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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/38430-8.txt b/38430-8.txt new file mode 100644 index 0000000..92c0875 --- /dev/null +++ b/38430-8.txt @@ -0,0 +1,3843 @@ +The Project Gutenberg EBook of The philosophy of B*rtr*nd R*ss*ll, by Various + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: The philosophy of B*rtr*nd R*ss*ll + +Author: Various + +Editor: Philip E. B. Jordain + +Release Date: December 28, 2011 [EBook #38430] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF B*RTR*ND R*SS*LL *** + + + + +Produced by Adrian Mastronardi and the Online Distributed +Proofreading Team at http://www.pgdp.net (This file was +produced from images generously made available by The +Internet Archive/Canadian Libraries) + + + + + + + + + + THE PHILOSOPHY OF + MR. B*RTR*ND R*SS*LL + + WITH AN APPENDIX OF LEADING + PASSAGES FROM CERTAIN OTHER WORKS + + EDITED BY + + PHILIP E. B. JOURDAIN + + [Illustration] + + LONDON: GEORGE ALLEN & UNWIN LTD. + RUSKIN HOUSE 40 MUSEUM STREET, W.C. 1 + CHICAGO: THE OPEN COURT PUBLISHING CO. + + + + _First published in 1918_ + + (_All rights reserved_) + + + + +EDITOR'S NOTE + + +When Mr. B*rtr*nd R*ss*ll, following the advice of Mr. W*ll**m J*m*s, +again "got into touch with reality" and in July 1911 was torn to pieces +by Anti-Suffragists, many of whom were political opponents of Mr. +R*ss*ll and held strong views on the Necessity of Protection of Trade +and person, a manuscript which was almost ready for the press was +fortunately saved from the flames on the occasion when a body of eager +champions of the Sacredness of Personal Property burnt the late Mr. +R*ss*ll's house. This manuscript, together with some further fragments +found in the late Mr. R*ss*ll's own interleaved copy of his _Prayer-Book +of Free Man's Worship_, which was fortunately rescued with a few of the +great author's other belongings, was first given to the world in the +_Monist_ for October 1911 and January 1916, and has here been arranged +and completed by some other hitherto undecipherable manuscripts. The +title of the above-mentioned _Prayer-Book_, it may perhaps be mentioned, +was apparently suggested to Mr. R*ss*ll by that of the Essay on +"The Free Man's Worship" in the _Philosophical Essays_ (London, 1910, +pp. 59-70[1]) of Mr. R*ss*ll's distinguished contemporary, Mr. Bertrand +Russell, from whom much of Mr. R*ss*ll's philosophy was derived. And, +indeed, the influence of Mr. Russell extended even beyond philosophical +views to arrangement and literary style. The method of arrangement of +the present work seems to have been borrowed from Mr. Russell's +_Philosophy of Leibniz_ of 1900; in the selection of subjects dealt +with, Mr. R*ss*ll seems to have been guided by Mr. Russell's _Principles +of Mathematics_ of 1903; while Mr. R*ss*ll's literary style fortunately +reminds us more of Mr. Russell's later clear and charming subtleties +than his earlier brilliant and no less subtle obscurities. But, on the +other hand, some important points of Mr. Russell's doctrine, which first +appeared in books published after Mr. R*ss*ll's death, were anticipated +in Mr. R*ss*ll's notes, and these anticipations, so interesting for +future historians of philosophy, have been provided by the editor with +references to the later works of Mr. Russell. All editorial notes are +enclosed in square brackets, to indicate that they were not written by +the late Mr. R*ss*ll. + +At the present time we have come to take a calm view of the question so +much debated seven years ago as to the legitimacy of logical arguments +in political discussions. No longer, fortunately, can that intense +feeling be roused which then found expression in the famous cry, +"Justice--right or wrong," and which played such a large part in the +politics of that time. Thus it will not be out of place in this +unimpassioned record of some of the truths and errors in the world to +refer briefly to Mr. R*ss*ll's short and stormy career. Before he was +torn to pieces, he had been forbidden to lecture on philosophy or +mathematics by some well-intentioned advocates of freedom in speech who +thought that the cause of freedom might be endangered by allowing Mr. +R*ss*ll to speak freely on points of logic, on the grounds, apparently, +that logic is both harmful and unnecessary and might be applied to +politics unless strong measures were taken for its suppression. On much +the same grounds, his liberty was taken from him by those who remarked +that, if necessary, they would die in defence of the sacred principle of +liberty; and it was in prison that the greater part of the present work +was written. Shortly after his liberation, which, like all actions of +public bodies, was brought about by the combined honour and interests of +those in authority, occurred his lamentable death to which we have +referred above. + +Mr. R*ss*ll maintained that the chief use of "implication" in politics +is to draw conclusions, which are thought to be true, and which are +consequently false, from identical propositions, and we can see these +views expressed in Chapters III and XIX of the present work. These +chapters were apparently written before the Government, in the spring +of 1910, arrived at the famous secret decision that only "certain +implications" are permitted in discussion. Naturally the secret decision +gave rise to much speculation among logicians as to which kinds of +implication were barred, and Mr. R*ss*ll and Mr. Bertrand Russell had +many arguments on the subject, which naturally could not be published at +the time. However, after Mr. R*ss*ll's death, successive prosecutions +which were made by the Government at last made it quite clear that the +opinion held by Mr. R*ss*ll was the correct one. There had been numerous +prosecutions of people who, from true but not identical premisses, had +deduced true conclusions, so that the possible legitimate forms of +"implication" were reduced. Further, the other doubtful cases were +cleared up in course of time by the prosecution of (1) members of the +Aristotelian Society for deducing true conclusions from false premisses; +(2) members of the _Mind_ Association for deducing false conclusions +from false premisses; and also by the attempted prosecution of an +eminent lady for deducing true conclusions from identities. Fortunately +this lady was able to defend herself successfully by pleading that one +eminent philosopher believed them to be true--which, of course, means +that the conclusions are false. Thus appeared the true nature of +legitimate political arguments. + +FOOTNOTES: + +[1] [This Essay is also reprinted in Mr. Russell's _Mysticism and +Logic_, London and New York, 1918, pp. 46-57.