but for the present we will only consider +its application to a single straight line and to the ordering of the +points on a straight line.<a id="FNanchor_11_1"></a><a href="#Footnote_11_1" class="fnanchor">[11]</a> +Taking any two points <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> the line <img style="vertical-align: -0.566ex; width: 3.928ex; height: 2.262ex;" src="images/87.svg" alt="" data-tex="(ab)"> +consists of three parts (besides <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> themselves): +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_11_1"></a><a href="#FNanchor_11_1"><span class="label">[11]</span></a>Cf. <i>Rivista di Matematica</i>, IV. pp. 55 ff.; <i>Principles of Mathematics</i>, p. 394 +(§ 375).</p></div> + +<p class="hanging2"> +(1) Points between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">.</span> +</p> +<p class="hanging2"> +(2) Points <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> such that <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is between <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">.</span> +</p> +<p class="hanging2"> +(3) Points <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> such that <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> is between <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">.</span> +</p> +<p> +Thus the line <img style="vertical-align: -0.566ex; width: 3.928ex; height: 2.262ex;" src="images/87.svg" alt="" data-tex="(ab)"> can be defined in terms of the relation +"between." +</p> +<p> +In order that this relation "between" may arrange the points +of the line in an order from left to right, we need certain assumptions, +namely, the following:— +</p> +<p class="hanging2"> +(1) If anything is between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> are not identical. +</p> +<p class="hanging2"> +(2) Anything between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> is also between <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> and <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">.</span> +</p> +<p class="hanging2"> +(3) Anything between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> is not identical with <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> (nor, +consequently, with <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> in virtue of (2)). +</p> +<p class="hanging2"> +(4) If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> anything between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is also +between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">.</span> +</p> +<p class="hanging2"> +(5) If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> and <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> is between <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> then <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> is +between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">.</span> +</p> +<p class="hanging2"> +(6) If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> then either <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are +identical, or <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> or <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is between <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">.</span> +</p> +<p class="hanging2"> +(7) If <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> is between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and also between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> then either +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are identical, or <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is between <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> or <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is between +<img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> +</p> +<p> +These seven properties are obviously verified in the case of points +on a straight line in ordinary space. Any three-term relation +which verifies them gives rise to series, as may be seen from the +following definitions. For the sake of definiteness, let us assume +<span class="pagenum" id="Page_39">[Pg 39]</span> +that <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is to the left of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">.</span> Then the points of the line <img style="vertical-align: -0.566ex; width: 3.928ex; height: 2.262ex;" src="images/87.svg" alt="" data-tex="(ab)"> are (1) those +between which and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> lies—these we will call to the left +of <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">;</span> (2) <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> itself; (3) those between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">;</span> (4) <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> itself; (5) those +between which and <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> lies <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">—these we will call to the right +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">.</span> We may now define generally that of two points <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> on +the line <span class="nowrap"><img style="vertical-align: -0.566ex; width: 3.928ex; height: 2.262ex;" src="images/87.svg" alt="" data-tex="(ab)">,</span> we shall say that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is "to the left of" <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> in any +of the following cases:— +</p> +<p class="hanging2"> +(1) When <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are both to the left of <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is between +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">;</span> +</p> +<p class="hanging2"> +(2) When <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is to the left of <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> or <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> or between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and +<img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> or to the right of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">;</span> +</p> +<p class="hanging2"> +(3) When <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> or is <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> or is to the +right of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">;</span> +</p> +<p class="hanging2"> +(4) When <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are both between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is between +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">;</span> +</p> +<p class="hanging2"> +(5) When <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is between <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> or to the right of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">;</span> +</p> +<p class="hanging2"> +(6) When <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is to the right of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">;</span> +</p> +<p class="hanging2"> +(7) When <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are both to the right of <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is between +<img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">.</span> +</p> +<p> +It will be found that, from the seven properties which we have +assigned to the relation "between," it can be deduced that the +relation "to the left of," as above defined, is a <i>serial</i> relation as +we defined that term. It is important to notice that nothing +in the definitions or the argument depends upon our meaning +by "between" the actual relation of that name which occurs in +empirical space: any three-term relation having the above seven +purely formal properties will serve the purpose of the argument +equally well. +</p> +<p> +Cyclic order, such as that of the points on a circle, cannot be +generated by means of three-term relations of "between." We +need a relation of four terms, which may be called "separation +of couples." The point may be illustrated by considering a +journey round the world. One may go from England to New +Zealand by way of Suez or by way of San Francisco; we cannot +<span class="pagenum" id="Page_40">[Pg 40]</span> +say definitely that either of these two places is "between" +England and New Zealand. But if a man chooses that route +to go round the world, whichever way round he goes, his times in +England and New Zealand are separated from each other by his +times in Suez and San Francisco, and conversely. Generalising, +if we take any four points on a circle, we can separate them into +two couples, say <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> such that, in order to get +from <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> to <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> one must pass through either <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> or <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> and in order to +get from <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> to <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> one must pass through either <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> or <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">.</span> Under these +circumstances we say that the couple <img style="vertical-align: -0.566ex; width: 4.934ex; height: 2.262ex;" src="images/88.svg" alt="" data-tex="(a, b)"> are "separated" by +the couple <span class="nowrap"><img style="vertical-align: -0.566ex; width: 5.169ex; height: 2.262ex;" src="images/89.svg" alt="" data-tex="(x, y)">.</span> Out of this relation a cyclic order can be generated, +in a way resembling that in which we generated an open +order from "between," but somewhat more complicated.<a id="FNanchor_12_1"></a><a href="#Footnote_12_1" class="fnanchor">[12]</a> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_12_1"></a><a href="#FNanchor_12_1"><span class="label">[12]</span></a>Cf. <i>Principles of Mathematics</i>, p. 205 (§ 194), and references there given.</p></div> + +<p> +The purpose of the latter half of this chapter has been to suggest +the subject which one may call "generation of serial relations." +When such relations have been defined, the generation of them +from other relations possessing only some of the properties +required for series becomes very important, especially in the +philosophy of geometry and physics. But we cannot, within +the limits of the present volume, do more than make the reader +aware that such a subject exists. +<span class="pagenum" id="Page_41">[Pg 41]</span> +</p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='V: KINDS OF RELATIONS'><a id="chap05"></a>CHAPTER V +<br><br> +KINDS OF RELATIONS</h2> + +<p class="nind"> +A great part of the philosophy of mathematics is concerned with +<i>relations</i>, and many different kinds of relations have different +kinds of uses. It often happens that a property which belongs +to <i>all</i> relations is only important as regards relations of certain +sorts; in these cases the reader will not see the bearing of the +proposition asserting such a property unless he has in mind the +sorts of relations for which it is useful. For reasons of this +description, as well as from the intrinsic interest of the subject, +it is well to have in our minds a rough list of the more +mathematically serviceable varieties of relations. +</p> +<p> +We dealt in the preceding chapter with a supremely important +class, namely, <i>serial</i> relations. Each of the three properties which +we combined in defining series—namely, <i>asymmetry</i>, <i>transitiveness</i>, +and <i>connexity</i>—has its own importance. We will begin by saying +something on each of these three. +</p> +<p> +<i>Asymmetry</i>, <i>i.e.</i> the property of being incompatible with the +converse, is a characteristic of the very greatest interest and +importance. In order to develop its functions, we will consider +various examples. The relation <i>husband</i> is asymmetrical, and +so is the relation <i>wife</i>; <i>i.e.</i> if <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is husband of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> +<img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> cannot be husband +of <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> and similarly in the case of <i>wife</i>. On the other hand, the +relation "spouse" is symmetrical: if <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is spouse of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> then <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> is +spouse of <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">.</span> Suppose now we are given the relation <i>spouse</i>, and +we wish to derive the relation <i>husband</i>. <i>Husband</i> is the same as +<i>male spouse</i> or <i>spouse of a female</i>; thus the relation <i>husband</i> can +<span class="pagenum" id="Page_42">[Pg 42]</span> +be derived from <i>spouse</i> either by limiting the domain to males +or by limiting the converse to females. We see from this instance +that, when a symmetrical relation is given, it is sometimes possible, +without the help of any further relation, to separate it into two +asymmetrical relations. But the cases where this is possible are +rare and exceptional: they are cases where there are two mutually +exclusive classes, say <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">,</span> such that whenever the relation +holds between two terms, one of the terms is a member of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and +the other is a member of <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">—as, in the case of <i>spouse</i>, one term +of the relation belongs to the class of males and one to the class +of females. In such a case, the relation with its domain confined +to <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> will be asymmetrical, and so will the relation with its domain +confined to <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">.</span> But such cases are not of the sort that occur +when we are dealing with series of more than two terms; for in +a series, all terms, except the first and last (if these exist), belong +both to the domain and to the converse domain of the generating +relation, so that a relation like <i>husband</i>, where the domain and +converse domain do not overlap, is excluded. +</p> +<p> +The question how to <i>construct</i> relations having some useful +property by means of operations upon relations which only have +rudiments of the property is one of considerable importance. +Transitiveness and connexity are easily constructed in many cases +where the originally given relation does not possess them: for +example, if <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is any relation whatever, the ancestral relation +derived from <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> by generalised induction is transitive; and if <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is +a many-one relation, the ancestral relation will be connected +if confined to the posterity of a given term. But asymmetry is +a much more difficult property to secure by construction. The +method by which we derived <i>husband</i> from <i>spouse</i> is, as we have +seen, not available in the most important cases, such as <i>greater</i>, +<i>before</i>, <i>to the right of</i>, where domain and converse domain overlap. +In all these cases, we can of course obtain a symmetrical relation +by adding together the given relation and its converse, but we +cannot pass back from this symmetrical relation to the original +asymmetrical relation except by the help of some asymmetrical +<span class="pagenum" id="Page_43">[Pg 43]</span> +relation. Take, for example, the relation <i>greater</i>: the relation +<i>greater or less</i>—<i>i.e.</i> <i>unequal</i>—is symmetrical, but there is nothing +in this relation to show that it is the sum of two asymmetrical +relations. Take such a relation as "differing in shape." This +is not the sum of an asymmetrical relation and its converse, since +shapes do not form a single series; but there is nothing to show +that it differs from "differing in magnitude" if we did not already +know that magnitudes have relations of greater and less. This +illustrates the fundamental character of asymmetry as a property +of relations. +</p> +<p> +From the point of view of the classification of relations, being +asymmetrical is a much more important characteristic than +implying diversity. Asymmetrical relations imply diversity, +but the converse is not the case. "Unequal," for example, +implies diversity, but is symmetrical. Broadly speaking, we +may say that, if we wished as far as possible to dispense with +relational propositions and replace them by such as ascribed +predicates to subjects, we could succeed in this so long as we +confined ourselves to <i>symmetrical</i> relations: those that do not +imply diversity, if they are transitive, may be regarded as asserting +a common predicate, while those that do imply diversity +may be regarded as asserting incompatible predicates. For +example, consider the relation of <i>similarity between classes</i>, +by means of which we defined numbers. This relation is symmetrical +and transitive and does not imply diversity. It would +be possible, though less simple than the procedure we adopted, +to regard the number of a collection as a predicate of the collection: +then two similar classes will be two that have the same +numerical predicate, while two that are not similar will be two +that have different numerical predicates. Such a method of +replacing relations by predicates is formally possible (though +often very inconvenient) so long as the relations concerned are +symmetrical; but it is formally impossible when the relations +are asymmetrical, because both sameness and difference of predicates +are symmetrical. Asymmetrical relations are, we may +<span class="pagenum" id="Page_44">[Pg 44]</span> +say, the most characteristically relational of relations, and the +most important to the philosopher who wishes to study the +ultimate logical nature of relations. +</p> +<p> +Another class of relations that is of the greatest use is the +class of one-many relations, <i>i.e.</i> relations which at most one +term can have to a given term. Such are father, mother, +husband (except in Tibet), square of, sine of, and so on. But +parent, square root, and so on, are not one-many. It is possible, +formally, to replace all relations by one-many relations by means +of a device. Take (say) the relation <i>less</i> among the inductive +numbers. Given any number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> greater than 1, there will not +be only one number having the relation <i>less</i> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> but we can +form the whole class of numbers that are less than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">.</span> This +is one class, and its relation to <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is not shared by any other class. +We may call the class of numbers that are less than <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> the "proper +ancestry" of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> in the sense in which we spoke of ancestry and +posterity in connection with mathematical induction. Then +"proper ancestry" is a one-many relation (<i>one-many</i> will always +be used so as to include <i>one-one</i>), since each number determines +a single class of numbers as constituting its proper ancestry. +Thus the relation <i>less than</i> can be replaced by <i>being a member of +the proper ancestry of</i>. In this way a one-many relation in which +the one is a class, together with membership of this class, can +always formally replace a relation which is not one-many. Peano, +who for some reason always instinctively conceives of a relation +as one-many, deals in this way with those that are naturally +not so. Reduction to one-many relations by this method, +however, though possible as a matter of form, does not represent +a technical simplification, and there is every reason to think +that it does not represent a philosophical analysis, if only because +classes must be regarded as "logical fictions." We shall therefore +continue to regard one-many relations as a special kind of +relations. +</p> +<p> +One-many relations are involved in all phrases of the form +"the so-and-so of such-and-such." "The King of England," +<span class="pagenum" id="Page_45">[Pg 45]</span> +"the wife of Socrates," "the father of John Stuart Mill," and +so on, all describe some person by means of a one-many relation +to a given term. A person cannot have more than one father, +therefore "the father of John Stuart Mill" described some one +person, even if we did not know whom. There is much to +say on the subject of descriptions, but for the present it is +relations that we are concerned with, and descriptions are only +relevant as exemplifying the uses of one-many relations. It +should be observed that all mathematical functions result from +one-many relations: the logarithm of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> the cosine of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> etc., +are, like the father of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> terms described by means of a one-many +relation (logarithm, cosine, etc.) to a given term (<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">)</span>. The +notion of <i>function</i> need not be confined to numbers, or to the +uses to which mathematicians have accustomed us; it can be +extended to all cases of one-many relations, and "the father of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" +is just as legitimately a function of which <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is the argument as +is "the logarithm of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span>" Functions in this sense are <i>descriptive</i> +functions. As we shall see later, there are functions of a still +more general and more fundamental sort, namely, <i>propositional</i> +functions; but for the present we shall confine our attention +to descriptive functions, <i>i.e.</i> "the term having the +relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> +to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span>" or, for short, "the <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span>" where <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is any one-many +relation. +</p> +<p> +It will be observed that if "the <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" is to describe a definite +term, <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> must be a term to which something has the relation <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">,</span> +and there must not be more than one term having the relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> +to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> since "the," correctly used, must imply uniqueness. +Thus we may speak of "the father of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is any human being +except Adam and Eve; but we cannot speak of "the father +of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a table or a chair or anything else that does not +have a father. We shall say that the <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> "exists" when +there is just one term, and no more, having the relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> +Thus if <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is a one-many relation, the <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> exists whenever +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> belongs to the converse domain of <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">,</span> and not otherwise. +Regarding "the <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" as a function in the mathematical +<span class="pagenum" id="Page_46">[Pg 46]</span> +sense, we say that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is the "argument" of the function, and if +<img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is the term which has the relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> <i>i.e.</i> +if <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is the <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> +then <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is the "value" of the function for the argument <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> If +<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is a one-many relation, the range of possible arguments to +the function is the converse domain of <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">,</span> and the range of values +is the domain. Thus the range of possible arguments to the +function "the father of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" is all who have fathers, <i>i.e.</i> the converse +domain of the relation <i>father</i>, while the range of possible +values for the function is all fathers, <i>i.e.</i> the domain of the relation. +</p> +<p> +Many of the most important notions in the logic of relations +are descriptive functions, for example: <i>converse</i>, <i>domain</i>, <i>converse +domain</i>, <i>field</i>. Other examples will occur as we proceed. +</p> +<p> +Among one-many relations, <i>one-one</i> relations are a specially +important class. We have already had occasion to speak of +one-one relations in connection with the definition of number, +but it is necessary to be familiar with them, and not merely +to know their formal definition. Their formal definition may +be derived from that of one-many relations: they may be +defined as one-many relations which are also the converses of +one-many relations, <i>i.e.</i> as relations which are both one-many +and many-one. One-many relations may be defined as relations +such that, if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation in question to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> there is no other +term <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">'</span> which also has the relation to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">.</span> Or, again, they may +be defined as follows: Given two terms <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">'</span>, the terms to +which <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the given relation and those to which <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">'</span> has it have +no member in common. Or, again, they may be defined as +relations such that the relative product of one of them and +its converse implies identity, where the "relative product" +of two relations <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> and <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is that relation which holds between +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> when there is an intermediate term <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> such that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has +the relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> has the relation <img style="vertical-align: -0.025ex; width: 0.891ex; height: 1.038ex;" src="images/91.svg" alt="" data-tex="\mathrm s"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">.</span> Thus, for +example, if <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is the relation of father to son, the relative product +of <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> and its converse will be the relation which holds between +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and a man <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> when there is a person <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> such that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is the father +of <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is the son of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">.</span> It is obvious that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> must be +<span class="pagenum" id="Page_47">[Pg 47]</span> +the same person. If, on the other hand, we take the relation +of parent and child, which is not one-many, we can no longer +argue that, if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a parent of <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is a child of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">,</span> <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> must +be the same person, because one may be the father of <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and the +other the mother. This illustrates that it is characteristic of +one-many relations when the relative product of a relation and +its converse implies identity. In the case of one-one relations +this happens, and also the relative product of the converse and +the relation implies identity. Given a relation <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">,</span> it is convenient, +if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> to think of <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> as being reached from <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> +by an "<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-step" or an "<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-vector." In the same case <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> will +be reached from <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> by a "backward <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-step." Thus we may +state the characteristic of one-many relations with which we +have been dealing by saying that an <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-step followed by a backward +<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">-step must bring us back to our starting-point. With +other relations, this is by no means the case; for example, if +<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is the relation of child to parent, the relative product of <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> and +its converse is the relation "self or brother or sister," and if <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is +the relation of grandchild to grandparent, the relative product +of <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> and its converse is "self or brother or sister or first cousin." +It will be observed that the relative product of two relations +is not in general commutative, <i>i.e.</i> the relative product of <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> +and <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is not in general the same relation as the relative product +of <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> and <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">.</span> <i>E.g.</i> the relative product of parent and brother is +uncle, but the relative product of brother and parent is parent. +</p> +<p> +One-one relations give a correlation of two classes, term for +term, so that each term in either class has its correlate in the +other. Such correlations are simplest to grasp when the two +classes have no members in common, like the class of husbands +and the class of wives; for in that case we know at once whether +a term is to be considered as one <i>from</i> which the correlating +relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> goes, or as one <i>to</i> which it goes. It is convenient +to use the word <i>referent</i> for the term <i>from</i> which the relation +goes, and the term <i>relatum</i> for the term <i>to</i> which it goes. Thus +if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are husband and wife, then, with respect to the relation +<span class="pagenum" id="Page_48">[Pg 48]</span> +"husband," <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is referent and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> relatum, but with respect to the +relation "wife," <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is referent and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> relatum. We say that a +relation and its converse have opposite "senses"; thus the +"sense" of a relation that goes from <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> to <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is the opposite of +that of the corresponding relation from <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> The fact that a +relation has a "sense" is fundamental, and is part of the reason +why order can be generated by suitable relations. It will be +observed that the class of all possible referents to a given relation +is its domain, and the class of all possible relata is its converse +domain. +</p> +<p> +But it very often happens that the domain and converse +domain of a one-one relation overlap. Take, for example, +the first ten integers (excluding 0), and add 1 to each; thus +instead of the first ten integers we now have the integers +<span class="align-center"><img style="vertical-align: -0.439ex; width: 28.348ex; height: 1.971ex;" src="images/10.svg" alt="" data-tex=" +2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 11. +"></span> +These are the same as those we had before, except that 1 has +been cut off at the beginning and 11 has been joined on at the +end. There are still ten integers: they are correlated with +the previous ten by the relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">,</span> which is a one-one +relation. Or, again, instead of adding 1 to each of our original +ten integers, we could have doubled each of them, thus obtaining +the integers +<span class="align-center"><img style="vertical-align: -0.439ex; width: 32.873ex; height: 1.971ex;" src="images/11.svg" alt="" data-tex=" +2,\ 4,\ 6,\ 8,\ 10,\ 12,\ 14,\ 16,\ 18,\ 20. +"></span> +Here we still have five of our previous set of integers, namely, +2, 4, 6, 8, 10. The correlating relation in this case is the relation +of a number to its double, which is again a one-one relation. +Or we might have replaced each number by its square, thus +obtaining the set +<span class="align-center"><img style="vertical-align: -0.439ex; width: 35.136ex; height: 1.971ex;" src="images/12.svg" alt="" data-tex=" +1,\ 4,\ 9,\ 16,\ 25,\ 36,\ 49,\ 64,\ 81,\ 100. +"></span> +On this occasion only three of our original set are left, namely, +1, 4, 9. Such processes of correlation may be varied endlessly. +</p> +<p> +The most interesting case of the above kind is the case where +our one-one relation has a converse domain which is part, but +<span class="pagenum" id="Page_49">[Pg 49]</span> +not the whole, of the domain. If, instead of confining the domain +to the first ten integers, we had considered the whole of the +inductive numbers, the above instances would have illustrated +this case. We may place the numbers concerned in two rows, +putting the correlate directly under the number whose correlate +it is. Thus when the correlator is the relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">,</span> we +have the two rows: +<span class="align-center"><img style="vertical-align: -2.036ex; width: 28.346ex; height: 5.204ex;" src="images/13.svg" alt="" data-tex=" +\begin{align*} +&1,\ 2,\ 3,\ 4,\ 5,\ \dots\ n,\ \dots \\ +&2,\ 3,\ 4,\ 5,\ 6,\ \dots\ n + 1,\ \dots. +\end{align*} +"></span> +When the correlator is the relation of a number to its double, +we have the two rows: +<span class="align-center"><img style="vertical-align: -2.036ex; width: 26.712ex; height: 5.204ex;" src="images/14.svg" alt="" data-tex=" +\begin{align*} +&1,\ 2,\ 3,\ 4,\,\,\,\, 5,\ \dots\ n,\ \dots \\ +&2,\ 4,\ 6,\ 8,\ 10,\ \dots\ 2n,\ \dots. +\end{align*} +"></span> +When the correlator is the relation of a number to its square, +the rows are: +<span class="align-center"><img style="vertical-align: -2.188ex; width: 27.7ex; height: 5.507ex;" src="images/15.svg" alt="" data-tex=" +\begin{align*} +&1,\ 2,\ 3,\ \,4,\ \,\,\,\,5,\ \dots\ n,\ \dots \\ +&1,\ 4,\ 9,\ 16,\ 25,\ \dots\ n^{2},\ \dots. +\end{align*} +"></span> +In all these cases, all inductive numbers occur in the top row, +and only some in the bottom row. +</p> +<p> +Cases of this sort, where the converse domain is a "proper +part" of the domain (<i>i.e.</i> a part not the whole), will occupy us +again when we come to deal with infinity. For the present, we +wish only to note that they exist and demand consideration. +</p> +<p> +Another class of correlations which are often important is +the class called "permutations," where the domain and converse +domain are identical. Consider, for example, the six possible +arrangements of three letters: +<span class="align-center"><img style="vertical-align: -7.919ex; width: 6.919ex; height: 16.968ex;" src="images/16.svg" alt="" data-tex=" +\begin{align*} +a,\ b,\ c; \\ +a,\ c,\ b; \\ +b,\ c,\ a; \\ +b,\ a,\ c; \\ +c,\ a,\ b; \\ +c,\ b,\ a. +\end{align*} +"></span> +<span class="pagenum" id="Page_50">[Pg 50]</span> +Each of these can be obtained from any one of the others by +means of a correlation. Take, for example, the first and last, +<img style="vertical-align: -0.566ex; width: 6.919ex; height: 2.262ex;" src="images/92.svg" alt="" data-tex="(a, b, c)"> and <span class="nowrap"><img style="vertical-align: -0.566ex; width: 6.919ex; height: 2.262ex;" src="images/93.svg" alt="" data-tex="(c, b, a)">.</span> Here <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is correlated with <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">,</span> <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> with itself, +and <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> with <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">.</span> It is obvious that the combination of two permutations +is again a permutation, <i>i.e.</i> the permutations of a given +class form what is called a "group." +</p> +<p> +These various kinds of correlations have importance in various +connections, some for one purpose, some for another. The +general notion of one-one correlations has boundless importance +in the philosophy of mathematics, as we have partly seen already, +but shall see much more fully as we proceed. One of its uses +will occupy us in our next chapter. +<span class="pagenum" id="Page_51">[Pg 51]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='VI: SIMILARITY OF RELATIONS'><a id="chap06"></a>CHAPTER VI +<br><br> +SIMILARITY OF RELATIONS</h2> + +<p class="nind"> +WE saw in Chapter II. that two classes have the same number +of terms when they are "similar," <i>i.e.</i> when there is a one-one +relation whose domain is the one class and whose converse +domain is the other. In such a case we say that there is a +"one-one correlation" between the two classes. +</p> +<p> +In the present chapter we have to define a relation between +relations, which will play the same part for them that similarity +of classes plays for classes. We will call this relation "similarity +of relations," or "likeness" when it seems desirable to use a +different word from that which we use for classes. How is +likeness to be defined? +</p> +<p> +We shall employ still the notion of correlation: we shall +assume that the domain of the one relation can be correlated +with the domain of the other, and the converse domain with the +converse domain; but that is not enough for the sort of resemblance +which we desire to have between our two relations. +What we desire is that, whenever either relation holds between +two terms, the other relation shall hold between the correlates +of these two terms. The easiest example of the sort of thing +we desire is a map. When one place is north of another, the +place on the map corresponding to the one is above the place +on the map corresponding to the other; when one place is west +of another, the place on the map corresponding to the one is +to the left of the place on the map corresponding to the other; +and so on. The structure of the map corresponds with that of +<span class="pagenum" id="Page_52">[Pg 52]</span> +the country of which it is a map. The space-relations in the +map have "likeness" to the space-relations in the country +mapped. It is this kind of connection between relations that +we wish to define. +</p> +<p> +We may, in the first place, profitably introduce a certain +restriction. We will confine ourselves, in defining likeness, to +such relations as have "fields," <i>i.e.</i> to such as permit of the +formation of a single class out of the domain and the converse +domain. This is not always the case. Take, for example, +the relation "domain," <i>i.e.</i> the relation which the domain of a +relation has to the relation. This relation has all classes for its +domain, since every class is the domain of some relation; and +it has all relations for its converse domain, since every relation +has a domain. But classes and relations cannot be added together +to form a new single class, because they are of different +logical "types." We do not need to enter upon the difficult +doctrine of types, but it is well to know when we are abstaining +from entering upon it. We may say, without entering upon +the grounds for the assertion, that a relation only has a "field" +when it is what we call "homogeneous," <i>i.e.</i> when its domain +and converse domain are of the same logical type; and as a +rough-and-ready indication of what we mean by a "type," +we may say that individuals, classes of individuals, relations +between individuals, relations between classes, relations of +classes to individuals, and so on, are different types. Now the +notion of likeness is not very useful as applied to relations that +are not homogeneous; we shall, therefore, in defining likeness, +simplify our problem by speaking of the "field" of one of the +relations concerned. This somewhat limits the generality of +our definition, but the limitation is not of any practical importance. +And having been stated, it need no longer be remembered. +We may define two relations <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> as "similar," or as +having "likeness," when there is a one-one relation <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> whose +domain is the field of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and whose converse domain is the field +of <span class="nowrap"><img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">,</span> and which is such that, if one term has the relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> +<span class="pagenum" id="Page_53">[Pg 53]</span> +to another, the correlate of the one has the relation <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> to the +correlate of the other, and <i>vice versa</i>. +</p> +<p class="nind"><img src="images/figure01.jpg" class='floatleft' alt='fig1'> +A figure will make this +clearer. Let <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> be two +terms having the relation <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">.</span> +Then there are to be two terms +<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;" src="images/96.svg" alt="" data-tex="w">,</span> such that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> +to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">,</span> <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> has the relation <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> +to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;" src="images/96.svg" alt="" data-tex="w">,</span> and <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> has the relation <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> +to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;" src="images/96.svg" alt="" data-tex="w">.</span> If this happens with +every pair of terms such as <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> +and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> and if the converse happens with every pair of terms such +as <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;" src="images/96.svg" alt="" data-tex="w">,</span> it is clear that for every instance in which the relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> +holds there is a corresponding instance in which the relation <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> +holds, and <i>vice versa</i>; and this is what we desire to secure by +our definition. We can eliminate some redundancies in the +above sketch of a definition, by observing that, when the above +conditions are realised, the relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> is the same as the relative +product of <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> and <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> and the converse of <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">,</span> +<i>i.e.</i> the <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">-step from +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> to <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> may be replaced by the succession of the <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">-step from +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">,</span> the <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">-step from <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;" src="images/96.svg" alt="" data-tex="w">,</span> and the backward <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">-step from +<img style="vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;" src="images/96.svg" alt="" data-tex="w"> to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">.</span> Thus we may set up the following definitions:— +</p> +<p> +A relation <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is said to be a "correlator" or an "ordinal +correlator" of two relations <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> if <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is one-one, has the +field of <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> for its converse domain, and is such that <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> is the +relative product of <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> and <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> and the converse of <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">.</span> +</p> +<p> +Two relations <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> are said to be "similar," or to have +"likeness," when there is at least one correlator of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and <span class="nowrap"><img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">.</span> +</p> +<p> +These definitions will be found to yield what we above decided +to be necessary. +</p> +<p> +It will be found that, when two relations are similar, they +share all properties which do not depend upon the actual terms +in their fields. For instance, if one implies diversity, so does +the other; if one is transitive, so is the other; if one is connected, +so is the other. Hence if one is serial, so is the other. +Again, if one is one-many or one-one, the other is one-many +<span class="pagenum" id="Page_54">[Pg 54]</span> +or one-one; and so on, through all the general properties of +relations. Even statements involving the actual terms of the +field of a relation, though they may not be true as they stand +when applied to a similar relation, will always be capable of +translation into statements that are analogous. We are led +by such considerations to a problem which has, in mathematical +philosophy, an importance by no means adequately recognised +hitherto. Our problem may be stated as follows:— +</p> +<p> +Given some statement in a language of which we know the +grammar and the syntax, but not the vocabulary, what are the +possible meanings of such a statement, and what are the meanings +of the unknown words that would make it true? +</p> +<p> +The reason that this question is important is that it represents, +much more nearly than might be supposed, the state of our +knowledge of nature. We know that certain scientific propositions—which, +in the most advanced sciences, are expressed +in mathematical symbols—are more or less true of the world, +but we are very much at sea as to the interpretation to be put +upon the terms which occur in these propositions. We know +much more (to use, for a moment, an old-fashioned pair of +terms) about the <i>form</i> of nature than about the <i>matter</i>. +Accordingly, what we really know when we enunciate a law +of nature is only that there is probably <i>some</i> interpretation of +our terms which will make the law approximately true. Thus +great importance attaches to the question: What are the +possible meanings of a law expressed in terms of which we do +not know the substantive meaning, but only the grammar and +syntax? And this question is the one suggested above. +</p> +<p> +For the present we will ignore the general question, which +will occupy us again at a later stage; the subject of likeness +itself must first be further investigated. +</p> +<p> +Owing to the fact that, when two relations are similar, their +properties are the same except when they depend upon the +fields being composed of just the terms of which they are composed, +it is desirable to have a nomenclature which collects +<span class="pagenum" id="Page_55">[Pg 55]</span> +together all the relations that are similar to a given relation. +Just as we called the set of those classes that are similar to a +given class the "number" of that class, so we may call the set +of all those relations that are similar to a given relation the +"number" of that relation. But in order to avoid confusion with +the numbers appropriate to classes, we will speak, in this case, of +a "relation-number." Thus we have the following definitions:— +</p> +<p> +The "relation-number" of a given relation is the class of all +those relations that are similar to the given relation. +</p> +<p> +"Relation-numbers" are the set of all those classes of relations +that are relation-numbers of various relations; or, what comes to +the same thing, a relation number is a class of relations consisting +of all those relations that are similar to one member of the class. +</p> +<p> +When it is necessary to speak of the numbers of classes in +a way which makes it impossible to confuse them with relation-numbers, +we shall call them "cardinal numbers." Thus cardinal +numbers are the numbers appropriate to classes. These include +the ordinary integers of daily life, and also certain infinite +numbers, of which we shall speak later. When we speak of +"numbers" without qualification, we are to be understood as +meaning <i>cardinal</i> numbers. The definition of a cardinal number, +it will be remembered, is as follows:— +</p> +<p> +The "cardinal number" of a given class is the set of all +those classes that are similar to the given class. +</p> +<p> +The most obvious application of relation-numbers is to <i>series</i>. +Two series may be regarded as equally long when they have +the same relation-number. Two <i>finite</i> series will have the +same relation-number when their fields have the same cardinal +number of terms, and only then—<i>i.e.</i> a series of (say) 15 terms +will have the same relation-number as any other series of fifteen +terms, but will not have the same relation-number as a series +of 14 or 16 terms, nor, of course, the same relation-number +as a relation which is not serial. Thus, in the quite special case +of finite series, there is parallelism between cardinal and relation-numbers. +The relation-numbers applicable to series may be +<span class="pagenum" id="Page_56">[Pg 56]</span> +called "serial numbers" (what are commonly called "ordinal +numbers" are a sub-class of these); thus a finite serial number +is determinate when we know the cardinal number of terms +in the field of a series having the serial number in question. +If <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is a finite cardinal number, the relation-number of a series +which has <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> terms is called the "ordinal" number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">.</span> (There +are also infinite ordinal numbers, but of them we shall speak +in a later chapter.) When the cardinal number of terms in +the field of a series is infinite, the relation-number of the series +is not determined merely by the cardinal number, indeed an +infinite number of relation-numbers exist for one infinite cardinal +number, as we shall see when we come to consider infinite series. +When a series is infinite, what we may call its "length," <i>i.e.</i> +its relation-number, may vary without change in the cardinal +number; but when a series is finite, this cannot happen. +</p> +<p> +We can define addition and multiplication for relation-numbers +as well as for cardinal numbers, and a whole arithmetic +of relation-numbers can be developed. The manner in which +this is to be done is easily seen by considering the case of series. +Suppose, for example, that we wish to define the sum of two +non-overlapping series in such a way that the relation-number +of the sum shall be capable of being defined as the sum of the +relation-numbers of the two series. In the first place, it is clear +that there is an <i>order</i> involved as between the two series: one +of them must be placed before the other. Thus if <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> +are the generating relations of the two series, in the series which +is their sum with <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> put before <span class="nowrap"><img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">,</span> every member of the field of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> +will precede every member of the field of <span class="nowrap"><img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">.</span> Thus the serial +relation which is to be defined as the sum of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> is not +"<img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> or <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">" simply, but "<img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> or <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> or the relation of any member +of the field of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> to any member of the field of <span class="nowrap"><img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">.</span>" Assuming +that <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> do not overlap, this relation is serial, +but "<img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> or <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">" +is not serial, being not connected, since it does not hold between +a member of the field of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and a member of the field of <span class="nowrap"><img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">.</span> Thus +the sum of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and <span class="nowrap"><img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">,</span> as above defined, is what we need in order +<span class="pagenum" id="Page_57">[Pg 57]</span> +to define the sum of two relation-numbers. Similar modifications +are needed for products and powers. The resulting arithmetic +does not obey the commutative law: the sum or product +which they are taken. But it obeys the associative law, one +form of the distributive law, and two of the formal laws for +powers, not only as applied to serial numbers, but as applied to +relation-numbers generally. Relation-arithmetic, in fact, though +recent, is a thoroughly respectable branch of mathematics. +</p> +<p> +It must not be supposed, merely because series afford the +most obvious application of the idea of likeness, that there are +no other applications that are important. We have already +mentioned maps, and we might extend our thoughts from this +illustration to geometry generally. If the system of relations +by which a geometry is applied to a certain set of terms can be +brought fully into relations of likeness with a system applying +to another set of terms, then the geometry of the two sets is +indistinguishable from the mathematical point of view, <i>i.e.</i> all +the propositions are the same, except for the fact that they are +applied in one case to one set of terms and in the other to another. +We may illustrate this by the relations of the sort that may be +called "between," which we considered in Chapter IV. We +there saw that, provided a three-term relation has certain formal +logical properties, it will give rise to series, and may be called +a "between-relation." Given any two points, we can use the +between-relation to define the straight line determined by those +two points; it consists of <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> together with all points <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> +such that the between-relation holds between the three points +<span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> in some order or other. It has been shown by O. Veblen +that we may regard our whole space as the field of a three-term +between-relation, and define our geometry by the properties we +assign to our between-relation.<a id="FNanchor_13_1"></a><a href="#Footnote_13_1" class="fnanchor">[13]</a> +Now likeness is just as easily +<span class="pagenum" id="Page_58">[Pg 58]</span> +definable between three-term relations as between two-term +relations. If <img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B"> and <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">'</span> are two between-relations, so that +"<img style="vertical-align: -0.566ex; width: 7.823ex; height: 2.262ex;" src="images/97.svg" alt="" data-tex="x\mathrm B(y, z)">" means "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is between <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> with respect to <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">,</span>" +we shall call <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> a correlator of <img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B"> and <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">'</span> if +it has the field of <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">'</span> +for its converse domain, and is such that the relation <img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B"> holds +between three terms when <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">'</span> holds between their <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">-correlates, +and only then. And we shall say that <img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B"> is like <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">'</span> when there +is at least one correlator of <img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B"> with <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">'</span>. The reader can easily +convince himself that, if <img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B"> is like <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">'</span> in this sense, there can be +no difference between the geometry generated by<img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B"> and that +generated by <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">'</span>. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_13_1"></a><a href="#FNanchor_13_1"><span class="label">[13]</span></a>This does not apply to elliptic space, but only to spaces in which +the straight line is an open series. <i>Modern Mathematics</i>, edited by +J. W. A. Young, pp. 3-51 (monograph by O. Veblen on "The Foundations of +Geometry").</p></div> + +<p> +It follows from this that the mathematician need not concern +himself with the particular being or intrinsic nature of his points, +lines, and planes, even when he is speculating as an <i>applied</i> +mathematician. We may say that there is empirical evidence +of the approximate truth of such parts of geometry as are not +matters of definition. But there is no empirical evidence as to +what a "point" is to be. It has to be something that as nearly +as possible satisfies our axioms, but it does not have to be "very +small" or "without parts." Whether or not it is those things +is a matter of indifference, so long as it satisfies the axioms. If +we can, out of empirical material, construct a logical structure, +no matter how complicated, which will satisfy our geometrical +axioms, that structure may legitimately be called a "point." +We must not say that there is nothing else that could legitimately +be called a "point"; we must only say: "This object we have +constructed is sufficient for the geometer; it may be one of +many objects, any of which would be sufficient, but that is no +concern of ours, since this object is enough to vindicate the +empirical truth of geometry, in so far as geometry is not a +matter of definition." This is only an illustration of the general +principle that what matters in mathematics, and to a very great +extent in physical science, is not the intrinsic nature of our +terms, but the logical nature of their interrelations. +</p> +<p> +We may say, of two similar relations, that they have the same +<span class="pagenum" id="Page_59">[Pg 59]</span> +"structure." For mathematical purposes (though not for those +of pure philosophy) the only thing of importance about a relation +is the cases in which it holds, not its intrinsic nature. Just as a +class may be defined by various different but co-extensive concepts—<i>e.g.</i> +"man" and "featherless biped,"—so two relations which +are conceptually different may hold in the same set of instances. +An "instance" in which a relation holds is to be conceived as a +couple of terms, with an order, so that one of the terms comes +first and the other second; the couple is to be, of course, +such that its first term has the relation in question to its second. +Take (say) the relation "father": we can define what we may +call the "extension" of this relation as the class of all ordered +couples <img style="vertical-align: -0.566ex; width: 5.169ex; height: 2.262ex;" src="images/89.svg" alt="" data-tex="(x, y)"> which are such that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is the father of <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">.</span> From +the mathematical point of view, the only thing of importance +about the relation "father" is that it defines this set of ordered +couples. Speaking generally, we say: +</p> +<p> +The "extension" of a relation is the class of those ordered +couples <img style="vertical-align: -0.566ex; width: 5.169ex; height: 2.262ex;" src="images/89.svg" alt="" data-tex="(x, y)"> which are such that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation in question +to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">.</span> +</p> +<p> +We can now go a step further in the process of abstraction, +and consider what we mean by "structure." Given any relation, +we can, if it is a sufficiently simple one, construct a map of it. +For the sake of definiteness, let us take a relation of which the +extension is the following couples: <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.167ex; height: 1.595ex;" src="images/98.svg" alt="" data-tex="ab">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.176ex; height: 1.025ex;" src="images/99.svg" alt="" data-tex="ac">,</span> <span class="nowrap"><img style="vertical-align: -0.023ex; width: 2.373ex; height: 1.593ex;" src="images/100.svg" alt="" data-tex="ad">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.95ex; height: 1.595ex;" src="images/101.svg" alt="" data-tex="bc">,</span> +<span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.034ex; height: 1.025ex;" src="images/102.svg" alt="" data-tex="ce">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.156ex; height: 1.595ex;" src="images/103.svg" alt="" data-tex="dc">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.231ex; height: 1.595ex;" src="images/104.svg" alt="" data-tex="de">,</span> where +<span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">,</span> <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;" src="images/105.svg" alt="" data-tex="d">,</span> <img style="vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;" src="images/106.svg" alt="" data-tex="e"> are five terms, no matter what. +</p> +<p class="nind"><img src="images/figure02.jpg" class='floatleft' alt='fig2'> +We may make a +"map" of this relation by taking five points +on a plane and connecting them by arrows, +as in the accompanying figure. What is +revealed by the map is what we call the +"structure" of the relation. +</p> +<p> +It is clear that the "structure" of the +relation does not depend upon the particular +terms that make up the field of the relation. +The field may be changed without changing the structure, and +the structure may be changed without changing the field—for +<span class="pagenum" id="Page_60">[Pg 60]</span> +example, if we were to add the couple <img style="vertical-align: -0.025ex; width: 2.251ex; height: 1.025ex;" src="images/107.svg" alt="" data-tex="ae"> in the above illustration +we should alter the structure but not the field. Two relations +have the same "structure," we shall say, when the same map +will do for both—or, what comes to the same thing, when either +can be a map for the other (since every relation can be its own +map). And that, as a moment's reflection shows, is the very +same thing as what we have called "likeness." That is to say, +two relations have the same structure when they have likeness, +<i>i.e.</i> when they have the same relation-number. Thus what we +defined as the "relation-number" is the very same thing as is +obscurely intended by the word "structure"—a word which, +important as it is, is never (so far as we know) defined in precise +terms by those who use it. +</p> +<p> +There has been a great deal of speculation in traditional +philosophy which might have been avoided if the importance of +structure, and the difficulty of getting behind it, had been realised. +For example, it is often said that space and time are subjective, +but they have objective counterparts; or that phenomena are +subjective, but are caused by things in themselves, which must +have differences <i>inter se</i> corresponding with the differences in +the phenomena to which they give rise. Where such hypotheses +are made, it is generally supposed that we can know very little +about the objective counterparts. In actual fact, however, if +the hypotheses as stated were correct, the objective counterparts +would form a world having the same structure as the phenomenal +world, and allowing us to infer from phenomena the truth of all +propositions that can be stated in abstract terms and are known +to be true of phenomena. If the phenomenal world has three +dimensions, so must the world behind phenomena; if the phenomenal +world is Euclidean, so must the other be; and so on. +In short, every proposition having a communicable significance +must be true of both worlds or of neither: the only difference +must lie in just that essence of individuality which always eludes +words and baffles description, but which, for that very reason, +is irrelevant to science. Now the only purpose that philosophers +<span class="pagenum" id="Page_61">[Pg 61]</span> +have in view in condemning phenomena is in order to persuade +themselves and others that the real world is very different from +the world of appearance. We can all sympathise with their wish +to prove such a very desirable proposition, but we cannot congratulate +them on their success. It is true that many of them +do not assert objective counterparts to phenomena, and these +escape from the above argument. Those who do assert counterparts +are, as a rule, very reticent on the subject, probably because +they feel instinctively that, if pursued, it will bring about too +much of a <i>rapprochement</i> between the real and the phenomenal +world. If they were to pursue the topic, they could hardly avoid +the conclusions which we have been suggesting. In such ways, +as well as in many others, the notion of structure or relation-number +is important. +<span class="pagenum" id="Page_62">[Pg 62]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='VII: RATIONAL, REAL, AND COMPLEX NUMBERS'><a id="chap07"></a>CHAPTER VII +<br><br> +RATIONAL, REAL, AND COMPLEX NUMBERS</h2> + +<p class="nind"> +WE have now seen how to define cardinal numbers, and also +relation-numbers, of which what are commonly called ordinal +numbers are a particular species. It will be found that each +of these kinds of number may be infinite just as well as finite. +But neither is capable, as it stands, of the more familiar extensions +of the idea of number, namely, the extensions to negative, +fractional, irrational, and complex numbers. In the present +chapter we shall briefly supply logical definitions of these various +extensions. +</p> +<p> +One of the mistakes that have delayed the discovery of correct +definitions in this region is the common idea that each extension +of number included the previous sorts as special cases. It was +thought that, in dealing with positive and negative integers, the +positive integers might be identified with the original signless +integers. Again it was thought that a fraction whose denominator +is 1 may be identified with the natural number which is its +numerator. And the irrational numbers, such as the square +root of 2, were supposed to find their place among rational fractions, +as being greater than some of them and less than the others, +so that rational and irrational numbers could be taken together +as one class, called "real numbers." And when the idea of +number was further extended so as to include "complex" +numbers, <i>i.e.</i> numbers involving the square root of -1, it was +thought that real numbers could be regarded as those among +complex numbers in which the imaginary part (<i>i.e.</i> the part +<span class="pagenum" id="Page_63">[Pg 63]</span> +which was a multiple of the square root of -1) was zero. All +these suppositions were erroneous, and must be discarded, as we +shall find, if correct definitions are to be given. +</p> +<p> +Let us begin with <i>positive and negative integers</i>. It is obvious +on a moment's consideration that +1 and -1 must both be +relations, and in fact must be each other's converses. The +obvious and sufficient definition is that +1 is the relation of +<img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> and -1 is the relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">.</span> +Generally, if <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> is +any inductive number, <img style="vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;" src="images/108.svg" alt="" data-tex="+m"> will be the relation of <img style="vertical-align: -0.186ex; width: 6.11ex; height: 1.505ex;" src="images/109.svg" alt="" data-tex="n + m"> to <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> +(for any <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">)</span>, and <img style="vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;" src="images/110.svg" alt="" data-tex="-m"> will be the relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 6.11ex; height: 1.505ex;" src="images/109.svg" alt="" data-tex="n + m">.</span> According +to this definition, <img style="vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;" src="images/108.svg" alt="" data-tex="+m"> is a relation which is one-one so +long as <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is a cardinal number (finite or infinite) and <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> is an +inductive cardinal number. But <img style="vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;" src="images/108.svg" alt="" data-tex="+m"> is under no circumstances +capable of being identified with <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m">,</span> which is not a relation, but +a class of classes. Indeed, <img style="vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;" src="images/108.svg" alt="" data-tex="+m"> is every bit as distinct from <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> +as <img style="vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;" src="images/110.svg" alt="" data-tex="-m"> is. +</p> +<p> +<i>Fractions</i> are more interesting than positive or negative integers. +We need fractions for many purposes, but perhaps most obviously +for purposes of measurement. My friend and collaborator Dr +A. N. Whitehead has developed a theory of fractions specially +adapted for their application to measurement, which is set forth +in <i>Principia Mathematica</i>.<a id="FNanchor_14_1"></a><a href="#Footnote_14_1" class="fnanchor">[14]</a> +But if all that is needed is to define +objects having the required purely mathematical properties, this +purpose can be achieved by a simpler method, which we shall +here adopt. We shall define the fraction <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> as being that +relation which holds between two inductive numbers <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> when +<span class="nowrap"><img style="vertical-align: -0.464ex; width: 8.764ex; height: 1.783ex;" src="images/112.svg" alt="" data-tex="xn = ym">.</span> This definition enables us to prove that <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> is a one-one +relation, provided neither <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> or <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is zero. And of course <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/113.svg" alt="" data-tex="n/m"> is +the converse relation to <span class="nowrap"><img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n">.</span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_14_1"></a><a href="#FNanchor_14_1"><span class="label">[14]</span></a>Vol. III. * 300 ff., especially 303.</p></div> + +<p> +From the above definition it is clear that the fraction <img style="vertical-align: -0.566ex; width: 4.249ex; height: 2.262ex;" src="images/114.svg" alt="" data-tex="m/1"> is +that relation between two integers <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> which consists in the +fact that <span class="nowrap"><img style="vertical-align: -0.464ex; width: 7.406ex; height: 1.783ex;" src="images/115.svg" alt="" data-tex="x = my">.</span> This relation, like the relation <span class="nowrap"><img style="vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;" src="images/108.svg" alt="" data-tex="+m">,</span> is by no +means capable of being identified with the inductive cardinal +number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m">,</span> because a relation and a class of classes are objects +<span class="pagenum" id="Page_64">[Pg 64]</span> +of utterly different kinds.<a id="FNanchor_15_1"></a><a href="#Footnote_15_1" class="fnanchor">[15]</a> +It will be seen that <img style="vertical-align: -0.566ex; width: 3.62ex; height: 2.262ex;" src="images/116.svg" alt="" data-tex="0/n"> is always the +same relation, whatever inductive number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> may be; it is, in short, +the relation of 0 to any other inductive cardinal. We may call +this the zero of rational numbers; it is not, of course, identical +with the cardinal number 0. Conversely, the relation <img style="vertical-align: -0.566ex; width: 4.249ex; height: 2.262ex;" src="images/117.svg" alt="" data-tex="m/0"> is +always the same, whatever inductive number <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> may be. There +is not any inductive cardinal to correspond to <span class="nowrap"><img style="vertical-align: -0.566ex; width: 4.249ex; height: 2.262ex;" src="images/117.svg" alt="" data-tex="m/0">.</span> We may call +it "the infinity of rationals." It is an instance of the sort of +infinite that is traditional in mathematics, and that is represented +by "<span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.262ex; height: 1.025ex;" src="images/118.svg" alt="" data-tex="\infty">.</span>" This is a totally different sort from the true Cantorian +infinite, which we shall consider in our next chapter. The infinity +of rationals does not demand, for its definition or use, any +infinite classes or infinite integers. It is not, in actual fact, a +very important notion, and we could dispense with it altogether +if there were any object in doing so. The Cantorian infinite, on +the other hand, is of the greatest and most fundamental importance; +the understanding of it opens the way to whole new realms +of mathematics and philosophy. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_15_1"></a><a href="#FNanchor_15_1"><span class="label">[15]</span></a>Of course in practice we shall continue to speak of a fraction as (say) +greater or less than 1, meaning greater or less than the ratio <span class="nowrap"><img style="vertical-align: -0.566ex; width: 3.394ex; height: 2.262ex;" src="images/119.svg" alt="" data-tex="1/1">.</span> So +long as it is understood that the ratio <img style="vertical-align: -0.566ex; width: 3.394ex; height: 2.262ex;" src="images/119.svg" alt="" data-tex="1/1"> and the cardinal number 1 are +different, it is not necessary to be always pedantic in emphasising the +difference.</p></div> + +<p> +It will be observed that zero and infinity, alone among ratios, +are not one-one. Zero is one-many, and infinity is many-one. +</p> +<p> +There is not any difficulty in defining <i>greater</i> and <i>less</i> among +ratios (or fractions). Given two ratios <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> and <span class="nowrap"><img style="vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;" src="images/120.svg" alt="" data-tex="p/q">,</span> we shall say +that <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> is <i>less</i> than <img style="vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;" src="images/120.svg" alt="" data-tex="p/q"> if <img style="vertical-align: -0.439ex; width: 3.027ex; height: 1.439ex;" src="images/121.svg" alt="" data-tex="mq"> is less than <span class="nowrap"><img style="vertical-align: -0.439ex; width: 2.495ex; height: 1.439ex;" src="images/122.svg" alt="" data-tex="pn">.</span> There is no +difficulty in proving that the relation "less than," so defined, is +serial, so that the ratios form a series in order of magnitude. In +this series, zero is the smallest term and infinity is the largest. +If we omit zero and infinity from our series, there is no longer +any smallest or largest ratio; it is obvious that if <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> is any ratio +other than zero and infinity, <img style="vertical-align: -0.566ex; width: 5.606ex; height: 2.262ex;" src="images/123.svg" alt="" data-tex="m/2n"> is smaller and <img style="vertical-align: -0.566ex; width: 5.606ex; height: 2.262ex;" src="images/124.svg" alt="" data-tex="2m/n"> is larger, +though neither is zero or infinity, so that <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> is neither the smallest +<span class="pagenum" id="Page_65">[Pg 65]</span> +nor the largest ratio, and therefore (when zero and infinity are +omitted) there is no smallest or largest, since <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> was chosen +arbitrarily. In like manner we can prove that however nearly +equal two fractions may be, there are always other fractions +between them. For, let <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> and <img style="vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;" src="images/120.svg" alt="" data-tex="p/q"> be two fractions, of which +<img style="vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;" src="images/120.svg" alt="" data-tex="p/q">is the greater. Then it is easy to see (or to prove) that +<img style="vertical-align: -0.566ex; width: 15.706ex; height: 2.262ex;" src="images/125.svg" alt="" data-tex="(m + p)/(n + q)"> will be greater than <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> and less than <span class="nowrap"><img style="vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;" src="images/120.svg" alt="" data-tex="p/q">.</span> Thus +the series of ratios is one in which no two terms are consecutive, +but there are always other terms between any two. Since there +are other terms between these others, and so on <i>ad infinitum</i>, it +is obvious that there are an infinite number of ratios between +any two, however nearly equal these two may be.<a id="FNanchor_16_1"></a><a href="#Footnote_16_1" class="fnanchor">[16]</a> +A series having the property that there are always other terms between +any two, so that no two are consecutive, is called "compact." +Thus the ratios in order of magnitude form a "compact" series. +Such series have many important properties, and it is important +to observe that ratios afford an instance of a compact series +generated purely logically, without any appeal to space or time +or any other empirical datum. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_16_1"></a><a href="#FNanchor_16_1"><span class="label">[16]</span></a>Strictly speaking, this statement, as well as those following to the end +of the paragraph, involves what is called the "axiom of infinity," which +will be discussed in a later chapter.</p></div> + +<p> +Positive and negative ratios can be defined in a way analogous +to that in which we defined positive and negative integers. +Having first defined the sum of two ratios <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> and <img style="vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;" src="images/120.svg" alt="" data-tex="p/q"> as +<span class="nowrap"><img style="vertical-align: -0.566ex; width: 13.578ex; height: 2.262ex;" src="images/126.svg" alt="" data-tex="(mq + pn)/nq">,</span> we define <img style="vertical-align: -0.566ex; width: 5.07ex; height: 2.262ex;" src="images/127.svg" alt="" data-tex="+p/q"> as the relation of <img style="vertical-align: -0.566ex; width: 10.551ex; height: 2.262ex;" src="images/128.svg" alt="" data-tex="m/n + p/q"> to <span class="nowrap"><img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n">,</span> +where <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> is any ratio; and <img style="vertical-align: -0.566ex; width: 5.07ex; height: 2.262ex;" src="images/129.svg" alt="" data-tex="-p/q"> is of course the converse of <span class="nowrap"><img style="vertical-align: -0.566ex; width: 5.07ex; height: 2.262ex;" src="images/127.svg" alt="" data-tex="+p/q">.</span> +This is not the only possible way of defining positive and +negative ratios, but it is a way which, for our purpose, has the +merit of being an obvious adaptation of the way we adopted in +the case of integers. +</p> +<p> +We come now to a more interesting extension of the idea of +number, <i>i.e.</i> the extension to what are called "real" numbers, +which are the kind that embrace irrationals. In Chapter I. we +had occasion to mention "incommensurables" and their discovery +<span class="pagenum" id="Page_66">[Pg 66]</span> +by Pythagoras. It was through them, <i>i.e.</i> through +geometry, that irrational numbers were first thought of. A +square of which the side is one inch long will have a diagonal of +which the length is the square root of 2 inches. But, as the +ancients discovered, there is no fraction of which the square is 2. +This proposition is proved in the tenth book of Euclid, which is +one of those books that schoolboys supposed to be fortunately lost +in the days when Euclid was still used as a text-book. The proof +is extraordinarily simple. If possible, let <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> be the square root +of 2, so that <span class="nowrap"><img style="vertical-align: -0.566ex; width: 10.599ex; height: 2.452ex;" src="images/130.svg" alt="" data-tex="m^{2}/n^{2} = 2">,</span> <i>i.e.</i> <span class="nowrap"><img style="vertical-align: -0.186ex; width: 9.468ex; height: 2.072ex;" src="images/131.svg" alt="" data-tex="m^{2} = 2n^{2}">.</span> +Thus <img style="vertical-align: -0.025ex; width: 2.974ex; height: 1.912ex;" src="images/132.svg" alt="" data-tex="m^{2}"> is an even number, +and therefore <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> must be an even number, because the square of +an odd number is odd. Now if <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> is even, <img style="vertical-align: -0.025ex; width: 2.974ex; height: 1.912ex;" src="images/132.svg" alt="" data-tex="m^{2}"> must divide by 4, +for if <span class="nowrap"><img style="vertical-align: -0.439ex; width: 7.273ex; height: 1.946ex;" src="images/133.svg" alt="" data-tex="m = 2p">,</span> then <span class="nowrap"><img style="vertical-align: -0.439ex; width: 9.248ex; height: 2.326ex;" src="images/134.svg" alt="" data-tex="m^{2} = 4p^{2}">.</span> Thus we shall have <span class="nowrap"><img style="vertical-align: -0.439ex; width: 9.75ex; height: 2.326ex;" src="images/135.svg" alt="" data-tex="4p^{2} = 2n^{2}">,</span> where +<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is half of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m">.</span> Hence <span class="nowrap"><img style="vertical-align: -0.439ex; width: 8.619ex; height: 2.326ex;" src="images/137.svg" alt="" data-tex="2p^{2} = n^{2}">,</span> and therefore <img style="vertical-align: -0.566ex; width: 3.627ex; height: 2.262ex;" src="images/138.svg" alt="" data-tex="n/p"> will also be the +square root of 2. But then we can repeat the argument: if +<span class="nowrap"><img style="vertical-align: -0.439ex; width: 6.547ex; height: 1.946ex;" src="images/139.svg" alt="" data-tex="n = 2q">,</span> <img style="vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;" src="images/120.svg" alt="" data-tex="p/q"> will also be the square root of 2, and so on, through +an unending series of numbers that are each half of its predecessor. +But this is impossible; if we divide a number by 2, and then +halve the half, and so on, we must reach an odd number after a +finite number of steps. Or we may put the argument even more +simply by assuming that the <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> we start with is in its lowest +terms; in that case, <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> and <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> cannot both be even; yet we have +seen that, if <span class="nowrap"><img style="vertical-align: -0.566ex; width: 10.599ex; height: 2.452ex;" src="images/130.svg" alt="" data-tex="m^{2}/n^{2} = 2">,</span> they must be. Thus there cannot be any +fraction <img style="vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;" src="images/111.svg" alt="" data-tex="m/n"> whose square is 2. +</p> +<p> +Thus no fraction will express exactly the length of the diagonal +of a square whose side is one inch long. This seems like a +challenge thrown out by nature to arithmetic. However the +arithmetician may boast (as Pythagoras did) about the power +of numbers, nature seems able to baffle him by exhibiting lengths +which no numbers can estimate in terms of the unit. But the +problem did not remain in this geometrical form. As soon as +algebra was invented, the same problem arose as regards the +solution of equations, though here it took on a wider form, +since it also involved complex numbers. +</p> +<p> +It is clear that fractions can be found which approach nearer +<span class="pagenum" id="Page_67">[Pg 67]</span> +and nearer to having their square equal to 2. We can form an +ascending series of fractions all of which have their squares +less than 2, but differing from 2 in their later members by +less than any assigned amount. That is to say, suppose I assign +some small amount in advance, say one-billionth, it will be +found that all the terms of our series after a certain one, say the +tenth, have squares that differ from 2 by less than this amount. +And if I had assigned a still smaller amount, it might have been +necessary to go further along the series, but we should have +reached sooner or later a term in the series, say the twentieth, +after which all terms would have had squares differing from 2 +by less than this still smaller amount. If we set to work to +extract the square root of 2 by the usual arithmetical rule, we +shall obtain an unending decimal which, taken to so-and-so +many places, exactly fulfils the above conditions. We can +equally well form a descending series of fractions whose squares +are all greater than 2, but greater by continually smaller amounts +as we come to later terms of the series, and differing, sooner or +later, by less than any assigned amount. In this way we seem +to be drawing a cordon round the square root of 2, and it may +seem difficult to believe that it can permanently escape us. +Nevertheless, it is not by this method that we shall actually +reach the square root of 2. +</p> +<p> +If we divide <i>all</i> ratios into two classes, according as their +squares are less than 2 or not, we find that, among those whose +squares are <i>not</i> less than 2, all have their squares greater than 2. +There is no maximum to the ratios whose square is less than 2, +and no minimum to those whose square is greater than 2. There +is no lower limit short of zero to the difference between the +numbers whose square is a little less than 2 and the numbers +whose square is a little greater than 2. We can, in short, divide +<i>all</i> ratios into two classes such that all the terms in one class +are less than all in the other, there is no maximum to the one +class, and there is no minimum to the other. Between these +two classes, where <img style="vertical-align: -0.225ex; width: 3.061ex; height: 2.398ex;" src="images/140.svg" alt="" data-tex="\sqrt{2}"> ought to be, there is nothing. Thus our +<span class="pagenum" id="Page_68">[Pg 68]</span> +cordon, though we have drawn it as tight as possible, has been +drawn in the wrong place, and has not caught <span class="nowrap"><img style="vertical-align: -0.225ex; width: 3.061ex; height: 2.398ex;" src="images/140.svg" alt="" data-tex="\sqrt{2}">.</span> +</p> +<p> +The above method of dividing all the terms of a series into +two classes, of which the one wholly precedes the other, was +brought into prominence by Dedekind,<a id="FNanchor_17_1"></a><a href="#Footnote_17_1" class="fnanchor">[17]</a> +and is therefore called +a "Dedekind cut." With respect to what happens at the point +of section, there are four possibilities: (1) there may be a +maximum to the lower section and a minimum to the upper +section, (2) there may be a maximum to the one and no minimum +to the other, (3) there may be no maximum to the one, but a +minimum to the other, (4) there may be neither a maximum to +the one nor a minimum to the other. Of these four cases, the +first is illustrated by any series in which there are consecutive +terms: in the series of integers, for instance, a lower section +must end with some number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> and the upper section must +then begin with <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">.</span> The second case will be illustrated +in the series of ratios if we take as our lower section all ratios +up to and including 1, and in our upper section all ratios greater +than 1. The third case is illustrated if we take for our lower +section all ratios less than 1, and for our upper section all ratios +from 1 upward (including 1 itself). The fourth case, as we have +seen, is illustrated if we put in our lower section all ratios whose +square is less than 2, and in our upper section all ratios whose +square is greater than 2. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_17_1"></a><a href="#FNanchor_17_1"><span class="label">[17]</span></a><i>Stetigkeit und irrationale Zahlen</i>, 2nd edition, Brunswick, 1892.</p></div> + +<p> +We may neglect the first of our four cases, since it only arises +in series where there are consecutive terms. In the second of +our four cases, we say that the maximum of the lower section +is the <i>lower limit</i> of the upper section, or of any set of terms +chosen out of the upper section in such a way that no term of +the upper section is before all of them. In the third of our +four cases, we say that the minimum of the upper section is the +upper limit of the lower section, or of any set of terms chosen +out of the lower section in such a way that no term of the lower +section is after all of them. In the fourth case, we say that +<span class="pagenum" id="Page_69">[Pg 69]</span> +there is a "gap": neither the upper section nor the lower has +a limit or a last term. In this case, we may also say that we +have an "irrational section," since sections of the series of ratios +have "gaps" when they correspond to irrationals. +</p> +<p> +What delayed the true theory of irrationals was a mistaken +belief that there must be "limits" of series of ratios. The +notion of "limit" is of the utmost importance, and before +proceeding further it will be well to define it. +</p> +<p> +A term <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is said to be an "upper limit" of a class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> with +respect to a relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> if (1) <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> has no maximum +in <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span> (2) every +member of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> which belongs to the field of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> precedes <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> (3) every +member of the field of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> which precedes <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> precedes some member +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> (By "precedes" we mean "has the relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> to.") +</p> +<p> +This presupposes the following definition of a "maximum":— +</p> +<p> +A term <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is said to be a "maximum" of a class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> with respect +to a relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a member of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and +of the field of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and does +not have the relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> to any other member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> +</p> +<p> +These definitions do not demand that the terms to which +they are applied should be quantitative. For example, given +a series of moments of time arranged by earlier and later, their +"maximum" (if any) will be the last of the moments; but if +they are arranged by later and earlier, their "maximum" (if +any) will be the first of the moments. +</p> +<p> +The "minimum" of a class with respect to <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> is its maximum +with respect to the converse of <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">;</span> and the "lower limit" with +respect to <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> is the upper limit with respect to the +converse of <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">.</span> +</p> +<p> +The notions of limit and maximum do not essentially demand +that the relation in respect to which they are defined should +be serial, but they have few important applications except to +cases when the relation is serial or quasi-serial. A notion which +is often important is the notion "upper limit or maximum," +to which we may give the name "upper boundary." Thus the +"upper boundary" of a set of terms chosen out of a series is +their last member if they have one, but, if not, it is the first +term after all of them, if there is such a term. If there is neither +<span class="pagenum" id="Page_70">[Pg 70]</span> +a maximum nor a limit, there is no upper boundary. The +"lower boundary" is the lower limit or minimum. +</p> +<p> +Reverting to the four kinds of Dedekind section, we see that +in the case of the first three kinds each section has a boundary +(upper or lower as the case may be), while in the fourth kind +neither has a boundary. It is also clear that, whenever the +lower section has an upper boundary, the upper section has +a lower boundary. In the second and third cases, the two +boundaries are identical; in the first, they are consecutive +terms of the series. +</p> +<p> +A series is called "Dedekindian" when every section has a +boundary, upper or lower as the case may be. +</p> +<p> +We have seen that the series of ratios in order of magnitude +is not Dedekindian. +</p> +<p> +From the habit of being influenced by spatial imagination, +people have supposed that series <i>must</i> have limits in cases where +it seems odd if they do not. Thus, perceiving that there was +no <i>rational</i> limit to the ratios whose square is less than 2, they +allowed themselves to "postulate" an <i>irrational</i> limit, which +was to fill the Dedekind gap. Dedekind, in the above-mentioned +work, set up the axiom that the gap must always be filled, <i>i.e.</i> +that every section must have a boundary. It is for this reason +that series where his axiom is verified are called "Dedekindian." +But there are an infinite number of series for which it is not +verified. +</p> +<p> +The method of "postulating" what we want has many advantages; +they are the same as the advantages of theft over honest +toil. Let us leave them to others and proceed with our honest toil. +</p> +<p> +It is clear that an irrational Dedekind cut in some way "represents" +an irrational. In order to make use of this, which to +begin with is no more than a vague feeling, we must find some +way of eliciting from it a precise definition; and in order to do +this, we must disabuse our minds of the notion that an irrational +must be the limit of a set of ratios. Just as ratios whose denominator +is 1 are not identical with integers, so those rational +<span class="pagenum" id="Page_71">[Pg 71]</span> +numbers which can be greater or less than irrationals, or can +have irrationals as their limits, must not be identified with ratios. +We have to define a new kind of numbers called "real numbers," +of which some will be rational and some irrational. Those that +are rational "correspond" to ratios, in the same kind of way +in which the ratio <img style="vertical-align: -0.566ex; width: 3.62ex; height: 2.262ex;" src="images/141.svg" alt="" data-tex="n/1"> corresponds to the integer <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">;</span> but they are +not the same as ratios. In order to decide what they are to be, +let us observe that an irrational is represented by an irrational +cut, and a cut is represented by its lower section. Let us confine +ourselves to cuts in which the lower section has no maximum; +in this case we will call the lower section a "segment." Then +those segments that correspond to ratios are those that consist +of all ratios less than the ratio they correspond to, which is +their boundary; while those that represent irrationals are those +that have no boundary. Segments, both those that have +boundaries and those that do not, are such that, of any two +pertaining to one series, one must be part of the other; hence +they can all be arranged in a series by the relation of whole and +part. A series in which there are Dedekind gaps, <i>i.e.</i> in which +there are segments that have no boundary, will give rise to more +segments than it has terms, since each term will define a segment +having that term for boundary, and then the segments without +boundaries will be extra. +</p> +<p> +We are now in a position to define a real number and an +irrational number. +</p> +<p> +A "real number" is a segment of the series of ratios in order +of magnitude. +</p> +<p> +An "irrational number" is a segment of the series of ratios +which has no boundary. +</p> +<p> +A "rational real number" is a segment of the series of ratios +which has a boundary. +</p> +<p> +Thus a rational real number consists of all ratios less than a +certain ratio, and it is the rational real number corresponding +to that ratio. The real number 1, for instance, is the class of +proper fractions. +<span class="pagenum" id="Page_72">[Pg 72]</span> +</p> +<p> +In the cases in which we naturally supposed that an irrational +must be the limit of a set of ratios, the truth is that it is the limit +of the corresponding set of rational real numbers in the series +of segments ordered by whole and part. For example, <img style="vertical-align: -0.225ex; width: 3.061ex; height: 2.398ex;" src="images/140.svg" alt="" data-tex="\sqrt{2}"> is +the upper limit of all those segments of the series of ratios that +correspond to ratios whose square is less than 2. More simply +still, <img style="vertical-align: -0.225ex; width: 3.061ex; height: 2.398ex;" src="images/140.svg" alt="" data-tex="\sqrt{2}"> is the segment <i>consisting</i> of all +those ratios whose square is less than 2. +</p> +<p> +It is easy to prove that the series of segments of any series +is Dedekindian. For, given any set of segments, their boundary +will be their logical sum, <i>i.e.</i> the class of all those terms that +belong to at least one segment of the set.<a id="FNanchor_18_1"></a><a href="#Footnote_18_1" class="fnanchor">[18]</a> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_18_1"></a><a href="#FNanchor_18_1"><span class="label">[18]</span></a>For a fuller treatment of the subject of segments and Dedekindian +relations, see <i>Principia Mathematica</i>, vol. II. * 210-214. For a fuller +treatment of real numbers, see <i>ibid.</i>, vol. III. * 310 ff., and +<i>Principles of Mathematics</i>, chaps. XXXIII. and XXXIV.</p></div> + +<p> +The above definition of real numbers is an example of "construction" +as against "postulation," of which we had another +example in the definition of cardinal numbers. The great +advantage of this method is that it requires no new assumptions, +but enables us to proceed deductively from the original apparatus +of logic. +</p> +<p> +There is no difficulty in defining addition and multiplication +for real numbers as above defined. Given two real numbers +<img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> and <span class="nowrap"><img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu">,</span> each being a class of ratios, take any member of <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> and +any member of <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> and add them together according to the rule +for the addition of ratios. Form the class of all such sums +obtainable by varying the selected members of <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> and <span class="nowrap"><img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu">.</span> This +gives a new class of ratios, and it is easy to prove that this new +class is a segment of the series of ratios. We define it as the +sum of <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> and <span class="nowrap"><img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu">.</span> We may state the definition more shortly as +follows:— +</p> +<p> +The <i>arithmetical sum of two real numbers</i> is the class of the +arithmetical sums of a member of the one and a member of the +other chosen in all possible ways. +<span class="pagenum" id="Page_73">[Pg 73]</span> +</p> +<p> +We can define the arithmetical product of two real numbers +in exactly the same way, by multiplying a member of the one by +a member of the other in all possible ways. The class of ratios +thus generated is defined as the product of the two real numbers. +(In all such definitions, the series of ratios is to be defined as +excluding 0 and infinity.) +</p> +<p> +There is no difficulty in extending our definitions to positive +and negative real numbers and their addition and multiplication. +</p> +<p> +It remains to give the definition of complex numbers. +</p> +<p> +Complex numbers, though capable of a geometrical interpretation, +are not demanded by geometry in the same imperative way +in which irrationals are demanded. A "complex" number means +a number involving the square root of a negative number, whether +integral, fractional, or real. Since the square of a negative +number is positive, a number whose square is to be negative has +to be a new sort of number. Using the letter <img style="vertical-align: -0.025ex; width: 0.781ex; height: 1.52ex;" src="images/144.svg" alt="" data-tex="i"> for the square +root of <span class="nowrap"><img style="vertical-align: -0.186ex; width: 2.891ex; height: 1.692ex;" src="images/145.svg" alt="" data-tex="-1">,</span> any number involving the square root of a negative +number can be expressed in the form <span class="nowrap"><img style="vertical-align: -0.464ex; width: 5.949ex; height: 1.959ex;" src="images/146.svg" alt="" data-tex="x + yi">,</span> where <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are +real. The part <img style="vertical-align: -0.464ex; width: 1.889ex; height: 1.959ex;" src="images/147.svg" alt="" data-tex="yi"> is called the "imaginary" part of this number, +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> being the "real" part. (The reason for the phrase "real +numbers" is that they are contrasted with such as are "imaginary.") +Complex numbers have been for a long time habitually +used by mathematicians, in spite of the absence of any precise +definition. It has been simply assumed that they would obey +the usual arithmetical rules, and on this assumption their employment +has been found profitable. They are required less for +geometry than for algebra and analysis. We desire, for example, +to be able to say that every quadratic equation has two roots, +and every cubic equation has three, and so on. But if we are +confined to real numbers, such an equation as <img style="vertical-align: -0.186ex; width: 10.327ex; height: 2.072ex;" src="images/148.svg" alt="" data-tex="x^{2} + 1 = 0"> has no +roots, and such an equation as <img style="vertical-align: -0.186ex; width: 10.327ex; height: 2.071ex;" src="images/149.svg" alt="" data-tex="x^{3} - 1 = 0"> has only one. Every +generalisation of number has first presented itself as needed for +some simple problem: negative numbers were needed in order +that subtraction might be always possible, since otherwise <img style="vertical-align: -0.186ex; width: 4.933ex; height: 1.756ex;" src="images/150.svg" alt="" data-tex="a - b"> +would be meaningless if <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> were less than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">;</span> fractions were needed +<span class="pagenum" id="Page_74">[Pg 74]</span> +in order that division might be always possible; and complex +numbers are needed in order that extraction of roots and solution +of equations may be always possible. But extensions of +number are not <i>created</i> by the mere need for them: they are +created by the definition, and it is to the definition of complex +numbers that we must now turn our attention. +</p> +<p> +A complex number may be regarded and defined as simply an +ordered couple of real numbers. Here, as elsewhere, many +definitions are possible. All that is necessary is that the definitions +adopted shall lead to certain properties. In the case of +complex numbers, if they are defined as ordered couples of real +numbers, we secure at once some of the properties required, +namely, that two real numbers are required to determine a complex +number, and that among these we can distinguish a first +and a second, and that two complex numbers are only identical +when the first real number involved in the one is equal to the +first involved in the other, and the second to the second. What +is needed further can be secured by defining the rules of addition +and multiplication. We are to have +<span class="align-center"><img style="vertical-align: -2.17ex; width: 47.805ex; height: 5.47ex;" src="images/17.svg" alt="" data-tex=" +\begin{alignat*}{2} +&(x + yi) + (x' + y'i) &&= (x + x') + (y + y')i, \\ +&(x + yi) (x' + y'i) &&= (xx' - yy') + (xy' + x'y)i. +\end{alignat*} +"></span> +Thus we shall define that, given two ordered couples of real +numbers, <img style="vertical-align: -0.566ex; width: 5.169ex; height: 2.262ex;" src="images/89.svg" alt="" data-tex="(x, y)"> and <span class="nowrap"><img style="vertical-align: -0.566ex; width: 6.424ex; height: 2.283ex;" src="images/151.svg" alt="" data-tex="(x', y')">,</span> their sum is to be the couple +<span class="nowrap"><img style="vertical-align: -0.566ex; width: 14.359ex; height: 2.283ex;" src="images/152.svg" alt="" data-tex="(x + x', y + y')">,</span> +and their product is to be the couple <span class="nowrap"><img style="vertical-align: -0.566ex; width: 20.419ex; height: 2.283ex;" src="images/153.svg" alt="" data-tex="(xx' - yy', xy' + x'y)">.</span> +By these definitions we shall secure that our ordered couples +shall have the properties we desire. For example, take the +product of the two couples <img style="vertical-align: -0.566ex; width: 5.006ex; height: 2.262ex;" src="images/154.svg" alt="" data-tex="(0, y)"> and <span class="nowrap"><img style="vertical-align: -0.566ex; width: 5.634ex; height: 2.283ex;" src="images/155.svg" alt="" data-tex="(0, y')">.</span> This will, by the +above rule, be the couple <span class="nowrap"><img style="vertical-align: -0.566ex; width: 8.503ex; height: 2.283ex;" src="images/156.svg" alt="" data-tex="(-yy', 0)">.</span> Thus the square of the +couple <img style="vertical-align: -0.566ex; width: 5.029ex; height: 2.262ex;" src="images/157.svg" alt="" data-tex="(0, 1)"> will be the couple <span class="nowrap"><img style="vertical-align: -0.566ex; width: 6.789ex; height: 2.262ex;" src="images/158.svg" alt="" data-tex="(-1, 0)">.</span> Now those couples in +which the second term is 0 are those which, according to the usual +nomenclature, have their imaginary part zero; in the notation +<span class="nowrap"><img style="vertical-align: -0.464ex; width: 5.949ex; height: 1.959ex;" src="images/146.svg" alt="" data-tex="x + yi">,</span> they are <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.972ex; height: 1.692ex;" src="images/159.svg" alt="" data-tex="x + 0i">,</span> which it is natural to write simply <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> Just +as it is natural (but erroneous) to identify ratios whose denominator +is unity with integers, so it is natural (but erroneous) +<span class="pagenum" id="Page_75">[Pg 75]</span> +to identify complex numbers whose imaginary part is zero with +real numbers. Although this is an error in theory, it is a convenience +in practice; "<img style="vertical-align: -0.186ex; width: 5.972ex; height: 1.692ex;" src="images/159.svg" alt="" data-tex="x + 0i">" may be replaced simply by "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" +and "<img style="vertical-align: -0.464ex; width: 5.786ex; height: 1.971ex;" src="images/160.svg" alt="" data-tex="0 + yi">" by "<span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.889ex; height: 1.959ex;" src="images/147.svg" alt="" data-tex="yi">,</span>" provided we remember that the "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" is +not really a real number, but a special case of a complex number. +And when <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is 1, "<img style="vertical-align: -0.464ex; width: 1.889ex; height: 1.959ex;" src="images/147.svg" alt="" data-tex="yi">" may of course be replaced by "<span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.781ex; height: 1.52ex;" src="images/144.svg" alt="" data-tex="i">.</span>" Thus +the couple <img style="vertical-align: -0.566ex; width: 5.029ex; height: 2.262ex;" src="images/157.svg" alt="" data-tex="(0, 1)"> is represented by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.781ex; height: 1.52ex;" src="images/144.svg" alt="" data-tex="i">,</span> and the couple <img style="vertical-align: -0.566ex; width: 6.789ex; height: 2.262ex;" src="images/158.svg" alt="" data-tex="(-1, 0)"> is +represented by -1. Now our rules of multiplication make the +square of <img style="vertical-align: -0.566ex; width: 5.029ex; height: 2.262ex;" src="images/157.svg" alt="" data-tex="(0, 1)"> equal to <span class="nowrap"><img style="vertical-align: -0.566ex; width: 6.789ex; height: 2.262ex;" src="images/158.svg" alt="" data-tex="(-1, 0)">,</span> <i>i.e.</i> the +square of <img style="vertical-align: -0.025ex; width: 0.781ex; height: 1.52ex;" src="images/144.svg" alt="" data-tex="i"> is -1. This +is what we desired to secure. Thus our definitions serve all +necessary purposes. +</p> +<p> +It is easy to give a geometrical interpretation of complex +numbers in the geometry of the plane. This subject was agreeably +expounded by W. K. Clifford in his <i>Common Sense of the +Exact Sciences</i>, a book of great merit, but written before the +importance of purely logical definitions had been realised. +</p> +<p> +Complex numbers of a higher order, though much less useful +and important than those what we have been defining, have +certain uses that are not without importance in geometry, as +may be seen, for example, in Dr Whitehead's <i>Universal Algebra</i>. +The definition of complex numbers of order <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is obtained by an +obvious extension of the definition we have given. We define a +complex number of order <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> as a one-many relation whose domain +consists of certain real numbers and whose converse domain +consists of the integers from 1 to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">.</span><a id="FNanchor_19_1"></a><a href="#Footnote_19_1" class="fnanchor">[19]</a> +This is what would ordinarily +be indicated by the notation <span class="nowrap"><img style="vertical-align: -0.566ex; width: 18.1ex; height: 2.262ex;" src="images/161.svg" alt="" data-tex="(x_{1}, x_{2}, x_{3}, \dots, x_{n})">,</span> where the +suffixes denote correlation with the integers used as suffixes, and +the correlation is one-many, not necessarily one-one, because <img style="vertical-align: -0.357ex; width: 2.203ex; height: 1.357ex;" src="images/162.svg" alt="" data-tex="x_{r}"> +and <img style="vertical-align: -0.355ex; width: 2.232ex; height: 1.355ex;" src="images/163.svg" alt="" data-tex="x_{s}"> may be equal when <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r"> and <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s"> are not equal. The above +definition, with a suitable rule of multiplication, will serve all +purposes for which complex numbers of higher orders are needed. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_19_1"></a><a href="#FNanchor_19_1"><span class="label">[19]</span></a>Cf. <i>Principles of Mathematics</i>, § 360, p. 379.</p></div> + +<p> +We have now completed our review of those extensions of +number which do not involve infinity. The application of number +to infinite collections must be our next topic. +<span class="pagenum" id="Page_76">[Pg 76]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='VIII: INFINITE CARDINAL NUMBERS'><a id="chap08"></a>CHAPTER VIII +<br><br> +INFINITE CARDINAL NUMBERS</h2> + +<p class="nind"> +THE definition of cardinal numbers which we gave in Chapter II. +was applied in Chapter III. to finite numbers, <i>i.e.</i> to the ordinary +natural numbers. To these we gave the name "inductive +numbers," because we found that they are to be defined as +numbers which obey mathematical induction starting from 0. +But we have not yet considered collections which do not have an +inductive number of terms, nor have we inquired whether such +collections can be said to have a number at all. This is an +ancient problem, which has been solved in our own day, chiefly +by Georg Cantor. In the present chapter we shall attempt to +explain the theory of transfinite or infinite cardinal numbers as +it results from a combination of his discoveries with those of +Frege on the logical theory of numbers. +</p> +<p> +It cannot be said to be <i>certain</i> that there are in fact any infinite +collections in the world. The assumption that there are is what +we call the "axiom of infinity." Although various ways suggest +themselves by which we might hope to prove this axiom, there +is reason to fear that they are all fallacious, and that there is no +conclusive logical reason for believing it to be true. At the same +time, there is certainly no logical reason <i>against</i> infinite collections, +and we are therefore justified, in logic, in investigating the hypothesis +that there are such collections. The practical form of this +hypothesis, for our present purposes, is the assumption that, if +<img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is any inductive number, <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is not equal to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">.</span> Various +subtleties arise in identifying this form of our assumption with +<span class="pagenum" id="Page_77">[Pg 77]</span> +the form that asserts the existence of infinite collections; but +we will leave these out of account until, in a later chapter, we +come to consider the axiom of infinity on its own account. For +the present we shall merely assume that, if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is an inductive +number, <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is not equal to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">.</span> This is involved in Peano's +assumption that no two inductive numbers have the same successor; +for, if <span class="nowrap"><img style="vertical-align: -0.186ex; width: 9.629ex; height: 1.692ex;" src="images/166.svg" alt="" data-tex="n = n + 1">,</span> then <img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/167.svg" alt="" data-tex="n - 1"> and <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> have the same successor, +namely <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">.</span> Thus we are assuming nothing that was not involved +in Peano's primitive propositions. +</p> +<p> +Let us now consider the collection of the inductive numbers +themselves. This is a perfectly well-defined class. In the first +place, a cardinal number is a set of classes which are all similar +to each other and are not similar to anything except each other. +We then define as the "inductive numbers" those among +cardinals which belong to the posterity of 0 with respect to the +relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">,</span> <i>i.e.</i> those which possess every property +possessed by 0 and by the successors of possessors, meaning by +the "successor" of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> the number <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">.</span> Thus the class of +"inductive numbers" is perfectly definite. By our general +definition of cardinal numbers, the number of terms in the class +of inductive numbers is to be defined as "all those classes that +are similar to the class of inductive numbers"—<i>i.e.</i> this set of +classes <i>is</i> the number of the inductive numbers according to our +definitions. +</p> +<p> +Now it is easy to see that this number is not one of the inductive +numbers. If <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is any inductive number, the number of numbers +from 0 to <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> (both included) is <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">;</span> therefore the total number +of inductive numbers is greater than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> no matter which of the +inductive numbers <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> may be. If we arrange the inductive +numbers in a series in order of magnitude, this series has no last +term; but if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is an inductive number, every series whose field +has <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> terms has a last term, as it is easy to prove. Such differences +might be multiplied <i>ad lib</i>. Thus the number of inductive +numbers is a new number, different from all of them, not possessing +all inductive properties. It may happen that 0 has a certain +<span class="pagenum" id="Page_78">[Pg 78]</span> +property, and that if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> has it so has <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">,</span> and yet that this new +number does not have it. The difficulties that so long delayed +the theory of infinite numbers were largely due to the fact that +some, at least, of the inductive properties were wrongly judged +to be such as <i>must</i> belong to all numbers; indeed it was thought +that they could not be denied without contradiction. The first +step in understanding infinite numbers consists in realising the +mistakenness of this view. +</p> +<p> +The most noteworthy and astonishing difference between an +inductive number and this new number is that this new number +is unchanged by adding 1 or subtracting 1 or doubling or halving +or any of a number of other operations which we think of as +necessarily making a number larger or smaller. The fact of being +not altered by the addition of 1 is used by Cantor for the definition +of what he calls "transfinite" cardinal numbers; but for +various reasons, some of which will appear as we proceed, it is +better to define an infinite cardinal number as one which does +not possess all inductive properties, <i>i.e.</i> simply as one which is +not an inductive number. Nevertheless, the property of being +unchanged by the addition of 1 is a very important one, and we +must dwell on it for a time. +</p> +<p> +To say that a class has a number which is not altered by the +addition of 1 is the same thing as to say that, if we take a term <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> +which does not belong to the class, we can find a one-one relation +whose domain is the class and whose converse domain is obtained +by adding <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> to the class. For in that case, the class is similar +to the sum of itself and the term <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> <i>i.e.</i> to a class having one extra +term; so that it has the same number as a class with one extra +term, so that if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is this number, <span class="nowrap"><img style="vertical-align: -0.186ex; width: 9.629ex; height: 1.692ex;" src="images/166.svg" alt="" data-tex="n = n + 1">.</span> In this case, we shall +also have <span class="nowrap"><img style="vertical-align: -0.186ex; width: 9.629ex; height: 1.692ex;" src="images/168.svg" alt="" data-tex="n = n - 1">,</span> <i>i.e.</i> there will be one-one relations whose +domains consist of the whole class and whose converse domains +consist of just one term short of the whole class. It can be shown +that the cases in which this happens are the same as the apparently +more general cases in which <i>some</i> part (short of the whole) can be +put into one-one relation with the whole. When this can be done, +<span class="pagenum" id="Page_79">[Pg 79]</span> +the correlator by which it is done may be said to "reflect" the +whole class into a part of itself; for this reason, such classes will +be called "reflexive." Thus: +</p> +<p> +A "reflexive" class is one which is similar to a proper part +of itself. (A "proper part" is a part short of the whole.) +</p> +<p> +A "reflexive" cardinal number is the cardinal number of a +reflexive class. +</p> +<p> +We have now to consider this property of reflexiveness. +</p> +<p> +One of the most striking instances of a "reflexion" is Royce's +illustration of the map: he imagines it decided to make a map +of England upon a part of the surface of England. A map, if +it is accurate, has a perfect one-one correspondence with its +original; thus our map, which is part, is in one-one relation with +the whole, and must contain the same number of points as the +whole, which must therefore be a reflexive number. Royce is +interested in the fact that the map, if it is correct, must contain +a map of the map, which must in turn contain a map of the map +of the map, and so on <i>ad infinitum</i>. This point is interesting, +but need not occupy us at this moment. In fact, we shall do +well to pass from picturesque illustrations to such as are more +completely definite, and for this purpose we cannot do better +than consider the number-series itself. +</p> +<p> +The relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">,</span> confined to inductive numbers, is +one-one, has the whole of the inductive numbers for its domain, +and all except 0 for its converse domain. Thus the whole class +of inductive numbers is similar to what the same class becomes +when we omit 0. Consequently it is a "reflexive" class according +to the definition, and the number of its terms is a "reflexive" +number. Again, the relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.489ex; height: 1.532ex;" src="images/59.svg" alt="" data-tex="2n">,</span> confined to inductive +numbers, is one-one, has the whole of the inductive numbers for +its domain, and the even inductive numbers alone for its converse +domain. Hence the total number of inductive numbers is the +same as the number of even inductive numbers. This property +was used by Leibniz (and many others) as a proof that infinite +numbers are impossible; it was thought self-contradictory that +<span class="pagenum" id="Page_80">[Pg 80]</span> +"the part should be equal to the whole." But this is one of those +phrases that depend for their plausibility upon an unperceived +vagueness: the word "equal" has many meanings, but if it is +taken to mean what we have called "similar," there is no contradiction, +since an infinite collection can perfectly well have parts +similar to itself. Those who regard this as impossible have, +unconsciously as a rule, attributed to numbers in general properties +which can only be proved by mathematical induction, +and which only their familiarity makes us regard, mistakenly, +as true beyond the region of the finite. +</p> +<p> +Whenever we can "reflect" a class into a part of itself, the +same relation will necessarily reflect that part into a smaller +part, and so on <i>ad infinitum</i>. For example, we can reflect, +as we have just seen, all the inductive numbers into the even +numbers; we can, by the same relation (that of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.489ex; height: 1.532ex;" src="images/59.svg" alt="" data-tex="2n">)</span> reflect +the even numbers into the multiples of 4, these into the multiples +of 8, and so on. This is an abstract analogue to Royce's problem +of the map. The even numbers are a "map" of all the inductive +numbers; the multiples of 4 are a map of the map; the multiples +of 8 are a map of the map of the map; and so on. If we had +applied the same process to the relation of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">,</span> our "map" +would have consisted of all the inductive numbers except 0; +the map of the map would have consisted of all from 2 onward, +the map of the map of the map of all from 3 onward; and so on. +The chief use of such illustrations is in order to become familiar +with the idea of reflexive classes, so that apparently paradoxical +arithmetical propositions can be readily translated into the +language of reflexions and classes, in which the air of paradox +is much less. +</p> +<p> +It will be useful to give a definition of the number which is +that of the inductive cardinals. For this purpose we will +first define the kind of series exemplified by the inductive cardinals +in order of magnitude. The kind of series which is called a +"progression" has already been considered in Chapter I. It is a +series which can be generated by a relation of consecutiveness: +<span class="pagenum" id="Page_81">[Pg 81]</span> +every member of the series is to have a successor, but there is +to be just one which has no predecessor, and every member of +the series is to be in the posterity of this term with respect to +the relation "immediate predecessor." These characteristics +may be summed up in the following definition:<a id="FNanchor_20_1"></a><a href="#Footnote_20_1" class="fnanchor">[20]</a>— +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_20_1"></a><a href="#FNanchor_20_1"><span class="label">[20]</span></a>Cf. <i>Principia Mathematica</i>, vol. II. * 123.