--ED.] + + + + + "Even a joke should have some meaning...." + + (The Red Queen, _T. L. G._, p. 105). + + + + +CONTENTS + + + PAGE + EDITOR'S NOTE 3 + ABBREVIATIONS 9 + CHAPTER + I. THE INDEFINABLES OF LOGIC 11 + II. OBJECTIVE VALIDITY OF THE "LAWS OF THOUGHT" 15 + III. IDENTITY 16 + IV. IDENTITY OF CLASSES 18 + V. ETHICAL APPLICATIONS OF THE LAW OF IDENTITY 19 + VI. THE LAW OF CONTRADICTION IN MODERN LOGIC 21 + VII. SYMBOLISM AND MEANING 22 + VIII. NOMINALISM 24 + IX. AMBIGUITY AND SYMBOLIC LOGIC 26 + X. LOGICAL ADDITION AND THE UTILITY OF SYMBOLISM 27 + XI. CRITICISM 29 + XII. HISTORICAL CRITICISM 30 + XIII. IS THE MIND IN THE HEAD? 31 + XIV. THE PRAGMATIST THEORY OF TRUTH 32 + XV. ASSERTION 34 + XVI. THE COMMUTATIVE LAW 35 + XVII. UNIVERSAL AND PARTICULAR PROPOSITIONS 36 + XVIII. DENIAL OF GENERALITY AND GENERALITY OF DENIAL 37 + XIX. IMPLICATION 39 + XX. DIGNITY 43 + XXI. THE SYNTHETIC NATURE OF DEDUCTION 45 + XXII. THE MORTALITY OF SOCRATES 48 + XXIII. DENOTING 53 + XXIV. THE 54 + XXV. NON-ENTITY 56 + XXVI. IS 58 + XXVII. AND AND OR 59 + XXVIII. THE CONVERSION OF RELATIONS 60 + XXIX. PREVIOUS PHILOSOPHICAL THEORIES OF MATHEMATICS 61 + XXX. FINITE AND INFINITE 63 + XXXI. THE MATHEMATICAL ATTAINMENTS OF TRISTRAM SHANDY 64 + XXXII. THE HARDSHIPS OF A MAN WITH AN UNLIMITED INCOME 66 + XXXIII. THE RELATIONS OF MAGNITUDE OF CARDINAL NUMBERS 69 + XXXIV. THE UNKNOWABLE 70 + XXXV. MR. SPENCER, THE ATHANASIAN CREED, AND THE ARTICLES 73 + XXXVI. THE HUMOUR OF MATHEMATICIANS 74 + XXXVII. THE PARADOXES OF LOGIC 75 + XXXVIII. MODERN LOGIC AND SOME PHILOSOPHICAL ARGUMENTS 79 + XXXIX. THE HIERARCHY OF JOKES 81 + XL. THE EVIDENCE OF GEOMETRICAL PROPOSITIONS 83 + XLI. ABSOLUTE AND RELATIVE POSITION 84 + XLII. LAUGHTER 86 + XLIII. "GEDANKENEXPERIMENTE" AND EVOLUTIONARY ETHICS 88 + APPENDIXES 89 + + + + +ABBREVIATIONS + + + _A. A. W._ Lewis Carroll: _Alice's Adventures in Wonderland_, London, + 1908. [This book was first published much earlier, but + this was the edition used by Mr. R*ss*ll. The same applies + to _H. S._ and _T. L. G._] + + _A. C. P._ John Henry Blunt (ed. by): _The Annotated Book of Common + Prayer_, London, new edition, 1888. + + _A. d. L._ Ernst Schröder: _Vorlesungen über die Algebra der Logik, + Leipzig_, vol. i., 1890; vol. ii. (two parts), 1891 and + 1905; vol. iii.: _Algebra und Logik der Relative_, 1895. + + _E. N._ Richard Dedekind: _Essays on the Theory of Numbers_, + Chicago and London, 1901. + + _E. L. L._ William Stanley Jevons: _Elementary Lessons in Logic, + Deductive and Inductive. With copious Questions and + Examples, and a Vocabulary of Logical Terms_, London, + 24th ed., 1907 [first published in 1870]. + + _E. u. I._ Ernst Mach: _Erkenntnis und Irrtum: Skizzen zur + Psychologie der Forschung_, Leipzig, 1906. + + _F. L._ Augustus De Morgan: _Formal Logic: or The Calculus of + Inference, Necessary and Probable_, London, 1847. + + _Fm. L._ John Neville Keynes: _Studies and Exercises in Formal + Logic_, 4th ed., London, 1906. + + _Gg._ Gottlob Frege: _Grundgesetze der Arithmetik + begriffschriftlich abgeleitet_, Jena, vol. i., 1893; + vol. ii., 1903. + + _Gl._ Gottlob Frege: _Die Grundlagen der Arithmetik: eine + logisch-mathematische Untersuchung über den Begriff der + Zahl_, Breslau, 1884. + + _G. u. E._ G. Heymans: _Die Gesetze und Elemente des + wisenschaftlichen Denkens_, Leiden, vol. i., 1890; + vol. ii., 1894. + + _H. J._ _The Hibbert Journal: a Quarterly Review of Religion, + Theology and Philosophy_, London and New York. + + _H. S._ Lewis Carroll: _The Hunting of the Snark: an Agony in + Eight Fits_, London, 1911. + + _M._ _The Monist: a Quarterly Magazine Devoted to Science and + Philosophy_, Chicago and London. + + _Md._ _Mind: a Quarterly Review of Psychology and Philosophy_, + London and New York. + + _Pa. Ma._ Alfred North Whitehead and Bertrand Russell: _Principia + Mathematica_, vol. i., Cambridge, 1910. [Other volumes + were published in 1912 and 1913.] + + _P. E._ Bertrand Russell: _Philosophical Essays_, London and New + York, 1910. + + _Ph. L._ Bertrand Russell: _A Critical Exposition of the Philosophy + of Leibniz, with an Appendix of Leading Passages_, + Cambridge, 1900. + + _P. M._ Bertrand Russell: _The Principles of Mathematics_, + vol. i., Cambridge, 1903. + + _R. M. M._ _Revue de Métaphysique et de Morale_, Paris. + + _S. B._ Lewis Carroll: _Sylvie and Bruno_, London, 1889. + + _S. L._ John Venn: _Symbolic Logic_, London, 1881; 2nd ed., 1894. + + _S. o. S._ William Stanley Jevons: _The Substitution of Similars, the + True Principle of Reasoning derived from a Modification of + Aristotle's Dictum_, London, 1869. + + _T. L. G._ Lewis Carroll: _Through the Looking-Glass, and what Alice + found there_, London, 1911. + + _Z. S._ Gottlob Frege: _Ueber die Zahlen des Herrn H. Schubert_, + Jena, 1899. + + + + +CHAPTER I + +THE INDEFINABLES OF LOGIC + + +The view that the fundamental principles of logic consist solely of the +law of identity was held by Leibniz,[2] Drobisch, Uberweg,[3] and +Tweedledee. Tweedledee, it may be remembered,[4] remarked that certain +identities "are" logic. Now, there is some doubt as to whether he, like +Jevons,[5] understood "are" to mean what mathematicians mean by "=," or, +like Schröder[6] and most logicians, to have the same meaning as the +relation of subsumption. The first alternative alone would justify our +contention; and we may, I