</p></div> + +<p> +A "progession" is a one-one relation such that there is just +one term belonging to the domain but not to the converse domain, +and the domain is identical with the posterity of this one term. +</p> +<p> +It is easy to see that a progression, so defined, satisfies Peano's +five axioms. The term belonging to the domain but not to the +converse domain will be what he calls "0"; the term to which +a term has the one-one relation will be the "successor" of the +term; and the domain of the one-one relation will be what +he calls "number." Taking his five axioms in turn, we have +the following translations:— +</p> +<p> +(1) "0 is a number" becomes: "The member of the domain +which is not a member of the converse domain is a member of +the domain." This is equivalent to the existence of such a +member, which is given in our definition. We will call this +member "the first term." +</p> +<p> +(2) "The successor of any number is a number" becomes: +"The term to which a given member of the domain has the relation +in question is again a member of the domain." This is +proved as follows: By the definition, every member of the +domain is a member of the posterity of the first term; hence +the successor of a member of the domain must be a member of +the posterity of the first term (because the posterity of a term +always contains its own successors, by the general definition of +posterity), and therefore a member of the domain, because by +the definition the posterity of the first term is the same as the +domain. +</p> +<p> +(3) "No two numbers have the same successor." This is +only to say that the relation is one-many, which it is by definition +(being one-one). +<span class="pagenum" id="Page_82">[Pg 82]</span> +</p> +<p> +(4) "0 is not the successor of any number" becomes: "The +first term is not a member of the converse domain," which is +again an immediate result of the definition. +</p> +<p> +(5) This is mathematical induction, and becomes: "Every +member of the domain belongs to the posterity of the first term," +which was part of our definition. +</p> +<p> +Thus progressions as we have defined them have the five +formal properties from which Peano deduces arithmetic. It is +easy to show that two progessions are "similar" in the sense +defined for similarity of relations in Chapter VI. We can, of +course, derive a relation which is serial from the one-one relation +by which we define a progression: the method used is that +explained in Chapter IV., and the relation is that of a term to +a member of its proper posterity with respect to the original +one-one relation. +</p> +<p> +Two transitive asymmetrical relations which generate progressions +are similar, for the same reasons for which the corresponding +one-one relations are similar. The class of all such +transitive generators of progressions is a "serial number" in +the sense of Chapter VI.; it is in fact the smallest of infinite +serial numbers, the number to which Cantor has given the name <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/169.svg" alt="" data-tex="\omega">,</span> +by which he has made it famous. +</p> +<p> +But we are concerned, for the moment, with <i>cardinal</i> numbers. +Since two progressions are similar relations, it follows that their +domains (or their fields, which are the same as their domains) +are similar classes. The domains of progressions form a cardinal +number, since every class which is similar to the domain of a +progression is easily shown to be itself the domain of a progression. +This cardinal number is the smallest of the infinite cardinal +numbers; it is the one to which Cantor has appropriated the +Hebrew Aleph with the suffix 0, to distinguish it from larger +infinite cardinals, which have other suffixes. Thus the name of +the smallest of infinite cardinals is <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span> +</p> +<p> +To say that a class has <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms is the same thing as to say +that it is a member of <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">,</span> and this is the same thing as to say +<span class="pagenum" id="Page_83">[Pg 83]</span> +that the members of the class can be arranged in a progression. +It is obvious that any progression remains a progression if we +omit a finite number of terms from it, or every other term, or +all except every tenth term or every hundredth term. These +methods of thinning out a progression do not make it cease to +be a progression, and therefore do not diminish the number of +its terms, which remains <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span> In fact, any selection from a progression +is a progression if it has no last term, however sparsely +it may be distributed. Take (say) inductive numbers of the form <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.505ex; height: 1.553ex;" src="images/171.svg" alt="" data-tex="n^{n}">,</span> +or <span class="nowrap"><img style="vertical-align: -0.025ex; width: 3.317ex; height: 1.927ex;" src="images/172.svg" alt="" data-tex="n^{n^{n}}">.</span> Such numbers grow very rare in the higher parts +of the number series, and yet there are just as many of them as +there are inductive numbers altogether, namely, <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span> +</p> +<p> +Conversely, we can add terms to the inductive numbers without +increasing their number. Take, for example, ratios. One +might be inclined to think that there must be many more ratios +than integers, since ratios whose denominator is 1 correspond +to the integers, and seem to be only an infinitesimal proportion +of ratios. But in actual fact the number of ratios (or fractions) +is exactly the same as the number of inductive numbers, namely, <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span> +This is easily seen by arranging ratios in a series on the +following plan: If the sum of numerator and denominator in +one is less than in the other, put the one before the other; if +the sum is equal in the two, put first the one with the smaller +numerator. This gives us the series +<span class="align-center"><img style="vertical-align: -0.566ex; width: 44.638ex; height: 2.262ex;" src="images/18.svg" alt="" data-tex=" +1,\ 1/2,\ 2,\ 1/3,\ 3,\ 1/4,\ 2/3,\ 3/2,\ 4,\ 1/5,\ \dots. +"></span> +This series is a progression, and all ratios occur in it sooner or +later. Hence we can arrange all ratios in a progression, and +their number is therefore <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span> +</p> +<p> +It is not the case, however, that <i>all</i> infinite collections have +<img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms. The number of real numbers, for example, is greater +than <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">;</span> it is, in fact, <span class="nowrap"><img style="vertical-align: 0; width: 2.995ex; height: 1.932ex;" src="images/173.svg" alt="" data-tex="2^{\aleph_{0}}">,</span> and it is not +hard to prove that <img style="vertical-align: 0; width: 2.279ex; height: 1.528ex;" src="images/174.svg" alt="" data-tex="2^{n}"> is +greater than <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> even when <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is infinite. The easiest way of +proving this is to prove, first, that if a class has <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> members, it +contains <img style="vertical-align: 0; width: 2.279ex; height: 1.528ex;" src="images/174.svg" alt="" data-tex="2^{n}"> sub-classes—in other words, that there are <img style="vertical-align: 0; width: 2.279ex; height: 1.528ex;" src="images/174.svg" alt="" data-tex="2^{n}"> ways +<span class="pagenum" id="Page_84">[Pg 84]</span> +of selecting some of its members (including the extreme cases +where we select all or none); and secondly, that the number of +sub-classes contained in a class is always greater than the number +of members of the class. Of these two propositions, the first +is familiar in the case of finite numbers, and is not hard to extend +to infinite numbers. The proof of the second is so simple and +so instructive that we shall give it: +</p> +<p> +In the first place, it is clear that the number of sub-classes +of a given class (say <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">)</span> is at least as great as the number of +members, since each member constitutes a sub-class, and we thus +have a correlation of all the members with some of the sub-classes. +Hence it follows that, if the number of sub-classes is +not <i>equal</i> to the number of members, it must be <i>greater</i>. Now +it is easy to prove that the number is not equal, by showing that, +given any one-one relation whose domain is the members and +whose converse domain is contained among the set of sub-classes, +there must be at least one sub-class not belonging to +the converse domain. The proof is as follows:<a id="FNanchor_21_1"></a><a href="#Footnote_21_1" class="fnanchor">[21]</a> +When a one-one +correlation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is established between all the members of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> +and some of the sub-classes, it may happen that a given member <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> +is correlated with a sub-class of which it is a member; or, +again, it may happen that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is correlated with a sub-class of +which it is not a member. Let us form the whole class, <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> say, +of those members <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> which are correlated with sub-classes of which +they are not members. This is a sub-class of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> and it is not +correlated with any member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> For, taking first the members +of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">,</span> each of them is (by the definition of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">)</span> correlated with +some sub-class of which it is not a member, and is therefore not +correlated with <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">.</span> Taking next the terms which are not members +of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">,</span> each of them (by the definition of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">)</span> is correlated with +some sub-class of which it is a member, and therefore again +is not correlated with <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">.</span> Thus no member of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is correlated +with <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">.</span> Since <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> was <i>any</i> one-one correlation of all members +<span class="pagenum" id="Page_85">[Pg 85]</span> +with some sub-classes, it follows that there is no correlation +of all members with <i>all</i> sub-classes. It does not matter to the +proof if <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> has no members: all that happens in that case is that +the sub-class which is shown to be omitted is the null-class. +Hence in any case the number of sub-classes is not equal to the +number of members, and therefore, by what was said earlier, +it is greater. Combining this with the proposition that, if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is +the number of members, <img style="vertical-align: 0; width: 2.279ex; height: 1.528ex;" src="images/174.svg" alt="" data-tex="2^{n}"> is the number of sub-classes, we have +the theorem that <img style="vertical-align: 0; width: 2.279ex; height: 1.528ex;" src="images/174.svg" alt="" data-tex="2^{n}"> is always greater than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> even when <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is +infinite. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_21_1"></a><a href="#FNanchor_21_1"><span class="label">[21]</span></a>This proof is taken from Cantor, with some simplifications: see +<i>Jahresbericht der deutschen Mathematiker-Vereinigung</i>, I. (1892), p. 77.</p></div> + +<p> +It follows from this proposition that there is no maximum +to the infinite cardinal numbers. However great an infinite +number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> may be, <img style="vertical-align: 0; width: 2.279ex; height: 1.528ex;" src="images/174.svg" alt="" data-tex="2^{n}"> will be still greater. The arithmetic of +infinite numbers is somewhat surprising until one becomes +accustomed to it. We have, for example, +<span class="align-center"><img style="vertical-align: -3.672ex; width: 45.257ex; height: 8.476ex;" src="images/19.svg" alt="" data-tex=" +\begin{align*} +\aleph_{0} + 1 &= \aleph_{0}, \\ +\aleph_{0} + n &= \aleph_{0}, \text{where $n$ is any inductive number,} \\ +\aleph_{0}^{2} &= \aleph_{0}. +\end{align*} +"></span> +(This follows from the case of the ratios, for, since a ratio is +determined by a pair of inductive numbers, it is easy to see that +the number of ratios is the square of the number of inductive +numbers, <i>i.e.</i> it is <span class="nowrap"><img style="vertical-align: -0.687ex; width: 2.37ex; height: 2.573ex;" src="images/175.svg" alt="" data-tex="\aleph_{0}^{2}">;</span> but we saw that it +is also <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span>) +<span class="align-center"><img style="vertical-align: -3.908ex; width: 59.455ex; height: 8.947ex;" src="images/20.svg" alt="" data-tex=" +\begin{alignat*}{2} +&&\llap{$\aleph_{0}^{n}$} &= \aleph_{0}, \text{where $n$ is any inductive number.} \\ +&\text{(This follows from } \aleph_{0}^{2} &&= \aleph_{0} \text{ by induction; for if $\aleph_{0}^{n} = \aleph_{0}$,} \\ +&\text{then} &\llap{$\aleph_{0}^{n+1}} &= \aleph_{0}^{2} = \aleph_{0}.) +\end{alignat*} +"></span> +But +<span class="align-center"><img style="vertical-align: -0.375ex; width: 9.011ex; height: 2.419ex;" src="images/21.svg" alt="" data-tex=" +2^{\aleph_{0}} > \aleph_{0}. +"></span> +In fact, as we shall see later, <img style="vertical-align: 0; width: 2.995ex; height: 1.932ex;" src="images/173.svg" alt="" data-tex="2^{\aleph_{0}}"> is a very important number, +namely, the number of terms in a series which has "continuity" +in the sense in which this word is used by Cantor. Assuming +space and time to be continuous in this sense (as we commonly +do in analytical geometry and kinematics), this will be the +number of points in space or of instants in time; it will also be +the number of points in any finite portion of space, whether +<span class="pagenum" id="Page_86">[Pg 86]</span> +line, area, or volume. After <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">,</span> <img style="vertical-align: 0; width: 2.995ex; height: 1.932ex;" src="images/173.svg" alt="" data-tex="2^{\aleph_{0}}"> is the most important and +interesting of infinite cardinal numbers. +</p> +<p> +Although addition and multiplication are always possible +with infinite cardinals, subtraction and division no longer give +definite results, and cannot therefore be employed as they are +employed in elementary arithmetic. Take subtraction to begin +with: so long as the number subtracted is finite, all goes well; +if the other number is reflexive, it remains unchanged. Thus +<span class="nowrap"><img style="vertical-align: -0.375ex; width: 11.88ex; height: 1.945ex;" src="images/176.svg" alt="" data-tex="\aleph_{0} - n = \aleph_{0}">,</span> if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is finite; so far, subtraction gives a perfectly +definite result. But it is otherwise when we subtract <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> from +itself; we may then get any result, from 0 up to <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span> This is +easily seen by examples. From the inductive, numbers, take +away the following collections of <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms:— +</p> +<p> +(1) All the inductive numbers—remainder, zero. +</p> +<p> +(2) All the inductive numbers from <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> onwards—remainder, +the numbers from 0 to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/167.svg" alt="" data-tex="n - 1">,</span> numbering <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> terms in all. +</p> +<p> +(3) All the odd numbers—remainder, all the even numbers, +numbering <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms. +</p> +<p> +All these are different ways of subtracting <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> from <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">,</span> and +all give different results. +</p> +<p> +As regards division, very similar results follow from the fact +that <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> is unchanged when multiplied by 2 or 3 or any finite +number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> or by <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span> It follows that <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> divided +by <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> may have +any value from 1 up to <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span> +</p> +<p> +From the ambiguity of subtraction and division it results +that negative numbers and ratios cannot be extended to infinite +numbers. Addition, multiplication, and exponentiation proceed +quite satisfactorily, but the inverse operations—subtraction, +division, and extraction of roots—are ambiguous, and the notions +that depend upon them fail when infinite numbers are concerned. +</p> +<p> +The characteristic by which we defined finitude was mathematical +induction, <i>i.e.</i> we defined a number as finite when it +obeys mathematical induction starting from 0, and a class as +finite when its number is finite. This definition yields the sort +of result that a definition ought to yield, namely, that the finite +<span class="pagenum" id="Page_87">[Pg 87]</span> +numbers are those that occur in the ordinary number-series +0, 1, 2, 3, ... But in the present chapter, the infinite numbers +we have discussed have not merely been non-inductive: +they have also been <i>reflexive</i>. Cantor used reflexiveness as the +<i>definition</i> of the infinite, and believes that it is equivalent to +non-inductiveness; that is to say, he believes that every class +and every cardinal is either inductive or reflexive. This may be +true, and may very possibly be capable of proof; but the proofs +hitherto offered by Cantor and others (including the present +author in former days) are fallacious, for reasons which will be +explained when we come to consider the "multiplicative axiom." +At present, it is not known whether there are classes and cardinals +which are neither reflexive nor inductive. If <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> were such a +cardinal, we should not have <span class="nowrap"><img style="vertical-align: -0.186ex; width: 9.629ex; height: 1.692ex;" src="images/166.svg" alt="" data-tex="n = n + 1">,</span> but <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> would not be one +of the "natural numbers," and would be lacking in some of the +inductive properties. All <i>known</i> infinite classes and cardinals +are reflexive; but for the present it is well to preserve an open +mind as to whether there are instances, hitherto unknown, of +classes and cardinals which are neither reflexive nor inductive. +Meanwhile, we adopt the following definitions:— +</p> +<p> +A <i>finite</i> class or cardinal is one which is <i>inductive</i>. +</p> +<p> +An <i>infinite</i> class or cardinal is one which is <i>not inductive</i>. +All <i>reflexive</i> classes and cardinals are infinite; but it is not known +at present whether all infinite classes and cardinals are reflexive. +We shall return to this subject in Chapter XII. +<span class="pagenum" id="Page_88">[Pg 88]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='IX: INFINITE SERIES AND ORDINALS'><a id="chap09"></a>CHAPTER IX +<br><br> +INFINITE SERIES AND ORDINALS</h2> + +<p class="nind"> +AN "infinite series" may be defined as a series of which the field +is an infinite class. We have already had occasion to consider +one kind of infinite series, namely, progressions. In this chapter +we shall consider the subject more generally. +</p> +<p> +The most noteworthy characteristic of an infinite series is +that its serial number can be altered by merely re-arranging +its terms. In this respect there is a certain oppositeness between +cardinal and serial numbers. It is possible to keep the cardinal +number of a reflexive class unchanged in spite of adding terms +to it; on the other hand, it is possible to change the serial +number of a series without adding or taking away any terms, +by mere re-arrangement. At the same time, in the case of any +infinite series it is also possible, as with cardinals, to add terms +without altering the serial number: everything depends upon +the way in which they are added. +</p> +<p> +In order to make matters clear, it will be best to begin with +examples. Let us first consider various different kinds of series +which can be made out of the inductive numbers arranged on +various plans. We start with the series +<span class="align-center"><img style="vertical-align: -0.439ex; width: 21.747ex; height: 1.971ex;" src="images/22.svg" alt="" data-tex=" +1,\ 2,\ 3,\ 4,\ \dots\ n,\ \dots, +"></span> +which, as we have already seen, represents the smallest of infinite +serial numbers, the sort that Cantor calls <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/169.svg" alt="" data-tex="\omega">.</span> Let us +proceed to thin out this series by repeatedly performing the +<span class="pagenum" id="Page_89">[Pg 89]</span> +operation of removing to the end the first even number that +occurs. We thus obtain in succession the various series: +<span class="align-center"><img style="vertical-align: -3.507ex; width: 32.746ex; height: 8.145ex;" src="images/23.svg" alt="" data-tex=" +\begin{align*} +&1,\ 3,\ 4,\ 5,\ \dots\ n,\ \dots\ 2, \\ +&1,\ 3,\ 5,\ 6,\ \dots\ n + 1,\ \dots\ 2,\ 4, \\ +&1,\ 3,\ 5,\ 7,\ \dots\ n + 2,\ \dots\ 2,\ 4,\ 6, +\end{align*} +"></span> +and so on. If we imagine this process carried on as long as +possible, we finally reach the series +<span class="align-center"><img style="vertical-align: -0.439ex; width: 49.589ex; height: 1.971ex;" src="images/24.svg" alt="" data-tex=" +1,\ 3,\ 5,\ 7,\ \dots\ 2n + 1,\ \dots\ 2,\ 4,\ 6,\ 8,\ \dots\ 2n,\ \dots, +"></span> +in which we have first all the odd numbers and then all the even +numbers. +</p> +<p> +The serial numbers of these various series are <span class="nowrap"><img style="vertical-align: -0.439ex; width: 25.063ex; height: 1.946ex;" src="images/177.svg" alt="" data-tex="\omega + 1, \omega + 2, +\omega + 3, \dots\ 2\omega">.</span> Each of these numbers is "greater" than any +of its predecessors, in the following sense:— +</p> +<p> +One serial number is said to be "greater" than another if +any series having the first number contains a part having the +second number, but no series having the second number contains +a part having the first number. +</p> +<p> +If we compare the two series +<span class="align-center"><img style="vertical-align: -2.036ex; width: 27.34ex; height: 5.204ex;" src="images/25.svg" alt="" data-tex=" +\begin{align*} +&1,\ 2,\ 3,\ 4,\ \dots\ n,\ \dots, \\ +&1,\ 3,\ 4,\ 5,\ \dots\ n + 1,\ \dots\ 2, +\end{align*} +"></span> +we see that the first is similar to the part of the second which +omits the last term, namely, the number 2, but the second is +not similar to any part of the first. (This is obvious, but is +easily demonstrated.) Thus the second series has a greater +serial number than the first, according to the definition—<i>i.e.</i> +<img style="vertical-align: -0.186ex; width: 5.304ex; height: 1.692ex;" src="images/178.svg" alt="" data-tex="\omega + 1"> is greater than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/169.svg" alt="" data-tex="\omega">.</span> But if we add a term at the beginning +of a progression instead of the end, we still have a progression. +Thus <span class="nowrap"><img style="vertical-align: -0.186ex; width: 9.729ex; height: 1.692ex;" src="images/179.svg" alt="" data-tex="1 + \omega = \omega">.</span> Thus <img style="vertical-align: -0.186ex; width: 5.304ex; height: 1.692ex;" src="images/180.svg" alt="" data-tex="1 + \omega"> is not equal to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.304ex; height: 1.692ex;" src="images/178.svg" alt="" data-tex="\omega + 1">.</span> This is +characteristic of relation-arithmetic generally: if <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> are +two relation-numbers, the general rule is that <img style="vertical-align: -0.489ex; width: 5.329ex; height: 1.808ex;" src="images/181.svg" alt="" data-tex="\mu + \nu"> is not equal +to <span class="nowrap"><img style="vertical-align: -0.489ex; width: 5.329ex; height: 1.808ex;" src="images/182.svg" alt="" data-tex="\nu + \mu">.</span> The case of finite ordinals, in which there is equality, +is quite exceptional. +</p> +<p> +The series we finally reached just now consisted of first all the +odd numbers and then all the even numbers, and its serial +<span class="pagenum" id="Page_90">[Pg 90]</span> +number is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.538ex; height: 1.532ex;" src="images/183.svg" alt="" data-tex="2\omega">.</span> This number is greater than <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/169.svg" alt="" data-tex="\omega"> or <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.53ex; height: 1.505ex;" src="images/184.svg" alt="" data-tex="\omega + n">,</span> where +<img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is finite. It is to be observed that, in accordance with the +general definition of order, each of these arrangements of integers +is to be regarded as resulting from some definite relation. <i>E.g.</i> +the one which merely removes 2 to the end will be defined by +the following relation: "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are finite integers, and either +<img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is 2 and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is not 2, or neither is 2 and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is less than <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">.</span>" The +one which puts first all the odd numbers and then all the even +ones will be defined by: "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are finite integers, and either +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is odd and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is even or <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is less than <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> and both are odd or both +are even." We shall not trouble, as a rule, to give these formulæ +in future; but the fact that they <i>could</i> be given is essential. +</p> +<p> +The number which we have called <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.538ex; height: 1.532ex;" src="images/183.svg" alt="" data-tex="2\omega">,</span> namely, the number of +a series consisting of two progressions, is sometimes called <span class="nowrap"><img style="vertical-align: -0.025ex; width: 4.173ex; height: 1.532ex;" src="images/185.svg" alt="" data-tex="\omega · 2">.</span> +Multiplication, like addition, depends upon the order of the +factors: a progression of couples gives a series such as +<span class="align-center"><img style="vertical-align: -0.464ex; width: 38.411ex; height: 1.464ex;" src="images/26.svg" alt="" data-tex=" +x_{1},\ y_{1},\ x_{2},\ y_{2},\ x_{3},\ y_{3},\ \dots\ x_{n},\ y_{n},\ \dots, +"></span> +which is itself a progression; but a couple of progressions gives +a series which is twice as long as a progression. It is therefore +necessary to distinguish between <img style="vertical-align: -0.025ex; width: 2.538ex; height: 1.532ex;" src="images/183.svg" alt="" data-tex="2\omega"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 4.173ex; height: 1.532ex;" src="images/185.svg" alt="" data-tex="\omega · 2">.</span> Usage is variable; +we shall use <img style="vertical-align: -0.025ex; width: 2.538ex; height: 1.532ex;" src="images/183.svg" alt="" data-tex="2\omega"> for a couple of progressions and <img style="vertical-align: -0.025ex; width: 4.173ex; height: 1.532ex;" src="images/185.svg" alt="" data-tex="\omega · 2"> for a progression +of couples, and this decision of course governs our +general interpretation of "<img style="vertical-align: -0.439ex; width: 4.363ex; height: 2.034ex;" src="images/186.svg" alt="" data-tex="\alpha · \beta">" when <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> are relation-numbers: +"<img style="vertical-align: -0.439ex; width: 4.363ex; height: 2.034ex;" src="images/186.svg" alt="" data-tex="\alpha · \beta">" will have to stand for a suitably constructed +sum of a relations each having <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> terms. +</p> +<p> +We can proceed indefinitely with the process of thinning +out the inductive numbers. For example, we can place first +the odd numbers, then their doubles, then the doubles of these, +and so on. We thus obtain the series +<span class="align-center"><img style="vertical-align: -2.036ex; width: 55.474ex; height: 5.204ex;" src="images/27.svg" alt="" data-tex=" +\begin{align*} +1,\ 3,\ 5,\ 7,\ \dots;\quad +2,\ 6,\ 10,\ 14,\ \dots;\quad +4,\ 12,\ 20,\ 28,\ \dots; \\ +8,\ 24,\ 40,\ 56,\ \dots, +\end{align*} +"></span> +of which the number is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.395ex; height: 1.912ex;" src="images/187.svg" alt="" data-tex="\omega^{2}">,</span> since it is a progression of progressions. +Any one of the progressions in this new series can of course be +<span class="pagenum" id="Page_91">[Pg 91]</span> +thinned out as we thinned out our original progression. We can +proceed to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.395ex; height: 1.91ex;" src="images/188.svg" alt="" data-tex="\omega^{3}">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.395ex; height: 1.929ex;" src="images/189.svg" alt="" data-tex="\omega^{4}">,</span> ... <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.59ex; height: 1.555ex;" src="images/190.svg" alt="" data-tex="\omega^{\omega}">,</span> and so on; however far we have gone, +we can always go further. +</p> +<p> +The series of all the ordinals that can be obtained in this way, +<i>i.e.</i> all that can be obtained by thinning out a progression, is +itself longer than any series that can be obtained by re-arranging +the terms of a progression. (This is not difficult to prove.) +The cardinal number of the class of such ordinals can be shown +to be greater than <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">;</span> it is the number which Cantor calls <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;" src="images/191.svg" alt="" data-tex="\aleph_{1}">.</span> +The ordinal number of the series of all ordinals that can +be made out of an <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">,</span> taken in order of magnitude, is called <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;" src="images/192.svg" alt="" data-tex="\omega_{1}">.</span> +Thus a series whose ordinal number is <img style="vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;" src="images/192.svg" alt="" data-tex="\omega_{1}"> has a field whose +cardinal number is <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;" src="images/191.svg" alt="" data-tex="\aleph_{1}">.</span> +</p> +<p> +We can proceed from <img style="vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;" src="images/192.svg" alt="" data-tex="\omega_{1}"> and <img style="vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;" src="images/191.svg" alt="" data-tex="\aleph_{1}"> to <img style="vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;" src="images/193.svg" alt="" data-tex="\omega_{2}"> and <img style="vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;" src="images/194.svg" alt="" data-tex="\aleph_{2}"> by a process +exactly analogous to that by which we advanced from <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/169.svg" alt="" data-tex="\omega"> and <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> +to <img style="vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;" src="images/192.svg" alt="" data-tex="\omega_{1}"> and <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;" src="images/191.svg" alt="" data-tex="\aleph_{1}">.</span> And there is nothing to prevent us from advancing +indefinitely in this way to new cardinals and new ordinals. It +is not known whether <img style="vertical-align: 0; width: 2.995ex; height: 1.932ex;" src="images/173.svg" alt="" data-tex="2^{\aleph_{0}}"> is equal to any of the cardinals in the +series of Alephs. It is not even known whether it is comparable +with them in magnitude; for aught we know, it may be neither +equal to nor greater nor less than any one of the Alephs. This +question is connected with the multiplicative axiom, of which +we shall treat later. +</p> +<p> +All the series we have been considering so far in this chapter +have been what is called "well-ordered." A well-ordered +series is one which has a beginning, and has consecutive terms, +and has a term <i>next</i> after any selection of its terms, provided +there are any terms after the selection. This excludes, on the +one hand, compact series, in which there are terms between +any two, and on the other hand series which have no beginning, +or in which there are subordinate parts having no beginning. +The series of negative integers in order of magnitude, having +no beginning, but ending with -1, is not well-ordered; but +taken in the reverse order, beginning with -1, it is well-ordered, +being in fact a progression. The definition is: +<span class="pagenum" id="Page_92">[Pg 92]</span> +</p> +<p> +A "well-ordered" series is one in which every sub-class +(except, of course, the null-class) has a first term. +</p> +<p> +An "ordinal" number means the relation-number of a well-ordered +series. It is thus a species of serial number. +</p> +<p> +Among well-ordered series, a generalised form of mathematical +induction applies. A property may be said to be "transfinitely +hereditary" if, when it belongs to a certain selection of the +terms in a series, it belongs to their immediate successor provided +they have one. In a well-ordered series, a transfinitely +hereditary property belonging to the first term of the series +belongs to the whole series. This makes it possible to prove +many propositions concerning well-ordered series which are not +true of all series. +</p> +<p> +It is easy to arrange the inductive numbers in series which +are not well-ordered, and even to arrange them in compact +series. For example, we can adopt the following plan: consider +the decimals from .1 (inclusive) to 1 (exclusive), arranged in order +of magnitude. These form a compact series; between any +two there are always an infinite number of others. Now omit +the dot at the beginning of each, and we have a compact series +consisting of all finite integers except such as divide by 10. If +we wish to include those that divide by 10, there is no difficulty; +instead of starting with .1, we will include all decimals less than 1, +but when we remove the dot, we will transfer to the right any +0's that occur at the beginning of our decimal. Omitting these, +and returning to the ones that have no 0's at the beginning, +we can state the rule for the arrangement of our integers as +follows: Of two integers that do not begin with the same digit, +the one that begins with the smaller digit comes first. Of two +that do begin with the same digit, but differ at the second digit, +the one with the smaller second digit comes first, but first of all +the one with no second digit; and so on. Generally, if two +integers agree as regards the first <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> digits, but not as regards +the <span class="nowrap"><img style="vertical-align: -0.566ex; width: 8.701ex; height: 2.497ex;" src="images/195.svg" alt="" data-tex="(n + 1)^\mathord{th}">,</span> that one comes first which has either no +<img style="vertical-align: -0.566ex; width: 8.701ex; height: 2.497ex;" src="images/195.svg" alt="" data-tex="(n + 1)^\mathord{th}"> +digit or a smaller one than the other. This rule of arrangement, +<span class="pagenum" id="Page_93">[Pg 93]</span> +as the reader can easily convince himself, gives rise to a compact +series containing all the integers not divisible by 10; and, +as we saw, there is no difficulty about including those +that are divisible by 10. It follows from this example that +it is possible to construct compact series having <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms. +In fact, we have already seen that there are <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> ratios, and +ratios in order of magnitude form a compact series; thus +we have here another example. We shall resume this topic +in the next chapter. +</p> +<p> +Of the usual formal laws of addition, multiplication, and exponentiation, +all are obeyed by transfinite cardinals, but only +some are obeyed by transfinite ordinals, and those that are obeyed +by them are obeyed by all relation-numbers. By the "usual +formal laws" we mean the following:— +</p> +<p> +I. The commutative law: +<span class="align-center"><img style="vertical-align: -0.439ex; width: 34.801ex; height: 2.034ex;" src="images/28.svg" alt="" data-tex=" +\alpha + \beta = \beta + \alpha \quad\text{and}\quad +\alpha × \beta = \beta × \alpha. +"></span> +</p> +<p> +II. The associative law: +<span class="align-center"><img style="vertical-align: -0.566ex; width: 55.807ex; height: 2.262ex;" src="images/29.svg" alt="" data-tex=" +(\alpha + \beta) + \gamma = \alpha + (\beta + \gamma) \quad\text{and}\quad +(\alpha × \beta) × \gamma = \alpha × (\beta × \gamma). +"></span> +</p> +<p> +III. The distributive law: +<span class="align-center"><img style="vertical-align: -0.566ex; width: 20.3ex; height: 2.262ex;" src="images/30.svg" alt="" data-tex=" +\alpha(\beta + \gamma) = \alpha\beta + \alpha\gamma. +"></span> +</p> +<p> +When the commutative law does not hold, the above form +of the distributive law must be distinguished from +<span class="align-center"><img style="vertical-align: -0.566ex; width: 20.3ex; height: 2.262ex;" src="images/31.svg" alt="" data-tex=" +(\beta + \gamma)\alpha = \beta\alpha + \gamma\alpha. +"></span> +As we shall see immediately, one form may be true and the +other false. +</p> +<p> +IV. The laws of exponentiation: +<span class="align-center"><img style="vertical-align: -0.566ex; width: 48.341ex; height: 2.628ex;" src="images/32.svg" alt="" data-tex=" +\alpha^{\beta} · \alpha^{\gamma} = \alpha^{\beta + \gamma},\quad +\alpha^{\gamma} · \beta^{\gamma} = (\alpha\beta)^{\gamma},\quad +(\alpha^{\beta})^{\gamma} = \alpha^{\beta\gamma}. +"></span> +</p> +<p> +All these laws hold for cardinals, whether finite or infinite, +and for <i>finite</i> ordinals. But when we come to infinite ordinals, +or indeed to relation-numbers in general, some hold and some +do not. The commutative law does not hold; the associative +law does hold; the distributive law (adopting the convention +<span class="pagenum" id="Page_94">[Pg 94]</span> +we have adopted above as regards the order of the factors in a +product) holds in the form +<span class="align-center"><img style="vertical-align: -0.566ex; width: 20.3ex; height: 2.262ex;" src="images/33.svg" alt="" data-tex=" +(\beta + \gamma)\alpha = \beta\alpha + \gamma\alpha, +"></span> +but not in the form +<span class="align-center"><img style="vertical-align: -0.566ex; width: 20.3ex; height: 2.262ex;" src="images/34.svg" alt="" data-tex=" +\alpha(\beta + \gamma) = \alpha\beta + \alpha\gamma; +"></span> +the exponential laws +<span class="align-center"><img style="vertical-align: -0.566ex; width: 35.315ex; height: 2.628ex;" src="images/35.svg" alt="" data-tex=" +\alpha^{\beta} · \alpha^{\gamma} = \alpha^{\beta + \gamma}, +\quad\text{and}\quad +(\alpha^{\beta})^{\gamma} = \alpha^{\beta\gamma} +"></span> +still hold, but not the law +<span class="align-center"><img style="vertical-align: -0.566ex; width: 15.667ex; height: 2.262ex;" src="images/36.svg" alt="" data-tex=" +\alpha^{\gamma} · \beta^{\gamma} = (\alpha\beta)^{\gamma}, +"></span> +which is obviously connected with the commutative law for +multiplication. +</p> +<p> +The definitions of multiplication and exponentiation that +are assumed in the above propositions are somewhat complicated. +The reader who wishes to know what they are and how the +above laws are proved must consult the second volume of +<i>Principia Mathematica</i>, * 172-176. +</p> +<p> +Ordinal transfinite arithmetic was developed by Cantor at +an earlier stage than cardinal transfinite arithmetic, because it +has various technical mathematical uses which led him to it. +But from the point of view of the philosophy of mathematics +it is less important and less fundamental than the theory of +transfinite cardinals. Cardinals are essentially simpler than +ordinals, and it is a curious historical accident that they first +appeared as an abstraction from the latter, and only gradually +came to be studied on their own account. This does not apply +to Frege's work, in which cardinals, finite and transfinite, were +treated in complete independence of ordinals; but it was +Cantor's work that made the world aware of the subject, while +Frege's remained almost unknown, probably in the main on +account of the difficulty of his symbolism. And mathematicians, +like other people, have more difficulty in understanding and +using notions which are comparatively "simple" in the logical +sense than in manipulating more complex notions which are +<span class="pagenum" id="Page_95">[Pg 95]</span> +more akin to their ordinary practice. For these reasons, it was +only gradually that the true importance of cardinals in mathematical +philosophy was recognised. The importance of ordinals, +though by no means small, is distinctly less than that of cardinals, +and is very largely merged in that of the more general conception +of relation-numbers. +<span class="pagenum" id="Page_96">[Pg 96]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='X: LIMITS AND CONTINUITY'><a id="chap10"></a>CHAPTER X +<br><br> +LIMITS AND CONTINUITY</h2> + +<p class="nind"> +THE conception of a "limit" is one of which the importance in +mathematics has been found continually greater than had been +thought. The whole of the differential and integral calculus, +indeed practically everything in higher mathematics, depends +upon limits. Formerly, it was supposed that infinitesimals were +involved in the foundations of these subjects, but Weierstrass +showed that this is an error: wherever infinitesimals were thought +to occur, what really occurs is a set of finite quantities having +zero for their lower limit. It used to be thought that "limit" +was an essentially quantitative notion, namely, the notion of a +quantity to which others approached nearer and nearer, so that +among those others there would be some differing by less than any +assigned quantity. But in fact the notion of "limit" is a purely +ordinal notion, not involving quantity at all (except by accident +when the series concerned happens to be quantitative). A given +point on a line may be the limit of a set of points on the line, +without its being necessary to bring in co-ordinates or measurement +or anything quantitative. The cardinal number <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> is the +limit (in the order of magnitude) of the cardinal numbers 1, 2, +3, ... <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> ..., although the numerical difference between <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> +and a finite cardinal is constant and infinite: from a quantitative +point of view, finite numbers get no nearer to <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> as they grow +larger. What makes <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> the limit of the finite numbers is the +fact that, in the series, it comes immediately after them, which +is an <i>ordinal</i> fact, not a quantitative fact. +<span class="pagenum" id="Page_97">[Pg 97]</span> +</p> +<p> +There are various forms of the notion of "limit," of increasing +complexity. The simplest and most fundamental form, +from which the rest are derived, has been already defined, but +we will here repeat the definitions which lead to it, in a general +form in which they do not demand that the relation concerned +shall be serial. The definitions are as follows:— +</p> +<p> +The "minima" of a class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> with respect to a relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> are +those members of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and the field of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> (if any) to which no member +of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> has the relation <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">.</span> +</p> +<p> +The "maxima" with respect to <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> are the minima with respect +to the converse of <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">.</span> +</p> +<p> +The "sequents" of a class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> with respect to a relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> are +the minima of the "successors" of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> and the "successors" of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> +are those members of the field of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> to which every member of +the common part of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and the field of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> has the relation <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">.</span> +</p> +<p> +The "precedents" with respect to <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> are the sequents with +respect to the converse of <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">.</span> +</p> +<p> +The "upper limits" of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> with respect to <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> are the sequents +provided <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> has no maximum; but if <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> has a maximum, it has no +upper limits. +</p> +<p> +The "lower limits" with respect to <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> are the upper limits with +respect to the converse of <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">.</span> +</p> +<p> +Whenever <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> has connexity, a class can have at most one +maximum, one minimum, one sequent, etc. Thus, in the cases +we are concerned with in practice, we can speak of "<i>the</i> limit" +(if any). +</p> +<p> +When <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> is a serial relation, we can greatly simplify the above +definition of a limit. We can, in that case, define first the +"boundary" of a class <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> <i>i.e.</i> its limits or maximum, and then +proceed to distinguish the case where the boundary is the limit +from the case where it is a maximum. For this purpose it is +best to use the notion of "segment." +</p> +<p> +We will speak of the "segment of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> defined by a class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">" as +all those terms that have the relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> to some one or more of +the members of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> This will be a segment in the sense defined +<span class="pagenum" id="Page_98">[Pg 98]</span> +in Chapter VII.; indeed, every segment in the sense there defined +is the segment defined by <i>some</i> class <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> If <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> is serial, the +segment defined by <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> consists of all the terms that precede +some term or other of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> If <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> has a maximum, the segment will +be all the predecessors of the maximum. But if <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> has no +maximum, every member of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> precedes some other member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> +and the whole of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is therefore included in the segment defined +by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> Take, for example, the class consisting of the fractions +<span class="align-center"><img style="vertical-align: -0.816ex; width: 18.303ex; height: 2.789ex;" src="images/37.svg" alt="" data-tex=" +\tfrac{1}{2},\ \tfrac{3}{4},\ \tfrac{7}{8},\ \tfrac{15}{16},\ \dots, +"></span> +<i>i.e.</i> of all fractions of the form <img style="vertical-align: -1.552ex; width: 7.171ex; height: 4.588ex;" src="images/196.svg" alt="" data-tex="1 - \dfrac{1}{2^{n}}"> for different finite values +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">.</span> This series of fractions has no maximum, and it is clear +that the segment which it defines (in the whole series of fractions +in order of magnitude) is the class of all proper fractions. Or, +again, consider the prime numbers, considered as a selection from +the cardinals (finite and infinite) in order of magnitude. In this +case the segment defined consists of all finite integers. +</p> +<p> +Assuming that <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> is serial, the "boundary" of a class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> will be +the term <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> (if it exists) whose predecessors are the segment +defined by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> +</p> +<p> +A "maximum" of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is a boundary which is a member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> +</p> +<p> +An "upper limit" of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is a boundary which is not a member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> +</p> +<p> +If a class has no boundary, it has neither maximum nor limit. +This is the case of an "irrational" Dedekind cut, or of what is +called a "gap." +</p> +<p> +Thus the "upper limit" of a set of terms <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> with respect to a +series <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> is that term <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> (if it exists) which comes after all the <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">'</span>s, +but is such that every earlier term comes before some of the <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">'</span>s. +</p> +<p> +We may define all the "upper limiting-points" of a set of +terms <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> as all those that are the upper limits of sets of terms +chosen out of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">.</span> We shall, of course, have to distinguish upper +limiting-points from lower limiting-points. If we consider, for +example, the series of ordinal numbers: +<span class="align-center"><img style="vertical-align: -0.439ex; width: 62.044ex; height: 2.439ex;" src="images/38.svg" alt="" data-tex=" +1,\ 2,\ 3,\ \dots\ +\omega,\ \omega + 1,\ \dots\ +2\omega,\ 2\omega + 1,\ +3\omega,\ \dots\ \omega^{2},\ \dots\ \omega^{3},\ \dots, +"></span> +<span class="pagenum" id="Page_99">[Pg 99]</span> +the upper limiting-points of the field of this series are those that +have no immediate predecessors, <i>i.e.</i> +<span class="align-center"><img style="vertical-align: -0.439ex; width: 49.009ex; height: 2.439ex;" src="images/39.svg" alt="" data-tex=" +1,\ \omega,\ 2\omega,\ 3\omega,\ \dots\ +\omega^{2},\ \omega^{2} + \omega,\ \dots\ 2\omega^{2},\ \dots\ +\omega^{3}\ \dots +"></span> +The upper limiting-points of the field of this new series will be +<span class="align-center"><img style="vertical-align: -0.439ex; width: 30.855ex; height: 2.439ex;" src="images/40.svg" alt="" data-tex=" +1,\ \omega^{2},\ 2\omega^{2},\ \dots\ +\omega^{3},\ \omega^{3} + \omega^{2}\ \dots +"></span> +On the other hand, the series of ordinals—and indeed every well-ordered +series—has no lower limiting-points, because there are +no terms except the last that have no immediate successors. But +if we consider such a series as the series of ratios, every member +of this series is both an upper and a lower limiting-point for +suitably chosen sets. If we consider the series of real numbers, +and select out of it the rational real numbers, this set (the +rationals) will have all the real numbers as upper and lower +limiting-points. The limiting-points of a set are called its "first +derivative," and the limiting-points of the first derivative are +called the second derivative, and so on. +</p> +<p> +With regard to limits, we may distinguish various grades of +what may be called "continuity" in a series. The word "continuity" +had been used for a long time, but had remained without +any precise definition until the time of Dedekind and Cantor. +Each of these two men gave a precise significance to the term, +but Cantor's definition is narrower than Dedekind's: a series +which has Cantorian continuity must have Dedekindian continuity, +but the converse does not hold. +</p> +<p> +The first definition that would naturally occur to a man seeking +a precise meaning for the continuity of series would be to define +it as consisting in what we have called "compactness," <i>i.e.</i> in the +fact that between any two terms of the series there are others. +But this would be an inadequate definition, because of the +existence of "gaps" in series such as the series of ratios. We +saw in Chapter VII. that there are innumerable ways in which +the series of ratios can be divided into two parts, of which one +wholly precedes the other, and of which the first has no last term, +<span class="pagenum" id="Page_100">[Pg 100]</span> +while the second has no first term. Such a state of affairs seems +contrary to the vague feeling we have as to what should characterise +"continuity," and, what is more, it shows that the series of +ratios is not the sort of series that is needed for many mathematical +purposes. Take geometry, for example: we wish to be able to +say that when two straight lines cross each other they have a +point in common, but if the series of points on a line were similar +to the series of ratios, the two lines might cross in a "gap" and +have no point in common. This is a crude example, but many +others might be given to show that compactness is inadequate as +a mathematical definition of continuity. +</p> +<p> +It was the needs of geometry, as much as anything, that led +to the definition of "Dedekindian" continuity. It will be remembered +that we defined a series as Dedekindian when every +sub-class of the field has a boundary. (It is sufficient to assume +that there is always an <i>upper</i> boundary, or that there is always +a <i>lower</i> boundary. If one of these is assumed, the other can be +deduced.) That is to say, a series is Dedekindian when there +are no gaps. The absence of gaps may arise either through +terms having successors, or through the existence of limits in the +absence of maxima. Thus a finite series or a well-ordered series +is Dedekindian, and so is the series of real numbers. The former +sort of Dedekindian series is excluded by assuming that our +series is compact; in that case our series must have a property +which may, for many purposes, be fittingly called continuity. +Thus we are led to the definition: +</p> +<p> +A series has "Dedekindian continuity" when it is Dedekindian +and compact. +</p> +<p> +But this definition is still too wide for many purposes. Suppose, +for example, that we desire to be able to assign such properties +to geometrical space as shall make it certain that every point +can be specified by means of co-ordinates which are real numbers: +this is not insured by Dedekindian continuity alone. We want +to be sure that every point which cannot be specified by <i>rational</i> +co-ordinates can be specified as the limit of a <i>progression</i> of points +<span class="pagenum" id="Page_101">[Pg 101]</span> +whose co-ordinates are rational, and this is a further property +which our definition does not enable us to deduce. +</p> +<p> +We are thus led to a closer investigation of series with respect +to limits. This investigation was made by Cantor and formed +the basis of his definition of continuity, although, in its simplest +form, this definition somewhat conceals the considerations which +have given rise to it. We shall, therefore, first travel through +some of Cantor's conceptions in this subject before giving his +definition of continuity. +</p> +<p> +Cantor defines a series as "perfect" when all its points are +limiting-points and all its limiting-points belong to it. But this +definition does not express quite accurately what he means. +There is no correction required so far as concerns the property +that all its points are to be limiting-points; this is a property +belonging to compact series, and to no others if all points are to +be upper limiting- or all lower limiting-points. But if it is only +assumed that they are limiting-points one way, without specifying +which, there will be other series that will have the property +in question—for example, the series of decimals in which a decimal +ending in a recurring 9 is distinguished from the corresponding +terminating decimal and placed immediately before it. Such a +series is very nearly compact, but has exceptional terms which +are consecutive, and of which the first has no immediate predecessor, +while the second has no immediate successor. Apart from +such series, the series in which every point is a limiting-point +are compact series; and this holds without qualification if it is +specified that every point is to be an upper limiting-point (or +that every point is to be a lower limiting-point). +</p> +<p> +Although Cantor does not explicitly consider the matter, we +must distinguish different kinds of limiting-points according to +the nature of the smallest sub-series by which they can be defined. +Cantor assumes that they are to be defined by progressions, or +by regressions (which are the converses of progressions). When +every member of our series is the limit of a progression or regression, +Cantor calls our series "condensed in itself" (<i>insichdicht</i>). +<span class="pagenum" id="Page_102">[Pg 102]</span> +</p> +<p> +We come now to the second property by which perfection was +to be defined, namely, the property which Cantor calls that of +being "closed" (<i>abgeschlossen</i>). This, as we saw, was first defined +as consisting in the fact that all the limiting-points of a series +belong to it. But this only has any effective significance if our +series is <i>given</i> as contained in some other larger series (as is the +case, <i>e.g.</i>, with a selection of real numbers), and limiting-points +are taken in relation to the larger series. Otherwise, if a series +is considered simply on its own account, it cannot fail to contain +its limiting-points. What Cantor <i>means</i> is not exactly what +he says; indeed, on other occasions he says something rather +different, which <i>is</i> what he means. What he really means is that +every subordinate series which is of the sort that might be expected +to have a limit does have a limit within the given series; +<i>i.e.</i> every subordinate series which has no maximum has a limit, +<i>i.e.</i> every subordinate series has a boundary. But Cantor does +not state this for <i>every</i> subordinate series, but only for progressions +and regressions. (It is not clear how far he recognises that +this is a limitation.) Thus, finally, we find that the definition we +want is the following:— +</p> +<p> +A series is said to be "closed" (<i>abgeschlossen</i>) when every progression +or regression contained in the series has a limit in the +series. +</p> +<p> +We then have the further definition:— +</p> +<p> +A series is "perfect" when it is <i>condensed in itself</i> and <i>closed</i>, +<i>i.e.</i> when every term is the limit of a progression or regression, +and every progression or regression contained in the series has a +limit in the series. +</p> +<p> +In seeking a definition of continuity, what Cantor has in mind +is the search for a definition which shall apply to the series of +real numbers and to any series similar to that, but to no others. +For this purpose we have to add a further property. Among +the real numbers some are rational, some are irrational; although +the number of irrationals is greater than the number of rationals, +yet there are rationals between any two real numbers, however +<span class="pagenum" id="Page_103">[Pg 103]</span> +little the two may differ. The number of rationals, as we saw, +is <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span> This gives a further property which suffices to characterise +continuity completely, namely, the property of containing a class +of <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> members in such a way that some of this class occur +between any two terms of our series, however near together. +This property, added to perfection, suffices to define a class of +series which are all similar and are in fact a serial number. This +class Cantor defines as that of continuous series. +</p> +<p> +We may slightly simplify his definition. To begin with, +we say: +</p> +<p> +A "median class" of a series is a sub-class of the field such +that members of it are to be found between any two terms of +the series. +</p> +<p> +Thus the rationals are a median class in the series of real +numbers. It is obvious that there cannot be median classes +except in compact series. +</p> +<p> +We then find that Cantor's definition is equivalent to the +following:— +</p> +<p> +A series is "continuous" when (1) it is Dedekindian, (2) it +contains a median class having <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms. +</p> +<p> +To avoid confusion, we shall speak of this kind as "Cantorian +continuity." It will be seen that it implies Dedekindian continuity, +but the converse is not the case. All series having +Cantorian continuity are similar, but not all series having +Dedekindian continuity. +</p> +<p> +The notions of <i>limit</i> and <i>continuity</i> which we have been defining +must not be confounded with the notions of the limit of a function +for approaches to a given argument, or the continuity of a function +in the neighbourhood of a given argument. These are different +notions, very important, but derivative from the above and more +complicated. The continuity of motion (if motion is continuous) +is an instance of the continuity of a function; on the other hand, +the continuity of space and time (if they are continuous) is an +instance of the continuity of series, or (to speak more cautiously) +of a kind of continuity which can, by sufficient mathematical +<span class="pagenum" id="Page_104">[Pg 104]</span> +manipulation, be reduced to the continuity of series. In view +of the fundamental importance of motion in applied mathematics, +as well as for other reasons, it will be well to deal +briefly with the notions of limits and continuity as applied +to functions; but this subject will be best reserved for a +separate chapter. +</p> +<p> +The definitions of continuity which we have been considering, +namely, those of Dedekind and Cantor, do not correspond very +closely to the vague idea which is associated with the word in +the mind of the man in the street or the philosopher. They +conceive continuity rather as absence of separateness, the sort +of general obliteration of distinctions which characterises a thick +fog. A fog gives an impression of vastness without definite +multiplicity or division. It is this sort of thing that a metaphysician +means by "continuity," declaring it, very truly, +to be characteristic of his mental life and of that of children +and animals. +</p> +<p> +The general idea vaguely indicated by the word "continuity" +when so employed, or by the word "flux," is one which is certainly +quite different from that which we have been defining. Take, +for example, the series of real numbers. Each is what it is, +quite definitely and uncompromisingly; it does not pass over +by imperceptible degrees into another; it is a hard, separate +unit, and its distance from every other unit is finite, though +it can be made less than any given finite amount assigned in +advance. The question of the relation between the kind of +continuity existing among the real numbers and the kind exhibited, +<i>e.g.</i> by what we see at a given time, is a difficult and +intricate one. It is not to be maintained that the two kinds +are simply identical, but it may, I think, be very well maintained +that the mathematical conception which we have been +considering in this chapter gives the abstract logical scheme to +which it must be possible to bring empirical material by suitable +manipulation, if that material is to be called "continuous" +in any precisely definable sense. It would be quite impossible +<span class="pagenum" id="Page_105">[Pg 105]</span> +to justify this thesis within the limits of the present volume. +The reader who is interested may read an attempt to justify +it as regards <i>time</i> in particular by the present author in the +<i>Monist</i> for 1914-5, as well as in parts of <i>Our Knowledge of the +External World</i>. With these indications, we must leave this +problem, interesting as it is, in order to return to topics more +closely connected with mathematics. +<span class="pagenum" id="Page_106">[Pg 106]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='XI: LIMITS AND CONTINUITY OF FUNCTIONS'><a id="chap11"></a>CHAPTER XI +<br><br> +LIMITS AND CONTINUITY OF FUNCTIONS</h2> + +<p class="nind"> +IN this chapter we shall be concerned with the definition of the +limit of a function (if any) as the argument approaches a given +value, and also with the definition of what is meant by a "continuous +function." Both of these ideas are somewhat technical, +and would hardly demand treatment in a mere introduction +to mathematical philosophy but for the fact that, especially +through the so-called infinitesimal calculus, wrong views upon +our present topics have become so firmly embedded in the minds +of professional philosophers that a prolonged and considerable +effort is required for their uprooting. It has been thought +ever since the time of Leibniz that the differential and integral +calculus required infinitesimal quantities. Mathematicians +(especially Weierstrass) proved that this is an error; but errors +incorporated, <i>e.g.</i> in what Hegel has to say about mathematics, +die hard, and philosophers have tended to ignore the work of +such men as Weierstrass. +</p> +<p> +Limits and continuity of functions, in works on ordinary +mathematics, are defined in terms involving number. This is +not essential, as Dr Whitehead has shown.<a id="FNanchor_22_1"></a><a href="#Footnote_22_1" class="fnanchor">[22]</a> +We will, however, +begin with the definitions in the text-books, and proceed afterwards +to show how these definitions can be generalised so as to +apply to series in general, and not only to such as are numerical +or numerically measurable. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_22_1"></a><a href="#FNanchor_22_1"><span class="label">[22]</span></a>See <i>Principia Mathematica</i>, vol. II. * 230-234.</p></div> + +<p> +Let us consider any ordinary mathematical function <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;" src="images/197.svg" alt="" data-tex="fx">,</span> where +<span class="pagenum" id="Page_107">[Pg 107]</span> +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;" src="images/197.svg" alt="" data-tex="fx"> are both real numbers, and <img style="vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;" src="images/197.svg" alt="" data-tex="fx"> is one-valued—<i>i.e.</i> when +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is given, there is only one value that <img style="vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;" src="images/197.svg" alt="" data-tex="fx"> can have. We call <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> +the "argument," and <img style="vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;" src="images/197.svg" alt="" data-tex="fx"> the "value for the argument <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span>" When +a function is what we call "continuous," the rough idea for which +we are seeking a precise definition is that small differences in <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> +shall correspond to small differences in <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;" src="images/197.svg" alt="" data-tex="fx">,</span> and if we make the +differences in <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> small enough, we can make the differences in <img style="vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;" src="images/197.svg" alt="" data-tex="fx"> +fall below any assigned amount. We do not want, if a function +is to be continuous, that there shall be sudden jumps, so that, +for some value of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> any change, however small, will make a +change in <img style="vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;" src="images/197.svg" alt="" data-tex="fx"> which exceeds some assigned finite amount. The +ordinary simple functions of mathematics have this property: +it belongs, for example, to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.282ex; height: 1.912ex;" src="images/198.svg" alt="" data-tex="x^{2}">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.282ex; height: 1.91ex;" src="images/199.svg" alt="" data-tex="x^{3}">,</span> ... <span class="nowrap"><img style="vertical-align: -0.466ex; width: 4.563ex; height: 2.036ex;" src="images/200.svg" alt="" data-tex="\log x">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 4.449ex; height: 1.538ex;" src="images/201.svg" alt="" data-tex="\sin x">,</span> and so on. +But it is not at all difficult to define discontinuous functions. +Take, as a non-mathematical example, "the place of birth of +the youngest person living at time <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/202.svg" alt="" data-tex="t">.</span>" This is a function of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/202.svg" alt="" data-tex="t">;</span> +its value is constant from the time of one person's birth to the +time of the next birth, and then the value changes <i>suddenly</i> +from one birthplace to the other. An analogous mathematical +example would be "the integer next below <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span>" where <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a real +number. This function remains constant from one integer to +the next, and then gives a sudden jump. The actual fact is +that, though continuous functions are more familiar, they are +the exceptions: there are infinitely more discontinuous functions +than continuous ones. +</p> +<p> +Many functions are discontinuous for one or several values of +the variable, but continuous for all other values. Take as an +example <span class="nowrap"><img style="vertical-align: -0.566ex; width: 6.712ex; height: 2.262ex;" src="images/203.svg" alt="" data-tex="\sin 1/x">.</span> The function <img style="vertical-align: -0.025ex; width: 4.216ex; height: 1.62ex;" src="images/204.svg" alt="" data-tex="\sin \theta"> passes through all values +from -1 to 1 every time that <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.618ex;" src="images/205.svg" alt="" data-tex="\theta"> passes from <img style="vertical-align: -0.566ex; width: 5.312ex; height: 2.262ex;" src="images/206.svg" alt="" data-tex="-\pi/2"> to <span class="nowrap"><img style="vertical-align: -0.566ex; width: 3.552ex; height: 2.262ex;" src="images/207.svg" alt="" data-tex="\pi/2">,</span> or from +<img style="vertical-align: -0.566ex; width: 3.552ex; height: 2.262ex;" src="images/207.svg" alt="" data-tex="\pi/2"> to <span class="nowrap"><img style="vertical-align: -0.566ex; width: 4.683ex; height: 2.262ex;" src="images/208.svg" alt="" data-tex="3\pi/2">,</span> or generally from <img style="vertical-align: -0.566ex; width: 11.698ex; height: 2.262ex;" src="images/209.svg" alt="" data-tex="(2n - 1)\pi/2"> to <span class="nowrap"><img style="vertical-align: -0.566ex; width: 11.698ex; height: 2.262ex;" src="images/210.svg" alt="" data-tex="(2n + 1)\pi/2">,</span> where +<img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is any integer. Now if we consider <img style="vertical-align: -0.566ex; width: 3.557ex; height: 2.262ex;" src="images/211.svg" alt="" data-tex="1/x"> when <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is very small, +we see that as <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> diminishes <img style="vertical-align: -0.566ex; width: 3.557ex; height: 2.262ex;" src="images/211.svg" alt="" data-tex="1/x"> grows faster and faster, so that +it passes more and more quickly through the cycle of values from +one multiple of <img style="vertical-align: -0.566ex; width: 3.552ex; height: 2.262ex;" src="images/207.svg" alt="" data-tex="\pi/2"> to another as <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> becomes smaller and smaller. +Consequently <img style="vertical-align: -0.566ex; width: 6.712ex; height: 2.262ex;" src="images/203.svg" alt="" data-tex="\sin 1/x"> passes more and more quickly from -1 +<span class="pagenum" id="Page_108">[Pg 108]</span> +to 1 and back again, as <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> grows smaller. In fact, if we take +any interval containing 0, say the interval from <img style="vertical-align: -0.186ex; width: 2.679ex; height: 1.505ex;" src="images/212.svg" alt="" data-tex="-\epsilon"> to <img style="vertical-align: -0.186ex; width: 2.679ex; height: 1.505ex;" src="images/213.svg" alt="" data-tex="+\epsilon"> where +<img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon"> is some very small number, <img style="vertical-align: -0.566ex; width: 6.712ex; height: 2.262ex;" src="images/203.svg" alt="" data-tex="\sin 1/x"> will go through an infinite +number of oscillations in this interval, and we cannot diminish +the oscillations by making the interval smaller. Thus round +about the argument 0 the function is discontinuous. It is easy +to manufacture functions which are discontinuous in several +places, or in <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> places, or everywhere. Examples will be found +in any book on the theory of functions of a real variable. +</p> +<p> +Proceeding now to seek a precise definition of what is meant +by saying that a function is continuous for a given argument, +when argument and value are both real numbers, let us first +define a "neighbourhood" of a number <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> as all the numbers +from <img style="vertical-align: -0.186ex; width: 4.978ex; height: 1.505ex;" src="images/215.svg" alt="" data-tex="x - \epsilon"> to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 4.978ex; height: 1.505ex;" src="images/216.svg" alt="" data-tex="x + \epsilon">,</span> where <img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon"> is some number which, in important +cases, will be very small. It is clear that continuity at a given +point has to do with what happens in <i>any</i> neighbourhood of that +point, however small. +</p> +<p> +What we desire is this: If <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is the argument for which we wish +our function to be continuous, let us first define a neighbourhood +(<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> say) containing the value <img style="vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;" src="images/217.svg" alt="" data-tex="fa"> which the function has for the +argument <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">;</span> we desire that, if we take a sufficiently small +neighbourhood containing <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> all values for arguments throughout +this neighbourhood shall be contained in the neighbourhood <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> +no matter how small we may have made <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> That is to say, if +we decree that our function is not to differ from <img style="vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;" src="images/217.svg" alt="" data-tex="fa"> by more than +some very tiny amount, we can always find a stretch of real +numbers, having <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> in the middle of it, such that throughout +this stretch <img style="vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;" src="images/197.svg" alt="" data-tex="fx"> will not differ from <img style="vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;" src="images/217.svg" alt="" data-tex="fa"> by more than the prescribed +tiny amount. And this is to remain true whatever +tiny amount we may select. Hence we are led to the following +definition:— +</p> +<p> +The function <img style="vertical-align: -0.566ex; width: 4.299ex; height: 2.262ex;" src="images/218.svg" alt="" data-tex="f(x)"> is said to be "continuous" for the argument <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> +if, for every positive number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon">,</span> different from 0, but as +small as we please, there exists a positive number <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;" src="images/219.svg" alt="" data-tex="\delta">,</span> different +from 0, such that, for all values of <img style="vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;" src="images/219.svg" alt="" data-tex="\delta"> which are numerically +<span class="pagenum" id="Page_109">[Pg 109]</span> +less<a id="FNanchor_23_1"></a><a href="#Footnote_23_1" class="fnanchor">[23]</a> +than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon">,</span> the difference <img style="vertical-align: -0.566ex; width: 15.819ex; height: 2.262ex;" src="images/220.svg" alt="" data-tex="f(a) + \delta) - f(a)"> is numerically less +than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/221.svg" alt="" data-tex="\sigma">.</span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_23_1"></a><a href="#FNanchor_23_1"><span class="label">[23]</span></a>A number is said to be "numerically less" than <img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon"> when it lies between +<img style="vertical-align: -0.186ex; width: 2.679ex; height: 1.505ex;" src="images/212.svg" alt="" data-tex="-\epsilon"> and <span class="nowrap"><img style="vertical-align: -0.186ex; width: 2.679ex; height: 1.505ex;" src="images/213.svg" alt="" data-tex="+\epsilon">.</span></p></div> + +<p> +In this definition, <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/221.svg" alt="" data-tex="\sigma"> first defines a neighbourhood of <span class="nowrap"><img style="vertical-align: -0.566ex; width: 4.201ex; height: 2.262ex;" src="images/222.svg" alt="" data-tex="f(a)">,</span> +namely, the neighbourhood from <img style="vertical-align: -0.566ex; width: 8.259ex; height: 2.262ex;" src="images/223.svg" alt="" data-tex="f(a) - \sigma"> to <span class="nowrap"><img style="vertical-align: -0.566ex; width: 7.886ex; height: 2.262ex;" src="images/224.svg" alt="" data-tex="f(a) + \epsilon">.</span> The definition +then proceeds to say that we can (by means of <img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon"> define a +neighbourhood, namely, that from <img style="vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;" src="images/225.svg" alt="" data-tex="a - \epsilon"> to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;" src="images/226.svg" alt="" data-tex="a + \epsilon">,</span> such that, for +all arguments within this neighbourhood, the value of the function +lies within the neighbourhood horn <img style="vertical-align: -0.566ex; width: 8.259ex; height: 2.262ex;" src="images/223.svg" alt="" data-tex="f(a) - \sigma"> to <span class="nowrap"><img style="vertical-align: -0.566ex; width: 8.259ex; height: 2.262ex;" src="images/227.svg" alt="" data-tex="f(a) + \sigma">.</span> If this +can be done, however <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/221.svg" alt="" data-tex="\sigma"> may be chosen, the function is "continuous" +for the argument <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">.</span> +</p> +<p> +So far we have not defined the "limit" of a function for a +given argument. If we had done so, we could have defined the +continuity of a function differently: a function is continuous +at a point where its value is the same as the limit of its value for +approaches either from above or from below. But it is only +the exceptionally "tame" function that has a definite limit as +the argument approaches a given point. The general rule is +that a function oscillates, and that, given any neighbourhood +of a given argument, however small, a whole stretch of values +will occur for arguments within this neighbourhood. As this +is the general rule, let us consider it first. +</p> +<p> +Let us consider what may happen as the argument approaches +some value <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> from below. That is to say, we wish to consider +what happens for arguments contained in the interval from +<img style="vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;" src="images/225.svg" alt="" data-tex="a - \epsilon"> to <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> where <img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon"> is some number which, in important cases, +will be very small. +</p> +<p> +The values of the function for arguments from <img style="vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;" src="images/225.svg" alt="" data-tex="a - \epsilon"> to <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> (<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> excluded) +will be a set of real numbers which will define a certain +section of the set of real numbers, namely, the section consisting +of those numbers that are not greater than <i>all</i> the values for +arguments from <img style="vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;" src="images/225.svg" alt="" data-tex="a - \epsilon"> to <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">.</span> Given any number in this section, +there are values at least as great as this number for arguments +between <img style="vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;" src="images/225.svg" alt="" data-tex="a - \epsilon"> and <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> <i>i.e.</i> for arguments that fall very little short +<span class="pagenum" id="Page_110">[Pg 110]</span> +of <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> (if <img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon"> is very small). Let us take all possible <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon">'</span>s and all +possible corresponding sections. The common part of all these +sections we will call the "ultimate section" as the argument +approaches <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">.</span> To say that a number <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> belongs to the ultimate +section is to say that, however small we may make <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon">,</span> there are +arguments between <img style="vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;" src="images/225.svg" alt="" data-tex="a - \epsilon"> and <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> for which the value of the function +is not less than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">.</span> +</p> +<p> +We may apply exactly the same process to upper sections, +<i>i.e.</i> to sections that go from some point up to the top, instead of +from the bottom up to some point. Here we take those numbers +that are not <i>less</i> than all the values for arguments from <img style="vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;" src="images/225.svg" alt="" data-tex="a - \epsilon"> +to <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">;</span> this defines an upper section which will vary as <img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon"> varies. +Taking the common part of all such sections for all possible <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon">'</span>s, +we obtain the "ultimate upper section." To say that a number <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> +belongs to the ultimate upper section is to say that, however +small we make <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon">,</span> there are arguments between <img style="vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;" src="images/225.svg" alt="" data-tex="a - \epsilon"> and <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> for +which the value of the function is not <i>greater</i> than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z">.</span> +</p> +<p> +If a term <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> belongs both to the ultimate section and to the +ultimate upper section, we shall say that it belongs to the +"ultimate oscillation." We may illustrate the matter by considering +once more the function <img style="vertical-align: -0.566ex; width: 6.712ex; height: 2.262ex;" src="images/203.svg" alt="" data-tex="\sin 1/x"> as <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/228.svg" alt="" data-tex="x "> approaches the +value 0. We shall assume, in order to fit in with the above +definitions, that this value is approached from below. +</p> +<p> +Let us begin with the "ultimate section." Between <img style="vertical-align: -0.186ex; width: 2.679ex; height: 1.505ex;" src="images/212.svg" alt="" data-tex="-\epsilon"> +and 0, whatever <img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon"> may be, the function will assume the value 1 +for certain arguments, but will never assume any greater value. +Hence the ultimate section consists of all real numbers, positive +and negative, up to and including 1; <i>i.e.</i> it consists of all negative +numbers together with 0, together with the positive numbers +up to and including 1. +</p> +<p> +Similarly the "ultimate upper section" consists of all positive +numbers together with 0, together with the negative numbers +down to and including -1. +</p> +<p> +Thus the "ultimate oscillation" consists of all real numbers +from -1 to 1, both included. +<span class="pagenum" id="Page_111">[Pg 111]</span> +</p> +<p> +We may say generally that the "ultimate oscillation" of +a function as the argument approaches <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> from below consists +of all those numbers <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> which are such that, however near we +come to <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> we shall still find values as great as <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and values as +small as <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> +</p> +<p> +The ultimate oscillation may contain no terms, or one term, +or many terms. In the first two cases the function has a definite +limit for approaches from below. If the ultimate oscillation +has one term, this is fairly obvious. It is equally true if it has +none; for it is not difficult to prove that, if the ultimate oscillation +is null, the boundary of the ultimate section is the same as +that of the ultimate upper section, and may be defined as the +limit of the function for approaches from below. But if the +ultimate oscillation has many terms, there is no definite limit to +the function for approaches from below. In this case we can +take the lower and upper boundaries of the ultimate oscillation +(<i>i.e.</i> the lower boundary of the ultimate upper section and the +upper boundary of the ultimate section) as the lower and upper +limits of its "ultimate" values for approaches from below. +Similarly we obtain lower and upper limits of the "ultimate" +values for approaches from above. Thus we have, in the general +case, <i>four</i> limits to a function for approaches to a given argument. +<i>The</i> limit for a given argument <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> only exists when all these four +are equal, and is then their common value. If it is also the +<i>value</i> for the argument <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> the function is continuous for this +argument. This may be taken as defining continuity: it is +equivalent to our former definition. +</p> +<p> +We can define the limit of a function for a given argument +(if it exists) without passing through the ultimate oscillation +and the four limits of the general case. The definition proceeds, +in that case, just as the earlier definition of continuity proceeded. +Let us define the limit for approaches from below. If there is to +be a definite limit for approaches to <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> from below, it is necessary +and sufficient that, given any small number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/221.svg" alt="" data-tex="\sigma">,</span> two values for +arguments sufficiently near to <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> (but both less than <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">)</span> will differ +<span class="pagenum" id="Page_112">[Pg 112]</span> +by less than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/221.svg" alt="" data-tex="\sigma">;</span> <i>i.e.</i> if <img style="vertical-align: -0.025ex; width: 0.919ex; height: 1ex;" src="images/214.svg" alt="" data-tex="\epsilon"> is sufficiently small, and our arguments +both lie between <img style="vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;" src="images/225.svg" alt="" data-tex="a - \epsilon"> and <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> (<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> excluded), then the difference +between the values for these arguments will be less than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/221.svg" alt="" data-tex="\sigma">.</span> +This is to hold for any <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/221.svg" alt="" data-tex="\sigma">,</span> however small; in that case the +function has a limit for approaches from below. Similarly +we define the case when there is a limit for approaches from +above. These two limits, even when both exist, need not be +identical; and if they are identical, they still need not be identical +with the <i>value</i> for the argument <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">.</span> It is only in this last case +that we call the function <i>continuous</i> for the argument <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">.</span> +</p> +<p> +A function is called "continuous" (without qualification) +when it is continuous for every argument. +</p> +<p> +Another slightly different method of reaching the definition +of continuity is the following:— +</p> +<p> +Let us say that a function "ultimately converges into a +class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">" if there is some real number such that, for this argument +and all arguments greater than this, the value of the function +is a member of the class <span class="nowrap"><img style="vertical-align: -0.439ex; width: 5.509ex; height: 2.009ex;" src="images/229.svg" alt="" data-tex="alpha">.</span> Similarly we shall say that a function +"converges into <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> as the argument approaches <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> from below" +if there is some argument <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> less than <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> such that throughout +the interval from <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> (included) to <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> (excluded) the function has +values which are members of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 5.509ex; height: 2.009ex;" src="images/229.svg" alt="" data-tex="alpha">.</span> We may now say that a +function is continuous for the argument <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> for which it has the +value <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;" src="images/217.svg" alt="" data-tex="fa">,</span> if it satisfies four conditions, namely:— +</p> +<p> +(1) Given any real number less than <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;" src="images/217.svg" alt="" data-tex="fa">,</span> the function converges +into the successors of this number as the argument +approaches <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> from below; +</p> +<p> +(2) Given any real number greater than <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;" src="images/217.svg" alt="" data-tex="fa">,</span> the function converges +into the predecessors of this number as the argument +approaches <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> from below; +</p> +<p> +(3) and (4) Similar conditions for approaches to <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> from above. +</p> +<p> +The advantages of this form of definition is that it analyses +the conditions of continuity into four, derived from considering +arguments and values respectively greater or less than the +argument and value for which continuity is to be defined. +<span class="pagenum" id="Page_113">[Pg 113]</span> +</p> +<p> +We may now generalise our definitions so as to apply to series +which are not numerical or known to be numerically measurable. +The case of motion is a convenient one to bear in mind. There +is a story by H. G. Wells which will illustrate, from the case of +motion, the difference between the limit of a function for a given +argument and its value for the same argument. The hero of +the story, who possessed, without his knowledge, the power of +realising his wishes, was being attacked by a policeman, but on +ejaculating "Go to——" he found that the policeman disappeared. +If <img style="vertical-align: -0.566ex; width: 3.821ex; height: 2.262ex;" src="images/230.svg" alt="" data-tex="f(t)"> was the policeman's position at time <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/202.svg" alt="" data-tex="t">,</span> and <img style="vertical-align: -0.375ex; width: 1.804ex; height: 1.791ex;" src="images/231.svg" alt="" data-tex="t_{0}"> the moment +of the ejaculation, the limit of the policeman's positions as <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/202.svg" alt="" data-tex="t"> approached +to <img style="vertical-align: -0.375ex; width: 1.804ex; height: 1.791ex;" src="images/231.svg" alt="" data-tex="t_{0}"> from below would be in contact with the hero, +whereas the value for the argument <img style="vertical-align: -0.375ex; width: 1.804ex; height: 1.791ex;" src="images/231.svg" alt="" data-tex="t_{0}"> was —. But such occurrences +are supposed to be rare in the real world, and it is assumed, +though without adequate evidence, that all motions are continuous, +<i>i.e.</i> that, given any body, if <img style="vertical-align: -0.566ex; width: 3.821ex; height: 2.262ex;" src="images/230.svg" alt="" data-tex="f(t)"> is its position at time <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/202.svg" alt="" data-tex="t">,</span> <img style="vertical-align: -0.566ex; width: 3.821ex; height: 2.262ex;" src="images/230.svg" alt="" data-tex="f(t)"> is +a continuous function of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/202.svg" alt="" data-tex="t">.</span> It is the meaning of "continuity" +involved in such statements which we now wish to define as +simply as possible. +</p> +<p> +The definitions given for the case of functions where argument +and value are real numbers can readily be adapted for more +general use. +</p> +<p> +Let <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> and <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> be two relations, which it is well to imagine +serial, though it is not necessary to our definitions that they +should be so. Let <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> be a one-many relation whose domain +is contained in the field of <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span> while its converse domain is contained +in the field of <span class="nowrap"><img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">.</span> Then <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is (in a generalised sense) a +function, whose arguments belong to the field of <span class="nowrap"><img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">,</span> while its +values belong to the field of <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">.</span> Suppose, for example, that we +are dealing with a particle moving on a line: let <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> be the time-series, +<img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> the series of points on our line from left to right, <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> the +relation of the position of our particle on the line at time <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> to +the time <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> so that "the <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> of <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">" is its position at time <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">.</span> This +illustration may be borne in mind throughout our definitions. +</p> +<p> +We shall say that the function <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is continuous for the argument <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> +<span class="pagenum" id="Page_114">[Pg 114]</span> +if, given any interval <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> on the <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">-series containing the value +of the function for the argument <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> there is an interval on the +<img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">-series containing <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> not as an end-point and such that, throughout +this interval, the function has values which are members +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> (We mean by an "interval" all the terms between any +two; <i>i.e.</i> if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are two members of the field of <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span> and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has +the relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> we shall mean by the "<img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">-interval <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> to <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">" +all terms <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> such that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> to <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/79.svg" alt="" data-tex="z"> has the relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> +to <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">—together, when so stated, with <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> or <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> themselves.) +</p> +<p> +We can easily define the "ultimate section" and the "ultimate +oscillation." To define the "ultimate section" for +approaches to the argument <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> from below, take any argument <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> +which precedes <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> (<i>i.e.</i> has the relation <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> to <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">)</span>, take the values +of the function for all arguments up to and including <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> and +form the section of <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> defined by these values, <i>i.e.</i> those members +of the <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">-series which are earlier than or identical with some of +these values. Form all such sections for all <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">'</span>s that precede <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> +and take their common part; this will be the ultimate section. +The ultimate upper section and the ultimate oscillation are then +defined exactly as in the previous case. +</p> +<p> +The adaptation of the definition of convergence and the +resulting alternative definition of continuity offers no difficulty +of any kind. +</p> +<p> +We say that a function <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is "ultimately <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">-convergent into <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">" +if there is a member <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> of the converse domain of <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> and the +field of <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> such that the value of the function for the argument <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> +and for any argument to which <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> has the relation <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is a member +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> We say that <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> "<img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">-converges into <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> as the argument +approaches a given argument <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">" if there is a term <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> having +the relation <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> to <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and belonging to the converse domain of <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> +and such that the value of the function for any argument in the +<img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">-interval from <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> (inclusive) to <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> (exclusive) belongs to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> +</p> +<p> +Of the four conditions that a function must fulfil in order +to be continuous for the argument <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> the first is, putting <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> for +the value for the argument <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">:</span> +<span class="pagenum" id="Page_115">[Pg 115]</span> +</p> +<p> +Given any term having the relation <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b">,</span> <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">-converges +into the successors of <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> (with respect to <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">)</span> as the argument +approaches <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> from below. +</p> +<p> +The second condition is obtained by replacing <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> by its +converse; the third and fourth are obtained from the first and +second by replacing <img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q"> by its converse. +</p> +<p> +There is thus nothing, in the notions of the limit of a function +or the continuity of a function, that essentially involves number. +Both can be defined generally, and many propositions about +them can be proved for any two series (one being the argument-series +and the other the value-series). It will be seen that the +definitions do not involve infinitesimals. They involve infinite +classes of intervals, growing smaller without any limit short of +zero, but they do not involve any intervals that are not finite. +This is analogous to the fact that if a line an inch long be halved, +then halved again, and so on indefinitely, we never reach infinitesimals +in this way: after <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> bisections, the length of our bit is +<img style="vertical-align: -1.552ex; width: 3.274ex; height: 4.588ex;" src="images/232.svg" alt="" data-tex="\dfrac{1}{2^{n}}"> of an inch; and this is finite whatever finite number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> may +be. The process of successive bisection does not lead to +divisions whose ordinal number is infinite, since it is essentially +a one-by-one process. Thus infinitesimals are not to be reached +in this way. Confusions on such topics have had much to do +with the difficulties which have been found in the discussion of +infinity and continuity. +<span class="pagenum" id="Page_116">[Pg 116]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='XII: SELECTIONS AND THE MULTIPLICATIVE AXIOM'><a id="chap12"></a>CHAPTER XII +<br><br> +SELECTIONS AND THE MULTIPLICATIVE AXIOM</h2> + +<p class="nind"> +IN this chapter we have to consider an axiom which can be +enunciated, but not proved, in terms of logic, and which is convenient, +though not indispensable, in certain portions of mathematics. +It is convenient, in the sense that many interesting +propositions, which it seems natural to suppose true, cannot +be proved without its help; but it is not indispensable, because +even without those propositions the subjects in which they +occur still exist, though in a somewhat mutilated form. +</p> +<p> +Before enunciating the multiplicative axiom, we must first +explain the theory of selections, and the definition of multiplication +when the number of factors may be infinite. +</p> +<p> +In defining the arithmetical operations, the only correct procedure +is to construct an actual class (or relation, in the case +of relation-numbers) having the required number of terms. +This sometimes demands a certain amount of ingenuity, but +it is essential in order to prove the existence of the number +defined. Take, as the simplest example, the case of addition. +Suppose we are given a cardinal number <span class="nowrap"><img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu">,</span> and a class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> which +has <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms. How shall we define <span class="nowrap"><img style="vertical-align: -0.489ex; width: 5.494ex; height: 1.808ex;" src="images/233.svg" alt="" data-tex="\mu + \mu">?</span> For this purpose +we must have <i>two</i> classes having <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms, and they must not +overlap. We can construct such classes from <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> in various ways, +of which the following is perhaps the simplest: Form first all +the ordered couples whose first term is a class consisting of a +single member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> and whose second term is the null-class; +then, secondly, form all the ordered couples whose first term is +<span class="pagenum" id="Page_117">[Pg 117]</span> +the null-class and whose second term is a class consisting of a +single member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> These two classes of couples have no +member in common, and the logical sum of the two classes will +have <img style="vertical-align: -0.489ex; width: 5.494ex; height: 1.808ex;" src="images/233.svg" alt="" data-tex="\mu + \mu"> terms. Exactly analogously we can define <span class="nowrap"><img style="vertical-align: -0.489ex; width: 5.329ex; height: 1.808ex;" src="images/181.svg" alt="" data-tex="\mu + \nu">,</span> +given that <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is the number of some class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> is the number +of some class <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">.</span> +</p> +<p> +Such definitions, as a rule, are merely a question of a suitable +technical device. But in the case of multiplication, where the +number of factors may be infinite, important problems arise out +of the definition. +</p> +<p> +Multiplication when the number of factors is finite offers no +difficulty. Given two classes <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">,</span> of which the first has +<img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms and the second <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> terms, we can define <img style="vertical-align: -0.489ex; width: 4.324ex; height: 1.6ex;" src="images/234.svg" alt="" data-tex="\mu × \nu"> as the number +of ordered couples that can be formed by choosing the first term +out of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and the second out of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">.</span> It will be seen that this definition +does not require that <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> should not overlap; it +even remains adequate when <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> are identical. For example, +let <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> be the class whose members are <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}">,</span> <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/236.svg" alt="" data-tex="x_{2}">,</span> <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/237.svg" alt="" data-tex="x_{3}">.</span> Then the class +which is used to define the product <img style="vertical-align: -0.489ex; width: 4.489ex; height: 1.6ex;" src="images/238.svg" alt="" data-tex="\mu × \mu"> is the class of couples: +<span class="align-center"><img style="vertical-align: -3.507ex; width: 25.762ex; height: 8.145ex;" src="images/41.svg" alt="" data-tex=" +\begin{align*} +(x_{1}, x_{1}),\ (x_{1}, x_{2}),\ (x_{1}, x_{3}); \\ +(x_{2}, x_{1}),\ (x_{2}, x_{2}),\ (x_{2}, x_{3}); \\ +(x_{3}, x_{1}),\ (x_{3}, x_{2}),\ (x_{3}, x_{3}). +\end{align*} +"></span> +This definition remains applicable when <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> or <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> or both are +infinite, and it can be extended step by step to three or four or +any finite number of factors. No difficulty arises as regards +this definition, except that it cannot be extended to an <i>infinite</i> +number of factors. +</p> +<p> +The problem of multiplication when the number of factors +may be infinite arises in this way: Suppose we have a class <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> +consisting of classes; suppose the number of terms in each of +these classes is given. How shall we define the product of all +these numbers? If we can frame our definition generally, it +will be applicable whether <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> is finite or infinite. It is to be +observed that the problem is to be able to deal with the case +when <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> is infinite, not with the case when its members are. If +<span class="pagenum" id="Page_118">[Pg 118]</span> +<img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> is not infinite, the method defined above is just as applicable +when its members are infinite as when they are finite. It is +the case when <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> is infinite, even though its members may be +finite, that we have to find a way of dealing with. +</p> +<p> +The following method of defining multiplication generally is +due to Dr Whitehead. It is explained and treated at length in +<i>Principia Mathematica</i>, vol. I. * 80 ff., and vol. II. * 114. +</p> +<p> +Let us suppose to begin with that <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> is a class of classes no two +of which overlap—say the constituencies in a country where +there is no plural voting, each constituency being considered +as a class of voters. Let us now set to work to choose one term +out of each class to be its <i>representative</i>, as constituencies do +when they elect members of Parliament, assuming that by law +each constituency has to elect a man who is a voter in that +constituency. We thus arrive at a class of representatives, who +make up our Parliament, one being selected out of each constituency. +How many different possible ways of choosing a +Parliament are there? Each constituency can select any one +of its voters, and therefore if there are <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> voters in a constituency, +it can make <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> choices. The choices of the different constituencies +are independent; thus it is obvious that, when the total number +of constituencies is finite, the number of possible Parliaments +is obtained by multiplying together the numbers of voters in the +various constituencies. When we do not know whether the +number of constituencies is finite or infinite, we may take the +number of possible Parliaments as <i>defining</i> the product of the +numbers of the separate constituencies. This is the method +by which infinite products are defined. We must now drop our +illustration, and proceed to exact statements. +</p> +<p> +Let <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> be a class of classes, and let us assume to begin with that +no two members of <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> overlap, <i>i.e.</i> that if <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> are two different +members of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">,</span> then no member of the one is a member of the +other. We shall call a class a "selection" from <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> when it consists +of just one term from each member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">;</span> <i>i.e.</i> <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> is a "selection" +from <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> if every member of <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> belongs to some member +<span class="pagenum" id="Page_119">[Pg 119]</span> +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">,</span> and if <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> be any member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">,</span> <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> and <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> have exactly one term +in common. The class of all "selections" from <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> we shall call +the "multiplicative class" of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">.</span> The number of terms in the +multiplicative class of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">,</span> <i>i.e.</i> the number of possible selections +from <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">,</span> is defined as the product of the numbers of the members +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">.</span> This definition is equally applicable whether <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> is finite +or infinite. +</p> +<p> +Before we can be wholly satisfied with these definitions, we +must remove the restriction that no two members of <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> are to +overlap. For this purpose, instead of defining first a class +called a "selection," we will define first a relation which we will +call a "selector." A relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> will be called a "selector" +from <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> if, from every member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">,</span> it picks out one term as the +representative of that member, <i>i.e.</i> if, given any member <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">,</span> +there is just one term <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> which is a member of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and has the +relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">;</span> and this is to be all that <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> does. The formal +definition is: +</p> +<p> +A "selector" from a class of classes <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> is a one-many relation, +having <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> for its converse domain, and such that, if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the +relation to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> then <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> +</p> +<p> +If <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> is a selector from <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">,</span> and <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is a member +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">,</span> and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is the +term which has the relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> we call <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> the "representative" +of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> in respect of the relation <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">.</span> +</p> +<p> +A "selection" from <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> will now be defined as the domain of a +selector; and the multiplicative class, as before, will be the class +of selections. +</p> +<p> +But when the members of <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> overlap, there may be more selectors +than selections, since a term <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> which belongs to two classes <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> +and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> may be selected once to represent <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and once +to represent <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">,</span> +giving rise to different selectors in the two cases, but to the same +selection. For purposes of defining multiplication, it is the +selectors we require rather than the selections. Thus we define: +</p> +<p> +"The product of the numbers of the members of a class of +classes <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">" is the number of selectors from <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">.</span> +</p> +<p> +We can define exponentiation by an adaptation of the above +<span class="pagenum" id="Page_120">[Pg 120]</span> +plan. We might, of course, define <img style="vertical-align: -0.489ex; width: 2.4ex; height: 2.017ex;" src="images/240.svg" alt="" data-tex="\mu^{\nu}"> as the number of selectors +from <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> classes, each of which has <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms. But there are +objections to this definition, derived from the fact that the +multiplicative axiom (of which we shall speak shortly) is unnecessarily +involved if it is adopted. We adopt instead the following +construction:— +</p> +<p> +Let <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> be a class having <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms, and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> a +class having <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> terms. +</p> +<p> +Let <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> be a member of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">,</span> and form the class of all ordered +couples that have <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> for their second term and a member of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> for +their first term. There will be <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> such couples for a given <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> since +any member of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> may be chosen for the first term, and <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> has <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> members. +If we now form all the classes of this sort that result +from varying <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span> we obtain altogether <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> classes, since <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> may be +any member of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">,</span> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> has <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> members. These <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> classes are each +of them a class of couples, namely, all the couples that can be +formed of a variable member of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and a fixed member of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">.</span> We +define <img style="vertical-align: -0.489ex; width: 2.4ex; height: 2.017ex;" src="images/240.svg" alt="" data-tex="\mu^{\nu}"> as the number of selectors from the class consisting of +these <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> classes. Or we may equally well define <img style="vertical-align: -0.489ex; width: 2.4ex; height: 2.017ex;" src="images/240.svg" alt="" data-tex="\mu^{\nu}"> as the number of +selections, for, since our classes of couples are mutually exclusive, +the number of selectors is the same as the number of selections. +A selection from our class of classes will be a set of ordered couples, +of which there will be exactly one having any given member of <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> +for its second term, and the first term may be any member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> +Thus <img style="vertical-align: -0.489ex; width: 2.4ex; height: 2.017ex;" src="images/240.svg" alt="" data-tex="\mu^{\nu}"> is defined by the selectors from a certain set of <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> classes +each having <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms, but the set is one having a certain structure +and a more manageable composition than is the case in general. +The relevance of this to the multiplicative axiom will appear +shortly. +</p> +<p> +What applies to exponentiation applies also to the product of +two cardinals. We might define "<img style="vertical-align: -0.489ex; width: 4.324ex; height: 1.6ex;" src="images/234.svg" alt="" data-tex="\mu × \nu">" as the sum of the +numbers of <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> classes each having <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms, but we prefer to define +it as the number of ordered couples to be formed consisting of a +member of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> followed by a member of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">,</span> where +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> has <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms +and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> has <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> terms. This definition, also, is designed to evade the +necessity of assuming the multiplicative axiom. +<span class="pagenum" id="Page_121">[Pg 121]</span> +</p> +<p> +With our definitions, we can prove the usual formal laws of +multiplication and exponentiation. But there is one thing we +cannot prove: we cannot prove that a product is only zero when +one of its factors is zero. We can prove this when the number +of factors is finite, but not when it is infinite. In other words, +we cannot prove that, given a class of classes none of which is +null, there must be selectors from them; or that, given a class +of mutually exclusive classes, there must be at least one class +consisting of one term out of each of the given classes. These +things cannot be proved; and although, at first sight, they seem +obviously true, yet reflection brings gradually increasing doubt, +until at last we become content to register the assumption and +its consequences, as we register the axiom of parallels, without +assuming that we can know whether it is true or false. The +assumption, loosely worded, is that selectors and selections exist +when we should expect them. There are many equivalent ways +of stating it precisely. We may begin with the following:— +</p> +<p> +"Given any class of mutually exclusive classes, of which none +is null, there is at least one class which has exactly one term in +common with each of the given classes." +</p> +<p> +This proposition we will call the "multiplicative axiom."<a id="FNanchor_24_1"></a><a href="#Footnote_24_1" class="fnanchor">[24]</a> +We will first give various equivalent forms of the proposition, +and then consider certain ways in which its truth or falsehood +is of interest to mathematics. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_24_1"></a><a href="#FNanchor_24_1"><span class="label">[24]</span></a><i>Principia Mathematica</i>, vol. I. * 88. Also vol. III. * 257-258.</p></div> + +<p> +The multiplicative axiom is equivalent to the proposition that +a product is only zero when at least one of its factors is zero; +<i>i.e.</i> that, if any number of cardinal numbers be multiplied together, +the result cannot be 0 unless one of the numbers concerned is 0. +</p> +<p> +The multiplicative axiom is equivalent to the proposition that, +if <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> be any relation, and <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> any class contained in the converse +domain of <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">,</span> then there is at least one one-many relation implying <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> +and having <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> for its converse domain. +</p> +<p> +The multiplicative axiom is equivalent to the assumption that +if <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> be any class, and <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> all the sub-classes of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> with the exception +<span class="pagenum" id="Page_122">[Pg 122]</span> +of the null-class, then there is at least one selector from <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">.</span> This +is the form in which the axiom was first brought to the notice of +the learned world by Zermelo, in his "Beweis, dass jede Menge +wohlgeordnet werden kann."<a id="FNanchor_25_1"></a><a href="#Footnote_25_1" class="fnanchor">[25]</a> +Zermelo regards the axiom as an +unquestionable truth. It must be confessed that, until he made +it explicit, mathematicians had used it without a qualm; but it +would seem that they had done so unconsciously. And the credit +due to Zermelo for having made it explicit is entirely independent +of the question whether it is true or false. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_25_1"></a><a href="#FNanchor_25_1"><span class="label">[25]</span></a><i>Mathematische Annalen</i>, vol. LIX. pp. 514-6. In this form we shall +speak of it as Zermelo's axiom.</p></div> + +<p> +The multiplicative axiom has been shown by Zermelo, in the +above-mentioned proof, to be equivalent to the proposition that +every class can be well-ordered, <i>i.e.</i> can be arranged in a series in +which every sub-class has a first term (except, of course, the null-class). +The full proof of this proposition is difficult, but it is not +difficult to see the general principle upon which it proceeds. It +uses the form which we call "Zermelo's axiom," <i>i.e.</i> it assumes +that, given any class <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> there is at least one one-many relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> +whose converse domain consists of all existent sub-classes of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> +and which is such that, if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;" src="images/241.svg" alt="" data-tex="\xi">,</span> then <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a +member of <span class="nowrap"><img style="vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;" src="images/241.svg" alt="" data-tex="\xi">.</span> Such a relation picks out a "representative" +from each sub-class; of course, it will often happen that two +sub-classes have the same representative. What Zermelo does, +in effect, is to count off the members of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> one by one, by means +of <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> and transfinite induction. We put first the representative +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">;</span> call it <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}">.</span> Then take the representative of the class consisting +of all of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> except <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}">;</span> call it <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/236.svg" alt="" data-tex="x_{2}">.</span> It must be different from <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}">,</span> +because every representative is a member of its class, and <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}">is +shut out from this class. Proceed similarly to take away <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/236.svg" alt="" data-tex="x_{2}">,</span> and +let <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/237.svg" alt="" data-tex="x_{3}"> be the representative of what is left. In this way we first +obtain a progression <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}">,</span> <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/236.svg" alt="" data-tex="x_{2}">,</span> ... <span class="nowrap"><img style="vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;" src="images/52.svg" alt="" data-tex="x_{n}">,</span> ..., assuming +that <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is not +finite. We then take away the whole progression; let <img style="vertical-align: -0.357ex; width: 2.477ex; height: 1.357ex;" src="images/242.svg" alt="" data-tex="x_{\omega}"> be the +representative of what is left of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> In this way we can go on +until nothing is left. The successive representatives will form a +<span class="pagenum" id="Page_123">[Pg 123]</span> +well-ordered series containing all the members of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> (The above +is, of course, only a hint of the general lines of the proof.) This +proposition is called "Zermelo's theorem." +</p> +<p> +The multiplicative axiom is also equivalent to the assumption +that of any two cardinals which are not equal, one must be the +greater. If the axiom is false, there will be cardinals <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> +such that <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> is neither less than, equal to, nor greater than <span class="nowrap"><img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu">.</span> We +have seen that <img style="vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;" src="images/191.svg" alt="" data-tex="\aleph_{1}"> and <img style="vertical-align: 0; width: 2.995ex; height: 1.932ex;" src="images/173.svg" alt="" data-tex="2^{\aleph_{0}}"> possibly form an instance of such a pair. +</p> +<p> +Many other forms of the axiom might be given, but the above +are the most important of the forms known at present. As to +the truth or falsehood of the axiom in any of its forms, nothing +is known at present. +</p> +<p> +The propositions that depend upon the axiom, without being +known to be equivalent to it, are numerous and important. Take +first the connection of addition and multiplication. We naturally +think that the sum of <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> mutually exclusive classes, each having +<img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms, must have <img style="vertical-align: -0.489ex; width: 4.324ex; height: 1.6ex;" src="images/234.svg" alt="" data-tex="\mu × \nu"> terms. When <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> is finite, this can be +proved. But when <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> is infinite, it cannot be proved without the +multiplicative axiom, except where, owing to some special circumstance, +the existence of certain selectors can be proved. The +way the multiplicative axiom enters in is as follows: Suppose +we have two sets of <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> mutually exclusive classes, each having <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms, +and we wish to prove that the sum of one set has as many +terms as the sum of the other. In order to prove this, we must +establish a one-one relation. Now, since there are in each case +<img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> classes, there is some one-one relation between the two sets of +classes; but what we want is a one-one relation between their +terms. Let us consider some one-one relation <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> between the +classes. Then if <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> and <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/243.svg" alt="" data-tex="\lambda"> are the two sets of classes, and <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is some +member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa">,</span> there will be a member <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> of <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/243.svg" alt="" data-tex="\lambda"> which will be the +correlate of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> with respect to <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">.