think, conclude from an opposition to +authority that may have been indicated by Tweedledee's frequent use of +the word "contrariwise" that he did not follow the majority of +logicians, but held, like Jevons,[7] the mistaken[8] view that the +quantification of the predicate is relevant to symbolic logic. + +It may be mentioned, by the way, that it is probable that +Humpty-Dumpty's "is" is the "is" of identity. In fact, it is not +unlikely that Humpty-Dumpty was a Hegelian; for, although his ability +for clear explanation may seem to militate against this, yet his +inability to understand mathematics,[9] together with his synthesis of a +cravat and a belt, which usually serve different purposes,[10] and his +proclivity towards riddles seem to make out a good case for those who +hold that he was in fact a Hegelian. Indeed, riddles are very closely +allied to puns, and it was upon a pun, consisting of the confusion of +the "is" of predication with the "is" of identity--so that, for example, +"Socrates" was identified with "mortal" and more generally the +particular with the universal--that Hegel's system of philosophy was +founded.[11] But the question of Humpty-Dumpty's philosophical opinions +must be left for final verification to the historians of philosophy: +here I am only concerned with an _a priori_ logical construction of what +his views might have been if they formed a consistent whole.[12] + +If the principle of identity were indeed the sole principle of logic, +the principles of logic could hardly be said to be, as in fact they are, +a body of propositions whose consistency it is impossible to prove.[13] +This characteristic is important and one of the marks of the greatest +possible security. For example, while a great achievement of late years +has been to prove the consistency of the principles of arithmetic, a +science which is unreservedly accepted except by some empiricists,[14] +it can be proved formally that one foundation of arithmetic is +shattered.[15] It is true that, quite lately, it has been shown that +this conclusion may be avoided, and, by a re-moulding of logic, we can +draw instead the paradoxical conclusion that the opinions held by +common-sense for so many years are, in part, justified. But it is quite +certain that, with the principles of logic, no such proof of +consistency, and no such paradoxical result of further investigations is +to be feared. + +Still, this re-moulding has had the result of bringing logic into a +fuller agreement with common-sense than might be expected. There were +only two alternatives: if we chose principles in accordance with +common-sense, we arrived at conclusions which shocked common-sense; by +starting with paradoxical principles, we arrived at ordinary +conclusions. Like the White Knight, we have dyed our whiskers an unusual +colour and then hidden them.[16] + +The quaint name of "Laws of Thought," which is often applied to the +principles of Logic, has given rise to confusion in two ways: in the +first place, the "Laws," unlike other laws, cannot be broken, even in +thought; and, in the second place, people think that the "Laws" have +something to do with holding for the operations of their minds, just as +laws of nature hold for events in the world around us.[17] But that the +laws are not psychological laws follows from the facts that a thing may +be true even if nobody believes it, and something else may be false if +everybody believes it. Such, it may be remarked, is usually the case. + +Perhaps the most frequent instance of the assumption that the laws of +logic are mental is the treatment of an identity as if its validity were +an affair of our permission. Some people suggest to others that they +should "let bygones be bygones." Another important piece of evidence +that the truth of propositions has nothing to do with mind is given by +the phrase "it is morally certain that such-and-such a proposition is +true." Now, in the first place, morality, curiously enough, seems to be +closely associated with mental acts: we have professorships and +lectureships of, and examinations in, "mental and moral philosophy." In +the second place, it is plain that a "morally certain" proposition is a +highly doubtful one. Thus it is as vain to expect any information about +our minds from a study of the "Laws of Thought" as it would be to expect +a description of a certain social event from Miss E. E. C. Jones's book +_An Introduction to General Logic_. + +Fortunately, the principles or laws of Logic are not a matter of +philosophical discussion. Idealists like Tweedledum and Tweedledee, and +even practical idealists like the White Knight, explicitly accept laws +like the law of identity and the excluded middle.[18] In fact, +throughout all logic and mathematics, the existence of the human or any +other mind is totally irrelevant; mental processes are studied by means +of logic, but the subject-matter of logic does not presuppose mental +processes, and would be equally true if there were no mental processes. +It is true that, in that case, we should not know logic; but our +knowledge must not be confounded with the truths which we know.[19] An +apple is not confused with the eating of it except by savages, +idealists, and people who are too hungry to think. + +FOOTNOTES: + +[2] Russell, _Ph. L._, pp. 17, 19, 207-8. + +[3] Schröder, _A. d. L._, i. p. 4. + +[4] See Appendix A. This Appendix also illustrates the importance +attached to the Principle of Identity by the Professor and Bruno. + +[5] _S. o. S._, pp. 9-15. + +[6] _A. d. L._, i. p. 132. + +[7] Cf., besides the reference in the last note but one, _E. L. L._, pp. +183, 191. "Contrariwise," it may be remarked, is not a term used in +traditional logic. + +[8] _S. L._, 1881, pp. 173-5, 324-5; 1894, pp. 194-6. + +[9] Cf. Appendix C, and William Robertson Smith, "Hegel and the +Metaphysics of the Fluxional Calculus," _Trans. Roy. Soc., Edinb._, vol. +xxv., 1869, pp. 491-511. + +[10] See Appendix B. + +[11] [This is a remarkable anticipation of the note on pp. 39-40 of Mr. +Russell's book, published about three years after the death of Mr. +R*ss*ll, and entitled _Our Knowledge of the External World as a Field +for Scientific Method in Philosophy_, Chicago and London, 1914.--ED.] + +[12] Cf. _Ph. L._, pp. v.-vi. 3. + +[13] Cf. Pieri, _R. M. M._, March 1906, p. 199. + +[14] As a type of these, Humpty-Dumpty, with his inability to admit +anything not empirically given and his lack of comprehension of pure +mathematics, may be taken (see Appendix C). In his (correct) thesis that +definitions are nominal, too, Humpty-Dumpty reminds one of J. S. Mill +(see Appendix D). + +[15] See Frege, _Gg._, ii. p. 253. + +[16] See Appendix E. + +[17] See Frege, _Gg._, i. p. 15. + +[18] See the above references and also Appendix F. + +[19] Cf. B. Russell, _H. J._, July 1904, p. 812. + + + + +CHAPTER II + +OBJECTIVE VALIDITY OF THE "LAWS OF THOUGHT" + + +I once inquired of a maid-servant whether her mistress was at home. She +replied, in a doubtful fashion, that she _thought_ that her mistress was +in unless she was out. I concluded that the maid was uncertain as to the +objective validity of the law of excluded middle, and remarked that to +her mistress. But since I used the phrase "laws of thought," the +mistress perhaps supposed that a "law of thought" has something to do +with thinking and seemed to imagine that I wished to impute to the maid +some moral defect of an unimportant nature. Thus she remonstrated with +me in an amused way, since she probably imagined that I meant to find +fault with the maid's capacity for thinking. + + + + +CHAPTER III + +IDENTITY + + +In the first chapter we have noticed the opinion that identities are +fundamental to all logic. We will now consider some other views of the +value of identities. + +Identities are frequently used in common life by people who seem to +imagine that they can draw important conclusions respecting conduct or +matters of fact from them. I have heard of a man who gained the double +reputation of being a philosopher and a fatalist by the repeated +enunciation of the identity "Whatever will be, will be"; and the Italian +equivalent of this makes up an appreciable part of one of Mr. Robert +Hichens' novels. Further, the identity "Life is Life" has not only been +often accepted as an explanation for a particular way of living but has +even been considered by an authoress who calls herself "Zack" to be an +appropriate title for a novel; while "Business is Business" is +frequently thought to provide an excuse for dishonesty in trading, for +which purpose it is plainly inadequate. + +Another example is given by a poem of Mr. Kipling, where he seems to +assert that "East is East" and "West is West" imply that "never the +twain shall meet." The conclusion, now, is false; for, since the world +is round--as geography books still maintain by arguments which strike +every intelligent child as invalid[20]--what is called the "West" does, +in fact, merge into the "East." Even if we are to take the statement +metaphorically, it is still untrue, as the Japanese nation has shown. + +The law-courts are often rightly blamed for being strenuous opponents of +the spread of modern logic: the frequent misuse of _and_, _or_, _the_, +and _provided that_ in them is notorious. But the fault seems partly to +lie in the uncomplicated nature of the logical problems which are dealt +with in them. Thus it is no uncommon thing for somebody to appear there +who is unable to establish his own identity, or for A to assert that B +was "not himself" when he made a will leaving his money to C. + +The chief use of identities is in implication. Since, in logic, we so +understand _implication_[21] that any true proposition implies and is +implied by any other true proposition; if one is convinced of the truth +of the proposition Q, it is advisable to choose one or more identities +P, whose truth is undoubted, and say that P implies Q. Thus, Mr. Austen +Chamberlain, according to _The Times_ of March 27, 1909, professed to +deduce the conclusion that it is not right that women should have votes +from the premisses that "man is man" and "woman is woman." This method +requires that one should have made up one's mind about the conclusion +before discovering the premisses--by what, no doubt, Jevons would call +an "inverse or inductive method." Thus the method is of use only in +speeches and in giving good advice. + +Mr. Austen Chamberlain afterwards rather destroyed one's belief in the +truth of his premisses by putting limits to the validity of the +principle of identity. In the course of the Debate on the Budget of +1909, he maintained, against Mr. Lloyd George, that a joke was a joke +except when it was an untruth: Mr. Lloyd George, apparently, being of +the plausible opinion that a joke is a joke under all circumstances. + +FOOTNOTES: + +[20] The argument about the hull of a ship disappearing first is not +convincing, since it would equally well prove that the surface of the +earth was, for example, corrugated on a large scale. If the common-sense +of the reader were supposed to dismiss the possibility of water clinging +to such corrugations, it might equally be supposed to dismiss the +possibility of water clinging to a spherical earth. Traditional +geography books, no doubt, gave rise to the opinions held by Lady Blount +and the Zetetic Society. + +[21] The subject of Implication will be further considered in Chapter +XIX. + + + + +CHAPTER IV + +IDENTITY OF CLASSES + + +I once heard of a meritorious lady who was extremely conventional; on +the slender grounds of carefully acquired habits of preferring the word +"woman" to the word "lady" and of going to the post-office without a +hat, imagined that she was unconventional and altogether a remarkable +person; and who once remarked with great satisfaction that she was a +"very queer person," and that nothing shocked her "except, of course, +bad form." + +Thus, she asserted that all the things which shocked her were actions in +bad form; and she would undoubtedly agree, though she did not actually +state