</span> Now <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and +<img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> each have <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms, +and are therefore similar. There are, accordingly, one-one correlations +of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">.</span> The trouble is that there are so many. In +order to obtain a one-one correlation of the sum of <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> with the +sum of <span class="nowrap"><img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/243.svg" alt="" data-tex="\lambda">,</span> we have to pick out <i>one selection</i> from a set of classes +<span class="pagenum" id="Page_124">[Pg 124]</span> +of correlators, one class of the set being all the one-one correlators +of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> with <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">.</span> If <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/239.svg" alt="" data-tex="\kappa"> and <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/243.svg" alt="" data-tex="\lambda"> are infinite, we cannot in general know +that such a selection exists, unless we can know that the multiplicative +axiom is true. Hence we cannot establish the usual +kind of connection between addition and multiplication. +</p> +<p> +This fact has various curious consequences. To begin with, +we know that <span class="nowrap"><img style="vertical-align: -0.687ex; width: 17.274ex; height: 2.573ex;" src="images/244.svg" alt="" data-tex="\aleph_{0}^{2} = \aleph_{0} × \aleph_{0} = \aleph_{0}">.</span> It is commonly inferred from +this that the sum of <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> classes each having <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> members must +itself have <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> members, but this inference is fallacious, since we +do not know that the number of terms in such a sum is <span class="nowrap"><img style="vertical-align: -0.375ex; width: 6.5ex; height: 1.945ex;" src="images/245.svg" alt="" data-tex="\aleph_{0} × \aleph_{0}">,</span> +nor consequently that it is <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span> This has a bearing upon the theory +of transfinite ordinals. It is easy to prove that an ordinal which +has <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> predecessors must be one of what Cantor calls the "second +class," <i>i.e.</i> such that a series having this ordinal number will have +<img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms in its field. It is also easy to see that, if we take any +progression of ordinals of the second class, the predecessors of +their limit form at most the sum of <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> classes each having <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms. +It is inferred thence—fallaciously, unless the multiplicative +axiom is true—that the predecessors of the limit are <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> in +number, and therefore that the limit is a number of the "second +class." That is to say, it is supposed to be proved that any progression +of ordinals of the second class has a limit which is again +an ordinal of the second class. This proposition, with the corollary +that <img style="vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;" src="images/192.svg" alt="" data-tex="\omega_{1}"> (the smallest ordinal of the third class) is not the +limit of any progression, is involved in most of the recognised +theory of ordinals of the second class. In view of the way in +which the multiplicative axiom is involved, the proposition and +its corollary cannot be regarded as proved. They may be true, +or they may not. All that can be said at present is that we do +not know. Thus the greater part of the theory of ordinals of +the second class must be regarded as unproved. +</p> +<p> +Another illustration may help to make the point clearer. We +know that <span class="nowrap"><img style="vertical-align: -0.375ex; width: 10.649ex; height: 1.945ex;" src="images/246.svg" alt="" data-tex="2 × \aleph_{0} = \aleph_{0}">.</span> Hence we might suppose that the sum +of <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> pairs must have <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms. But this, though we can prove +that it is sometimes the case, cannot be proved to happen <i>always</i> +<span class="pagenum" id="Page_125">[Pg 125]</span> +unless we assume the multiplicative axiom. This is illustrated +by the millionaire who bought a pair of socks whenever he bought +a pair of boots, and never at any other time, and who had such +a passion for buying both that at last he had <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> pairs of boots +and <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> pairs of socks. The problem is: How many boots had +he, and how many socks? One would naturally suppose that +he had twice as many boots and twice as many socks as he had +pairs of each, and that therefore he had <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> of each, since that +number is not increased by doubling. But this is an instance of +the difficulty, already noted, of connecting the sum of <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> classes +each having <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> terms with <span class="nowrap"><img style="vertical-align: -0.489ex; width: 4.324ex; height: 1.6ex;" src="images/234.svg" alt="" data-tex="\mu × \nu">.</span> Sometimes this can be done, +sometimes it cannot. In our case it can be done with the boots, +but not with the socks, except by some very artificial device. +The reason for the difference is this: Among boots we can distinguish +right and left, and therefore we can make a selection of +one out of each pair, namely, we can choose all the right boots or +all the left boots; but with socks no such principle of selection +suggests itself, and we cannot be sure, unless we assume the +multiplicative axiom, that there is any class consisting of one +sock out of each pair. Hence the problem. +</p> +<p> +We may put the matter in another way. To prove that a +class has <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms, it is necessary and sufficient to find some way +of arranging its terms in a progression. There is no difficulty in +doing this with the boots. The <i>pairs</i> are given as forming an <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">,</span> +and therefore as the field of a progression. Within each pair, +take the left boot first and the right second, keeping the order +of the pairs unchanged; in this way we obtain a progression of +all the boots. But with the socks we shall have to choose arbitrarily, +with each pair, which to put first; and an infinite number +of arbitrary choices is an impossibility. Unless we can find a +<i>rule</i> for selecting, <i>i.e.</i> a relation which is a selector, we do not know +that a selection is even theoretically possible. Of course, in the +case of objects in space, like socks, we always can find some +principle of selection. For example, take the centres of mass +of the socks: there will be points <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> in space such that, with any +<span class="pagenum" id="Page_126">[Pg 126]</span> +pair, the centres of mass of the two socks are not both at exactly +the same distance from <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">;</span> thus we can choose, from each pair, +that sock which has its centre of mass nearer to <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">.</span> But there is +no theoretical reason why a method of selection such as this +should always be possible, and the case of the socks, with a little +goodwill on the part of the reader, may serve to show how a +selection might be impossible. +</p> +<p> +It is to be observed that, if it <i>were</i> impossible to select one out +of each pair of socks, it would follow that the socks <i>could</i> not be +arranged in a progression, and therefore that there were not <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> of +them. This case illustrates that, if <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> is an infinite number, +one set of <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> pairs may not contain the same number of terms as +another set of <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> pairs; for, given <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> pairs of boots, there are +certainly <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> boots, but we cannot be sure of this in the case of +the socks unless we assume the multiplicative axiom or fall back +upon some fortuitous geometrical method of selection such as +the above. +</p> +<p> +Another important problem involving the multiplicative +axiom is the relation of reflexiveness to non-inductiveness. It +will be remembered that in Chapter VIII. we pointed out that a +reflexive number must be non-inductive, but that the converse +(so far as is known at present) can only be proved if we assume +the multiplicative axiom. The way in which this comes about +is as follows:— +</p> +<p> +It is easy to prove that a reflexive class is one which contains +sub-classes having <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms. (The class may, of course, itself +have <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms.) Thus we have to prove, if we can, that, given +any non-inductive class, it is possible to choose a progression +out of its terms. Now there is no difficulty in showing that +a non-inductive class must contain more terms than any inductive +class, or, what comes to the same thing, that if <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is a non-inductive +class and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> is any inductive number, there are sub-classes +of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> that have <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> terms. Thus we can form sets of finite sub-classes +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">:</span> First one class having no terms, then classes having +1 term (as many as there are members of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">)</span>, then classes having +<span class="pagenum" id="Page_127">[Pg 127]</span> +2 terms, and so on. We thus get a progression of sets of sub-classes, +each set consisting of all those that have a certain given +finite number of terms. So far we have not used the multiplicative +axiom, but we have only proved that the number of collections +of sub-classes of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is a reflexive number, <i>i.e.</i> that, if <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> is +the number of members of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> so that <img style="vertical-align: 0; width: 2.284ex; height: 1.528ex;" src="images/247.svg" alt="" data-tex="2^{\mu}"> is the number of sub-classes +of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <img style="vertical-align: 0; width: 2.934ex; height: 1.902ex;" src="images/248.svg" alt="" data-tex="2^{2^{\mu}}"> is the number of collections of sub-classes, +then, provided <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> is not inductive, <img style="vertical-align: 0; width: 2.934ex; height: 1.902ex;" src="images/248.svg" alt="" data-tex="2^{2^{\mu}}"> must be reflexive. But +this is a long way from what we set out to prove. +</p> +<p> +In order to advance beyond this point, we must employ the +multiplicative axiom. From each set of sub-classes let us +choose out one, omitting the sub-class consisting of the null-class +alone. That is to say, we select one sub-class containing +one term, <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/249.svg" alt="" data-tex="\alpha_{1}">,</span> say; one containing two terms, <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/250.svg" alt="" data-tex="\alpha_{2}">,</span> say; one containing +three, <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;" src="images/251.svg" alt="" data-tex="\alpha_{3}">,</span> say; and so on. (We can do this if the multiplicative +axiom is assumed; otherwise, we do not know whether +we can always do it or not.) We have now a progression +<span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/249.svg" alt="" data-tex="\alpha_{1}">,</span> <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/250.svg" alt="" data-tex="\alpha_{2}">,</span> <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;" src="images/251.svg" alt="" data-tex="\alpha_{3}">,</span> ... sub-classes of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> instead of a progression of +collections of sub-classes; thus we are one step nearer to our +goal. We now know that, assuming the multiplicative axiom, +if <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> is a non-inductive number, <img style="vertical-align: 0; width: 2.284ex; height: 1.528ex;" src="images/247.svg" alt="" data-tex="2^{\mu}"> must be a reflexive number. +</p> +<p> +The next step is to notice that, although we cannot be sure +that new members of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> come in at any one specified stage in the +progression <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/249.svg" alt="" data-tex="\alpha_{1}">,</span> <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/250.svg" alt="" data-tex="\alpha_{2}">,</span> <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;" src="images/251.svg" alt="" data-tex="\alpha_{3}">,</span> ... we can be sure that new members +keep on coming in from time to time. Let us illustrate. +The class <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/249.svg" alt="" data-tex="\alpha_{1}">,</span> which consists of one term, is a new beginning; +let the one term be <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}">.</span> The class <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/250.svg" alt="" data-tex="\alpha_{2}">,</span> consisting of two terms, +may or may not contain <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}">;</span> if it does, it introduces one new +term; and if it does not, it must introduce two new terms, say +<span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/236.svg" alt="" data-tex="x_{2}">,</span> <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/237.svg" alt="" data-tex="x_{3}">.</span> In this case it is possible that <img style="vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;" src="images/251.svg" alt="" data-tex="\alpha_{3}"> consists +of <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}">,</span> <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/236.svg" alt="" data-tex="x_{2}">,</span> <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/237.svg" alt="" data-tex="x_{3}">,</span> +and so introduces no new terms, but in that case <img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/252.svg" alt="" data-tex="\alpha_{4}"> must introduce +a new term. The first <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> classes <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/249.svg" alt="" data-tex="\alpha_{1}">,</span> <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/250.svg" alt="" data-tex="\alpha_{2}">,</span> +<span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;" src="images/251.svg" alt="" data-tex="\alpha_{3}">,</span> ... <img style="vertical-align: -0.339ex; width: 2.484ex; height: 1.339ex;" src="images/253.svg" alt="" data-tex="\alpha_{\nu}"> contain, at +the very most, <img style="vertical-align: -0.186ex; width: 18.307ex; height: 1.692ex;" src="images/254.svg" alt="" data-tex="1 + 2 + 3 + \dots + \nu"> terms, <i>i.e.</i> +<img style="vertical-align: -0.566ex; width: 11.449ex; height: 2.262ex;" src="images/255.svg" alt="" data-tex="\nu/(\nu + 1)/2"> terms; +thus it would be possible, if there were no repetitions in the +first <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> classes, to go on with only repetitions from the <img style="vertical-align: -0.566ex; width: 8.543ex; height: 2.497ex;" src="images/256.svg" alt="" data-tex="(\nu + 1)^{th}"> +<span class="pagenum" id="Page_128">[Pg 128]</span> +class to the <img style="vertical-align: -0.566ex; width: 12.005ex; height: 2.497ex;" src="images/257.svg" alt="" data-tex="\nu(\nu + 1)/2^{th}"> class. But by that time the old terms +would no longer be sufficiently numerous to form a next class +with the right number of members, <i>i.e.</i> <span class="nowrap"><img style="vertical-align: -0.566ex; width: 14.215ex; height: 2.262ex;" src="images/258.svg" alt="" data-tex="\nu(\nu + 1)/2 + 1">,</span> therefore +new terms must come in at this point if not sooner. It +follows that, if we omit from our progression +<span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/249.svg" alt="" data-tex="\alpha_{1}">,</span> <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/250.svg" alt="" data-tex="\alpha_{2}">,</span> <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;" src="images/251.svg" alt="" data-tex="\alpha_{3}">,</span>... all +those classes that are composed entirely of members that have +occurred in previous classes, we shall still have a progression. +Let our new progression be called <span class="nowrap"><img style="vertical-align: -0.439ex; width: 2.268ex; height: 2.034ex;" src="images/259.svg" alt="" data-tex="\beta_{1}">,</span> <span class="nowrap"><img style="vertical-align: -0.439ex; width: 2.268ex; height: 2.034ex;" src="images/260.svg" alt="" data-tex="\beta_{2}">,</span> +<span class="nowrap"><img style="vertical-align: -0.439ex; width: 2.268ex; height: 2.034ex;" src="images/261.svg" alt="" data-tex="\beta_{3}">.</span>... (We shall +have <img style="vertical-align: -0.439ex; width: 7.721ex; height: 2.034ex;" src="images/262.svg" alt="" data-tex="\alpha_{1} = \beta_{1}"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 7.721ex; height: 2.034ex;" src="images/263.svg" alt="" data-tex="\alpha_{2} = \beta_{2}">,</span> +because <img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/249.svg" alt="" data-tex="\alpha_{1}"> and <img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/250.svg" alt="" data-tex="\alpha_{2}"> <i>must</i> introduce new +terms. We may or may not have <span class="nowrap"><img style="vertical-align: -0.439ex; width: 7.721ex; height: 2.034ex;" src="images/264.svg" alt="" data-tex="\alpha_{3} = \beta_{3}">,</span> but, speaking generally, +<img style="vertical-align: -0.439ex; width: 2.316ex; height: 2.034ex;" src="images/265.svg" alt="" data-tex="\beta_{\nu}"> will be <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.484ex; height: 1.339ex;" src="images/253.svg" alt="" data-tex="\alpha_{\nu}">,</span> where <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> is some number greater +than <span class="nowrap"><img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu">;</span> <i>i.e.</i> the +<span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">'</span>s are <i>some</i> of the <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">'</span>s.) Now these <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">'</span>s are such that any one +of them, say <span class="nowrap"><img style="vertical-align: -0.685ex; width: 2.433ex; height: 2.28ex;" src="images/266.svg" alt="" data-tex="\beta_{\mu}">,</span> contains members which have not occurred in +any of the previous <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">'</span>s. Let <img style="vertical-align: -0.685ex; width: 2.324ex; height: 1.683ex;" src="images/267.svg" alt="" data-tex="\gamma_{\mu}"> be the part of <img style="vertical-align: -0.685ex; width: 2.433ex; height: 2.28ex;" src="images/266.svg" alt="" data-tex="\beta_{\mu}"> which consists +of new members. Thus we get a new progression <span class="nowrap"><img style="vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;" src="images/268.svg" alt="" data-tex="\gamma_{1}">,</span> +<span class="nowrap"><img style="vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;" src="images/269.svg" alt="" data-tex="\gamma_{2}">,</span> <span class="nowrap"><img style="vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;" src="images/270.svg" alt="" data-tex="\gamma_{3}">,</span>... +(Again <img style="vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;" src="images/268.svg" alt="" data-tex="\gamma_{1}"> will be identical with <img style="vertical-align: -0.439ex; width: 2.268ex; height: 2.034ex;" src="images/259.svg" alt="" data-tex="\beta_{1}"> and with <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/249.svg" alt="" data-tex="\alpha_{1}">;</span> +if <img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/250.svg" alt="" data-tex="\alpha_{2}"> does not +contain the one member of <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/249.svg" alt="" data-tex="\alpha_{1}">,</span> we shall have <span class="nowrap"><img style="vertical-align: -0.489ex; width: 12.898ex; height: 2.084ex;" src="images/271.svg" alt="" data-tex="\gamma_{2} = \beta_{2} = \alpha_{2}">,</span> but if +<img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/250.svg" alt="" data-tex="\alpha_{2}"> does contain this one member, <img style="vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;" src="images/269.svg" alt="" data-tex="\gamma_{2}"> will consist of the other +member of <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;" src="images/250.svg" alt="" data-tex="\alpha_{2}">)</span>. This new progression of <span class="nowrap"><img style="vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;" src="images/65.svg" alt="" data-tex="\gamma">'</span>s consists of mutually +exclusive classes. Hence a selection from them will be a progression; +<i>i.e.</i> if <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}"> is the member of <span class="nowrap"><img style="vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;" src="images/268.svg" alt="" data-tex="\gamma_{1}">,</span> <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/236.svg" alt="" data-tex="x_{2}"> is +a member of <span class="nowrap"><img style="vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;" src="images/269.svg" alt="" data-tex="\gamma_{2}">,</span> <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/237.svg" alt="" data-tex="x_{3}"> is +a member of <span class="nowrap"><img style="vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;" src="images/270.svg" alt="" data-tex="\gamma_{3}">,</span> and so on; then <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}">,</span> <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/236.svg" alt="" data-tex="x_{2}">,</span> +<span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/237.svg" alt="" data-tex="x_{3}">,</span> ... is a progression, +and is a sub-class of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> Assuming the multiplicative axiom, +such a selection can be made. Thus by twice using this axiom +we can prove that, if the axiom is true, every non-inductive +cardinal must be reflexive. This could also be deduced from +Zermelo's theorem, that, if the axiom is true, every class can be +well ordered; for a well-ordered series must have either a finite +or a reflexive number of terms in its field. +</p> +<p> +There is one advantage in the above direct argument, as +against deduction from Zermelo's theorem, that the above +argument does not demand the universal truth of the multiplicative +axiom, but only its truth as applied to a set of <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> classes. +It may happen that the axiom holds for <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> classes, though not +for larger numbers of classes. For this reason it is better, when +<span class="pagenum" id="Page_129">[Pg 129]</span> +it is possible, to content ourselves with the more restricted +assumption. The assumption made in the above direct argument +is that a product of <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> factors is never zero unless one of +the factors is zero. We may state this assumption in the form: +"<img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> is a <i>multipliable</i> number," where a number <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> is defined as +"multipliable" when a product of <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/143.svg" alt="" data-tex="\nu"> factors is never zero unless +one of the factors is zero. We can <i>prove</i> that a <i>finite</i> number is +always multipliable, but we cannot prove that any infinite number +is so. The multiplicative axiom is equivalent to the assumption +that <i>all</i> cardinal numbers are multipliable. But in order to +identify the reflexive with the non-inductive, or to deal with the +problem of the boots and socks, or to show that any progression +of numbers of the second class is of the second class, we only +need the very much smaller assumption that <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> is multipliable. +</p> +<p> +It is not improbable that there is much to be discovered +in regard to the topics discussed in the present chapter. Cases +may be found where propositions which seem to involve the +multiplicative axiom can be proved without it. It is conceivable +that the multiplicative axiom in its general form may be shown +to be false. From this point of view, Zermelo's theorem offers +the best hope: the continuum or some still more dense series +<i>might</i> be proved to be incapable of having its terms well ordered, +which would prove the multiplicative axiom false, in virtue of +Zermelo's theorem. But so far, no method of obtaining such +results has been discovered, and the subject remains wrapped in +obscurity. +<span class="pagenum" id="Page_130">[Pg 130]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='XIII: THE AXIOM OF INFINITY AND LOGICAL TYPES'><a id="chap13"></a>CHAPTER XIII +<br><br> +THE AXIOM OF INFINITY AND LOGICAL TYPES</h2> + +<p class="nind"> +THE axiom of infinity is an assumption which may be enunciated +as follows:— +</p> +<p> +"If <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> be any inductive cardinal number, there is at least one +class of individuals having <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> terms." +</p> +<p> +If this is true, it follows, of course, that there are many classes +of individuals having <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> terms, and that the total number of +individuals in the world is not an inductive number. For, by +the axiom, there is at least one class having <img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1"> terms, from which +it follows that there are many classes of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> terms and that <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is +not the number of individuals in the world. Since <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is <i>any</i> +inductive number, it follows that the number of individuals +in the world must (if our axiom be true) exceed any inductive +number. In view of what we found in the preceding chapter, +about the possibility of cardinals which are neither inductive +nor reflexive, we cannot infer from our axiom that there are at +least <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> individuals, unless we assume the multiplicative axiom. +But we do know that there are at least <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> classes of classes, +since the inductive cardinals are classes of classes, and form a +progression if our axiom is true. The way in which the need +for this axiom arises may be explained as follows:—One of +Peano's assumptions is that no two inductive cardinals have the +same successor, <i>i.e.</i> that we shall not have <img style="vertical-align: -0.186ex; width: 14.155ex; height: 1.692ex;" src="images/68.svg" alt="" data-tex="m + 1 = n + 1"> unless +<span class="nowrap"><img style="vertical-align: -0.186ex; width: 6.361ex; height: 1.505ex;" src="images/69.svg" alt="" data-tex="m = n">,</span> if <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> and <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> are inductive cardinals. In Chapter VIII. we +had occasion to use what is virtually the same as the above +assumption of Peano's, namely, that, if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is an inductive cardinal, +<span class="pagenum" id="Page_131">[Pg 131]</span> +<img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is not equal to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">.</span> It might be thought that this could be +proved. We can prove that, if <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is an inductive class, and <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is +the number of members of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> then <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is not equal to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">.</span> +This proposition is easily proved by induction, and might be +thought to imply the other. But in fact it does not, since there +might be no such class as <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> What it does imply is this: If +<img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is an inductive cardinal such that there is at least one class +having <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> members, then <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is not equal to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">.</span> The axiom of +infinity assures us (whether truly or falsely) that there are classes +having <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> members, and thus enables us to assert that <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is not +equal to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">.</span> But without this axiom we should be left with +the possibility that <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> and <img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1"> might both be the null-class. +</p> +<p> +Let us illustrate this possibility by an example: Suppose +there were exactly nine individuals in the world. (As to what +is meant by the word "individual," I must ask the reader to +be patient.) Then the inductive cardinals from 0 up to 9 would +be such as we expect, but 10 (defined as <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.028ex; height: 1.692ex;" src="images/272.svg" alt="" data-tex="9 + 1">)</span> would be the +null-class. It will be remembered that <img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1"> may be defined as +follows: <img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1"> is the collection of all those classes which have a +term <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> such that, when <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is taken away, there remains a class +of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> terms. Now applying this definition, we see that, in the +case supposed, <img style="vertical-align: -0.186ex; width: 5.028ex; height: 1.692ex;" src="images/272.svg" alt="" data-tex="9 + 1"> is a class consisting of no classes, <i>i.e.</i> it is +the null-class. The same will be true of <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.028ex; height: 1.692ex;" src="images/273.svg" alt="" data-tex="9 + 2">,</span> or generally of +<span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/274.svg" alt="" data-tex="9 + n">,</span> unless <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is zero. Thus 10 and all subsequent inductive +cardinals will all be identical, since they will all be the null-class. +In such a case the inductive cardinals will not form a progression, +nor will it be true that no two have the same successor, for 9 +and 10 will both be succeeded by the null-class (10 being itself +the null-class). It is in order to prevent such arithmetical +catastrophes that we require the axiom of infinity. +</p> +<p> +As a matter of fact, so long as we are content with the arithmetic +of finite integers, and do not introduce either infinite +integers or infinite classes or series of finite integers or ratios, +it is possible to obtain all desired results without the axiom of +infinity. That is to say, we can deal with the addition, multiplication, +<span class="pagenum" id="Page_132">[Pg 132]</span> +and exponentiation of finite integers and of ratios, +but we cannot deal with infinite integers or with irrationals. +Thus the theory of the transfinite and the theory of real numbers +fails us. How these various results come about must now be +explained. +</p> +<p> +Assuming that the number of individuals in the world is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> +the number of classes of individuals will be <span class="nowrap"><img style="vertical-align: 0; width: 2.279ex; height: 1.528ex;" src="images/174.svg" alt="" data-tex="2^{n}">.</span> This is in virtue +of the general proposition mentioned in Chapter VIII. that the +number of classes contained in a class which has <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> members +is <span class="nowrap"><img style="vertical-align: 0; width: 2.279ex; height: 1.528ex;" src="images/174.svg" alt="" data-tex="2^{n}">.</span> Now <img style="vertical-align: 0; width: 2.279ex; height: 1.528ex;" src="images/174.svg" alt="" data-tex="2^{n}"> is always greater than <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">.</span> Hence the number +of classes in the world is greater than the number of individuals. +If, now, we suppose the number of individuals to be 9, as we did +just now, the number of classes will be <span class="nowrap"><img style="vertical-align: 0; width: 2.119ex; height: 1.887ex;" src="images/275.svg" alt="" data-tex="2^{9}">,</span> <i>i.e.</i> 512. Thus if we +take our numbers as being applied to the counting of classes +instead of to the counting of individuals, our arithmetic will +be normal until we reach 512: the first number to be null will +be 513. And if we advance to classes of classes we shall do still +better: the number of them will be <span class="nowrap"><img style="vertical-align: 0; width: 3.719ex; height: 1.887ex;" src="images/276.svg" alt="" data-tex="2^{512}">,</span> a number which is so +large as to stagger imagination, since it has about 153 digits. +And if we advance to classes of classes of classes, we shall obtain +a number represented by 2 raised to a power which has about +153 digits; the number of digits in this number will be about +three times <span class="nowrap"><img style="vertical-align: -0.05ex; width: 4.85ex; height: 2.005ex;" src="images/277.svg" alt="" data-tex="10^{152}">.</span> In a time of paper shortage it is undesirable +to write out this number, and if we want larger ones we can +obtain them by travelling further along the logical hierarchy. +In this way any assigned inductive cardinal can be made to +find its place among numbers which are not null, merely by +travelling along the hierarchy for a sufficient distance.<a id="FNanchor_26_1"></a><a href="#Footnote_26_1" class="fnanchor">[26]</a> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_26_1"></a><a href="#FNanchor_26_1"><span class="label">[26]</span></a>On this subject see <i>Principia Mathematica</i>, vol. II. * 120 ff. On the +corresponding problems as regards ratio, see <i>ibid.</i>, vol. III. * 303 ff.</p></div> + +<p> +As regards ratios, we have a very similar state of affairs. +If a ratio <img style="vertical-align: -0.566ex; width: 3.695ex; height: 2.262ex;" src="images/278.svg" alt="" data-tex="\mu/\nu"> is to have the expected properties, there must +be enough objects of whatever sort is being counted to insure +that the null-class does not suddenly obtrude itself. But this +can be insured, for any given ratio <span class="nowrap"><img style="vertical-align: -0.566ex; width: 3.695ex; height: 2.262ex;" src="images/278.svg" alt="" data-tex="\mu/\nu">,</span> without the axiom of +<span class="pagenum" id="Page_133">[Pg 133]</span> +infinity, by merely travelling up the hierarchy a sufficient distance. +If we cannot succeed by counting individuals, we can try counting +classes of individuals; if we still do not succeed, we can try +classes of classes, and so on. Ultimately, however few individuals +there may be in the world, we shall reach a stage where +there are many more than <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> objects, whatever inductive number +<img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/142.svg" alt="" data-tex="\mu"> may be. Even if there were no individuals at all, this would +still be true, for there would then be one class, namely, the null-class, +2 classes of classes (namely, the null-class of classes and the +class whose only member is the null-class of individuals), 4 classes +of classes of classes, 16 at the next stage, 65,536 at the next +stage, and so on. Thus no such assumption as the axiom of +infinity is required in order to reach any given ratio or any given +inductive cardinal. +</p> +<p> +It is when we wish to deal with the whole class or series of +inductive cardinals or of ratios that the axiom is required. We +need the whole class of inductive cardinals in order to establish +the existence of <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">,</span> and the whole series in order to establish +the existence of progressions: for these results, it is necessary +that we should be able to make a single class or series in which +no inductive cardinal is null. We need the whole series of ratios +in order of magnitude in order to define real numbers as segments: +this definition will not give the desired result unless the series +of ratios is compact, which it cannot be if the total number of +ratios, at the stage concerned, is finite. +</p> +<p> +It would be natural to suppose—as I supposed myself in former +days—that, by means of constructions such as we have been +considering, the axiom of infinity could be <i>proved</i>. It may be +said: Let us assume that the number of individuals is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> where +<img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> may be 0 without spoiling our argument; then if we form the +complete set of individuals, classes, classes of classes, etc., all +taken together, the number of terms in our whole set will be +<span class="align-center"><img style="vertical-align: -0.462ex; width: 25.132ex; height: 2.477ex;" src="images/42.svg" alt="" data-tex=" +n + 2^{n} + 2^{2^{n}} + \dots\ \mathit{ad\,inf.}, +"></span> +which is <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}">.</span> Thus taking all kinds of objects together, and not +<span class="pagenum" id="Page_134">[Pg 134]</span> +confining ourselves to objects of any one type, we shall certainly +obtain an infinite class, and shall therefore not need the axiom +of infinity. So it might be said. +</p> +<p> +Now, before going into this argument, the first thing to observe +is that there is an air of hocus-pocus about it: something reminds +one of the conjurer who brings things out of the hat. The man +who has lent his hat is quite sure there wasn't a live rabbit in it +before, but he is at a loss to say how the rabbit got there. So +the reader, if he has a robust sense of reality, will feel convinced +that it is impossible to manufacture an infinite collection out of +a finite collection of individuals, though he may be unable to +say where the flaw is in the above construction. It would be a +mistake to lay too much stress on such feelings of hocus-pocus; +like other emotions, they may easily lead us astray. But they +afford a <i>prima facie</i> ground for scrutinising very closely any +argument which arouses them. And when the above argument +is scrutinised it will, in my opinion, be found to be fallacious, +though the fallacy is a subtle one and by no means easy to avoid +consistently. +</p> +<p> +The fallacy involved is the fallacy which may be called "confusion +of types." To explain the subject of "types" fully would +require a whole volume; moreover, it is the purpose of this book +to avoid those parts of the subjects which are still obscure and +controversial, isolating, for the convenience of beginners, those +parts which can be accepted as embodying mathematically ascertained +truths. Now the theory of types emphatically does not +belong to the finished and certain part of our subject: much of +this theory is still inchoate, confused, and obscure. But the need +of <i>some</i> doctrine of types is less doubtful than the precise form +the doctrine should take; and in connection with the axiom of +infinity it is particularly easy to see the necessity of some such +doctrine. +</p> +<p> +This necessity results, for example, from the "contradiction of +the greatest cardinal." We saw in Chapter VIII. that the number +of classes contained in a given class is always greater than the +<span class="pagenum" id="Page_135">[Pg 135]</span> +number of members of the class, and we inferred that there is +no greatest cardinal number. But if we could, as we suggested +a moment ago, add together into one class the individuals, classes +of individuals, classes of classes of individuals, etc., we should +obtain a class of which its own sub-classes would be members. +The class consisting of all objects that can be counted, of whatever +sort, must, if there be such a class, have a cardinal number which +is the greatest possible. Since all its sub-classes will be members +of it, there cannot be more of them than there are members. +Hence we arrive at a contradiction. +</p> +<p> +When I first came upon this contradiction, in the year 1901, +I attempted to discover some flaw in Cantor's proof that there is +no greatest cardinal, which we gave in Chapter VIII. Applying +this proof to the supposed class of all imaginable objects, +I was led to a new and simpler contradiction, namely, the +following:— +</p> +<p> +The comprehensive class we are considering, which is to embrace +everything, must embrace itself as one of its members. In other +words, if there is such a thing as "everything," then "everything" +is something, and is a member of the class "everything." +But normally a class is not a member of itself. Mankind, for +example, is not a man. Form now the assemblage of all classes +which are not members of themselves. This is a class: is it a +member of itself or not? If it is, it is one of those classes that +are not members of themselves, <i>i.e.</i> it is not a member of itself. +If it is not, it is not one of those classes that are not members of +themselves, <i>i.e.</i> it is a member of itself. Thus of the two hypotheses—that +it is, and that it is not, a member of itself—each +implies its contradictory. This is a contradiction. +</p> +<p> +There is no difficulty in manufacturing similar contradictions +<i>ad lib</i>. The solution of such contradictions by the theory of +types is set forth fully in <i>Principia Mathematica</i>,<a id="FNanchor_27_1"></a><a href="#Footnote_27_1" class="fnanchor">[27]</a> +and also, more +briefly, in articles by the present author in the <i>American Journal +<span class="pagenum" id="Page_136">[Pg 136]</span> +of Mathematics</i>,<a id="FNanchor_28_1"></a><a href="#Footnote_28_1" class="fnanchor">[28]</a> +and in the <i>Revue de Metaphysique et de Morale</i>.<a id="FNanchor_29_1"></a><a href="#Footnote_29_1" class="fnanchor">[29]</a> +For the present an outline of the solution must suffice. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_27_1"></a><a href="#FNanchor_27_1"><span class="label">[27]</span></a>Vol. I., Introduction, chap. II., * 12 and * 20; vol. II., Prefatory +Statement</p></div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_28_1"></a><a href="#FNanchor_28_1"><span class="label">[28]</span></a>"Mathematical Logic as based on the Theory of Types," vol. XXX., +1908, pp. 222-262.</p></div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_29_1"></a><a href="#FNanchor_29_1"><span class="label">[29]</span></a>"Les paradoxes de la logique," 1906, pp. 627-650.</p></div> + +<p> +The fallacy consists in the formation of what we may call +"impure" classes, <i>i.e.</i> classes which are not pure as to "type." +As we shall see in a later chapter, classes are logical fictions, and +a statement which appears to be about a class will only be significant +if it is capable of translation into a form in which no mention +is made of the class. This places a limitation upon the ways in +which what are nominally, though not really, names for classes +can occur significantly: a sentence or set of symbols in which +such pseudo-names occur in wrong ways is not false, but strictly +devoid of meaning. The supposition that a class is, or that it +is not, a member of itself is meaningless in just this way. And +more generally, to suppose that one class of individuals is a +member, or is not a member, of another class of individuals +will be to suppose nonsense; and to construct symbolically any +class whose members are not all of the same grade in the logical +hierarchy is to use symbols in a way which makes them no +longer symbolise anything. +</p> +<p> +Thus if there are <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> individuals in the world, and <img style="vertical-align: 0; width: 2.279ex; height: 1.528ex;" src="images/174.svg" alt="" data-tex="2^{n}"> classes of +individuals, we cannot form a new class, consisting of both +individuals and classes and having <img style="vertical-align: -0.186ex; width: 6.402ex; height: 1.714ex;" src="images/279.svg" alt="" data-tex="n + 2^{n}"> members. In this way +the attempt to escape from the need for the axiom of infinity +breaks down. I do not pretend to have explained the doctrine +of types, or done more than indicate, in rough outline, why there +is need of such a doctrine. I have aimed only at saying just +so much as was required in order to show that we cannot <i>prove</i> +the existence of infinite numbers and classes by such conjurer's +methods as we have been examining. There remain, however, +certain other possible methods which must be considered. +</p> +<p> +Various arguments professing to prove the existence of infinite +classes are given in the <i>Principles of Mathematics</i>, § 339 (p. 357). +<span class="pagenum" id="Page_137">[Pg 137]</span> +In so far as these arguments assume that, if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is an inductive +cardinal, <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is not equal to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1">,</span> they have been already dealt +with. There is an argument, suggested by a passage in Plato's +<i>Parmenides</i>, to the effect that, if there is such a number as 1, +then 1 has being; but 1 is not identical with being, and therefore +1 and being are two, and therefore there is such a number as 2, +and 2 together with 1 and being gives a class of three terms, and +so on. This argument is fallacious, partly because "being" is +not a term having any definite meaning, and still more because, +if a definite meaning were invented for it, it would be found that +numbers do not have being—they are, in fact, what are called +"logical fictions," as we shall see when we come to consider +the definition of classes. +</p> +<p> +The argument that the number of numbers from 0 to <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> (both +inclusive) is <img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1"> depends upon the assumption that up to and +including <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> no number is equal to its successor, which, as we have +seen, will not be always true if the axiom of infinity is false. It +must be understood that the equation <span class="nowrap"><img style="vertical-align: -0.186ex; width: 9.629ex; height: 1.692ex;" src="images/166.svg" alt="" data-tex="n = n + 1">,</span> which might be +true for a finite <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> exceeded the total number of individuals +in the world, is quite different from the same equation as applied +to a reflexive number. As applied to a reflexive number, it +means that, given a class of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> terms, this class is "similar" to +that obtained by adding another term. But as applied to a +number which is too great for the actual world, it merely means +that there is no class of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> individuals, and no class of <img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1"> individuals; +it does not mean that, if we mount the hierarchy of +types sufficiently far to secure the existence of a class of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> terms, +we shall then find this class "similar" to one of <img style="vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;" src="images/58.svg" alt="" data-tex="n + 1"> terms, for +if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is inductive this will not be the case, quite independently of +the truth or falsehood of the axiom of infinity. +</p> +<p> +There is an argument employed by both Bolzano<a id="FNanchor_30_1"></a><a href="#Footnote_30_1" class="fnanchor">[30]</a> +and Dedekind<a id="FNanchor_31_1"></a><a href="#Footnote_31_1" class="fnanchor">[31]</a> +to prove the existence of reflexive classes. The argument, +in brief, is this: An object is not identical with the idea of the +<span class="pagenum" id="Page_138">[Pg 138]</span> +object, but there is (at least in the realm of being) an idea of any +object. The relation of an object to the idea of it is one-one, and +ideas are only some among objects. Hence the relation "idea +of" constitutes a reflexion of the whole class of objects into a +part of itself, namely, into that part which consists of ideas. +Accordingly, the class of objects and the class of ideas are both +infinite. This argument is interesting, not only on its own +account, but because the mistakes in it (or what I judge to be +mistakes) are of a kind which it is instructive to note. The +main error consists in assuming that there is an idea of every +object. It is, of course, exceedingly difficult to decide what is +meant by an "idea"; but let us assume that we know. We are +then to suppose that, starting (say) with Socrates, there is the +idea of Socrates, and so on <i>ad inf</i>. Now it is plain that this is not +the case in the sense that all these ideas have actual empirical +existence in people's minds. Beyond the third or fourth stage +they become mythical. If the argument is to be upheld, the +"ideas" intended must be Platonic ideas laid up in heaven, for +certainly they are not on earth. But then it at once becomes +doubtful whether there are such ideas. If we are to know that +there are, it must be on the basis of some logical theory, proving +that it is necessary to a thing that there should be an idea of it. +We certainly cannot obtain this result empirically, or apply it, +as Dedekind does, to "meine Gedankenwelt"—the world of my +thoughts. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_30_1"></a><a href="#FNanchor_30_1"><span class="label">[30]</span></a>Bolzano, <i>Paradoxien des Unendlichen</i>, 13.</p></div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_31_1"></a><a href="#FNanchor_31_1"><span class="label">[31]</span></a>Dedekind, <i>Was sind und was sollen die Zahlen?</i> No. 66.</p></div> + +<p> +If we were concerned to examine fully the relation of idea and +object, we should have to enter upon a number of psychological +and logical inquiries, which are not relevant to our main purpose. +But a few further points should be noted. If "idea" is to be +understood logically, it may be <i>identical</i> with the object, or it +may stand for a <i>description</i> (in the sense to be explained in a +subsequent chapter). In the former case the argument fails, +because it was essential to the proof of reflexiveness that object +and idea should be distinct. In the second case the argument +also fails, because the relation of object and description is not +<span class="pagenum" id="Page_139">[Pg 139]</span> +one-one: there are innumerable correct descriptions of any given +object. Socrates (<i>e.g.</i>) may be described as "the master of +Plato," or as "the philosopher who drank the hemlock," or as +"the husband of Xantippe." If—to take up the remaining +hypothesis—"idea" is to be interpreted psychologically, it must +be maintained that there is not any one definite psychological +entity which could be called <i>the</i> idea of the object: there are innumerable +beliefs and attitudes, each of which could be called <i>an</i> +idea of the object in the sense in which we might say "my idea +of Socrates is quite different from yours," but there is not any +central entity (except Socrates himself) to bind together various +"ideas of Socrates," and thus there is not any such one-one relation +of idea and object as the argument supposes. Nor, of course, +as we have already noted, is it true psychologically that there are +ideas (in however extended a sense) of more than a tiny proportion +of the things in the world. For all these reasons, the above +argument in favour of the logical existence of reflexive classes +must be rejected. +</p> +<p> +It might be thought that, whatever may be said of <i>logical</i> +arguments, the <i>empirical</i> arguments derivable from space and +time, the diversity of colours, etc., are quite sufficient to prove +the actual existence of an infinite number of particulars. I do +not believe this. We have no reason except prejudice for believing +in the infinite extent of space and time, at any rate in the sense +in which space and time are physical facts, not mathematical +fictions. We naturally regard space and time as continuous, or, +at least, as compact; but this again is mainly prejudice. The +theory of "quanta" in physics, whether true or false, illustrates +the fact that physics can never afford proof of continuity, though +it might quite possibly afford disproof. The senses are not +sufficiently exact to distinguish between continuous motion and +rapid discrete succession, as anyone may discover in a cinema. +A world in which all motion consisted of a series of small finite +jerks would be empirically indistinguishable from one in which +motion was continuous. It would take up too much space to +<span class="pagenum" id="Page_140">[Pg 140]</span> +defend these theses adequately; for the present I am merely +suggesting them for the reader's consideration. If they are valid, +it follows that there is no empirical reason for believing the +number of particulars in the world to be infinite, and that there +never can be; also that there is at present no empirical reason +to believe the number to be finite, though it is theoretically +conceivable that some day there might be evidence pointing, +though not conclusively, in that direction. +</p> +<p> +From the fact that the infinite is not self-contradictory, but is +also not demonstrable logically, we must conclude that nothing +can be known <i>a priori</i> as to whether the number of things +in the world is finite or infinite. The conclusion is, therefore, +to adopt a Leibnizian phraseology, that some of the possible +worlds are finite, some infinite, and we have no means of +knowing to which of these two kinds our actual world belongs. +The axiom of infinity will be true in some possible worlds +and false in others; whether it is true or false in this world, +we cannot tell. +</p> +<p> +Throughout this chapter the synonyms "individual" and +"particular" have been used without explanation. It would be +impossible to explain them adequately without a longer disquisition +on the theory of types than would be appropriate to the +present work, but a few words before we leave this topic may +do something to diminish the obscurity which would otherwise +envelop the meaning of these words. +</p> +<p> +In an ordinary statement we can distinguish a verb, expressing +an attribute or relation, from the substantives which express the +subject of the attribute or the terms of the relation. "Cæsar +lived" ascribes an attribute to Cæsar; "Brutus killed Cæsar" +expresses a relation between Brutus and Cæsar. Using the word +"subject" in a generalised sense, we may call both Brutus and +Cæsar subjects of this proposition: the fact that Brutus is grammatically +subject and Cæsar object is logically irrelevant, since +the same occurrence may be expressed in the words "Cæsar was +killed by Brutus," where Cæsar is the grammatical subject. +<span class="pagenum" id="Page_141">[Pg 141]</span> +Thus in the simpler sort of proposition we shall have an attribute +or relation holding of or between one, two or more "subjects" +in the extended sense. (A relation may have more than two +terms: <i>e.g.</i> "<img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A"> gives <img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B"> to <img style="vertical-align: -0.048ex; width: 1.633ex; height: 1.643ex;" src="images/76.svg" alt="" data-tex="\mathrm C">" +is a relation of <i>three</i> terms.) Now +it often happens that, on a closer scrutiny, the apparent subjects +are found to be not really subjects, but to be capable of analysis; +the only result of this, however, is that new subjects take their +places. It also happens that the verb may grammatically be +made subject: <i>e.g.</i> we may say, "Killing is a relation which +holds between Brutus and Cæsar." But in such cases the +grammar is misleading, and in a straightforward statement, +following the rules that should guide philosophical grammar, +Brutus and Cæsar will appear as the subjects and killing +as the verb. +</p> +<p> +We are thus led to the conception of terms which, when they +occur in propositions, can <i>only</i> occur as subjects, and never in +any other way. This is part of the old scholastic definition +of <i>substance</i>; but persistence through time, which belonged to +that notion, forms no part of the notion with which we are concerned. +We shall define "proper names" as those terms which +can only occur as <i>subjects</i> in propositions (using "subject" +in the extended sense just explained). We shall further define +"individuals" or "particulars" as the objects that can be +named by proper names. (It would be better to define them +directly, rather than by means of the kind of symbols by which +they are symbolised; but in order to do that we should have +to plunge deeper into metaphysics than is desirable here.) It +is, of course, possible that there is an endless regress: that +whatever appears as a particular is really, on closer scrutiny, +a class or some kind of complex. If this be the case, the axiom +of infinity must of course be true. But if it be not the case, +it must be theoretically possible for analysis to reach ultimate +subjects, and it is these that give the meaning of "particulars" +or "individuals." It is to the number of these that the axiom +of infinity is assumed to apply. If it is true of them, it is true +<span class="pagenum" id="Page_142">[Pg 142]</span> +of classes of them, and classes of classes of them, and so on; +similarly if it is false of them, it is false throughout this hierarchy. +Hence it is natural to enunciate the axiom concerning them rather +than concerning any other stage in the hierarchy. But whether +the axiom is true or false, there seems no known method of +discovering. +<span class="pagenum" id="Page_143">[Pg 143]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='XIV: INCOMPATIBILITY AND THE THEORY OF DEDUCTION'><a id="chap14"></a>CHAPTER XIV +<br><br> +INCOMPATIBILITY AND THE THEORY OF DEDUCTION</h2> + +<p class="nind"> +WE have now explored, somewhat hastily it is true, that part +of the philosophy of mathematics which does not demand a +critical examination of the idea of <i>class</i>. In the preceding +chapter, however, we found ourselves confronted by problems +which make such an examination imperative. Before we can +undertake it, we must consider certain other parts of the philosophy +of mathematics, which we have hitherto ignored. In a +synthetic treatment, the parts which we shall now be concerned +with come first: they are more fundamental than anything +that we have discussed hitherto. Three topics will concern us +before we reach the theory of classes, namely: (1) the theory +of deduction, (2) propositional functions, (3) descriptions. Of +these, the third is not logically presupposed in the theory of +classes, but it is a simpler example of the <i>kind</i> of theory that +is needed in dealing with classes. It is the first topic, the theory +of deduction, that will concern us in the present chapter. +</p> +<p> +Mathematics is a deductive science: starting from certain +premisses, it arrives, by a strict process of deduction, at the +various theorems which constitute it. It is true that, in the past, +mathematical deductions were often greatly lacking in rigour; +it is true also that perfect rigour is a scarcely attainable ideal. +Nevertheless, in so far as rigour is lacking in a mathematical +proof, the proof is defective; it is no defence to urge that common +sense shows the result to be correct, for if we were to rely upon +that, it would be better to dispense with argument altogether, +<span class="pagenum" id="Page_144">[Pg 144]</span> +rather than bring fallacy to the rescue of common sense. No +appeal to common sense, or "intuition," or anything except strict +deductive logic, ought to be needed in mathematics after the +premisses have been laid down. +</p> +<p> +Kant, having observed that the geometers of his day could +not prove their theorems by unaided argument, but required +an appeal to the figure, invented a theory of mathematical +reasoning according to which the inference is never strictly +logical, but always requires the support of what is called +"intuition." The whole trend of modern mathematics, with +its increased pursuit of rigour, has been against this Kantian +theory. The things in the mathematics of Kant's day which +cannot be <i>proved</i>, cannot be <i>known</i>—for example, the axiom of +parallels. What can be known, in mathematics and by mathematical +methods, is what can be deduced from pure logic. What +else is to belong to human knowledge must be ascertained otherwise—empirically, +through the senses or through experience in +some form, but not <i>a priori</i>. The positive grounds for this +thesis are to be found in <i>Principia Mathematica</i>, <i>passim</i>; a +controversial defence of it is given in the <i>Principles of Mathematics</i>. +We cannot here do more than refer the reader to those +works, since the subject is too vast for hasty treatment. Meanwhile, +we shall assume that all mathematics is deductive, and +proceed to inquire as to what is involved in deduction. +</p> +<p> +In deduction, we have one or more propositions called <i>premisses</i>, +from which we infer a proposition called the <i>conclusion</i>. +For our purposes, it will be convenient, when there are originally +several premisses, to amalgamate them into a single proposition, +so as to be able to speak of <i>the</i> premiss as well as of <i>the</i> conclusion. +Thus we may regard deduction as a process by which +we pass from knowledge of a certain proposition, the premiss, +to knowledge of a certain other proposition, the conclusion. +But we shall not regard such a process as <i>logical</i> deduction unless +it is <i>correct</i>, <i>i.e.</i> unless there is such a relation between premiss +and conclusion that we have a right to believe the conclusion +<span class="pagenum" id="Page_145">[Pg 145]</span> +if we know the premiss to be true. It is this relation that is +chiefly of interest in the logical theory of deduction. +</p> +<p> +In order to be able validly to infer the truth of a proposition, +we must know that some other proposition is true, and that +there is between the two a relation of the sort called "implication," +<i>i.e.