it, that all the things which were done in bad form would shock +her. Consequently she asserted that the class of things which shocked +her was the class of actions in bad form. Consequently the statement of +this lady that some or all of the actions done in bad form shocked her +is an identical proposition of the form "nothing shocks me, except, of +course, the things which do, in fact, shock me"; and this statement the +lady certainly did not intend to make. + +This excellent lady, had she but known it, was logically justified in +making any statement whatever about her unconventionality. For the class +of her unconventional actions was the null class. Thus she might +logically have made inconsistent statements about this class of actions. +As a matter of fact she did make inconsistent statements, but +unfortunately she justified them by stating that, "It is the privilege +of woman to be inconsistent." She was one of those persons who say +things like that. + + + + +CHAPTER V + +ETHICAL APPLICATIONS OF THE LAW OF IDENTITY + + +It may be remembered that Mr. Podsnap remarked, with sadness tempered by +satisfaction, that he regretted to say that "Foreign nations do as they +do do." Besides aiding the comforting expression of moral disapproval, +the law of identity has yet another useful purpose in practical ethics: +It serves the welcome purpose of providing an excuse for infractions of +the moral law. There was once a man who treated his wife badly, was +unfaithful to her, was dishonest in business, and was not particular in +his use of language; and yet his life on earth was described in the +lines: + + This man maintained a wife's a wife, + Men are as they are made, + Business is business, life is life; + And called a spade a spade. + +One of the objects of Dr. G. E. Moore's _Principia Ethica_[22] was to +argue that the word "good" means simply good, and not pleasant or +anything else. Appropriately enough, this book bore on its title-page +the quotation from the preface to the _Sermons_, published in 1726, of +Bishop Joseph Butler, the author of the _Analogy_: "Everything is what +it is and not another thing." + +But another famous Butler--Samuel Butler, the author of _Hudibras_--went +farther than this, and maintained that identities were the highest +attainment of metaphysics itself. At the beginning of the first Canto of +_Hudibras_, in the description of Hudibras himself, Butler wrote: + + He knew what's what, and that's as high + As metaphysic wit can fly. + +I once conducted what I imagined to be an æsthetic investigation for the +purpose of discovery, by the continual use of the word "Why?"[23] the +grounds upon which certain people choose to put milk into a tea-cup +before the tea. I was surprised to discover that it was an ethical, and +not an æsthetic problem; for I soon elicited the fact that it was done +because it was "right." A continuance of my patient questioning elicited +further evidence of the fundamental character of the principle of +identity in ethics; for it was right, I learned, because "right is +right." + +It appears that some people unconsciously think that the principle of +identity is the foundation, in certain religions, of the reasons which +can be alleged for moral conduct, and are surprised when this fact is +pointed out to them. The late Sir Leslie Stephen, when travelling by +railway, fell into conversation with an officer of the Salvation Army, +who tried hard to convert him. Failing in this laudable endeavour, the +Salvationist at last remarked: "But if you aren't saved, you can't go to +heaven!" "That, my friend," replied Stephen, "is an identical +proposition." + +FOOTNOTES: + +[22] Cambridge, 1903. + +[23] Cf. _P. E._, p. 2. + + + + +CHAPTER VI + +THE LAW OF CONTRADICTION IN MODERN LOGIC + + +Considering the important place assigned by philosophers and logicians +to the law of contradiction, the remark will naturally be resented by +many of the older schools of philosophy, and especially by Kantians, +that "in spite of its fame we have found few occasions for its use."[24] +Also in modern times, Benedetto Croce, an opponent of both traditional +logic and mathematical logic, began the preface of the book of 1908 on +Logic[25] by saying that that volume "is and is not" a certain memoir of +his which had been published in 1905. + +FOOTNOTES: + +[24] _Pa. Ma._, p. 116. + +[25] [English translation of the third Italian edition by Douglas +Ainslie, under the title: _Logic as the Science of the Pure Concept_, +London 1917.--ED.] + + + + +CHAPTER VII + +SYMBOLISM AND MEANING + + +When people write down any statement such as "The curfew tolls the knell +of parting day,"[26] which we will call "C" for shortness, what they +mean is not "C" but the _meaning_ of "C"; and not "the meaning of 'C'" +but the _meaning_ of "the meaning of 'C'." And so on, _ad infinitum_. +Thus, in writing or in speech, we always fail to state the meaning of +any proposition whatever. Sometimes, indeed, we succeed in _conveying_ +it; but there is danger in too great a disregard of statement and +preoccupation with conveyance of meaning. Thus many mathematicians have +been so anxious to convey to us a perfectly distinct and unmetaphysical +concept of number that they have stripped away from it everything that +they considered unessential (like its logical nature) and have finally +delivered it to us as a mere _sign_. By the labours of Helmholtz, +Kronecker, Heine, Stolz, Thomae, Pringsheim, and Schubert, many people +were persuaded that, when they said "'2' is a number" they were speaking +the truth, and hold that "Paris" is a town containing the letter "P." +When Frege pointed out[27] this difficulty he was almost universally +denounced in Germany as "_spitzfindig_." In fact, Germans seem to have +been influenced perhaps by that great contemner of "_Spitzfindigkeit_," +Kant, to reject the White Knight's[28] distinctions between words and +their denotations and to regard subtlety with disfavour to such a degree +that their only mathematical logician except Frege, namely Schröder--the +least subtle of mortals, by the way--seems to have been filled with such +fear of being thought subtle, that he made his books so prolix that +nobody has read them. + +Another term which, as we shall see when discussing the paradoxes of +logic, mathematicians are accustomed to apply to thought which is more +exact than any to which they are accustomed is "scholastic."[29] By +this, I suppose, they mean that the pursuits of certain acute people of +the Middle Ages are unimportant in contrast with the great achievements +of modern thought, as exemplified by a method of making plausible +guesses known as induction,[30] the bicycle, and the gramophone--all of +them instruments of doubtful merit. + +FOOTNOTES: + +[26] Cf. _Md_, N. S., vol. xiv., October 1905, p. 486. + +[27] In _Z. S._, for example. + +[28] See Appendix G. + +[29] Cf. Chapter XXXVII below. + +[30] Cf. _P. M._, p. 11, note. + + + + +CHAPTER VIII + +NOMINALISM + + +De Morgan[31] said that, "if all mankind had spoken one language, we +cannot doubt that there would have been a powerful, perhaps universal, +school of philosophers who would have believed in the inherent connexion +between names and things; who would have taken the sound _man_ to be the +mode of agitating the air which is essentially communicative of the +ideas of reason, cookery, bipedality, etc.... 'The French,' said the +sailor, 'call a cabbage a _shoe_; the fools! Why can't they call it a +cabbage, when they must know it is one?'" + +One of the chief differences between logicians and men of letters is +that the latter mean many different things by one word, whereas the +former do not--at least nowadays. Most mathematicians belong to the +class of men of letters. + +I once had a manservant who told me on a certain occasion that he "never +thought a word about it." I was doubtful whether to class him with such +eminent mathematicians as are mentioned in the last chapter, or as a +supporter of Max Müller's theory of the identity of thought and +language. However, since the man was very untruthful, and he told me +that he meant what he said and said what he meant,[32] the conclusion is +probably correct that he really believed that the meanings of his words +were not the words themselves. Thus I think it most probable that my +manservant had been a mathematician but had escaped by the aid of logic. + +As regards his remark that he meant what he said and said what he +meant, he plainly wished to pride himself on certain virtues which he +did not possess, and was not indifferent to applause, which, however, +was never evoked. The virtues, if so they be, and the applause were +withheld for other reasons than that the above statements are either +nonsensical or false. Suppose that "I say what I mean" expresses a +truth. What I say (or write) is always a symbol--words (or marks); and +what I mean by the symbol is the meaning of the symbol and not the +symbol itself. So the remark cannot express a truth, any more than the +name "Wellington" won the battle of Waterloo. + +FOOTNOTES: + +[31] _F. L._, pp. 246-7. + +[32] The Hatter (see Appendix H) pointed out that there is a difference +between these two assertions. Thus, he clearly showed that he was a +nominalist, and philosophically opposed to the March Hare who had +recommended Alice to say what she meant. + + + + +CHAPTER IX + +AMBIGUITY AND SYMBOLIC LOGIC + + +The universal use of some system of Symbolic Logic would not only enable +everybody easily to deal with exceedingly complicated arguments, but +would prevent ambiguous arguments. In denying the indispensability of +Symbolic Logic in the former state of things, Keynes[33] is probably +alone, against the need strongly felt by Alice when speaking to the +Duchess,[34] and most modern logicians. It may be noticed that the +Duchess is more consistent than Keynes, for Keynes really uses the signs +for logical multiplication and addition of Boole and Venn under the +different shapes of the words "and" and "or." + +As regards ambiguity, a translation of _Hymns Ancient and Modern_ into, +say, Peanesque, would prevent the puzzle of childhood as to whether the +"his" in + + And Satan trembles when he sees + The weakest saint upon his knees + +refers to the saint's knees or Satan's. + +FOOTNOTES: + +[33] In his _Fm. L._ + +[34] See Appendix I. + + + + +CHAPTER X + +LOGICAL ADDITION AND THE UTILITY OF SYMBOLISM + + +Frequently ordinary language contains subtle psychological implications +which cannot be translated into symbolic logic except at great length. +Thus if a man (say Mr. Jones) wishes to speak collectively of himself +and his wife, the order of mentioning the terms in the class considered +and the names applied to these terms are, logically speaking, +irrelevant. And yet more or less definite information is given about Mr. +Jones according as he talks to his friends of: + + (1) Mrs. Jones and I, + (2) I (or me) and my wife (or missus), + (3) My wife and I, + or (4) I (or me) and Mrs. Jones. + +In case (1) one is probably correct in placing Mr. Jones among the +clergy or the small professional men who make up the bulk of the +middle-class; in case (2) one would conclude that Mr. Jones belonged to +the lower middle-class; the form (3) would be used by Mr. Jones if he +were a member of the upper, upper middle, or lower class; while form (4) +is only used by retired shopkeepers of the lower middle-class, of which +a male member usually combines belief in the supremacy of man with +belief in the dignity of his wife as well as himself. A further +complication is introduced if a wife is referred to as "the wife."