</i> that (as we say) the premiss "implies" the conclusion. (We +shall define this relation shortly.) Or we may know that a certain +other proposition is false, and that there is a relation between +the two of the sort called "disjunction," expressed by "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span>"<a id="FNanchor_32_1"></a><a href="#Footnote_32_1" class="fnanchor">[32]</a> +so that the knowledge that the one is false allows us to infer +that the other is true. Again, what we wish to infer may be +the <i>falsehood</i> of some proposition, not its truth. This may be +inferred from the truth of another proposition, provided we know +that the two are "incompatible," <i>i.e.</i> that if one is true, the other +is false. It may also be inferred from the falsehood of another +proposition, in just the same circumstances in which the truth +of the other might have been inferred from the truth of the one; +<i>i.e.</i> from the falsehood of <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> we may infer the falsehood of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> when +<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">.</span> All these four are cases of inference. When our +minds are fixed upon inference, it seems natural to take "implication" +as the primitive fundamental relation, since this is the +relation which must hold between <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> if we are to be able +to infer the <i>truth</i> of <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> from the <i>truth</i> of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">.</span> But for technical +reasons this is not the best primitive idea to choose. Before +proceeding to primitive ideas and definitions, let us consider +further the various functions of propositions suggested by the +above-mentioned relations of propositions. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_32_1"></a><a href="#FNanchor_32_1"><span class="label">[32]</span></a>We shall use the letters <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">,</span> <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s">,</span> <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/202.svg" alt="" data-tex="t"> to denote variable propositions.</p></div> + +<p> +The simplest of such functions is the negative, "not-<span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">.</span>" +This is that function of <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> which is true when <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is false, and false +when <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is true. It is convenient to speak of the truth of a proposition, +or its falsehood, as its "truth-value"<a id="FNanchor_33_1"></a><a href="#Footnote_33_1" class="fnanchor">[33]</a>; +<i>i.e.</i> <i>truth</i> is +the "truth-value" of a true proposition, and <i>falsehood</i> of a false +one. Thus not <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> has the opposite truth-value to <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">.</span> +<span class="pagenum" id="Page_146">[Pg 146]</span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_33_1"></a><a href="#FNanchor_33_1"><span class="label">[33]</span></a>This term is due to Frege.</p></div> + +<p> +We may take next <i>disjunction</i>, "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span>" This is a function +whose truth-value is truth when <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is true and also when <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is true, +but is falsehood when both <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> are false. +</p> +<p> +Next we may take <i>conjunction</i>, "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" This has truth +for its truth-value when <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> are both true; otherwise it +has falsehood for its truth-value. +</p> +<p> +Take next <i>incompatibility</i>, <i>i.e.</i> "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> are not both true." +This is the negation of conjunction; it is also the disjunction +of the negations of <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> <i>i.e.</i> it is "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or not-<span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span>" Its truth-value +is truth when <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is false and likewise when <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is false; its +truth-value is falsehood when <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> are both true. +</p> +<p> +Last take <i>implication</i>, <i>i.e.</i> "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span>" or "if <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> then <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span>" +This is to be understood in the widest sense that will allow us +to infer the truth of <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> if we know the truth of <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> . Thus we interpret +it as meaning: "Unless <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is false, <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is true," or "either +<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is false or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is true." (The fact that "implies" is capable +of other meanings does not concern us; this is the meaning which +is convenient for us.) That is to say, "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" is to mean +"not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">": its truth-value is to be truth if <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is false, likewise +if <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is true, and is to be falsehood if <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is true and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is false. +</p> +<p> +We have thus five functions: negation, disjunction, conjunction, +incompatibility, and implication. We might have added others, +for example, joint falsehood, "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and not-<span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span>" but the above +five will suffice. Negation differs from the other four in being +a function of <i>one</i> proposition, whereas the others are functions +of <i>two</i>. But all five agree in this, that their truth-value depends +only upon that of the propositions which are their arguments. +Given the truth or falsehood of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> or of <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> (as the case may +be), we are given the truth or falsehood of the negation, disjunction, +conjunction, incompatibility, or implication. A function of +propositions which has this property is called a "truth-function." +</p> +<p> +The whole meaning of a truth-function is exhausted by the +statement of the circumstances under which it is true or false. +"Not-<span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span>" for example, is simply that function of <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> which is true +when <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is false, and false when <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is true: there is no further +<span class="pagenum" id="Page_147">[Pg 147]</span> +meaning to be assigned to it. The same applies to "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" +and the rest. It follows that two truth-functions which have +the same truth-value for all values of the argument are indistinguishable. +For example, "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" is the negation of +"not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or not-<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" and <i>vice versa</i>; thus either of these may be +<i>defined</i> as the negation of the other. There is no further meaning +in a truth-function over and above the conditions under which +it is true or false. +</p> +<p> +It is clear that the above five truth-functions are not all independent. +We can define some of them in terms of others. There +is no great difficulty in reducing the number to two; the two +chosen in <i>Principia Mathematica</i> are negation and disjunction. +Implication is then defined as "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">"; incompatibility +as "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or not-<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">"; conjunction as the negation of incompatibility. +But it has been shown by Sheffer<a id="FNanchor_34_1"></a><a href="#Footnote_34_1" class="fnanchor">[34]</a> +that we can be content +with <i>one</i> primitive idea for all five, and by Nicod<a id="FNanchor_35_1"></a><a href="#Footnote_35_1" class="fnanchor">[35]</a> +that this enables +us to reduce the primitive propositions required in the theory +of deduction to two non-formal principles and one formal one. +For this purpose, we may take as our one indefinable either +incompatibility or joint falsehood. We will choose the former. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_34_1"></a><a href="#FNanchor_34_1"><span class="label">[34]</span></a><i>Trans. Am. Math. Soc.</i>, vol. XIV. pp. 481-488.</p></div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_35_1"></a><a href="#FNanchor_35_1"><span class="label">[35]</span></a><i>Proc. Camb. Phil. Soc.</i>, vol. XIX., i., January 1917.</p></div> + +<p> +Our primitive idea, now, is a certain truth-function called +"incompatibility," which we will denote by <span class="nowrap"><img style="vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;" src="images/120.svg" alt="" data-tex="p/q">.</span> Negation +can be at once defined as the incompatibility of a proposition +with itself, <i>i.e.</i> "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">" is defined as "<span class="nowrap"><img style="vertical-align: -0.566ex; width: 3.407ex; height: 2.262ex;" src="images/281.svg" alt="" data-tex="p/p">.</span>" Disjunction is +the incompatibility of not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and not-<span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> <i>i.e.</i> it is <span class="nowrap"><img style="vertical-align: -0.566ex; width: 10.769ex; height: 2.262ex;" src="images/282.svg" alt="" data-tex="(p/p) | (q/q)">.</span> +Implication is the incompatibility of <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and not-<span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> <i>i.e.</i> <span class="nowrap"><img style="vertical-align: -0.566ex; width: 6.74ex; height: 2.262ex;" src="images/283.svg" alt="" data-tex="p | (q/q)">.</span> +Conjunction is the negation of incompatibility, <i>i.e.</i> it is <span class="nowrap"><img style="vertical-align: -0.566ex; width: 10.769ex; height: 2.262ex;" src="images/284.svg" alt="" data-tex="(p/q) | (p/q)">.</span> +Thus all our four other functions are defined in terms +of incompatibility. +</p> +<p> +It is obvious that there is no limit to the manufacture of truth-functions, +either by introducing more arguments or by repeating +arguments. What we are concerned with is the connection of +this subject with inference. +<span class="pagenum" id="Page_148">[Pg 148]</span> +</p> +<p> +If we know that <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is true and that <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> we can proceed +to assert <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span> There is always unavoidably <i>something</i> psychological +about inference: inference is a method by which we arrive +at new knowledge, and what is not psychological about it is the +relation which allows us to infer correctly; but the actual passage +from the assertion of <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> to the assertion of <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is a psychological +process, and we must not seek to represent it in purely logical +terms. +</p> +<p> +In mathematical practice, when we infer, we have always +some expression containing variable propositions, say <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> +which is known, in virtue of its form, to be true for all values +of <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">;</span> we have also some other expression, part of the former, +which is also known to be true for all values of <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">;</span> and in +virtue of the principles of inference, we are able to drop this part +of our original expression, and assert what is left. This somewhat +abstract account may be made clearer by a few examples. +</p> +<p> +Let us assume that we know the five formal principles of +deduction enumerated in <i>Principia Mathematica</i>. (M. Nicod has +reduced these to one, but as it is a complicated proposition, +we will begin with the five.) These five propositions are as +follows:— +</p> +<p> +(1) "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">" implies <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">—<i>i.e.</i> if either <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is true +or <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is true, then <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is true. +</p> +<p> +(2) <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> implies "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">"—<i>i.e.</i> the disjunction "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" +is true when one of its alternatives is true. +</p> +<p> +(3) "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" implies "<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> or <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">.</span>" This would not be required +if we had a theoretically more perfect notation, since in the +conception of disjunction there is no order involved, so that +"<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" and "<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> or <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">" should be identical. But since our +symbols, in any convenient form, inevitably introduce an order, +we need suitable assumptions for showing that the order is +irrelevant. +</p> +<p> +(4) If either <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is true or "<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> or <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">" is true, then either <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is true +or "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">" is true. (The twist in this proposition serves to +increase its deductive power.) +<span class="pagenum" id="Page_149">[Pg 149]</span> +</p> +<p> +(5) If <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> implies <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">,</span> then "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" implies "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">.</span>" +</p> +<p> +These are the <i>formal</i> principles of deduction employed in +<i>Principia Mathematica</i>. A formal principle of deduction has a +double use, and it is in order to make this clear that we have +cited the above five propositions. It has a use as the premiss +of an inference, and a use as establishing the fact that the premiss +implies the conclusion. In the schema of an inference +we have a proposition <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> and a proposition "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span>" from +which we infer <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span> Now when we are concerned with the principles +of deduction, our apparatus of primitive propositions has +to yield both the <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and the "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" of our inferences. +That is to say, our rules of deduction are to be used, not <i>only</i> as +<i>rules</i>, which is their use for establishing "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" but <i>also</i> +as substantive premisses, <i>i.e.</i> as the <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> of our schema. Suppose, +for example, we wish to prove that if <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> then if <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> implies <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r"> +it follows that <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">.</span> We have here a relation of +three propositions which state implications. Put +</p> +<p class="hanging2"> +<img style="vertical-align: -0.439ex; width: 6.281ex; height: 1.758ex;" src="images/285.svg" alt="" data-tex="p_{1} = p"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> +<img style="vertical-align: -0.439ex; width: 6.184ex; height: 1.758ex;" src="images/286.svg" alt="" data-tex="p_{2} = q"> implies <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">,</span> and +<img style="vertical-align: -0.439ex; width: 6.281ex; height: 1.758ex;" src="images/287.svg" alt="" data-tex="p_{3} = p"> implies <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">.</span> +</p> +<p class="nind"> +Then we have to prove that <img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/288.svg" alt="" data-tex="p_{1}"> implies that <img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/289.svg" alt="" data-tex="p_{2}"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/290.svg" alt="" data-tex="p_{3}">.</span> Now +take the fifth of our above principles, substitute not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> for <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> +and remember that "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" is by definition the same as +"<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span>" Thus our fifth principle yields: +</p> + +<p class="hanging2"> +"If <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> implies <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">,</span> then '<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">'</span> implies '<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">implies <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">,</span>'" +<i>i.e.</i> "<img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/289.svg" alt="" data-tex="p_{2}">implies that <img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/288.svg" alt="" data-tex="p_{1}"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/290.svg" alt="" data-tex="p_{3}">.</span>" Call this +proposition <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A">.</span> +</p> + +<p class="nind"> +But the fourth of our principles, when we substitute not-<span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> +not-<span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> for <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> and remember the definition of implication, +becomes: +</p> + +<p class="hanging2"> +"If <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies that <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> implies <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">,</span> then <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> implies +that <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">.</span>" +</p> + +<p class="nind"> +Writing <img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/289.svg" alt="" data-tex="p_{2}"> in place of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> <img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/288.svg" alt="" data-tex="p_{1}"> in place of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> and +<img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/290.svg" alt="" data-tex="p_{3}"> in place of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">,</span> this +becomes: +</p> + +<p class="hanging2"> +"If <img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/289.svg" alt="" data-tex="p_{2}"> implies that <img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/288.svg" alt="" data-tex="p_{1}"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/290.svg" alt="" data-tex="p_{3}">,</span> then +<img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/288.svg" alt="" data-tex="p_{1}"> implies that <img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/289.svg" alt="" data-tex="p_{2}"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/290.svg" alt="" data-tex="p_{3}">.</span>" +Call this <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">.</span> +</p> +<p><span class="pagenum" id="Page_150">[Pg 150]</span></p> + +<p class="nind"> +Now we proved by means of our fifth principle that +</p> + +<p class="hanging2"> +"<img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/289.svg" alt="" data-tex="p_{2}"> implies that <img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/288.svg" alt="" data-tex="p_{1}"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/290.svg" alt="" data-tex="p_{3}">,</span>" which was +what we called <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A">.</span> +</p> + +<p class="nind"> +Thus we have here an instance of the schema of inference, +since <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A"> represents the <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> of our scheme, and <img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B"> represents the +"<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span>" Hence we arrive at <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> namely, +</p> + +<p class="hanging2"> +"<img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/288.svg" alt="" data-tex="p_{1}"> implies that <img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/289.svg" alt="" data-tex="p_{2}"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;" src="images/290.svg" alt="" data-tex="p_{3}">,</span>" +</p> + +<p class="nind"> +which was the proposition to be proved. In this proof, the +adaptation of our fifth principle, which yields <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A">,</span> occurs as a +substantive premiss; while the adaptation of our fourth principle, +which yields <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">,</span> is used to give the <i>form</i> of the inference. The +formal and material employments of premisses in the theory +of deduction are closely intertwined, and it is not very important +to keep them separated, provided we realise that they are in +theory distinct. +</p> +<p> +The earliest method of arriving at new results from a premiss +is one which is illustrated in the above deduction, but which +itself can hardly be called deduction. The primitive propositions, +whatever they may be, are to be regarded as asserted for all +possible values of the variable propositions <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r"> which occur +in them. We may therefore substitute for (say) <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> any expression +whose value is always a proposition, <i>e.g.</i> not-<span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> +"<img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s"> implies <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/202.svg" alt="" data-tex="t">,</span>" +and so on. By means of such substitutions we really obtain +sets of special cases of our original proposition, but from a practical +point of view we obtain what are virtually new propositions. +The legitimacy of substitutions of this kind has to be insured by +means of a non-formal principle of inference.<a id="FNanchor_36_1"></a><a href="#Footnote_36_1" class="fnanchor">[36]</a> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_36_1"></a><a href="#FNanchor_36_1"><span class="label">[36]</span></a>No such principle is enunciated in <i>Principia Mathematica</i>, or in M. Nicod's +article mentioned above. But this would seem to be an omission.</p></div> + +<p> +We may now state the one formal principle of inference to +which M. Nicod has reduced the five given above. For this +purpose we will first show how certain truth-functions can be +defined in terms of incompatibility. We saw already that +</p> +<p class="hanging2"> +<img style="vertical-align: -0.566ex; width: 6.74ex; height: 2.262ex;" src="images/283.svg" alt="" data-tex="p | (q/q)"> means "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span>" +<span class="pagenum" id="Page_151">[Pg 151]</span> +</p> +<p class="nind"> +We now observe that +</p> +<p class="hanging2"> +<img style="vertical-align: -0.566ex; width: 6.719ex; height: 2.262ex;" src="images/291.svg" alt="" data-tex="p | (q/r)"> means "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies both <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">.</span>" +</p> +<p class="nind"> +For this expression means "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is incompatible with the incompatibility +of <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">,</span>" <i>i.e.</i> "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies that <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> and <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r"> are not incompatible," +<i>i.e.</i> "<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies that <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> and <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r"> are both true"—for, as +we saw, the conjunction of <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> and <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r"> is the negation of their +incompatibility. +</p> +<p> +Observe next that <img style="vertical-align: -0.566ex; width: 5.971ex; height: 2.262ex;" src="images/292.svg" alt="" data-tex="t | (t/t)"> means "<img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/202.svg" alt="" data-tex="t"> implies itself." This is a +particular case of <span class="nowrap"><img style="vertical-align: -0.566ex; width: 6.74ex; height: 2.262ex;" src="images/283.svg" alt="" data-tex="p | (q/q)">.</span> +</p> +<p> +Let us write <img style="vertical-align: -0.439ex; width: 1.138ex; height: 2.156ex;" src="images/293.svg" alt="" data-tex="\overline{p}"> for the negation of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">;</span> thus +<img style="vertical-align: -0.566ex; width: 3.33ex; height: 2.98ex;" src="images/294.svg" alt="" data-tex="\overline{p/s}"> will mean the +negation of <span class="nowrap"><img style="vertical-align: -0.566ex; width: 3.33ex; height: 2.262ex;" src="images/295.svg" alt="" data-tex="p/s">,</span> <i>i.e.</i> it will mean the conjunction of +<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s">.</span> It follows that +<span class="align-center"><img style="vertical-align: -0.566ex; width: 8.952ex; height: 2.98ex;" src="images/43.svg" alt="" data-tex=" +(s/q) | \overline{p/s} +"></span> +expresses the incompatibility of <img style="vertical-align: -0.566ex; width: 3.233ex; height: 2.262ex;" src="images/296.svg" alt="" data-tex="s/q"> with the conjunction of +<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s">;</span> in other words, it states that if <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s"> are both true, +<img style="vertical-align: -0.566ex; width: 3.233ex; height: 2.262ex;" src="images/296.svg" alt="" data-tex="s/q"> is false, <i>i.e.</i> <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s"> and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> are both true; in still simpler words, +it states that <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s"> jointly imply <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s"> and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> jointly. +</p> +<p> +Now, put +<span class="align-center"><img style="vertical-align: -3.865ex; width: 14.388ex; height: 8.862ex;" src="images/44.svg" alt="" data-tex=" +\begin{align*} + P &= p | (q/r), \\ + \pi &= t | (t/t), \\ + Q &= (s/q) | \overline{p/s}. +\end{align*} +"></span> +Then M. Nicod's sole formal principle of deduction is +<span class="align-center"><img style="vertical-align: -0.566ex; width: 7.167ex; height: 2.262ex;" src="images/45.svg" alt="" data-tex=" +P | \pi/Q, +"></span> +in other words, <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> implies both <img style="vertical-align: -0.025ex; width: 1.29ex; height: 1ex;" src="images/297.svg" alt="" data-tex="\pi"> and <span class="nowrap"><img style="vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;" src="images/95.svg" alt="" data-tex="\mathrm Q">.</span> +</p> +<p> +He employs in addition one non-formal principle belonging +to the theory of types (which need not concern us), and one +corresponding to the principle that, given <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> and given that +<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> we can assert <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span> This principle is: +</p> + +<p class="nind"> +"If <img style="vertical-align: -0.566ex; width: 6.719ex; height: 2.262ex;" src="images/298.svg" alt="" data-tex="p | (r/q)"> is true, and <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is true, then <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is true." +</p> + +<p class="nind"> +From this apparatus the whole theory of deduction follows, except +in so far as we are concerned with deduction from or to the +existence or the universal truth of "propositional functions," +which we shall consider in the next chapter. +</p> +<p> +There is? if I am not mistaken, a certain confusion in the +<span class="pagenum" id="Page_152">[Pg 152]</span> +minds of some authors as to the relation, between propositions, +in virtue of which an inference is valid. In order that it may +be <i>valid</i> to infer <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> from <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> it is only necessary that <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> should be +true and that the proposition "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" should be true. +Whenever this is the case, it is clear that <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> must be true. But +inference will only in fact take place when the proposition "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> +or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" is <i>known</i> otherwise than through knowledge of not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or +knowledge of <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span> Whenever <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> is false, "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" is true, +but is useless for inference, which requires that <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> should be true. +Whenever <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is already known to be true, "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" is of +course also known to be true, but is again useless for inference, +since <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is already known, and therefore does not need to be +inferred. In fact, inference only arises when "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" +can be known without our knowing already which of the two +alternatives it is that makes the disjunction true. Now, the +circumstances under which this occurs are those in which certain +relations of form exist between <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span> For example, we know +that if <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r"> implies the negation of <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s">,</span> then <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s"> implies the negation +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">.</span> Between "<img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r"> implies not-<img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s">" and "<img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/165.svg" alt="" data-tex="s"> implies not-<img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">" there +is a formal relation which enables us to <i>know</i> that the first implies +the second, without having first to know that the first is false +or to know that the second is true. It is under such circumstances +that the relation of implication is practically useful for +drawing inferences. +</p> +<p> +But this formal relation is only required in order that we may +be able to <i>know</i> that either the premiss is false or the conclusion +is true. It is the truth of "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" that is required for +the <i>validity</i> of the inference; what is required further is only +required for the practical feasibility of the inference. Professor +C. I. Lewis<a id="FNanchor_37_1"></a><a href="#Footnote_37_1" class="fnanchor">[37]</a> +has especially studied the narrower, formal relation +which we may call "formal deducibility." He urges that the +wider relation, that expressed by "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" should not be +called "implication." That is, however, a matter of words. +<span class="pagenum" id="Page_153">[Pg 153]</span> +Provided our use of words is consistent, it matters little how we +define them. The essential point of difference between the +theory which I advocate and the theory advocated by Professor +Lewis is this: He maintains that, when one proposition <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> is +"formally deducible" from another <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span> the relation which we +perceive between them is one which he calls "strict implication," +which is not the relation expressed by "not-<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">" but a narrower +relation, holding only when there are certain formal connections +between <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">.</span> I maintain that, whether or not there be +such a relation as he speaks of, it is in any case one that mathematics +does not need, and therefore one that, on general grounds +of economy, ought not to be admitted into our apparatus of +fundamental notions; that, whenever the relation of "formal +deducibility" holds between two propositions, it is the case that +we can see that either the first is false or the second true, and that +nothing beyond this fact is necessary to be admitted into our +premisses; and that, finally, the reasons of detail which Professor +Lewis adduces against the view which I advocate can all be met +in detail, and depend for their plausibility upon a covert and +unconscious assumption of the point of view which I reject. +I conclude, therefore, that there is no need to admit as a fundamental +notion any form of implication not expressible as a +truth-function. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_37_1"></a><a href="#FNanchor_37_1"><span class="label">[37]</span></a>See <i>Mind</i>, vol. XXI., 1912, pp. 522-531; and vol. XXIII., 1914, pp. 240-247.</p></div> + +<p><span class="pagenum" id="Page_154">[Pg 154]</span></p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='XV: PROPOSITIONAL FUNCTIONS'><a id="chap15"></a>CHAPTER XV +<br><br> +PROPOSITIONAL FUNCTIONS</h2> + +<p class="nind"> +WHEN, in the preceding chapter, we were discussing propositions, +we did not attempt to give a definition of the word "proposition." +But although the word cannot be formally defined, it is necessary +to say something as to its meaning, in order to avoid the very +common confusion with "propositional functions," which are to +be the topic of the present chapter. +</p> +<p> +We mean by a "proposition" primarily a form of words which +expresses what is either true or false. I say "primarily," +because I do not wish to exclude other than verbal symbols, or +even mere thoughts if they have a symbolic character. But I +think the word "proposition" should be limited to what may, +in some sense, be called "symbols," and further to such symbols +as give expression to truth and falsehood. Thus "two and two +are four" and "two and two are five" will be propositions, +and so will "Socrates is a man" and "Socrates is not a man." +The statement: "Whatever numbers <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> may be, +<img style="vertical-align: -0.566ex; width: 23.671ex; height: 2.452ex;" src="images/299.svg" alt="" data-tex="(a + b)^{2} = a^{2} + 2ab + b^{2}">" +is a proposition; but the bare formula "<img style="vertical-align: -0.566ex; width: 23.671ex; height: 2.452ex;" src="images/299.svg" alt="" data-tex="(a + b)^{2} = a^{2} + 2ab + b^{2}">" +alone is not, since it asserts nothing definite unless +we are further told, or led to suppose, that <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> and <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/86.svg" alt="" data-tex="b"> are to have +all possible values, or are to have such-and-such values. The +former of these is tacitly assumed, as a rule, in the enunciation +of mathematical formulæ, which thus become propositions; +but if no such assumption were made, they would be "propositional +functions." A "propositional function," in fact, is an +expression containing one or more undetermined constituents, +<span class="pagenum" id="Page_155">[Pg 155]</span> +such that, when values are assigned to these constituents, the +expression becomes a proposition. In other words, it is a function +whose values are propositions. But this latter definition must +be used with caution. A descriptive function, <i>e.g.</i> "the hardest +proposition in <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A">'</span>s mathematical treatise," will not be a propositional +function, although its values are propositions. But in +such a case the propositions are only described: in a propositional +function, the values must actually <i>enunciate</i> propositions. +</p> +<p> +Examples of propositional functions are easy to give: "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is +human" is a propositional function; so long as <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> remains +undetermined, it is neither true nor false, but when a value +is assigned to <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> it becomes a true or false proposition. Any +mathematical equation is a propositional function. So long as +the variables have no definite value, the equation is merely an +expression awaiting determination in order to become a true or +false proposition. If it is an equation containing one variable, +it becomes true when the variable is made equal to a root +of the equation, otherwise it becomes false; but if it is an +"identity" it will be true when the variable is any number. +The equation to a curve in a plane or to a surface in space is a +propositional function, true for values of the co-ordinates belonging +to points on the curve or surface, false for other values. +Expressions of traditional logic such as "all <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A"> is <img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">" are propositional +functions: <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A"> and <img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B"> have to be determined as definite +classes before such expressions become true or false. +</p> +<p> +The notion of "cases" or "instances" depends upon propositional +functions. Consider, for example, the kind of process +suggested by what is called "generalisation," and let us take +some very primitive example, say, "lightning is followed by +thunder." We have a number of "instances" of this, <i>i.e.</i> a +number of propositions such as: "this is a flash of lightning +and is followed by thunder." What are these occurrences +"instances" of? They are instances of the propositional +function: "If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a flash of lightning, <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is followed by thunder." +The process of generalisation (with whose validity we are fortunately +<span class="pagenum" id="Page_156">[Pg 156]</span> +not concerned) consists in passing from a number of such +instances to the <i>universal</i> truth of the propositional function: +"If <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a flash of lightning, <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is followed by thunder." It will +be found that, in an analogous way, propositional functions +are always involved whenever we talk of instances or cases or +examples. +</p> +<p> +We do not need to ask, or attempt to answer, the question: +"What <i>is</i> a propositional function?" A propositional function +standing all alone may be taken to be a mere schema, a mere +shell, an empty receptacle for meaning, not something already +significant. We are concerned with propositional functions, +broadly speaking, in two ways: first, as involved in the notions +"true in all cases" and "true in some cases"; secondly, as +involved in the theory of classes and relations. The second of +these topics we will postpone to a later chapter; the first must +occupy us now. +</p> +<p> +When we say that something is "always true" or "true in +all cases," it is clear that the "something" involved cannot be +a proposition. A proposition is just true or false, and there +is an end of the matter. There are no instances or cases of +"Socrates is a man" or "Napoleon died at St Helena." These +are propositions, and it would be meaningless to speak of their +being true "in all cases." This phrase is only applicable to +propositional <i>functions</i>. Take, for example, the sort of thing +that is often said when causation is being discussed. (We are +net concerned with the truth or falsehood of what is said, but +only with its logical analysis.) We are told that <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A"> is, in every +instance, followed by <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">.</span> Now if there are "instances" of <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A">,</span> +<img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A"> must be some general concept of which it is significant to say +"<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}"> is <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A">,</span>" "<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/236.svg" alt="" data-tex="x_{2}"> is <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A">,</span>" "<img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/237.svg" alt="" data-tex="x_{3}"> is <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A">,</span>" +and so on, where <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/235.svg" alt="" data-tex="x_{1}">,</span> <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/236.svg" alt="" data-tex="x_{2}">,</span> <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/237.svg" alt="" data-tex="x_{3}"> are +particulars which are not identical one with another. This +applies, <i>e.g.</i>, to our previous case of lightning. We say that +lightning (<span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A">)</span> is followed by thunder (<span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">)</span>. But the separate +flashes are particulars, not identical, but sharing the common +property of being lightning. The only way of expressing a +<span class="pagenum" id="Page_157">[Pg 157]</span> +common property generally is to say that a common property +of a number of objects is a propositional function which becomes +true when any one of these objects is taken as the value of the +variable. In this case all the objects are "instances" of the +truth of the propositional function—for a propositional function, +though it cannot itself be true or false, is true in certain instances +and false in certain others, unless it is "always true" or "always +false." When, to return to our example, we say that <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A"> is in +every instance followed by <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">,</span> we mean that, whatever <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> may be, +if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is an <span class="nowrap"><img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A">,</span> it is followed by a <span class="nowrap"><img style="vertical-align: 0; width: 1.602ex; height: 1.545ex;" src="images/75.svg" alt="" data-tex="\mathrm B">;</span> that is, +we are asserting that +a certain propositional function is "always true." +</p> +<p> +Sentences involving such words as "all," "every," "a," +"the," "some" require propositional functions for their interpretation. +The way in which propositional functions occur +can be explained by means of two of the above words, namely, +"all" and "some." +</p> +<p> +There are, in the last analysis, only two things that can be +done with a propositional function: one is to assert that it is +true in <i>all</i> cases, the other to assert that it is true in at least one +case, or in <i>some</i> cases (as we shall say, assuming that there is +to be no necessary implication of a plurality of cases). All the +other uses of propositional functions can be reduced to these two. +When we say that a propositional function is true "in all cases," +or "always" (as we shall also say, without any temporal suggestion), +we mean that all its values are true. If "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">" is the +function, and <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is the right sort of object to be an argument to "<span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">,</span>" +then <img style="vertical-align: -0.464ex; width: 2.545ex; height: 2.034ex;" src="images/301.svg" alt="" data-tex="\phi a"> is to be true, however <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> may have been chosen. +For example, "if <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is human, <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is mortal" is true whether <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is +human or not; in fact, every proposition of this form is true. +Thus the propositional function "if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human, <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is mortal" +is "always true," or "true in all cases." Or, again, the statement +"there are no unicorns" is the same as the statement +"the propositional function '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is not a unicorn' is true in all +cases." The assertions in the preceding chapter about propositions, +<i>e.g.</i> "'<img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">'</span> implies '<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> or <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p">,</span>'" are really assertions +<span class="pagenum" id="Page_158">[Pg 158]</span> +that certain propositional functions are true in all cases. We do +not assert the above principle, for example, as being true only +of this or that particular <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q">,</span> but as being true of +<i>any</i> <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> or <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> +concerning which it can be made significantly. The condition +that a function is to be <i>significant</i> for a given argument is the same +as the condition that it shall have a value for that argument, +either true or false. The study of the conditions of significance +belongs to the doctrine of types, which we shall not pursue +beyond the sketch given in the preceding chapter. +</p> +<p> +Not only the principles of deduction, but all the primitive +propositions of logic, consist of assertions that certain propositional +functions are always true. If this were not the case, they +would have to mention particular things or concepts—Socrates, +or redness, or east and west, or what not,—and clearly it is not +the province of logic to make assertions which are true concerning +one such thing or concept but not concerning another. It is +part of the definition of logic (but not the whole of its definition) +that all its propositions are completely general, <i>i.e.</i> they all +consist of the assertion that some propositional function containing +no constant terms is always true. We shall return in +our final chapter to the discussion of propositional functions +containing no constant terms. For the present we will proceed +to the other thing that is to be done with a propositional function, +namely, the assertion that it is "sometimes true," <i>i.e.</i> true in at +least one instance. +</p> +<p> +When we say "there are men," that means that the propositional +function "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a man" is sometimes true. When we +say "some men are Greeks," that means that the propositional +function "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a man and a Greek" is sometimes true. When we +say "cannibals still exist in Africa," that means that the propositional +function "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a cannibal now in Africa" is sometimes +true, <i>i.e.</i> is true for some values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> To say "there are at least +<img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> individuals in the world" is to say that the propositional +function "<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is a class of individuals and a member of the cardinal +number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">" is sometimes true, or, as we may say, is true for certain +<span class="pagenum" id="Page_159">[Pg 159]</span> +values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">.</span> This form of expression is more convenient when it +is necessary to indicate which is the variable constituent which +we are taking as the argument to our propositional function. +For example, the above propositional function, which we may +shorten to "<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is a class of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> individuals," contains two variables, +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">.</span> The axiom of infinity, in the language of propositional +functions, is: "The propositional function 'if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is an inductive +number, it is true for some values of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> that <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is a +class of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> individuals' is true for all possible values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">.</span>" +Here there is a +subordinate function, "<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is a class of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> individuals," which is +said to be, in respect of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> <i>sometimes</i> true; and the assertion +that this happens if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is an inductive number is said to be, in +respect of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> <i>always</i> true. +</p> +<p> +The statement that a function <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is always true is the negation +of the statement that not-<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is sometimes true, and the statement +that <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is sometimes true is the negation of the statement +that not-<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is always true. Thus the statement "all +men are mortals" is the negation of the statement that the +function "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is an immortal man" is sometimes true. And the +statement "there are unicorns" is the negation of the statement +that the function "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is not a unicorn" is always true.<a id="FNanchor_38_1"></a><a href="#Footnote_38_1" class="fnanchor">[38]</a> +We say that <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is "never true" or "always false" if not-<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is +always true. We can, if we choose, take one of the pair "always," +"sometimes" as a primitive idea, and define the other by means +of the one and negation. Thus if we choose "sometimes" as +our primitive idea, we can define: "'<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is always true' is to +mean 'it is false that not-<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is sometimes true.'"<a id="FNanchor_39_1"></a><a href="#Footnote_39_1" class="fnanchor">[39]</a> +But for +reasons connected with the theory of types it seems more correct +to take both "always" and "sometimes" as primitive ideas, +and define by their means the negation of propositions in which +they occur. That is to say, assuming that we have already +<span class="pagenum" id="Page_160">[Pg 160]</span> +defined (or adopted as a primitive idea) the negation of propositions +of the type to which <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> belongs, we define: "The +negation of '<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> always' is 'not-<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> sometimes'; and the negation +of '<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> sometimes' is 'not-<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> always.'" In like manner +we can re-define disjunction and the other truth-functions, +as applied to propositions containing apparent variables, in +terms of the definitions and primitive ideas for propositions +containing no apparent variables. Propositions containing no +apparent variables are called "elementary propositions." From +these we can mount up step by step, using such methods as have +just been indicated, to the theory of truth-functions as applied +to propositions containing one, two, three, ... variables, or any +number up to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> where <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is any assigned finite number. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_38_1"></a><a href="#FNanchor_38_1"><span class="label">[38]</span></a>The method of deduction is given in <i>Principia Mathematica</i>, +vol. I. * 9.</p></div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_39_1"></a><a href="#FNanchor_39_1"><span class="label">[39]</span></a>For linguistic reasons, to avoid suggesting either the plural or the +singular, it is often convenient to say "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">is not always false" rather +than "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> sometimes" or "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is sometimes true."</p></div> + +<p> +The forms which are taken as simplest in traditional formal +logic are really far from being so, and all involve the assertion +of all values or some values of a compound propositional function. +Take, to begin with, "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">.</span>" We will take it that <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is +defined by a propositional function <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">,</span> and <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> by a propositional +function <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">.</span> <i>E.g.</i>, if <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <i>men</i>, <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> +will be "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human"; if <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> is +<i>mortals</i>, <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x"> will be "there is a time at which <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> dies." Then +"all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" means: "'<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> implies <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">'</span> is always true." It is +to be observed that "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" does not apply only to those +terms that actually are <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">'</span>s; it says something equally about +terms which are not <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">'</span>s. Suppose we come across an <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> of which +we do not know whether it is an <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> or not; still, our statement +"all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" tells us something about <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> namely, +that if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is an <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">,</span> +then <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">.</span> And this is every bit as true when <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is not an <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> as +when <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is an <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">.</span> If it were not equally true in both cases, the +<i>reductio ad absurdum</i> would not be a valid method; for the +essence of this method consists in using implications in cases +where (as it afterwards turns out) the hypothesis is false. We may +put the matter another way. In order to understand "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span>" +it is not necessary to be able to enumerate what terms are <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">'</span>s; +provided we know what is meant by being an <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> and what by +being a <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span> we can understand completely what is actually affirmed +<span class="pagenum" id="Page_161">[Pg 161]</span> +by "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span>" however little we may know of actual instances +of either. This shows that it is not merely the actual terms that +are <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">'</span>s that are relevant in the statement "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span>" but all the +terms concerning which the supposition that they are <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">'</span>s is +significant, <i>i.e.</i> all the terms that are <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">'</span>s, together with all the +terms that are not <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">'</span>s—<i>i.e.</i> the whole of the appropriate logical +"type." What applies to statements about <i>all</i> applies also to +statements about <i>some</i>. "There are men," <i>e.g.</i>, means that +"<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human" is true for <i>some</i> values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> Here <i>all</i> values of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> +(<i>i.e.</i> all values for which "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human" is significant, whether +true or false) are relevant, and not only those that in fact are +human. (This becomes obvious if we consider how we could +prove such a statement to be <i>false</i>.) Every assertion about +"all" or "some" thus involves not only the arguments that +make a certain function true, but all that make it significant, +<i>i.e.</i> all for which it has a value at all, whether true or false. +</p> +<p> +We may now proceed with our interpretation of the traditional +forms of the old-fashioned formal logic. We assume that <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is +those terms <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> for which <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is true, and <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> is +those for which <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x"> is +true. (As we shall see in a later chapter, all classes are derived +in this way from propositional functions.) Then: +</p> + +<p class="hanging2"> +"All <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" means "'<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> implies +<span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">'</span> is always true." +</p> +<p class="hanging2"> +"Some <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" means "'<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> and +<span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">'</span> is sometimes true." +</p> +<p class="hanging2"> +"No <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" means "'<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> +implies not-<span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">'</span> is always true." +</p> +<p class="hanging2"> +"Some <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is not <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" means +"'<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> and not-<span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">'</span> is sometimes true." +</p> + +<p class="nind"> +It will be observed that the propositional functions which are +here asserted for all or some values are not <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> and <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x"> themselves, +but truth-functions of <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> and <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x"> for the <i>same</i> argument <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span> +The easiest way to conceive of the sort of thing that is +intended is to start not from <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> and <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x"> in general, but from +<img style="vertical-align: -0.464ex; width: 2.545ex; height: 2.034ex;" src="images/301.svg" alt="" data-tex="\phi a"> and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.67ex; height: 2.034ex;" src="images/303.svg" alt="" data-tex="\psi a">,</span> where <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is some constant. Suppose we are considering +"all men are mortal": we will begin with +</p> + +<p class="hanging2"> +"If Socrates is human, Socrates is mortal," +</p> + +<p><span class="pagenum" id="Page_162">[Pg 162]</span></p> + +<p class="nind"> +and then we will regard "Socrates" as replaced by a variable <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> +wherever "Socrates" occurs. The object to be secured is that, +although <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> remains a variable, without any definite value, yet +it is to have the same value in "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">" as in "<img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">" when we are +asserting that "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> implies <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">" is always true. This requires +that we shall start with a function whose values are such as +"<img style="vertical-align: -0.464ex; width: 2.545ex; height: 2.034ex;" src="images/301.svg" alt="" data-tex="\phi a"> implies <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.67ex; height: 2.034ex;" src="images/303.svg" alt="" data-tex="\psi a">,</span>" rather than with two separate functions <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> +and <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">;</span> for if we start with two separate functions we can +never secure that the <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> while remaining undetermined, shall +have the same value in both. +</p> +<p> +For brevity we say "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> always implies <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">" when we +mean that "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> implies <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">" is always true. Propositions +of the form "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> always implies <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">" are called "formal +implications"; this name is given equally if there are several +variables. +</p> +<p> +The above definitions show how far removed from the simplest +forms are such propositions as "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span>" with which traditional +logic begins. It is typical of the lack of analysis involved +that traditional logic treats "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" as a proposition of +the same form as "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">"—<i>e.g.</i>, it treats "all men are mortal" +as of the same form as "Socrates is mortal." As we have just +seen, the first is of the form "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> always implies <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">,</span>" while the +second is of the form "<span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">.