[35] +Cases (2) and (3) then each give rise to one more case. Cases (1) and +(4) do not, since nobody has hitherto referred to his wife as "the Mrs. +Jones"--at least without a qualifying adjective before the "Mrs." + +On the other hand, certain descriptive phrases and certain propositions +can be expressed more shortly and more accurately by means of symbolic +logic. Let us consider the proposition "No man marries his deceased +wife's sister." If we assume, as a first approximation, that all +marriages are fertile and that all children are legitimate, then, with +only four primitive ideas: the relation of parent to child (P) and the +three classes of males, females, and dead people, we can define "wife" +(a female who has the relation formed by taking the relative product of +P and [vP][36] to a male), "sister," "deceased wife," and "deceased +wife's sister" in terms of these ideas and of the fundamental notions of +logic. Then the proposition "No man marries his deceased wife's sister" +can be expressed unambiguously by about twenty-nine simple signs on +paper, whereas, in words, the unasserted statement consists of no less +than thirty-four letters. Although, legally speaking, we should have to +adopt somewhat different definitions and possibly increase the +complications of our proposition, it must be remembered that, on the +other hand, we always reduce the number of symbols in any proposition by +increasing the number of definitions in the preliminaries to it. + +But the utility of symbolic logic should not be estimated by the brevity +with which propositions may sometimes be expressed by its means. Logical +simplicity, in fact, can very often only be obtained by apparently +complicated statements. For example, the logical interpretation of "The +father of Charles II was executed" is, "It is not always false of _x_ +that _x_ begat Charles II, and that _x_ was executed and that 'if _y_ +begat Charles II, _y_ is identical with _x_' is always true of _y_."[37] +From the point of view of logic, we may say that the apparently simple +is most often very complicated, and, even if it is not so, symbolism +will make it seem so,[38] and thus draw attention to what might +otherwise easily be overlooked. + +FOOTNOTES: + +[35] Cf. Chapter XXIV below. + +[36] C. S. Peirce's notation for the relation "converse of P." + +[37] Russell, _Md._, N. S., vol. xiv., October 1905, p. 482. + +[38] Russell, _International Monthly_, vol. iv., 1901, pp. 85-6; cf. +_M._, vol. xxii., 1912, p. 153. [This essay is reprinted in _Mysticism +and Logic_, London and New York, 1918, pp. 74-96.--ED.] + + + + +CHAPTER XI + +CRITICISM + + +Those people who think that it is more godlike to seem to turn water +into wine than to seem to turn wine into water surprise me. I cannot +imagine an intolerable critic. It seems to me that, if A resents B's +criticism in trying to put his (A's) discovery in the right or wrong +place, A acts as if he thought he had some private property in truth. +The White Queen seems to have shared the popular misconception as to the +nature of criticism.[39] + +FOOTNOTES: + +[39] See Appendix J. + + + + +CHAPTER XII + +HISTORICAL CRITICISM + + +From a problem in Diophantus's _Arithmetic_ about the price of some wine +it would seem that the wine was of poor quality, and Paul Tannery has +suggested that the prices mentioned for such a wine are higher than were +usual until after the end of the second century. He therefore rejected +the view which was formerly held that Diophantus lived in that +century.[40] + +The same method applied to a problem given by the ancient Hindu +algebraist Brahmagupta, who lived in the seventh century after Christ, +might result in placing Brahmagupta in prehistoric times. This is the +problem:[41] "Two apes lived at the top of a cliff of height _h_, whose +base was distant _mh_ from a neighbouring village. One descended the +cliff and walked to the village, the other flew up a height _x_ and then +flew in a straight line to the village. The distance traversed by each +was the same. Find _x_." + +FOOTNOTES: + +[40] W. W. Rouse Ball, _A Short Account of the History of Mathematics_, +4th edition, London, 1908, p. 109. + +[41] _Ibid._, pp. 148-9. + + + + +CHAPTER XIII + +IS THE MIND IN THE HEAD? + + +The contrary opinion has been maintained by idealists and a certain +election agent with whom I once had to deal and who remarked that +something slipped his mind and then went out of his head altogether. At +some period, then, a remembrance was in his head and out of his mind; +his mind was not, then, wholly within his head. Also, one is sometimes +assured that with certain people "out of sight is out of mind." What is +in their minds is therefore in sight, and cannot therefore be inside +their heads. + + + + +CHAPTER XIV + +THE PRAGMATIST THEORY OF TRUTH + + +The pragmatist theory that "truth" is a belief which works well +sometimes conflicts with common-sense and not with logic. It is commonly +supposed that it is always better to be sometimes right than to be never +right. But this is by no means true. For example, consider the case of a +watch which has stopped; it is exactly right twice every day. A watch, +on the other hand, which is always five minutes slow is never exactly +right. And yet there can be no question but that a belief in the +accuracy of the watch which was never right would, on the whole, produce +better results than such a belief in the one which had altogether +stopped. The pragmatist would, then, conclude that the watch which was +always inaccurate gave truer results than the one which was sometimes +accurate. In this conclusion the pragmatist would seem to be correct, +and this is an instance of how the false premisses of pragmatism may +give rise to true conclusions. + +From the text written above the church clock in a certain English +village, "Be ye ready, for ye know not the time," it would be concluded +that the clock never stopped for a period as long as twelve hours. For +the text is rather a vague symbolical expression of a propositional +function which is asserted to be true at all instants. The proposition +that a presumably not illiterate and credulous observer of the clock at +any definite instant does not know the time implies, then, that the +clock is always wrong. Now, if the clock stopped for twelve hours, it +would be absolutely right at least once. It must be right twice if it +were right at the first instant it stopped or the last instant at which +it went;[42] but