</span>" The emphatic separation of these +two forms, which was effected by Peano and Frege, was a very +vital advance in symbolic logic. +</p> +<p> +It will be seen that "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" and +"no <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" do not +really differ in form, except by the substitution of not-<img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x"> for <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">,</span> +and that the same applies to "some <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" and "some <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is +not <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">.</span>" It should also be observed that the traditional rules +of conversion are faulty, if we adopt the view, which is the only +technically tolerable one, that such propositions as "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" +do not involve the "existence" of <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">'</span>s, <i>i.e.</i> do not require that +there should be terms which are <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">'</span>s. The above definitions +lead to the result that, if <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is always false, <i>i.e.</i> if there are no <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">'</span>s, +then "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" and "no <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" will both be true, whatever +<span class="pagenum" id="Page_163">[Pg 163]</span> +<img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P"> may be. For, according to the definition in the last +chapter, "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> implies <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">" means "not-<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> or <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">" which is +always true if not-<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is always true. At the first moment, +this result might lead the reader to desire different definitions, +but a little practical experience soon shows that any different +definitions would be inconvenient and would conceal the important +ideas. The proposition "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> always implies <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">,</span> and <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is +sometimes true" is essentially composite, and it would be +very awkward to give this as the definition of "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span>" +for then we should have no language left for "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> always implies <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">,</span>" +which is needed a hundred times for once that the other is +needed. But, with our definitions, "all <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">" does not imply +"some <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span>" since the first allows the non-existence of <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> and +the second does not; thus conversion <i>per accidens</i> becomes +invalid, and some moods of the syllogism are fallacious, <i>e.g.</i> +Darapti: "All <img style="vertical-align: 0; width: 2.075ex; height: 1.545ex;" src="images/304.svg" alt="" data-tex="\mathrm M"> is <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S">,</span> all <img style="vertical-align: 0; width: 2.075ex; height: 1.545ex;" src="images/304.svg" alt="" data-tex="\mathrm M"> is +<span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span> therefore some <img style="vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;" src="images/90.svg" alt="" data-tex="\mathrm S"> is <span class="nowrap"><img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/80.svg" alt="" data-tex="\mathrm P">,</span>" which +fails if there is no <span class="nowrap"><img style="vertical-align: 0; width: 2.075ex; height: 1.545ex;" src="images/304.svg" alt="" data-tex="\mathrm M">.</span> +</p> +<p> +The notion of "existence" has several forms, one of which +will occupy us in the next chapter; but the fundamental form +is that which is derived immediately from the notion of "sometimes +true." We say that an argument <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> "satisfies" a function <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> +if <img style="vertical-align: -0.464ex; width: 2.545ex; height: 2.034ex;" src="images/301.svg" alt="" data-tex="\phi a"> is true; this is the same sense in which the roots of an +equation are said to satisfy the equation. Now if <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is sometimes +true, we may say there are <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">'</span>s for which it is true, or we may say +"arguments satisfying <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> <i>exist</i>" This is the fundamental meaning +of the word "existence." Other meanings are either derived +from this, or embody mere confusion of thought. We may +correctly say "men exist," meaning that "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a man" is sometimes +true. But if we make a pseudo-syllogism: "Men exist, +Socrates is a man, therefore Socrates exists," we are talking +nonsense, since "Socrates" is not, like "men," merely an undetermined +argument to a given propositional function. The +fallacy is closely analogous to that of the argument: "Men are +numerous, Socrates is a man, therefore Socrates is numerous." +In this case it is obvious that the conclusion is nonsensical, but +<span class="pagenum" id="Page_164">[Pg 164]</span> +in the case of existence it is not obvious, for reasons which will +appear more fully in the next chapter. For the present let us +merely note the fact that, though it is correct to say "men exist," +it is incorrect, or rather meaningless, to ascribe existence to a +given particular <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> who happens to be a man. Generally, "terms +satisfying <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> exist" means "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is sometimes true"; but "<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> exists" +(where <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is a term satisfying <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">)</span> is a mere noise or shape, +devoid of significance. It will be found that by bearing in mind +this simple fallacy we can solve many ancient philosophical +puzzles concerning the meaning of existence. +</p> +<p> +Another set of notions as to which philosophy has allowed +itself to fall into hopeless confusions through not sufficiently +separating propositions and propositional functions are the +notions of "modality": <i>necessary</i>, <i>possible</i>, and <i>impossible</i>. +(Sometimes <i>contingent</i> or <i>assertoric</i> is used instead of <i>possible</i>.) +The traditional view was that, among true propositions, some +were necessary, while others were merely contingent or assertoric; +while among false propositions some were impossible, namely, +those whose contradictories were necessary, while others merely +happened not to be true. In fact, however, there was never +any clear account of what was added to truth by the conception +of necessity. In the case of propositional functions, the three-fold +division is obvious. If "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">" is an undetermined value of a +certain propositional function, it will be <i>necessary</i> if the function +is always true, <i>possible</i> if it is sometimes true, and <i>impossible</i> if +it is never true. This sort of situation arises in regard to probability, +for example. Suppose a ball <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is drawn from a bag +which contains a number of balls: if all the balls are white, +"<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is white" is necessary; if some are white, it is possible; +if none, it is impossible. Here all that is <i>known</i> about <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is that +it satisfies a certain propositional function, namely, "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> was a +ball in the bag." This is a situation which is general in probability +problems and not uncommon in practical life—<i>e.g.</i> when +a person calls of whom we know nothing except that he brings +a letter of introduction from our friend so-and-so. In all such +<span class="pagenum" id="Page_165">[Pg 165]</span> +cases, as in regard to modality in general, the propositional +function is relevant. For clear thinking, in many very diverse +directions, the habit of keeping propositional functions sharply +separated from propositions is of the utmost importance, and +the failure to do so in the past has been a disgrace to +philosophy. +<span class="pagenum" id="Page_166">[Pg 166]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='XVI: DESCRIPTIONS'><a id="chap16"></a>CHAPTER XVI +<br><br> +DESCRIPTIONS</h2> + +<p class="nind"> +We dealt in the preceding chapter with the words <i>all</i> and <i>some</i>; +in this chapter we shall consider the word <i>the</i> in the singular, +and in the next chapter we shall consider the word <i>the</i> in the +plural. It may be thought excessive to devote two chapters +to one word, but to the philosophical mathematician it is a +word of very great importance: like Browning's Grammarian +with the enclitic <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.923ex; height: 1.647ex;" src="images/305.svg" alt="" data-tex="\delta\epsilon">,</span> I would give the doctrine of this word if I +were "dead from the waist down" and not merely in a prison. +</p> +<p> +We have already had occasion to mention "descriptive +functions," <i>i.e.</i> such expressions as "the father of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" or "the sine +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span>" These are to be defined by first defining "descriptions." +</p> +<p> +A "description" may be of two sorts, definite and indefinite +(or ambiguous). An indefinite description is a phrase of the +form "a so-and-so," and a definite description is a phrase of +the form "the so-and-so" (in the singular). Let us begin with +the former. +</p> +<p> +"Who did you meet?" "I met a man." "That is a very +indefinite description." We are therefore not departing from +usage in our terminology. Our question is: What do I really +assert when I assert "I met a man"? Let us assume, for the +moment, that my assertion is true, and that in fact I met Jones. +It is clear that what I assert is <i>not</i> "I met Jones." I may say +"I met a man, but it was not Jones"; in that case, though I lie, +I do not contradict myself, as I should do if when I say I met a +<span class="pagenum" id="Page_167">[Pg 167]</span> +man I really mean that I met Jones. It is clear also that the +person to whom I am speaking can understand what I say, even +if he is a foreigner and has never heard of Jones. +</p> +<p> +But we may go further: not only Jones, but no actual man, +enters into my statement. This becomes obvious when the statement +is false, since then there is no more reason why Jones +should be supposed to enter into the proposition than why anyone +else should. Indeed the statement would remain significant, +though it could not possibly be true, even if there were no man +at all. "I met a unicorn" or "I met a sea-serpent" is a +perfectly significant assertion, if we know what it would be to +be a unicorn or a sea-serpent, <i>i.e.</i> what is the definition of these +fabulous monsters. Thus it is only what we may call the <i>concept</i> +that enters into the proposition. In the case of "unicorn," +for example, there is only the concept: there is not also, somewhere +among the shades, something unreal which may be called +"a unicorn." Therefore, since it is significant (though false) +to say "I met a unicorn," it is clear that this proposition, rightly +analysed, does not contain a constituent "a unicorn," though +it does contain the concept "unicorn." +</p> +<p> +The question of "unreality," which confronts us at this +point, is a very important one. Misled by grammar, the great +majority of those logicians who have dealt with this question +have dealt with it on mistaken lines. They have regarded +grammatical form as a surer guide in analysis than, in fact, +it is. And they have not known what differences in grammatical +form are important. "I met Jones" and "I met a +man" would count traditionally as propositions of the same form, +but in actual fact they are of quite different forms: the first +names an actual person, Jones; while the second involves a +propositional function, and becomes, when made explicit: "The +function 'I met <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human' is sometimes true." (It +will be remembered that we adopted the convention of using +"sometimes" as not implying more than once.) This proposition +is obviously not of the form "I met <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span>" which accounts +<span class="pagenum" id="Page_168">[Pg 168]</span> +for the existence of the proposition "I met a unicorn" in spite +of the fact that there is no such thing as "a unicorn." +</p> +<p> +For want of the apparatus of propositional functions, many +logicians have been driven to the conclusion that there are +unreal objects. It is argued, <i>e.g.</i> by Meinong,<a id="FNanchor_40_1"></a><a href="#Footnote_40_1" class="fnanchor">[40]</a> +that we can +speak about "the golden mountain," "the round square," +and so on; we can make true propositions of which these are +the subjects; hence they must have some kind of logical being, +since otherwise the propositions in which they occur would be +meaningless. In such theories, it seems to me, there is a failure +of that feeling for reality which ought to be preserved even in +the most abstract studies. Logic, I should maintain, must no +more admit a unicorn than zoology can; for logic is concerned +with the real world just as truly as zoology, though with its +more abstract and general features. To say that unicorns have +an existence in heraldry, or in literature, or in imagination, +is a most pitiful and paltry evasion. What exists in heraldry +is not an animal, made of flesh and blood, moving and breathing +of its own initiative. What exists is a picture, or a description +in words. Similarly, to maintain that Hamlet, for example, +exists in his own world, namely, in the world of Shakespeare's +imagination, just as truly as (say) Napoleon existed in the +ordinary world, is to say something deliberately confusing, or +else confused to a degree which is scarcely credible. There is +only one world, the "real" world: Shakespeare's imagination +is part of it, and the thoughts that he had in writing Hamlet +are real. So are the thoughts that we have in reading the play. +But it is of the very essence of fiction that only the thoughts, +feelings, etc., in Shakespeare and his readers are real, and that +there is not, in addition to them, an objective Hamlet. When +you have taken account of all the feelings roused by Napoleon +in writers and readers of history, you have not touched the actual +man; but in the case of Hamlet you have come to the end of +him. If no one thought about Hamlet, there would be nothing +<span class="pagenum" id="Page_169">[Pg 169]</span> +left of him; if no one had thought about Napoleon, he would +have soon seen to it that some one did. The sense of reality is +vital in logic, and whoever juggles with it by pretending that +Hamlet has another kind of reality is doing a disservice to +thought. A robust sense of reality is very necessary in framing +a correct analysis of propositions about unicorns, golden mountains, +round squares, and other such pseudo-objects. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_40_1"></a><a href="#FNanchor_40_1"><span class="label">[40]</span></a><i>Untersuchungen zur Gegenstandstheorie und Psychologie</i>, 1904.</p></div> + +<p> +In obedience to the feeling of reality, we shall insist that, +in the analysis of propositions, nothing "unreal" is to be +admitted. But, after all, if there <i>is</i> nothing unreal, how, it +may be asked, <i>could</i> we admit anything unreal? The reply +is that, in dealing with propositions, we are dealing in the first +instance with symbols, and if we attribute significance to groups +of symbols which have no significance, we shall fall into the +error of admitting unrealities, in the only sense in which this is +possible, namely, as objects described. In the proposition +"I met a unicorn," the whole four words together make a significant +proposition, and the word "unicorn" by itself is significant, +in just the same sense as the word "man." But the <i>two</i> words +"a unicorn" do not form a subordinate group having a meaning +of its own. Thus if we falsely attribute meaning to these two +words, we find ourselves saddled with "a unicorn," and with +the problem how there can be such a thing in a world where +there are no unicorns. "A unicorn" is an indefinite description +which describes nothing. It is not an indefinite description +which describes something unreal. Such a proposition as +"<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is unreal" only has meaning when "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" is a description, +definite or indefinite; in that case the proposition will be true +if "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" is a description which describes nothing. But whether +the description "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" describes something or describes nothing, +it is in any case not a constituent of the proposition in which it +occurs; like "a unicorn" just now, it is not a subordinate group +having a meaning of its own. All this results from the fact that, +when "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" is a description, "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is unreal" or "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> does not exist" +is not nonsense, but is always significant and sometimes true. +<span class="pagenum" id="Page_170">[Pg 170]</span> +</p> +<p> +We may now proceed to define generally the meaning of +propositions which contain ambiguous descriptions. Suppose +we wish to make some statement about "a so-and-so," where +"so-and-so's" are those objects that have a certain property <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi">,</span> +<i>i.e.</i> those objects <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> for which the propositional function <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is +true. (<i>E.g.</i> if we take "a man" as our instance of "a so-and-so," +<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> will be "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human.") Let us now wish to assert the property <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/307.svg" alt="" data-tex="\psi"> +of "a so-and-so," <i>i.e.</i> we wish to assert that "a so-and-so" has +that property which <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has when <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x"> is true. (<i>E.g.</i> in the case +of "I met a man," <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x"> will be "I met <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span>") Now the proposition +that "a so-and-so" has the property <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/307.svg" alt="" data-tex="\psi"> is <i>not</i> a proposition of +the form "<span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">.</span>" If it were, "a so-and-so" would have to be +identical with <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> for a suitable <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">;</span> and although (in a sense) this +may be true in some cases, it is certainly not true in such a case +as "a unicorn." It is just this fact, that the statement that a +so-and-so has the property <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/307.svg" alt="" data-tex="\psi"> is not of the form <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">,</span> which makes +it possible for "a so-and-so" to be, in a certain clearly definable +sense, "unreal." The definition is as follows:— +</p> +<p> +The statement that "an object having the property <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi"> has +the property <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/307.svg" alt="" data-tex="\psi">" +</p> +<p class="nind"> +means: +</p> +<p> +"The joint assertion of <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> and <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x"> is not always false." +</p> +<p> +So far as logic goes, this is the same proposition as might +be expressed by "some <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi">'</span>s are <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/307.svg" alt="" data-tex="\psi">'</span>s"; but rhetorically there is +a difference, because in the one case there is a suggestion of +singularity, and in the other case of plurality. This, however, +is not the important point. The important point is that, when +rightly analysed, propositions verbally about "a so-and-so" +are found to contain no constituent represented by this phrase. +And that is why such propositions can be significant even when +there is no such thing as a so-and-so. +</p> +<p> +The definition of <i>existence</i>, as applied to ambiguous descriptions, +results from what was said at the end of the preceding +chapter. We say that "men exist" or "a man exists" if the +<span class="pagenum" id="Page_171">[Pg 171]</span> +propositional function "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human" is sometimes true; and +generally "a so-and-so" exists if "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is so-and-so" is sometimes +true. We may put this in other language. The proposition +"Socrates is a man" is no doubt <i>equivalent</i> to "Socrates is +human," but it is not the very same proposition. The <i>is</i> of +"Socrates is human" expresses the relation of subject and +predicate; the <i>is</i> of "Socrates is a man" expresses identity. +It is a disgrace to the human race that it has chosen to employ +the same word "is" for these two entirely different ideas—a +disgrace which a symbolic logical language of course remedies. +The identity in "Socrates is a man" is identity between an +object named (accepting "Socrates" as a name, subject to +qualifications explained later) and an object ambiguously +described. An object ambiguously described will "exist" when +at least one such proposition is true, <i>i.e.</i> when there is at least +one true proposition of the form "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a so-and-so," where "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" is +a name. It is characteristic of ambiguous (as opposed to +definite) descriptions that there may be any number of true +propositions of the above form—Socrates is a man, Plato is a +man, etc. Thus "a man exists" follows from Socrates, or +Plato, or anyone else. With definite descriptions, on the other +hand, the corresponding form of proposition, namely, "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is the +so-and-so" (where "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" is a name), can only be true for one +value of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> at most. This brings us to the subject of definite +descriptions, which are to be defined in a way analogous to +that employed for ambiguous descriptions, but rather more +complicated. +</p> +<p> +We come now to the main subject of the present chapter, +namely, the definition of the word <i>the</i> (in the singular). One +very important point about the definition of "a so-and-so" +applies equally to "the so-and-so"; the definition to be sought +is a definition of propositions in which this phrase occurs, not a +definition of the phrase itself in isolation. In the case of "a +so-and-so," this is fairly obvious: no one could suppose that +"a man" was a definite object, which could be defined by itself. +<span class="pagenum" id="Page_172">[Pg 172]</span> +Socrates is a man, Plato is a man, Aristotle is a man, but we +cannot infer that "a man" means the same as "Socrates" +means and also the same as "Plato" means and also the same +as "Aristotle" means, since these three names have different +meanings. Nevertheless, when we have enumerated all the +men in the world, there is nothing left of which we can say, +"This is a man, and not only so, but it is <i>the</i> 'a man,' the quintessential +entity that is just an indefinite man without being anybody +in particular." It is of course quite clear that whatever +there is in the world is definite: if it is a man it is one definite +man and not any other. Thus there cannot be such an entity +as "a man" to be found in the world, as opposed to specific +man. And accordingly it is natural that we do not define "a +man" itself, but only the propositions in which it occurs. +</p> +<p> +In the case of "the so-and-so" this is equally true, though +at first sight less obvious. We may demonstrate that this must +be the case, by a consideration of the difference between a <i>name</i> +and a <i>definite description</i>. Take the proposition, "Scott is the +author of <i>Waverley</i>." We have here a name, "Scott," and a +description, "the author of <i>Waverley</i>," which are asserted to +apply to the same person. The distinction between a name and +all other symbols may be explained as follows:— +</p> +<p> +A name is a simple symbol whose meaning is something that +can only occur as subject, <i>i.e.</i> something of the kind that, in +Chapter XIII., we defined as an "individual" or a "particular." +And a "simple" symbol is one which has no parts that are +symbols. Thus "Scott" is a simple symbol, because, though it +has parts (namely, separate letters), these parts are not symbols. +On the other hand, "the author of <i>Waverley</i>" is not a simple +symbol, because the separate words that compose the phrase +are parts which are symbols. If, as may be the case, whatever +<i>seems</i> to be an "individual" is really capable of further analysis, +we shall have to content ourselves with what may be called +"relative individuals," which will be terms that, throughout +the context in question, are never analysed and never occur +<span class="pagenum" id="Page_173">[Pg 173]</span> +otherwise than as subjects. And in that case we shall have +correspondingly to content ourselves with "relative names." +From the standpoint of our present problem, namely, the definition +of descriptions, this problem, whether these are absolute +names or only relative names, may be ignored, since it concerns +different stages in the hierarchy of "types," whereas we +have to compare such couples as "Scott" and "the author of +<i>Waverley</i>," which both apply to the same object, and do not +raise the problem of types. We may, therefore, for the moment, +treat names as capable of being absolute; nothing that we shall +have to say will depend upon this assumption, but the wording +may be a little shortened by it. +</p> +<p> +We have, then, two things to compare: (1) a <i>name</i>, which +is a simple symbol, directly designating an individual which +is its meaning, and having this meaning in its own right, independently +of the meanings of all other words; (2) a <i>description</i>, +which consists of several words, whose meanings are already +fixed, and from which results whatever is to be taken as the +"meaning" of the description. +</p> +<p> +A proposition containing a description is not identical with +what that proposition becomes when a name is substituted, +even if the name names the same object as the description +describes. "Scott is the author of <i>Waverley</i>" is obviously a +different proposition from "Scott is Scott": the first is a fact +in literary history, the second a trivial truism. And if we put +anyone other than Scott in place of "the author of <i>Waverley</i>," +our proposition would become false, and would therefore certainly +no longer be the same proposition. But, it may be said, our +proposition is essentially of the same form as (say) "Scott is +Sir Walter," in which two names are said to apply to the same +person. The reply is that, if "Scott is Sir Walter" really means +"the person named 'Scott' is the person named 'Sir Walter,'" +then the names are being used as descriptions: <i>i.e.</i> the individual, +instead of being named, is being described as the person having +that name. This is a way in which names are frequently used +<span class="pagenum" id="Page_174">[Pg 174]</span> +in practice, and there will, as a rule, be nothing in the phraseology +to show whether they are being used in this way or <i>as</i> names. +When a name is used directly, merely to indicate what we are +speaking about, it is no part of the <i>fact</i> asserted, or of the falsehood +if our assertion happens to be false: it is merely part of the +symbolism by which we express our thought. What we want +to express is something which might (for example) be translated +into a foreign language; it is something for which the actual +words are a vehicle, but of which they are no part. On the other +hand, when we make a proposition about "the person called +'Scott,'" the actual name "Scott" enters into what we are +asserting, and not merely into the language used in making the +assertion. Our proposition will now be a different one if we +substitute "the person called 'Sir Walter.'" But so long as +we are using names <i>as</i> names, whether we say "Scott" or whether +we say "Sir Walter" is as irrelevant to what we are asserting +as whether we speak English or French. Thus so long as names +are used as names, "Scott is Sir Walter" is the same trivial +proposition as "Scott is Scott." This completes the proof that +"Scott is the author of <i>Waverley</i>" is not the same proposition +as results from substituting a name for "the author of <i>Waverley</i>," +no matter what name may be substituted. +</p> +<p> +When we use a variable, and speak of a propositional function, +<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> say, the process of applying general statements about <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> to +particular cases will consist in substituting a name for the letter "<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span>" +assuming that <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi"> is a function which has individuals for its +arguments. Suppose, for example, that <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is "always true"; +let it be, say, the "law of identity," <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.605ex; height: 1.505ex;" src="images/308.svg" alt="" data-tex="x = x">.</span> Then we may substitute +for "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" any name we choose, and we shall obtain a true +proposition. Assuming for the moment that "Socrates," +"Plato," and "Aristotle" are names (a very rash assumption), +we can infer from the law of identity that Socrates is Socrates, +Plato is Plato, and Aristotle is Aristotle. But we shall commit +a fallacy if we attempt to infer, without further premisses, that +the author of <i>Waverley</i> is the author of <i>Waverley</i>. This results +<span class="pagenum" id="Page_175">[Pg 175]</span> +from what we have just proved, that, if we substitute a name for +"the author of <i>Waverley</i>" in a proposition, the proposition +we obtain is a different one. That is to say, applying the result +to our present case: If "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" is a name, "<img style="vertical-align: -0.186ex; width: 5.605ex; height: 1.505ex;" src="images/308.svg" alt="" data-tex="x = x">" is not the same +proposition as "the author of <i>Waverley</i> is the author of <i>Waverley</i>," +no matter what name "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" may be. Thus from the fact that +all propositions of the form "<img style="vertical-align: -0.186ex; width: 5.605ex; height: 1.505ex;" src="images/308.svg" alt="" data-tex="x = x">" are true we cannot infer, +without more ado, that the author of <i>Waverley</i> is the author of +<i>Waverley</i>. In fact, propositions of the form "the so-and-so +is the so-and-so" are not always true: it is necessary that the +so-and-so should <i>exist</i> (a term which will be explained shortly). +It is false that the present King of France is the present King of +France, or that the round square is the round square. When we +substitute a description for a name, propositional functions +which are "always true" may become false, if the description +describes nothing. There is no mystery in this as soon as we +realise (what was proved in the preceding paragraph) that when +we substitute a description the result is not a value of the +propositional function in question. +</p> +<p> +We are now in a position to define propositions in which a +definite description occurs. The only thing that distinguishes +"the so-and-so" from "a so-and-so" is the implication of +uniqueness. We cannot speak of "<i>the</i> inhabitant of London," +because inhabiting London is an attribute which is not unique. +We cannot speak about "the present King of France," because +there is none; but we can speak about "the present King of +England." Thus propositions about "the so-and-so" always +imply the corresponding propositions about "a so-and-so," +with the addendum that there is not more than one so-and-so. +Such a proposition as "Scott is the author of <i>Waverly</i>" could +not be true if <i>Waverly</i> had never been written, or if several +people had written it; and no more could any other proposition +resulting from a propositional function <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> by the substitution +of "the author of <i>Waverly</i>" for "<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">.</span>" We may say that "the +author of <i>Waverly</i>" means "the value of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> for which '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> wrote +<span class="pagenum" id="Page_176">[Pg 176]</span> +<i>Waverly</i>' is true." Thus the proposition "the author of +<i>Waverly</i> was Scotch," for example, involves: +</p> +<p> +(1) "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> wrote <i>Waverly</i>" is not always false; +</p> +<p> +(2) "if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> wrote <i>Waverly</i>, <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> are identical" is +always true; +</p> +<p> +(3) "if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> wrote <i>Waverly</i>, <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> was Scotch" is always true. +</p> +<p class="nind"> +These three propositions, translated into ordinary language, +state: +</p> +<p> +(1) at least one person wrote <i>Waverly</i>; +</p> +<p> +(2) at most one person wrote <i>Waverly</i>; +</p> +<p> +(3) whoever wrote <i>Waverly</i> was Scotch. +</p> +<p class="nind"> +All these three are implied by "the author of <i>Waverly</i> was +Scotch." Conversely, the three together (but no two of them) +imply that the author of <i>Waverly</i> was Scotch. Hence the +three together may be taken as defining what is meant by the +proposition "the author of <i>Waverly</i> was Scotch." +</p> +<p> +We may somewhat simplify these three propositions. The +first and second together are equivalent to: "There is a term <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> +such that '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> wrote <i>Waverly</i>' is true when <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> and is false +when <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is not <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">.</span>" In other words, "There is a term <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> such that +'<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> wrote <i>Waverly</i>' is always equivalent to '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">.</span>'" (Two +propositions are "equivalent" when both are true or both are +false.) We have here, to begin with, two functions of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> wrote +<i>Waverly</i>" and "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">,</span>" and we form a function of <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> by +considering the equivalence of these two functions of <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> for all +values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">;</span> we then proceed to assert that the resulting function +of <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> is "sometimes true," <i>i.e.</i> that it is true for at least one value +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">.</span> (It obviously cannot be true for more than one value of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">.</span>) +These two conditions together are defined as giving the meaning +of "the author of <i>Waverly</i> exists." +</p> +<p> +We may now define "the term satisfying the function <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> +exists." This is the general form of which the above is a particular +case. "The author of <i>Waverly</i>" is "the term satisfying +the function '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> wrote <i>Waverly</i>.'" And "the so-and-so" will +<span class="pagenum" id="Page_177">[Pg 177]</span> +always involve reference to some propositional function, namely, +that which defines the property that makes a thing a so-and-so. +Our definition is as follows:— +</p> +<p> +"The term satisfying the function <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> exists" means: +</p> +<p> +"There is a term <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> such that <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is always equivalent +to '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">.</span>'" +</p> +<p> +In order to define "the author of <i>Waverly</i> was Scotch," +we have still to take account of the third of our three propositions, +namely, "Whoever wrote <i>Waverly</i> was Scotch." This +will be satisfied by merely adding that the <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> in question is to +be Scotch. Thus "the author of <i>Waverly</i> was Scotch" is: +</p> +<p class="hanging2"> +"There is a term <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> such that (1) '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> wrote <i>Waverly</i>' is always +equivalent to '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">,</span>' (2) <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> is Scotch." +</p> + +<p class="nind"> +And generally: "the term satisfying <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> satisfies <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">" is +defined as meaning: +</p> + +<p class="hanging2"> +"There is a term <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> such that (1) <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is always equivalent to +'<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">,</span>' (2) <img style="vertical-align: -0.464ex; width: 2.452ex; height: 2.034ex;" src="images/309.svg" alt="" data-tex="\psi c"> is true." +</p> + +<p class="nind"> +This is the definition of propositions in which descriptions occur. +</p> +<p> +It is possible to have much knowledge concerning a term +described, <i>i.e.</i> to know many propositions concerning "the so-and-so," +without actually knowing what the so-and-so is, <i>i.e.</i> +without knowing any proposition of the form "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is the so-and-so," +where "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" is a name. In a detective story propositions about +"the man who did the deed" are accumulated, in the hope +that ultimately they will suffice to demonstrate that it was <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/73.svg" alt="" data-tex="\mathrm A"> +who did the deed. We may even go so far as to say that, +in all such knowledge as can be expressed in words—with the +exception of "this" and "that" and a few other words of +which the meaning varies on different occasions—no names, +in the strict sense, occur, but what seem like names are really +descriptions. We may inquire significantly whether Homer +existed, which we could not do if "Homer" were a name. The +proposition "the so-and-so exists" is significant, whether +true or false; but if <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is the so-and-so (where "<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">" is a name), +the words "<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> exists" are meaningless. It is only of descriptions—definite +<span class="pagenum" id="Page_178">[Pg 178]</span> +or indefinite—that existence can be significantly +asserted; for, if "<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">" is a name, it <i>must</i> name something: what +does not name anything is not a name, and therefore, if intended +to be a name, is a symbol devoid of meaning, whereas a description, +like "the present King of France," does not become incapable +of occurring significantly merely on the ground that it +describes nothing, the reason being that it is a <i>complex</i> symbol, +of which the meaning is derived from that of its constituent +symbols. And so, when we ask whether Homer existed, we are +using the word "Homer" as an abbreviated description: we +may replace it by (say) "the author of the <i>Iliad</i> and the <i>Odyssey</i>." +The same considerations apply to almost all uses of what look +like proper names. +</p> +<p> +When descriptions occur in propositions, it is necessary to +distinguish what may be called "primary" and "secondary" +occurrences. The abstract distinction is as follows. A description +has a "primary" occurrence when the proposition in +which it occurs results from substituting the description for "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" +in some propositional function <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">;</span> a description has a +"secondary" occurrence when the result of substituting the +description for <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> in <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> gives only <i>part</i> of the proposition concerned. +An instance will make this clearer. Consider "the +present King of France is bald." Here "the present King of +France" has a primary occurrence, and the proposition is false. +Every proposition in which a description which describes nothing +has a primary occurrence is false. But now consider "the +present King of France is not bald." This is ambiguous. If +we are first to take "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is bald," then substitute "the present +King of France" for "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" and then deny the result, the occurrence +of "the present King of France" is secondary and our proposition +is true; but if we are to take "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is not bald" and substitute +"the present King of France" for "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" then "the present +King of France" has a primary occurrence and the proposition +is false. Confusion of primary and secondary occurrences is a +ready source of fallacies where descriptions are concerned. +<span class="pagenum" id="Page_179">[Pg 179]</span> +</p> +<p> +Descriptions occur in mathematics chiefly in the form of +<i>descriptive functions</i>, <i>i.e.</i> "the term having the +relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span>" +or "the <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> of <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">" as we may say, on the analogy of "the +father of <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">" and similar phrases. To say "the father of <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y"> is +rich," for example, is to say that the following propositional +function of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">:</span> "<img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> is rich, and '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> begat <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">'</span> is always equivalent +to '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">,</span>'" is "sometimes true," <i>i.e.</i> is true for at least one +value of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">.</span> It obviously cannot be true for more than one +value. +</p> +<p> +The theory of descriptions, briefly outlined in the present +chapter, is of the utmost importance both in logic and in theory +of knowledge. But for purposes of mathematics, the more +philosophical parts of the theory are not essential, and have +therefore been omitted in the above account, which has confined +itself to the barest mathematical requisites. +<span class="pagenum" id="Page_180">[Pg 180]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='XVII: CLASSES'><a id="chap17"></a>CHAPTER XVII +<br><br> +CLASSES</h2> + +<p class="nind"> +IN the present chapter we shall be concerned with <i>the</i> in the +plural: the inhabitants of London, the sons of rich men, and +so on. In other words, we shall be concerned with <i>classes</i>. We +saw in Chapter II. that a cardinal number is to be defined as a +class of classes, and in Chapter III. that the number 1 is to be +defined as the class of all unit classes, <i>i.e.</i> of all that have just +one member, as we should say but for the vicious circle. Of +course, when the number 1 is defined as the class of all unit +classes, "unit classes" must be defined so as not to assume +that we know what is meant by "one"; in fact, they are defined +in a way closely analogous to that used for descriptions, namely: +A class <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is said to be a "unit" class if the propositional function +"'<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is an <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">'</span> is always equivalent to '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">'</span>" (regarded as a +function of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">)</span> is not always false, <i>i.e.</i>, in more ordinary language, +if there is a term <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> such that <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> will be a member of +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> when <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> +but not otherwise. This gives us a definition of a unit class if we +already know what a class is in general. Hitherto we have, in +dealing with arithmetic, treated "class" as a primitive idea. +But, for the reasons set forth in Chapter XIII., if for no others, +we cannot accept "class" as a primitive idea. We must seek a +definition on the same lines as the definition of descriptions, +<i>i.e.</i> a definition which will assign a meaning to propositions in +whose verbal or symbolic expression words or symbols apparently +representing classes occur, but which will assign a meaning that +altogether eliminates all mention of classes from a right analysis +<span class="pagenum" id="Page_181">[Pg 181]</span> +of such propositions. We shall then be able to say that the +symbols for classes are mere conveniences, not representing +objects called "classes," and that classes are in fact, like descriptions, +logical fictions, or (as we say) "incomplete symbols." +</p> +<p> +The theory of classes is less complete than the theory of descriptions, +and there are reasons (which we shall give in outline) +for regarding the definition of classes that will be suggested as +not finally satisfactory. Some further subtlety appears to be +required; but the reasons for regarding the definition which +will be offered as being approximately correct and on the right +lines are overwhelming. +</p> +<p> +The first thing is to realise why classes cannot be regarded +as part of the ultimate furniture of the world. It is difficult +to explain precisely what one means by this statement, but one +consequence which it implies may be used to elucidate its meaning. +If we had a complete symbolic language, with a definition for +everything definable, and an undefined symbol for everything +indefinable, the undefined symbols in this language would represent +symbolically what I mean by "the ultimate furniture of +the world." I am maintaining that no symbols either for "class" +in general or for particular classes would be included in this +apparatus of undefined symbols. On the other hand, all the +particular things there are in the world would have to have +names which would be included among undefined symbols. +We might try to avoid this conclusion by the use of descriptions. +Take (say) "the last thing Cæsar saw before he died." This +is a description of some particular; we might use it as (in one +perfectly legitimate sense) a <i>definition</i> of that particular. But +if "<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">" is a <i>name</i> for the same particular, a proposition in which +"<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">" occurs is not (as we saw in the preceding chapter) identical +with what this proposition becomes when for "<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">" we substitute +"the last thing Cæsar saw before he died." If our language +does not contain the name "<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">" or some other name for the same +particular, we shall have no means of expressing the proposition +which we expressed by means of "<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">" as opposed to the one that +<span class="pagenum" id="Page_182">[Pg 182]</span> +we expressed by means of the description. Thus descriptions +would not enable a perfect language to dispense with names for +all particulars. In this respect, we are maintaining, classes +differ from particulars, and need not be represented by undefined +symbols. Our first business is to give the reasons for this opinion. +</p> +<p> +We have already seen that classes cannot be regarded as a +species of individuals, on account of the contradiction about +classes which are not members of themselves (explained in +Chapter XIII.), and because we can prove that the number of +classes is greater than the number of individuals. +</p> +<p> +We cannot take classes in the <i>pure</i> extensional way as simply +heaps or conglomerations. If we were to attempt to do that, +we should find it impossible to understand how there can be such +a class as the null-class, which has no members at all and cannot +be regarded as a "heap"; we should also find it very hard to +understand how it comes about that a class which has only one +member is not identical with that one member. I do not mean +to assert, or to deny, that there are such entities as "heaps." +As a mathematical logician, I am not called upon to have an +opinion on this point. All that I am maintaining is that, if there +are such things as heaps, we cannot identify them with the classes +composed of their constituents. +</p> +<p> +We shall come much nearer to a satisfactory theory if we +try to identify classes with propositional functions. Every +class, as we explained in Chapter II., is defined by some propositional +function which is true of the members of the class +and false of other things. But if a class can be defined by one +propositional function, it can equally well be defined by any +other which is true whenever the first is true and false whenever +the first is false. For this reason the class cannot be identified +with any one such propositional function rather than with +any other—and given a propositional function, there are always +many others which are true when it is true and false when it is +false. We say that two propositional functions are "formally +equivalent" when this happens. Two <i>propositions</i> are "equivalent" +<span class="pagenum" id="Page_183">[Pg 183]</span> +when both are true or both false; two propositional +functions <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">,</span> <img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x"> are "formally equivalent" when <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is always +equivalent to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;" src="images/302.svg" alt="" data-tex="\psi x">.</span> It is the fact that there are other functions +formally equivalent to a given function that makes it impossible +to identify a class with a function; for we wish classes to be such +that no two distinct classes have exactly the same members, +and therefore two formally equivalent functions will have to +determine the same class. +</p> +<p> +When we have decided that classes cannot be things of the +same sort as their members, that they cannot be just heaps or +aggregates, and also that they cannot be identified with propositional +functions, it becomes very difficult to see what they +can be, if they are to be more than symbolic fictions. And if +we can find any way of dealing with them as symbolic fictions, +we increase the logical security of our position, since we avoid +the need of assuming that there are classes without being compelled +to make the opposite assumption that there are no classes. +We merely abstain from both assumptions. This is an example +of Occam's razor, namely, "entities are not to be multiplied +without necessity." But when we refuse to assert that there +are classes, we must not be supposed to be asserting dogmatically +that there are none. We are merely agnostic as regards them: +like Laplace, we can say, "<i>je n'ai pas besoin de cette hypothèse.</i>" +</p> +<p> +Let us set forth the conditions that a symbol must fulfil if +it is to serve as a class. I think the following conditions will +be found necessary and sufficient:— +</p> +<p> +(1) Every propositional function must determine a class, +consisting of those arguments for which the function is true. +Given any proposition (true or false), say about Socrates, we +can imagine Socrates replaced by Plato or Aristotle or a gorilla +or the man in the moon or any other individual in the world. +In general, some of these substitutions will give a true proposition +and some a false one. The class determined will consist of all +those substitutions that give a true one. Of course, we have +still to decide what we mean by "all those which, etc." All that +<span class="pagenum" id="Page_184">[Pg 184]</span> +we are observing at present is that a class is rendered determinate +by a propositional function, and that every propositional function +determines an appropriate class. +</p> +<p> +(2) Two formally equivalent propositional functions must +determine the same class, and two which are not formally equivalent +must determine different classes. That is, a class is determined +by its membership, and no two different classes can have +the same membership. (If a class is determined by a function <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">,</span> +we say that <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is a "member" of the class if <img style="vertical-align: -0.464ex; width: 2.545ex; height: 2.034ex;" src="images/301.svg" alt="" data-tex="\phi a"> is true.) +</p> +<p> +(3) We must find some way of defining not only classes, but +classes of classes. We saw in Chapter II. that cardinal numbers +are to be defined as classes of classes. The ordinary phrase +of elementary mathematics, "The combinations of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> things +<img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> at a time" represents a class of classes, namely, the class of +all classes of <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/47.svg" alt="" data-tex="m"> terms that can be selected out of a given class +of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> terms. Without some symbolic method of dealing with +classes of classes, mathematical logic would break down. +</p> +<p> +(4) It must under all circumstances be meaningless (not false) +to suppose a class a member of itself or not a member of itself. +This results from the contradiction which we discussed in +Chapter XIII. +</p> +<p> +(5) Lastly—and this is the condition which is most difficult +of fulfilment,—it must be possible to make propositions about +<i>all</i> the classes that are composed of individuals, or about <i>all</i> the +classes that are composed of objects of any one logical "type." +If this were not the case, many uses of classes would go astray—for +example, mathematical induction. In defining the posterity +of a given term, we need to be able to say that a member of the +posterity belongs to <i>all</i> hereditary classes to which the given +term belongs, and this requires the sort of totality that is in +question. The reason there is a difficulty about this condition +is that it can be proved to be impossible to speak of <i>all</i> the propositional +functions that can have arguments of a given type. +</p> +<p> +We will, to begin with, ignore this last condition and the +problems which it raises. The first two conditions may be +<span class="pagenum" id="Page_185">[Pg 185]</span> +taken together. They state that there is to be one class, no +more and no less, for each group of formally equivalent propositional +functions; <i>e.g.</i> the class of men is to be the same as +that of featherless bipeds or rational animals or Yahoos or whatever +other characteristic may be preferred for defining a human +being. Now, when we say that two formally equivalent propositional +functions may be not identical, although they define +the same class, we may prove the truth of the assertion by pointing +out that a statement may be true of the one function and +false of the other; <i>e.g.</i> "I believe that all men are mortal" +may be true, while "I believe that all rational animals are +mortal" may be false, since I may believe falsely that the +Phoenix is an immortal rational animal. Thus we are led to +consider <i>statements about functions</i>, or (more correctly) <i>functions +of functions</i>. +</p> +<p> +Some of the things that may be said about a function may +be regarded as said about the class defined by the function, +whereas others cannot. The statement "all men are mortal" +involves the functions "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human" and "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is mortal"; or, +if we choose, we can say that it involves the classes <i>men</i> and +<i>mortals</i>. We can interpret the statement in either way, because +its truth-value is unchanged if we substitute for "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human" +or for "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is mortal" any formally equivalent function. But, +as we have just seen, the statement "I believe that all men are +mortal" cannot be regarded as being about the class determined +by either function, because its truth-value may be changed +by the substitution of a formally equivalent function (which +leaves the class unchanged). We will call a statement involving +a function <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> an "extensional" function of the function <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">,</span> if +it is like "all men are mortal," <i>i.e.</i> if its truth-value is unchanged +by the substitution of any formally equivalent function; and +when a function of a function is not extensional, we will call it +"intensional," so that "I believe that all men are mortal" +is an intensional function of "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human" or "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is mortal." +Thus <i>extensional</i> functions of a function <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> may, for practical +<span class="pagenum" id="Page_186">[Pg 186]</span> +purposes, be regarded as functions of the class determined by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span> +while <i>intensional</i> functions cannot be so regarded. +</p> +<p> +It is to be observed that all the <i>specific</i> functions of functions +that we have occasion to introduce in mathematical logic are +extensional. Thus, for example, the two fundamental functions +of functions are: "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is always true" and "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is sometimes +true." Each of these has its truth-value unchanged if any +formally equivalent function is substituted for <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">.</span> In the +language of classes, if <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is the class determined +by <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">,</span> "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is +always true" is equivalent to "everything is a member of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span>" +and "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is sometimes true" is equivalent to "<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> has members" +or (better) "<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> has at least one member." Take, again, the +condition, dealt with in the preceding chapter, for the existence +of "the term satisfying <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">.</span>" The condition is that there is a +term <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> such that <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is always equivalent to "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">.</span>" This +is obviously extensional. It is equivalent to the assertion +that the class defined by the function <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is a unit class, <i>i.e.</i> a +class having one member; in other words, a class which is a +member of 1. +</p> +<p> +Given a function of a function which may or may not be +extensional, we can always derive from it a connected and +certainly extensional function of the same function, by the +following plan: Let our original function of a function be one +which attributes to <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> the property <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/310.svg" alt="" data-tex="f">;</span> then consider the assertion +"there is a function having the property <img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/310.svg" alt="" data-tex="f"> and formally +equivalent to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">.</span>" This is an extensional function of <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">;</span> it +is true when our original statement is true, and it is formally +equivalent to the original function of <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> if this original function +is extensional; but when the original function is intensional, +the new one is more often true than the old one. For example, +consider again "I believe that all men are mortal," regarded +as a function of "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human." The derived extensional function +is: "There is a function formally equivalent to '<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human' +and such that I believe that whatever satisfies it is mortal." +This remains true when we substitute "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a rational animal" +<span class="pagenum" id="Page_187">[Pg 187]</span> +for "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is human," even if I believe falsely that the Phoenix is +rational and immortal. +</p> +<p> +We give the name of "derived extensional function" to the +function constructed as above, namely, to the function: "There +is a function having the property <img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/310.svg" alt="" data-tex="f"> and formally equivalent to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">,</span>" +where the original function was "the function <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> has +the property <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/310.svg" alt="" data-tex="f">.</span>" +</p> +<p> +We may regard the derived extensional function as having +for its argument the class determined by the function <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">,</span> and +as asserting <img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/310.svg" alt="" data-tex="f"> of this class. This may be taken as the definition +of a proposition about a class. <i>I.e.</i> we may define: +</p> +<p> +To assert that "the class determined by the function <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> +has the property <img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/310.svg" alt="" data-tex="f">" is to assert that <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> satisfies the extensional +function derived from <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/310.svg" alt="" data-tex="f">.</span> +</p> +<p> +This gives a meaning to any statement about a class which +can be made significantly about a function; and it will be +found that technically it yields the results which are required +in order to make a theory symbolically satisfactory.<a id="FNanchor_41_1"></a><a href="#Footnote_41_1" class="fnanchor">[41]</a> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_41_1"></a><a href="#FNanchor_41_1"><span class="label">[41]</span></a>See <i>Principia Mathematica</i>, vol. I. pp. 75-84 and * 20.</p></div> + +<p> +What we have said just now as regards the definition of +classes is sufficient to satisfy our first four conditions. The +way in which it secures the third and fourth, namely, the possibility +of classes of classes, and the impossibility of a class being +or not being a member of itself, is somewhat technical; it is +explained in <i>Principia Mathematica</i>, but may be taken for +granted here. It results that, but for our fifth condition, we +might regard our task as completed. But this condition—at +once the most important and the most difficult—is not fulfilled +in virtue of anything we have said as yet. The difficulty is +connected with the theory of types, and must be briefly discussed.<a id="FNanchor_42_1"></a><a href="#Footnote_42_1" class="fnanchor">[42]</a> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_42_1"></a><a href="#FNanchor_42_1"><span class="label">[42]</span></a>The reader who desires a fuller discussion should consult <i>Principia +Mathematica</i>, Introduction, chap. II.; also * 12.</p></div> + +<p> +We saw in Chapter XIII. that there is a hierarchy of logical +types, and that it is a fallacy to allow an object belonging to +one of these to be substituted for an object belonging to another. +<span class="pagenum" id="Page_188">[Pg 188]</span> +Now it is not difficult to show that the various functions which +can take a given object <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> as argument are not all of one type. +Let us call them all <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions. We may take first those among +them which do not involve reference to any collection of functions; +these we will call "predicative <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions." If we now proceed +to functions involving reference to the totality of predicative +<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions, we shall incur a fallacy if we regard these as of the +same type as the predicative <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions. Take such an everyday +statement as "<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> is a typical Frenchman." How shall +we define a "typical" Frenchman? We may define him as +one "possessing all qualities that are possessed by most French +men." But unless we confine "all qualities" to such as do not +involve a reference to any totality of qualities, we shall have to +observe that most Frenchmen are <i>not</i> typical in the above sense, +and therefore the definition shows that to be not typical is +essential to a typical Frenchman. This is not a logical contradiction, +since there is no reason why there should be any typical +Frenchmen; but it illustrates the need for separating off +qualities that involve reference to a totality of qualities from +those that do not. +</p> +<p> +Whenever, by statements about "all" or "some" of the +values that a variable can significantly take, we generate a +new object, this new object must not be among the values which +our previous variable could take, since, if it were, the totality +of values over which the variable could range would only be +definable in terms of itself, and we should be involved in a vicious +circle. For example, if I say "Napoleon had all the qualities +that make a great general," I must define "qualities" in such a +way that it will not include what I am now saying, <i>i.e.</i> "having +all the qualities that make a great general" must not be itself a +quality in the sense supposed. This is fairly obvious, and is +the principle which leads to the theory of types by which vicious-circle +paradoxes are avoided. As applied to <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions, we +may suppose that "qualities" is to mean "predicative functions." +Then when I say "Napoleon had all the qualities, etc.," I mean +<span class="pagenum" id="Page_189">[Pg 189]</span> +"Napoleon satisfied all the predicative functions, etc." This +statement attributes a property to Napoleon, but not a predicative +property; thus we escape the vicious circle. But +wherever "all functions which" occurs, the functions in question +must be limited to one type if a vicious circle is to be avoided; +and, as Napoleon and the typical Frenchman have shown, the +type is not rendered determinate by that of the argument. It +would require a much fuller discussion to set forth this point +fully, but what has been said may suffice to make it clear that +the functions which can take a given argument are of an infinite +series of types. We could, by various technical devices, construct +a variable which would run through the first <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> of these +types, where <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is finite, but we cannot construct a variable which +will run through them all, and, if we could, that mere fact would +at once generate a new type of function with the same arguments, +and would set the whole process going again. +</p> +<p> +We call predicative <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions the <i>first</i> type of <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions; +<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions involving reference to the totality of the first type +we call the <i>second</i> type; and so on. No variable <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-function +can run through all these different types: it must stop short at +some definite one. +</p> +<p> +These considerations are relevant to our definition of the +derived extensional function. We there spoke of "a function +formally equivalent to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">.</span>" It is necessary to decide upon +the type of our function. Any decision will do, but some decision +is unavoidable. Let us call the supposed formally equivalent +function <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/307.svg" alt="" data-tex="\psi">.</span> Then <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/307.svg" alt="" data-tex="\psi"> appears as a variable, and must be of +some determinate type. All that we know necessarily about +the type of <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi"> is that it takes arguments of a given type—that +it is (say) an <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-function. But this, as we have just seen, does +not determine its type. If we are to be able (as our fifth requisite +demands) to deal with <i>all</i> classes whose members are of the same +type as <span class="nowrap"><img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">,</span> we must be able to define all such classes by means of +functions of some one type; that is to say, there must be some +type of <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-function, say the <span class="nowrap"><img style="vertical-align: -0.025ex; width: 3.044ex; height: 1.956ex;" src="images/311.svg" alt="" data-tex="n^\mathord{th}">,</span> such that any <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-function is formally +<span class="pagenum" id="Page_190">[Pg 190]</span> +equivalent to some <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-function of the <img style="vertical-align: -0.025ex; width: 3.044ex; height: 1.956ex;" src="images/311.svg" alt="" data-tex="n^\mathord{th}"> type. If this is the case, +then any extensional function which holds of all <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions +of the <img style="vertical-align: -0.025ex; width: 3.044ex; height: 1.956ex;" src="images/311.svg" alt="" data-tex="n^\mathord{th}"> type will hold of any <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-function whatever. It is chiefly +as a technical means of embodying an assumption leading to +this result that classes are useful. The assumption is called the +"axiom of reducibility," and may be stated as follows:— +</p> +<p> +"There is a type (<img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/312.svg" alt="" data-tex="\tau"> say) of <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions such that, given any +<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-function, it is formally equivalent to some function of the type +in question." +</p> +<p> +If this axiom is assumed, we use functions of this type in +defining our associated extensional function. Statements about +all <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-classes (<i>i.e.</i> all classes defined by <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions) can be reduced +to statements about all <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions of the type <span class="nowrap"><img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/312.svg" alt="" data-tex="\tau">.</span> So long as +only extensional functions of functions are involved, this gives +us in practice results which would otherwise have required the +impossible notion of "all <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a">-functions." One particular region +where this is vital is mathematical induction. +</p> +<p> +The axiom of reducibility involves all that is really essential +in the theory of classes. It is therefore worth while to ask +whether there is any reason to suppose it true. +</p> +<p> +This axiom, like the multiplicative axiom and the axiom +of infinity, is necessary for certain results, but not for the bare +existence of deductive reasoning. The theory of deduction, +as explained in Chapter XIV., and the laws for propositions +involving "all" and "some," are of the very texture of mathematical +reasoning: without them, or something like them, +we should not merely not obtain the same results, but we should +not obtain any results at all. We cannot use them as hypotheses, +and deduce hypothetical consequences, for they are +rules of deduction as well as premisses. They must be absolutely +true, or else what we deduce according to them does not even +follow from the premisses. On the other hand, the axiom of +reducibility, like our two previous mathematical axioms, could +perfectly well be stated as an hypothesis whenever it is used, +instead of being assumed to be actually true. We can deduce +<span class="pagenum" id="Page_191">[Pg 191]</span> +its consequences hypothetically; we can also deduce the consequences +of supposing it false. It is therefore only convenient, +not necessary. And in view of the complication of the theory +of types, and of the uncertainty of all except its most general +principles, it is impossible as yet to say whether there may +not be some way of dispensing with the axiom of reducibility +altogether. However, assuming the correctness of the theory +outlined above, what can we say as to the truth or falsehood of +the axiom? +</p> +<p> +The axiom, we may observe, is a generalised form of Leibniz's +identity of indiscernibles. Leibniz assumed, as a logical principle, +that two different subjects must differ as to predicates. Now +predicates are only some among what we called "predicative +functions," which will include also relations to given terms, +and various properties not to be reckoned as predicates. Thus +Leibniz's assumption is a much stricter and narrower one than +ours. (Not, of course, according to <i>his</i> logic, which regarded +<i>all</i> propositions as reducible to the subject-predicate form.) +But there is no good reason for believing his form, so far as I can +see. There might quite well, as a matter of abstract logical +possibility, be two things which had exactly the same predicates, +in the narrow sense in which we have been using the word "predicate." +How does our axiom look when we pass beyond predicates +in this narrow sense? In the actual world there seems +no way of doubting its empirical truth as regards particulars, +owing to spatio-temporal differentiation: no two particulars +have exactly the same spatial and temporal relations to all other +particulars. But this is, as it were, an accident, a fact about +the world in which we happen to find ourselves. Pure logic, +and pure mathematics (which is the same thing), aims at being +true, in Leibnizian phraseology, in all possible worlds, not only +in this higgledy-piggledy job-lot of a world in which chance has +imprisoned us. There is a certain lordliness which the logician +should preserve: he must not condescend to derive arguments +from the things he sees about him. +<span class="pagenum" id="Page_192">[Pg 192]</span> +</p> +<p> +Viewed from this strictly logical point of view, I do not see +any reason to believe that the axiom of reducibility is logically +necessary, which is what would be meant by saying that it is +true in all possible worlds. The admission of this axiom into +a system of logic is therefore a defect, even if the axiom is empirically +true. It is for this reason that the theory of classes cannot +be regarded as being as complete as the theory of descriptions. +There is need of further work on the theory of types, in the hope +of arriving at a doctrine of classes which does not require such a +dubious assumption. But it is reasonable to regard the theory +outlined in the present chapter as right in its main lines, <i>i.e.</i> in +its reduction of propositions nominally about classes to propositions +about their defining functions. The avoidance of +classes as entities by this method must, it would seem, be sound +in principle, however the detail may still require adjustment. +It is because this seems indubitable that we have included the +theory of classes, in spite of our desire to exclude, as far as possible, +whatever seemed open to serious doubt. +</p> +<p> +The theory of classes, as above outlined, reduces itself to one +axiom and one definition. For the sake of definiteness, we will +here repeat them. The axiom is: +</p> +<p> +<i>There is a type <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/312.svg" alt="" data-tex="\tau"> such that if <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi"> is a function which can take a +given object</i> <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> <i>as argument, then there is a function <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/307.svg" alt="" data-tex="\psi"> +of the type <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/312.svg" alt="" data-tex="\tau"> which is formally equivalent to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi">.</span></i> +</p> +<p> +The definition is: +</p> +<p> +<i>If <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi"> is a function which can take a given object</i> <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/85.svg" alt="" data-tex="a"> <i>as argument, +and <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/312.svg" alt="" data-tex="\tau"> the type mentioned in the above axiom, then to say that +the class determined by <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi"> has the property <img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/310.svg" alt="" data-tex="f"> is to say that there +is a function of type <span class="nowrap"><img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/312.svg" alt="" data-tex="\tau">,</span> formally equivalent to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi">,</span> and having the +property <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/310.svg" alt="" data-tex="f">.</span></i> +<span class="pagenum" id="Page_193">[Pg 193]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='XVIII: MATHEMATICS AND LOGIC'><a id="chap18"></a>CHAPTER XVIII +<br><br> +MATHEMATICS AND LOGIC</h2> + +<p class="nind"> +MATHEMATICS and logic, historically speaking, have been entirely +distinct studies. Mathematics has been connected with science, +logic with Greek. But both have developed in modern times: +logic has become more mathematical and mathematics has +become more logical. The consequence is that it has now become +wholly impossible to draw a line between the two; in fact, the +two are one. They differ as boy and man: logic is the youth +of mathematics and mathematics is the manhood of logic. This +view is resented by logicians who, having spent their time in +the study of classical texts, are incapable of following a piece +of symbolic reasoning, and by mathematicians who have learnt +a technique without troubling to inquire into its meaning or +justification. Both types are now fortunately growing rarer. +So much of modern mathematical work is obviously on the +border-line of logic, so much of modern logic is symbolic and +formal, that the very close relationship of logic and mathematics +has become obvious to every instructed student. The proof +of their identity is, of course, a matter of detail: starting with +premisses which would be universally admitted to belong to +logic, and arriving by deduction at results which as obviously +belong to mathematics, we find that there is no point at which +a sharp line can be drawn, with logic to the left and mathematics +to the right. If there are still those who do not admit +the identity of logic and mathematics, we may challenge them +to indicate at what point, in the successive definitions and +<span class="pagenum" id="Page_194">[Pg 194]</span> +deductions of <i>Principia Mathematica</i>, they consider that logic +ends and mathematics begins. It will then be obvious that any +answer must be quite arbitrary. +</p> +<p> +In the earlier chapters of this book, starting from the natural +numbers, we have first defined "cardinal number" and shown +how to generalise the conception of number, and have then +analysed the conceptions involved in the definition, until we found +ourselves dealing with the fundamentals of logic. In a synthetic, +deductive treatment these fundamentals come first, and the +natural numbers are only reached after a long journey. Such +treatment, though formally more correct than that which we +have adopted, is more difficult for the reader, because the ultimate +logical concepts and propositions with which it starts are remote +and unfamiliar as compared with the natural numbers. Also +they represent the present frontier of knowledge, beyond which +is the still unknown; and the dominion of knowledge over them +is not as yet very secure. +</p> +<p> +It used to be said that mathematics is the science of "quantity." +"Quantity" is a vague word, but for the sake of argument +we may replace it by the word "number." The statement +that mathematics is the science of number would be untrue +in two different ways. On the one hand, there are recognised +branches of mathematics which have nothing to do with number—all +geometry that does not use co-ordinates or measurement, +for example: projective and descriptive geometry, down to +the point at which co-ordinates are introduced, does not have +to do with number, or even with quantity in the sense of <i>greater</i> +and <i>less</i>. On the other hand, through the definition of cardinals, +through the theory of induction and ancestral relations, through +the general theory of series, and through the definitions of the +arithmetical operations, it has become possible to generalise much +that used to be proved only in connection with numbers. The +result is that what was formerly the single study of Arithmetic +has now become divided into numbers of separate studies, no +one of which is specially concerned with numbers. The most +<span class="pagenum" id="Page_195">[Pg 195]</span> +elementary properties of numbers are concerned with one-one +relations, and similarity between classes. Addition is concerned +with the construction of mutually exclusive classes respectively +similar to a set of classes which are not known to be mutually +exclusive. Multiplication is merged in the theory of "selections," +<i>i.e.</i> of a certain kind of one-many relations. Finitude +is merged in the general study of ancestral relations, which yields +the whole theory of mathematical induction. The ordinal +properties of the various kinds of number-series, and the elements +of the theory of continuity of functions and the limits of functions, +can be generalised so as no longer to involve any essential reference +to numbers. It is a principle, in all formal reasoning, to generalise +to the utmost, since we thereby secure that a given process of +deduction shall have more widely applicable results; we are, +therefore, in thus generalising the reasoning of arithmetic, +merely following a precept which is universally admitted in +mathematics. And in thus generalising we have, in effect, +created a set of new deductive systems, in which traditional +arithmetic is at once dissolved and enlarged; but whether any +one of these new deductive systems—for example, the theory of +selections—is to be said to belong to logic or to arithmetic is +entirely arbitrary, and incapable of being decided rationally. +</p> +<p> +We are thus brought face to face with the question: What +is this subject, which may be called indifferently either mathematics +or logic? Is there any way in which we can define it? +</p> +<p> +Certain characteristics of the subject are clear. To begin +with, we do not, in this subject, deal with particular things or +particular properties: we deal formally with what can be said +about <i>any</i> thing or <i>any</i> property. We are prepared to say that +one and one are two, but not that Socrates and Plato are two, +because, in our capacity of logicians or pure mathematicians, +we have never heard of Socrates and Plato. A world in which +there were no such individuals would still be a world in which +one and one are two. It is not open to us, as pure mathematicians +or logicians, to mention anything at all, because, if we do so, +<span class="pagenum" id="Page_196">[Pg 196]</span> +we introduce something irrelevant and not formal. We may +make this clear by applying it to the case of the syllogism. +Traditional logic says: "All men are mortal, Socrates is a man, +therefore Socrates is mortal." Now it is clear that what we +<i>mean</i> to assert, to begin with, is only that the premisses imply +the conclusion, not that premisses and conclusion are actually +true; even the most traditional logic points out that the actual +truth of the premisses is irrelevant to logic. Thus the first +change to be made in the above traditional syllogism is to state +it in the form: "If all men are mortal and Socrates is a man, +then Socrates is mortal." We may now observe that it is intended +to convey that this argument is valid in virtue of its <i>form</i>, not +in virtue of the particular terms occurring in it. If we had +omitted "Socrates is a man" from our premisses, we should +have had a non-formal argument, only admissible because +Socrates is in fact a man; in that case we could not have generalised +the argument. But when, as above, the argument is <i>formal</i>, +nothing depends upon the terms that occur in it. Thus we may +substitute <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> for <i>men</i>, <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> for <i>mortals</i>, and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> for Socrates, where +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> are any classes whatever, and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is any individual. We +then arrive at the statement: "No matter what possible values +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> and <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> may have, if all <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">'</span>s are +<span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">'</span>s and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is an <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> then <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is +a <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">"; in other words, "the propositional function 'if all <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">'</span>s +are <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta"> and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is an <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span> then <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">'</span> is always true." Here at last +we have a proposition of logic—the one which is only <i>suggested</i> by +the traditional statement about Socrates and men and mortals. +</p> +<p> +It is clear that, if <i>formal</i> reasoning is what we are aiming at, +we shall always arrive ultimately at statements like the above, +in which no actual things or properties are mentioned; this +will happen through the mere desire not to waste our time proving +in a particular case what can be proved generally. It would be +ridiculous to go through a long argument about Socrates, and then +go through precisely the same argument again about Plato. If +our argument is one (say) which holds of all men, we shall prove +it concerning "<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">,</span>" with the hypothesis "if <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a man." With +<span class="pagenum" id="Page_197">[Pg 197]</span> +this hypothesis, the argument will retain its hypothetical validity +even when <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is not a man. But now we shall find that our argument +would still be valid if, instead of supposing <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> to be a man, +we were to suppose him to be a monkey or a goose or a Prime +Minister. We shall therefore not waste our time taking as our +premiss "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a man" but shall take "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is an <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">,</span>" +where <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> is any +class of individuals, or "<img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x">" where <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi"> is any propositional +function of some assigned type. Thus the absence of all mention +of particular things or properties in logic or pure mathematics +is a necessary result of the fact that this study is, as we say, +"purely formal." +</p> +<p> +At this point we find ourselves faced with a problem which +is easier to state than to solve. The problem is: "What are +the constituents of a logical proposition?" I do not know the +answer, but I propose to explain how the problem arises. +</p> +<p> +Take (say) the proposition "Socrates was before Aristotle." +Here it seems obvious that we have a relation between two terms, +and that the constituents of the proposition (as well as of the +corresponding fact) are simply the two terms and the relation, +<i>i.e.</i> Socrates, Aristotle, and <i>before</i>. (I ignore the fact that +Socrates and Aristotle are not simple; also the fact that what +appear to be their names are really truncated descriptions. +Neither of these facts is relevant to the present issue.) We may +represent the general form of such propositions by "<span class="nowrap"><img style="vertical-align: -0.464ex; width: 4.068ex; height: 2.009ex;" src="images/313.svg" alt="" data-tex="x\mathrm Ry">,</span>" +which may be read "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> has the relation <img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R"> to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">.</span>" This general +form may occur in logical propositions, but no particular instance +of it can occur. Are we to infer that the general form itself is a +constituent of such logical propositions? +</p> +<p> +Given a proposition, such as "Socrates is before Aristotle," +we have certain constituents and also a certain form. But the +form is not itself a new constituent; if it were, we should need a +new form to embrace both it and the other constituents. We +can, in fact, turn <i>all</i> the constituents of a proposition into +variables, while keeping the form unchanged. This is what we +do when we use such a schema as "<span class="nowrap"><img style="vertical-align: -0.464ex; width: 4.068ex; height: 2.009ex;" src="images/313.svg" alt="" data-tex="x\mathrm Ry">,</span>" which stands for any +<span class="pagenum" id="Page_198">[Pg 198]</span> +one of a certain class of propositions, namely, those asserting +relations between two terms. We can proceed to general assertions, +such as "<img style="vertical-align: -0.464ex; width: 4.068ex; height: 2.009ex;" src="images/313.svg" alt="" data-tex="x\mathrm Ry"> is sometimes true"—<i>i.e.</i> there are cases +where dual relations hold. This assertion will belong to logic +(or mathematics) in the sense in which we are using the word. +But in this assertion we do not mention any particular things +or particular relations; no particular things or relations can +ever enter into a proposition of pure logic. We are left with pure +<i>forms</i> as the only possible constituents of logical propositions. +</p> +<p> +I do not wish to assert positively that pure forms—<i>e.g.</i> the +form "<img style="vertical-align: -0.464ex; width: 4.068ex; height: 2.009ex;" src="images/313.svg" alt="" data-tex="x\mathrm Ry">"—do actually enter into propositions of the kind +we are considering. The question of the analysis of such propositions +is a difficult one, with conflicting considerations on the +one side and on the other. We cannot embark upon this question +now, but we may accept, as a first approximation, the view +that <i>forms</i> are what enter into logical propositions as their +constituents. And we may explain (though not formally define) +what we mean by the "form" of a proposition as follows:— +</p> +<p> +The "form" of a proposition is that, in it, that remains unchanged +when every constituent of the proposition is replaced +by another. +</p> +<p> +Thus "Socrates is earlier than Aristotle" has the same form +as "Napoleon is greater than Wellington," though every constituent +of the two propositions is different. +</p> +<p> +We may thus lay down, as a necessary (though not sufficient) +characteristic of logical or mathematical propositions, that they +are to be such as can be obtained from a proposition containing +no variables (<i>i.e.</i> no such words as <i>all</i>, <i>some</i>, <i>a</i>, <i>the</i>, etc.) by turning +every constituent into a variable and asserting that the result +is always true or sometimes true, or that it is always true in +respect of some of the variables that the result is sometimes true +in respect of the others, or any variant of these forms. And +another way of stating the same thing is to say that logic (or +mathematics) is concerned only with <i>forms</i>, and is concerned +with them only in the way of stating that they are always or +<span class="pagenum" id="Page_199">[Pg 199]</span> +sometimes true—with all the permutations of "always" and +"sometimes" that may occur. +</p> +<p> +There are in every language some words whose sole function is +to indicate form. These words, broadly speaking, are commonest +in languages having fewest inflections. Take "Socrates is +human." Here "is" is not a constituent of the proposition, +but merely indicates the subject-predicate form. Similarly +in "Socrates is earlier than Aristotle," "is" and "than" +merely indicate form; the proposition is the same as "Socrates +precedes Aristotle," in which these words have disappeared +and the form is otherwise indicated. Form, as a rule, <i>can</i> be +indicated otherwise than by specific words: the order of the +words can do most of what is wanted. But this principle +must not be pressed. For example, it is difficult to see how we +could conveniently express molecular forms of propositions +(<i>i.e.</i> what we call "truth-functions") without any word at all. +We saw in Chapter XIV. that one word or symbol is enough for +this purpose, namely, a word or symbol expressing <i>incompatibility</i>. +But without even one we should find ourselves in difficulties. +This, however, is not the point that is important for +our present purpose. What is important for us is to observe +that form may be the one concern of a general proposition, +even when no word or symbol in that proposition designates +the form. If we wish to speak about the form itself, we must +have a word for it; but if, as in mathematics, we wish to speak +about all propositions that have the form, a word for the form +will usually be found not indispensable; probably in theory it +is <i>never</i> indispensable. +</p> +<p> +Assuming—as I think we may—that the forms of propositions +can be represented by the forms of the propositions in which +they are expressed without any special word for forms, we should +arrive at a language in which everything formal belonged to +syntax and not to vocabulary. In such a language we could +express <i>all</i> the propositions of mathematics even if we did not +know one single word of the language. The language of mathematical +<span class="pagenum" id="Page_200">[Pg 200]</span> +logic, if it were perfected, would be such a language. +We should have symbols for variables, such as "<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x">" and "<img style="vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;" src="images/72.svg" alt="" data-tex="\mathrm R">" +and "<span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/57.svg" alt="" data-tex="y">,</span>" arranged in various ways; and the way of arrangement +would indicate that something was being said to be true of +all values or some values of the variables. We should not need +to know any words, because they would only be needed for giving +values to the variables, which is the business of the applied +mathematician, not of the pure mathematician or logician. +It is one of the marks of a proposition of logic that, given a +suitable language, such a proposition can be asserted in such a +language by a person who knows the syntax without knowing +a single word of the vocabulary. +</p> +<p> +But, after all, there are words that express form, such as "is" +and "than." And in every symbolism hitherto invented for +mathematical logic there are symbols having constant formal +meanings. We may take as an example the symbol for incompatibility +which is employed in building up truth-functions. +Such words or symbols may occur in logic. The question is: +How are we to define them? +</p> +<p> +Such words or symbols express what are called "logical +constants." Logical constants may be defined exactly as +we defined forms; in fact, they are in essence the same thing. +A fundamental logical constant will be that which is in common +among a number of propositions, any one of which can result +from any other by substitution of terms one for another. For +example, "Napoleon is greater than Wellington" results from +"Socrates is earlier than Aristotle" by the substitution of +"Napoleon" for "Socrates," "Wellington" for "Aristotle," +and "greater" for "earlier." Some propositions can be obtained +in this way from the prototype "Socrates is earlier than Aristotle" +and some cannot; those that can are those that are of +the form "<span class="nowrap"><img style="vertical-align: -0.464ex; width: 4.068ex; height: 2.009ex;" src="images/313.svg" alt="" data-tex="x\mathrm Ry">,</span>" <i>i.e.</i> express dual relations. We cannot obtain +from the above prototype by term-for-term substitution such +propositions as "Socrates is human" or "the Athenians gave +the hemlock to Socrates," because the first is of the subject-predicate +<span class="pagenum" id="Page_201">[Pg 201]</span> +form and the second expresses a three-term relation. +If we are to have any words in our pure logical language, they +must be such as express "logical constants," and "logical +constants" will always either be, or be derived from, what is in +common among a group of propositions derivable from each +other, in the above manner, by term-for-term substitution. And +this which is in common is what we call "form." +</p> +<p> +In this sense all the "constants" that occur in pure mathematics +are logical constants. The number 1, for example, is +derivative from propositions of the form: "There is a term <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c"> +such that <img style="vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;" src="images/300.svg" alt="" data-tex="\phi x"> is true when, and only when, <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/94.svg" alt="" data-tex="c">.</span>" This is a +function of <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi">,</span> and various different propositions result from +giving different values to <span class="nowrap"><img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi">.</span> We may (with a little omission +of intermediate steps not relevant to our present purpose) take +the above function of <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi"> as what is meant by "the class determined +by <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi"> is a unit class" or "the class determined by <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/306.svg" alt="" data-tex="\phi"> is a +member of 1" (1 being a class of classes). In this way, propositions +in which 1 occurs acquire a meaning which is derived from +a certain constant logical form. And the same will be found +to be the case with all mathematical constants: all are logical +constants, or symbolic abbreviations whose full use in a proper +context is defined by means of logical constants. +</p> +<p> +But although all logical (or mathematical) propositions can +be expressed wholly in terms of logical constants together with +variables, it is not the case that, conversely, all propositions +that can be expressed in this way are logical. We have found +so far a necessary but not a sufficient criterion of mathematical +propositions. We have sufficiently defined the character of the +primitive <i>ideas</i> in terms of which all the ideas of mathematics +can be <i>defined</i>, but not of the primitive <i>propositions</i> from which +all the propositions of mathematics can be <i>deduced</i>. This is a +more difficult matter, as to which it is not yet known what the +full answer is. +</p> +<p> +We may take the axiom of infinity as an example of a proposition +which, though it can be enunciated in logical terms, +<span class="pagenum" id="Page_202">[Pg 202]</span> +cannot be asserted by logic to be true. All the propositions of +logic have a characteristic which used to be expressed by saying +that they were analytic, or that their contradictories were self-contradictory. +This mode of statement, however, is not satisfactory. +The law of contradiction is merely one among logical +propositions; it has no special pre-eminence; and the proof +that the contradictory of some proposition is self-contradictory +is likely to require other principles of deduction besides the +law of contradiction. Nevertheless, the characteristic of logical +propositions that we are in search of is the one which was felt, +and intended to be defined, by those who said that it consisted +in deducibility from the law of contradiction. This characteristic, +which, for the moment, we may call <i>tautology</i>, obviously +does not belong to the assertion that the number of individuals +in the universe is <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n">,</span> whatever number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> may be. But for the +diversity of types, it would be possible to prove logically that +there are classes of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> terms, where <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> is any finite integer; or even +that there are classes of <img style="vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;" src="images/170.svg" alt="" data-tex="\aleph_{0}"> terms. But, owing to types, such +proofs, as we saw in Chapter XIII., are fallacious. We are left +to empirical observation to determine whether there are as many +as <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/46.svg" alt="" data-tex="n"> individuals in the world. Among "possible" worlds, +in the Leibnizian sense, there will be worlds having one, two, +three, ... individuals. There does not even seem any logical +necessity why there should be even one individual<a id="FNanchor_43_1"></a><a href="#Footnote_43_1" class="fnanchor">[43]</a>—why, in +fact, there should be any world at all. The ontological proof +of the existence of God, if it were valid, would establish the +logical necessity of at least one individual. But it is generally +recognised as invalid, and in fact rests upon a mistaken view of +existence—<i>i.e.</i> it fails to realise that existence can only be asserted +of something described, not of something named, so that it is +meaningless to argue from "this is the so-and-so" and "the +so-and-so exists" to "this exists." If we reject the ontological +<span class="pagenum" id="Page_203">[Pg 203]</span> +argument, we seem driven to conclude that the existence of a +world is an accident—<i>i.e.</i> it is not logically necessary. If that +be so, no principle of logic can assert "existence" except under +a hypothesis, <i>i.e.</i> none can be of the form "the propositional +function so-and-so is sometimes true." Propositions of this +form, when they occur in logic, will have to occur as hypotheses +or consequences of hypotheses, not as complete asserted propositions. +The complete asserted propositions of logic will all +be such as affirm that some propositional function is <i>always</i> true. +For example, it is always true that if <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> and <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/280.svg" alt="" data-tex="q"> implies <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r"> +then <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/136.svg" alt="" data-tex="p"> implies <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/164.svg" alt="" data-tex="r">,</span> or that, if all <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha">'</span>s are <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">'</span>s +and <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is an <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/63.svg" alt="" data-tex="\alpha"> then +<img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/56.svg" alt="" data-tex="x"> is a <span class="nowrap"><img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/64.svg" alt="" data-tex="\beta">.</span> Such propositions may occur in logic, and their truth +is independent of the existence of the universe. We may lay +it down that, if there were no universe, <i>all</i> general propositions +would be true; for the contradictory of a general proposition +(as we saw in Chapter XV.) is a proposition asserting existence, +and would therefore always be false if no universe existed. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_43_1"></a><a href="#FNanchor_43_1"><span class="label">[43]</span></a>The primitive propositions in <i>Principia Mathematica</i> are such as to +allow the inference that at least one individual exists. But I now view +this as a defect in logical purity.</p></div> + +<p> +Logical propositions are such as can be known <i>a priori</i>, without +study of the actual world. We only know from a study of +empirical facts that Socrates is a man, but we know the correctness +of the syllogism in its abstract form (<i>i.e.</i> when it is stated +in terms of variables) without needing any appeal to experience. +This is a characteristic, not of logical propositions in themselves, +but of the way in which we know them. It has, however, a +bearing upon the question what their nature may be, since there +are some kinds of propositions which it would be very difficult +to suppose we could know without experience. +</p> +<p> +It is clear that the definition of "logic" or "mathematics" +must be sought by trying to give a new definition of the old +notion of "analytic" propositions. Although we can no longer +be satisfied to define logical propositions as those that follow +from the law of contradiction, we can and must still admit that +they are a wholly different class of propositions from those that +we come to know empirically. They all have the characteristic +which, a moment ago, we agreed to call "tautology." This, +<span class="pagenum" id="Page_204">[Pg 204]</span> +combined with the fact that they can be expressed wholly in terms +of variables and logical constants (a logical constant being something +which remains constant in a proposition even when <i>all</i> +its constituents are changed)—will give the definition of logic +or pure mathematics. For the moment, I do not know how to +define "tautology."<a id="FNanchor_44_1"></a><a href="#Footnote_44_1" class="fnanchor">[44]</a> +It would be easy to offer a definition +which might seem satisfactory for a while; but I know of none +that I feel to be satisfactory, in spite of feeling thoroughly +familiar with the characteristic of which a definition is wanted. +At this point, therefore, for the moment, we reach the frontier +of knowledge on our backward journey into the logical foundations +of mathematics. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_44_1"></a><a href="#FNanchor_44_1"><span class="label">[44]</span></a>The importance of "tautology" for a definition of mathematics was +pointed out to me by my former pupil Ludwig Wittgenstein, who was +working on the problem. I do not know whether he has solved it, or even +whether he is alive or dead.</p></div> + +<p> +We have now come to an end of our somewhat summary introduction +to mathematical philosophy. It is impossible to convey +adequately the ideas that are concerned in this subject so long +as we abstain from the use of logical symbols. Since ordinary +language has no words that naturally express exactly what we +wish to express, it is necessary, so long as we adhere to ordinary +language, to strain words into unusual meanings; and the reader +is sure, after a time if not at first, to lapse into attaching the usual +meanings to words, thus arriving at wrong notions as to what is +intended to be said. Moreover, ordinary grammar and syntax +is extraordinarily misleading. This is the case, <i>e.g.</i>, as regards +numbers; "ten men" is grammatically the same form as +"white men," so that 10 might be thought to be an adjective +qualifying "men." It is the case, again, wherever propositional +functions are involved, and in particular as regards existence and +descriptions. Because language is misleading, as well as because +it is diffuse and inexact when applied to logic (for which it was +never intended), logical symbolism is absolutely necessary to +any exact or thorough treatment of our subject. Those readers, +<span class="pagenum" id="Page_205">[Pg 205]</span> +therefore, who wish to acquire a mastery of the principles of +mathematics, will, it is to be hoped, not shrink from the labour +of mastering the symbols—a labour which is, in fact, much less +than might be thought. As the above hasty survey must have +made evident, there are innumerable unsolved problems in the +subject, and much work needs to be done. If any student is +led into a serious study of mathematical logic by this little +book, it will have served the chief purpose for which it has been +written. +<span class="pagenum" id="Page_206">[Pg 206]</span> +</p> +</div> +<p><br><br><br></p> + +<div class='chapter'> +<h2 title='INDEX'><a id="INDEX"></a>INDEX</h2> +<p class="indx"> +Aggregates, <a href="#Page_12">12</a> +<br> +Alephs, <a href="#Page_83">83</a>, <a href="#Page_92">92</a>, <a href="#Page_97">97</a>, <a href="#Page_125">125</a> +<br> +Aliorelatives, <a href="#Page_32">32</a> +<br> +<i>All</i>, <a href="#Page_158">158</a> ff. +<br> +Analysis, <a href="#Page_4">4</a> +<br> +Ancestors, <a href="#Page_25">25</a>, <a href="#Page_33">33</a> +<br> +Argument of a function, <a href="#Page_47">47</a>, <a href="#Page_108">108</a> +<br> +Arithmetising of mathematics, <a href="#Page_4">4</a> +<br> +Associative law, <a href="#Page_58">58</a>, <a href="#Page_94">94</a> +<br> +Axioms, <a href="#Page_1">1</a> +<br> +<br> +Between, <a href="#Page_38">38</a> ff., <a href="#Page_58">58</a> +<br> +Bolzano, <a href="#Page_138">138</a> <i>n.</i> +<br> +Boots and socks, <a href="#Page_126">126</a> +<br> +Boundary, <a href="#Page_70">70</a>, <a href="#Page_98">98</a>, <a href="#Page_99">99</a> +<br> +<br> +Cantor, Georg, <a href="#Page_77">77</a>, <a href="#Page_79">79</a>, <a href="#Page_85">85</a> <i>n.</i>, <a href="#Page_86">86</a>, <a href="#Page_89">89</a>,<br> +<span style="margin-left: 1em;"><a href="#Page_95">95</a>, <a href="#Page_102">102</a>, <a href="#Page_136">136</a></span> +<br> +Classes, <a href="#Page_12">12</a>, <a href="#Page_137">137</a>, <a href="#Page_181">181</a> ff.;<br> +<span style="margin-left: 1em;">reflexive, <a href="#Page_80">80</a>, <a href="#Page_127">127</a>, <a href="#Page_138">138</a>;</span><br> +<span style="margin-left: 1em;">similar, <a href="#Page_15">15</a>, <a href="#Page_16">16</a></span> +<br> +Clifford, W. K., <a href="#Page_76">76</a> +<br> +Collections, infinite, <a href="#Page_13">13</a> +<br> +Commutative law, <a href="#Page_58">58</a>, <a href="#Page_94">94</a> +<br> +Conjunction, <a href="#Page_147">147</a> +<br> +Consecutiveness, <a href="#Page_37">37</a>, <a href="#Page_38">38</a>, <a href="#Page_81">81</a> +<br> +Constants, <a href="#Page_202">202</a> +<br> +Construction, method of, <a href="#Page_73">73</a> +<br> +Continuity, <a href="#Page_86">86</a>, <a href="#Page_97">97</a> ff.;<br> +<span style="margin-left: 1em;">Cantorian, <a href="#Page_102">102</a> ff.;</span><br> +<span style="margin-left: 1em;">Dedekindian, <a href="#Page_101">101</a> ff.;</span><br> +<span style="margin-left: 1em;">in philosophy, <a href="#Page_105">105</a>;</span><br> +<span style="margin-left: 1em;">of functions, <a href="#Page_106">106</a> ff.</span> +<br> +Contradictions, <a href="#Page_135">135</a> ff. +<br> +Convergence, <a href="#Page_115">115</a> +<br> +Converse, <a href="#Page_16">16</a>, <a href="#Page_32">32</a>, <a href="#Page_49">49</a> +<br> +Correlators, <a href="#Page_54">54</a> +<br> +Counterparts, objective, <a href="#Page_61">61</a> +<br> +Counting, <a href="#Page_14">14</a>, <a href="#Page_16">16</a> +<br> +<br> +Dedekind, <a href="#Page_69">69</a>, <a href="#Page_99">99</a>, <a href="#Page_138">138</a> <i>n.</i> +<br> +Deduction, <a href="#Page_144">144</a> ff. +<br> +Definition, <a href="#Page_3">3</a>;<br> +<span style="margin-left: 1em;">extensional and intensional, <a href="#Page_12">12</a></span> +<br> +Derivatives, <a href="#Page_100">100</a> +<br> +Descriptions, <a href="#Page_139">139</a>, <a href="#Page_144">144</a> +<br> +Descriptions, <a href="#Page_167">167</a> +<br> +Dimensions, <a href="#Page_29">29</a> +<br> +Disjunction, <a href="#Page_147">147</a> +<br> +Distributive law, <a href="#Page_58">58</a>, <a href="#Page_94">94</a> +<br> +Diversity, <a href="#Page_87">87</a> +<br> +Domain, <a href="#Page_16">16</a>, <a href="#Page_32">32</a>, <a href="#Page_49">49</a> +<br> +<br> +Equivalence, <a href="#Page_183">183</a> +<br> +Euclid, <a href="#Page_67">67</a> +<br> +Existence, <a href="#Page_164">164</a>, <a href="#Page_171">171</a>, <a href="#Page_177">177</a> +<br> +Exponentiation, <a href="#Page_94">94</a>, <a href="#Page_120">120</a> +<br> +Extension of a relation, <a href="#Page_60">60</a> +<br> +<br> +Fictions, logical, <a href="#Page_14">14</a> <i>n.</i>, <a href="#Page_45">45</a>, <a href="#Page_137">137</a> +<br> +Field of a relation, <a href="#Page_32">32</a>, <a href="#Page_53">53</a> +<br> +Finite, <a href="#Page_27">27</a> +<br> +Flux, <a href="#Page_105">105</a> +<br> +Form, <a href="#Page_198">198</a> +<br> +Fractions, <a href="#Page_37">37</a>, <a href="#Page_64">64</a> +<br> +Frege, <a href="#Page_7">7</a>, <a href="#Page_10">10</a>, <a href="#Page_25">25</a> <i>n.</i>, <a href="#Page_77">77</a>, <a href="#Page_95">95</a>, <a href="#Page_146">146</a> <i>n.</i> +<br> +Functions, <a href="#Page_46">46</a>;<br> +<span style="margin-left: 1em;">descriptive, <a href="#Page_46">46</a>, <a href="#Page_180">180</a>;</span><br> +<span style="margin-left: 1em;">intensional and extensional, <a href="#Page_186">186</a>;</span><br> +<span style="margin-left: 1em;">predicative, <a href="#Page_189">189</a>;</span><br> +<span style="margin-left: 1em;">propositional, <a href="#Page_46">46</a>, <a href="#Page_144">144</a>;</span><br> +<span style="margin-left: 1em;">propositional, <a href="#Page_155">155</a>;</span> +<br> +<br> +Gap, Dedekindian, <a href="#Page_70">70</a> ff., <a href="#Page_99">99</a> +<br> +Generalisation, <a href="#Page_156">156</a> +<br> +Geometry, <a href="#Page_29">29</a>, <a href="#Page_59">59</a>, <a href="#Page_67">67</a>, <a href="#Page_74">74</a>, <a href="#Page_100">100</a>, <a href="#Page_145">145</a>;<br> +<span style="margin-left: 1em;">analytical, <a href="#Page_4">4</a>, <a href="#Page_86">86</a></span> +<br> +Greater and less, <a href="#Page_65">65</a>, <a href="#Page_90">90</a> +<br> +<br> +Hegel, <a href="#Page_107">107</a> +<br> +Hereditary properties, <a href="#Page_21">21</a> +<br> +<br> +Implication, <a href="#Page_146">146</a>, <a href="#Page_153">153</a>;<br> +<span style="margin-left: 1em;">formal, <a href="#Page_163">163</a></span> +<br> +Incommensurables, <a href="#Page_4">4</a>, <a href="#Page_66">66</a> +<br> +Incompatibility, <a href="#Page_147">147</a> ff., <a href="#Page_200">200</a> +<br> +Incomplete symbols, <a href="#Page_182">182</a> +<br> +Indiscernibles, <a href="#Page_192">192</a> +<br> +Individuals, <a href="#Page_132">132</a>, <a href="#Page_141">141</a>, <a href="#Page_173">173</a> +<br> +Induction, mathematical, <a href="#Page_20">20</a> ff., <a href="#Page_87">87</a>, <a href="#Page_93">93</a>,<br> +<span style="margin-left: 1em;"><a href="#Page_185">185</a></span> +<br> +Inductive properties, <a href="#Page_21">21</a> +<br> +Inference, <a href="#Page_148">148</a> +<br> +Infinite, <a href="#Page_28">28</a>; of rationals, <a href="#Page_65">65</a>;<br> +<span style="margin-left: 1em;">Cantorian, <a href="#Page_65">65</a>;</span><br> +<span style="margin-left: 1em;">of cardinals, <a href="#Page_77">77</a> ff.;</span><br> +<span style="margin-left: 1em;">and series and ordinals, <a href="#Page_89">89</a> ff.</span> +<br> +Infinity, axiom of, <a href="#Page_66">66</a> <i>n.</i>, <a href="#Page_77">77</a>, <a href="#Page_131">131</a> ff.,<br> +<span style="margin-left: 1em;"><a href="#Page_202">202</a></span> +<br> +Instances, <a href="#Page_156">156</a> +<br> +Integers, positive and negative, <a href="#Page_64">64</a> +<br> +Intervals, <a href="#Page_115">115</a> +<br> +Intuition, <a href="#Page_145">145</a> +<br> +Irrationals, <a href="#Page_66">66</a>, <a href="#Page_72">72</a><br> +<span class="pagenum" id="Page_207">[Pg 207]</span> +<br> +<br> +Kant, <a href="#Page_145">145</a> +<br> +<br> +Leibniz, <a href="#Page_80">80</a>, <a href="#Page_107">107</a>, <a href="#Page_192">192</a> +<br> +Lewis, C. I., <a href="#Page_153">153</a>, <a href="#Page_154">154</a> +<br> +Likeness, <a href="#Page_52">52</a> +<br> +Limit, <a href="#Page_29">29</a>, <a href="#Page_69">69</a> ff., <a href="#Page_97">97</a> ff.;<br> +<span style="margin-left: 1em;">of functions, <a href="#Page_106">106</a> ff.</span> +<br> +Limiting points, <a href="#Page_99">99</a> +<br> +Logic, <a href="#Page_159">159</a>, <a href="#Page_65">65</a>, <a href="#Page_194">194</a> ff.;<br> +<span style="margin-left: 1em;">mathematical, <a href="#Page_v">v</a>, <a href="#Page_201">201</a>, <a href="#Page_206">206</a></span> +<br> +Logicising of mathematics, <a href="#Page_7">7</a> +<br> +<br> +Maps, <a href="#Page_52">52</a>, <a href="#Page_60">60</a> ff., <a href="#Page_80">80</a> +<br> +Mathematics, <a href="#Page_194">194</a> ff. +<br> +Maximum, <a href="#Page_70">70</a>, <a href="#Page_98">98</a> +<br> +Median class, <a href="#Page_104">104</a> +<br> +Meinong, <a href="#Page_169">169</a> +<br> +Method, <a href="#Page_vi">vi</a> +<br> +Minimum, <a href="#Page_70">70</a>, <a href="#Page_98">98</a> +<br> +Modality, <a href="#Page_165">165</a> +<br> +Multiplication, <a href="#Page_118">118</a> ff. +<br> +Multiplicative axiom, <a href="#Page_92">92</a>, <a href="#Page_117">117</a> ff. +<br> +<br> +Names, <a href="#Page_173">173</a>, <a href="#Page_182">182</a> +<br> +Necessity, <a href="#Page_165">165</a> +<br> +Neighbourhood, <a href="#Page_109">109</a> +<br> +Nicod, <a href="#Page_148">148</a>, <a href="#Page_149">149</a>, <a href="#Page_151">151</a> +<br> +Null-class, <a href="#Page_23">23</a>, <a href="#Page_132">132</a> +<br> +Number, cardinal, <a href="#Page_10">10</a> ff., <a href="#Page_56">56</a>, <a href="#Page_77">77</a> ff., <a href="#Page_95">95</a>;<br> +<span style="margin-left: 1em;">complex, <a href="#Page_74">74</a> ff.;</span><br> +<span style="margin-left: 1em;">finite, <a href="#Page_20">20</a> ff.;</span><br> +<span style="margin-left: 1em;">inductive, <a href="#Page_27">27</a>, <a href="#Page_78">78</a>, <a href="#Page_131">131</a>;</span><br> +<span style="margin-left: 1em;">infinite, <a href="#Page_77">77</a> ff.;</span><br> +<span style="margin-left: 1em;">irrational, <a href="#Page_66">66</a>, <a href="#Page_72">72</a>;</span><br> +<span style="margin-left: 1em;">maximum? <a href="#Page_135">135</a>;</span><br> +<span style="margin-left: 1em;">multipliable, <a href="#Page_130">130</a>;</span><br> +<span style="margin-left: 1em;">natural, <a href="#Page_2">2</a> ff., <a href="#Page_22">22</a>;</span><br> +<span style="margin-left: 1em;">non-inductive, <a href="#Page_88">88</a>, <a href="#Page_127">127</a>;</span><br> +<span style="margin-left: 1em;">real, <a href="#Page_66">66</a>, <a href="#Page_72">72</a>, <a href="#Page_84">84</a>;</span><br> +<span style="margin-left: 1em;">reflexive, <a href="#Page_80">80</a>, <a href="#Page_127">127</a>;</span><br> +<span style="margin-left: 1em;">relation, <a href="#Page_56">56</a>, <a href="#Page_94">94</a>;</span><br> +<span style="margin-left: 1em;">serial, <a href="#Page_57">57</a></span> +<br> +<br> +Occam, <a href="#Page_184">184</a> +<br> +Occurrences, primary and secondary,<br> +<span style="margin-left: 1em;"><a href="#Page_179">179</a></span> +<br> +Ontological proof, <a href="#Page_203">203</a> +<br> +Order 29ff.; cyclic, <a href="#Page_40">40</a> +<br> +Oscillation, ultimate, <a href="#Page_111">111</a> +<br> +<br> +<i>Parmenides</i>, <a href="#Page_138">138</a> +<br> +Particulars, <a href="#Page_140">140</a> ff., <a href="#Page_173">173</a> +<br> +Peano, <a href="#Page_5">5</a> ff., <a href="#Page_23">23</a>, <a href="#Page_24">24</a>, <a href="#Page_78">78</a>, <a href="#Page_81">81</a>, <a href="#Page_131">131</a>, <a href="#Page_163">163</a> +<br> +Peirce, <a href="#Page_32">32</a> <i>n.</i> +<br> +Permutations, <a href="#Page_50">50</a> +<br> +Philosophy, mathematical, <a href="#Page_v">v</a>, <a href="#Page_1">1</a> +<br> +Plato, <a href="#Page_138">138</a> +<br> +Plurality, <a href="#Page_10">10</a> +<br> +Poincaré, <a href="#Page_27">27</a> +<br> +Points, <a href="#Page_59">59</a> +<br> +Posterity, <a href="#Page_22">22</a> ff., <a href="#Page_32">32</a>; proper, <a href="#Page_36">36</a> +<br> +Postulates, <a href="#Page_71">71</a>, <a href="#Page_73">73</a> +<br> +Precedent, <a href="#Page_98">98</a> +<br> +Premisses of arithmetic, <a href="#Page_5">5</a> +<br> +Primitive ideas and propositions, <a href="#Page_5">5</a>, <a href="#Page_202">202</a> +<br> +Progressions, <a href="#Page_8">8</a>, <a href="#Page_81">81</a> ff. +<br> +Propositions, <a href="#Page_155">155</a>; analytic, <a href="#Page_204">204</a>;<br> +<span style="margin-left: 1em;">elementary, <a href="#Page_161">161</a></span> +<br> +Pythagoras, <a href="#Page_4">4</a>, <a href="#Page_67">67</a> +<br> +<br> +Quantity, <a href="#Page_97">97</a>, <a href="#Page_195">195</a> +<br> +<br> +Ratios, <a href="#Page_64">64</a>, <a href="#Page_71">71</a>, <a href="#Page_84">84</a>, <a href="#Page_133">133</a> +<br> +Reducibility, axiom of, <a href="#Page_191">191</a> +<br> +<i>Referent</i>, <a href="#Page_48">48</a> +<br> +Relation numbers, <a href="#Page_56">56</a> ff. +<br> +Relations, asymmetrical <a href="#Page_31">31</a>, <a href="#Page_42">42</a>;<br> +<span style="margin-left: 1em;">connected, <a href="#Page_32">32</a>;</span><br> +<span style="margin-left: 1em;">many-one, <a href="#Page_15">15</a>;</span><br> +<span style="margin-left: 1em;">one-many, <a href="#Page_15">15</a>, <a href="#Page_45">45</a>;</span><br> +<span style="margin-left: 1em;">one-one, <a href="#Page_15">15</a>, <a href="#Page_47">47</a>, <a href="#Page_79">79</a>;</span><br> +<span style="margin-left: 1em;">reflexive, <a href="#Page_16">16</a>;</span><br> +<span style="margin-left: 1em;">serial, <a href="#Page_34">34</a>;</span><br> +<span style="margin-left: 1em;">similar, <a href="#Page_52">52</a>;</span><br> +<span style="margin-left: 1em;">squares of, <a href="#Page_32">32</a>;</span><br> +<span style="margin-left: 1em;">symmetrical, <a href="#Page_16">16</a>, <a href="#Page_44">44</a>;</span><br> +<span style="margin-left: 1em;">transitive, <a href="#Page_16">16</a>, <a href="#Page_32">32</a></span> +<br> +<i>Relatum</i>, <a href="#Page_48">48</a> +<br> +Representatives, <a href="#Page_120">120</a> +<br> +Rigour, <a href="#Page_144">144</a> +<br> +Royce, <a href="#Page_80">80</a> +<br> +<br> +Section, Dedekindian, <a href="#Page_69">69</a> ff.;<br> +<span style="margin-left: 1em;">ultimate, <a href="#Page_111">111</a></span> +<br> +Segments, <a href="#Page_72">72</a>, <a href="#Page_98">98</a> +<br> +Selections, <a href="#Page_117">117</a> +<br> +Sequent, <a href="#Page_98">98</a> +<br> +Series, <a href="#Page_29">29</a> ff.; closed, <a href="#Page_103">103</a>;<br> +<span style="margin-left: 1em;">compact, <a href="#Page_66">66</a>, <a href="#Page_93">93</a>, <a href="#Page_100">100</a>;</span><br> +<span style="margin-left: 1em;">condensed in itself, <a href="#Page_102">102</a>;</span><br> +<span style="margin-left: 1em;">Dedekindian, <a href="#Page_71">71</a>, <a href="#Page_73">73</a>, <a href="#Page_101">101</a>;</span><br> +<span style="margin-left: 1em;">generation of, <a href="#Page_41">41</a>;</span><br> +<span style="margin-left: 1em;">infinite, <a href="#Page_89">89</a>;</span><br> +<span style="margin-left: 1em;">perfect, <a href="#Page_102">102</a>, <a href="#Page_103">103</a>;</span><br> +<span style="margin-left: 1em;">well-ordered, <a href="#Page_92">92</a>, <a href="#Page_123">123</a></span> +<br> +Sheffer, <a href="#Page_148">148</a> +<br> +Similarity, of classes, <a href="#Page_15">15</a> ff.;<br> +<span style="margin-left: 1em;">of relations, <a href="#Page_83">83</a>;</span><br> +<span style="margin-left: 1em;">of relations, <a href="#Page_52">52</a></span> +<br> +Some, <a href="#Page_158">158</a> ff. +<br> +Space, <a href="#Page_61">61</a>, <a href="#Page_86">86</a>, <a href="#Page_140">140</a> +<br> +Structure, <a href="#Page_60">60</a> ff. +<br> +Sub-classes, <a href="#Page_84">84</a> ff. +<br> +Subjects, <a href="#Page_142">142</a> +<br> +Subtraction, <a href="#Page_87">87</a> +<br> +Successor of a number, <a href="#Page_23">23</a>, <a href="#Page_35">35</a> +<br> +Syllogism, <a href="#Page_197">197</a> +<br> +<br> +Tautology, <a href="#Page_203">203</a>, <a href="#Page_205">205</a> +<br> +<i>The</i>, <a href="#Page_167">167</a>, <a href="#Page_172">172</a> ff. +<br> +Time, <a href="#Page_61">61</a>, <a href="#Page_86">86</a>, <a href="#Page_140">140</a> +<br> +Truth-function, <a href="#Page_147">147</a> +<br> +Truth-value, <a href="#Page_146">146</a> +<br> +Types, logical, <a href="#Page_53">53</a>, <a href="#Page_135">135</a> ff., <a href="#Page_185">185</a>, <a href="#Page_188">188</a> +<br> +<br> +Unreality, <a href="#Page_168">168</a> +<br> +<br> +Value of a function, <a href="#Page_47">47</a>, <a href="#Page_108">108</a> +<br> +Variables, <a href="#Page_10">10</a>, <a href="#Page_161">161</a>, <a href="#Page_199">199</a> +<br> +Veblen, <a href="#Page_58">58</a> +<br> +Verbs, <a href="#Page_141">141</a> +<br> +<br> +Weierstrass, <a href="#Page_97">97</a>, <a href="#Page_107">107</a> +<br> +Wells, H. G., <a href="#Page_114">114</a> +<br> +Whitehead, <a href="#Page_64">64</a>, <a href="#Page_76">76</a>, <a href="#Page_107">107</a>, <a href="#Page_119">119</a> +<br> +Wittgenstein, <a href="#Page_205">205</a> <i>n.</i> +<br> +<br> +Zermelo, <a href="#Page_123">123</a>, <a href="#Page_129">129</a> +<br> +Zero, <a href="#Page_65">65</a> +</p> +</div> +<p><br><br><br></p> + +<p class="center"> +PRINTED IN GREAT BRITAIN BY NEILL AND CO., LTD., EDINBURGH. +</p> + +<p><br><br></p> + +<div class="transnote"> +<p class="center"><b>TRANSCRIBER'S NOTES</b></p> + +<p> +Minor typographical corrections and presentational changes have been +made without comment. +</p> +<p> +This ebook was produced using OCR text generously provided by the +University of Toronto through the Internet Archive. +</p></div> + +<p><br><br><br></p> + +<div style='text-align:center'>*** END OF THE PROJECT GUTENBERG EBOOK 41654 ***</div> +</body> + +</html> + + diff --git a/41654-h/images/1.svg b/41654-h/images/1.svg new 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