the second possibility is excluded by hypothesis, and +the occurrence of the first possibility--or of the analogous possibility +of the stopped clock being right three times in twenty-four hours--does +not affect the present question. Hence the clock can never stop for +twelve hours. + +The pragmatist's criterion of truth appears to be far more difficult to +apply than the Bellman's,[43] that what he said three times is true, and +to give results just as insecure. + +FOOTNOTES: + +[42] Both cases cannot occur; the question is similar to that arising in +the discussion of the mortality of Socrates (see Chapter XXII). + +[43] See Appendix K. + + + + +CHAPTER XV + +ASSERTION + + +The subject of the present chapter must not be confused with the +assertion of ordinary life. Commonly, an unasserted proposition is +synonymous with a probably false statement, while an asserted +proposition is synonymous with one that is certainly false. But in logic +we apply assertion also to true propositions, and, as Lewis Carroll +showed in his version of "What the Tortoise said to Achilles,"[44] +usually pass over unconsciously an infinite series of implications in so +doing. If _p_ and _q_ are propositions, _p_ is true, and _p_ implies +_q_, then, at first sight, one would think that one might assert _q_. +But, from (A) _p_ is true, and (B) _p_ implies _q_, we must, in order to +deduce (Z) _q_ is true, accept the hypothetical: (C) If A and B are +true, Z must be true. And then, in order to deduce Z from A, B, and C, +we must accept another hypothetical: (D) If A, B, and C are true, Z must +be true; and so on _ad infinitum_. Thus, in deducing Z, we pass over an +infinite series of hypotheticals which increase in complexity. Thus we +need a new principle to be able to assert _q_. + +Frege was the first logician sharply to distinguish between an asserted +proposition, like "A is greater than B," and one which is merely +considered, like "A's being greater than B," although an analogous +distinction had been made in our common discourse on certain +psychological grounds, for long previously. In fact, soon after the +invention of speech, the necessity of distinguishing between a +considered proposition and an asserted one became evident, on account of +the state of things referred to at the beginning of this chapter. + +FOOTNOTES: + +[44] _Md._ N. S., vol. iv., 1895, pp. 278-80. Cf. Russell, _P. M._, p. +35. + + + + +CHAPTER XVI + +THE COMMUTATIVE LAW + + +Often the meaning of a sentence tacitly implies that the commutative law +does not hold. We are all familiar with the passage in which Macaulay +pointed out that, by using the commutative law because of exigencies of +metre, Robert Montgomery unintentionally made Creation tremble at the +Atheist's nod instead of the Almighty's. This use of the commutative law +by writers of verse renders it doubtful whether, in the hymn-line: + + The humble poor believe, + +we are to understand a statement about the humble poor, or a doubtful +maxim as to the attitude of our minds to statements made by the humble +poor. + +The non-commutativity of English titles offers difficulties to some +novelists and Americans who refer to Mary Lady So-and-So as Lady Mary +So-and-So, and _vice versa_. + + + + +CHAPTER XVII + +UNIVERSAL AND PARTICULAR PROPOSITIONS + + +People who are cynical as to the morality of the English are often +unpleasantly surprised to learn that "All trespassers will be +prosecuted" does not necessarily imply that "some trespassers will be +prosecuted." The view that universal propositions are non-existential is +now generally held: Bradley and Venn seem to have been the first to hold +this, while older logicians, such as De Morgan,[45] considered universal +propositions to be existential, like particular ones. + +If the Gnat[46] had been content to affirm his proposition about the +means of subsistence of Bread-and-Butter flies, in consequence of their +lack of which such flies always die, without pointing out such an insect +and thereby proving that the class of them is not null, Alice's doubt as +to the existence of the class in question, even if it were proved to be +well founded, would not have affected the validity of the proposition. + +This brings us to a great convenience in treating universal propositions +as non-existential: we can maintain that all _x_'s are _y_'s at the same +time as that no _x_'s are _y_'s, if only _x_ is the null-class. Thus, +when Mr. MacColl[47] objected to other symbolic logicians that their +premisses imply that all Centaurs are flower-pots, they could reply that +their premisses also imply the more usual view that Centaurs are not +flower-pots. + +FOOTNOTES: + +[45] Cf., e.g., _F. L._, p. 4. + +[46] See Appendix L. + +[47] Cf., e.g., _Md._, N. S., vol. xiv., July, 1905, pp. 399-400. + + + + +CHAPTER XVIII + +DENIAL OF GENERALITY AND GENERALITY OF DENIAL + + +The conclusion of a certain song[48] about a young man who poisoned his +sweetheart with sheep's-head broth, and was frightened to death by a +voice exclaiming: + + "Where's that young maid + What you did poison with my head?" + +at his bedside, gives rise to difficulties which are readily solved by a +symbolism that brings into relief the principle that the denial of a +universal and non-existential proposition is a particular and +existential one. The conclusion of the song is: + + Now all young men, both high and low, + Take warning by this dismal go! + For if he'd never done nobody no wrong, + He might have been here to have heard this song. + +It is an obvious error, say Whitehead and Russell,[49] though one easy +to commit, to assume that the cases: (1) all the propositions of a +certain class are true; and (2) no proposition of the class is true; are +each other's contradictories. However, in the modification[50] of +Frege's symbolism which was used by Russell + + (1) is (_x_). _x_, + and (2) is (_x_). not _x_; + +while the contradictory of (1) is: + + not (_x_). _x_. + +The last line but one of the above verse may, then, be written: + + (_t_). not (_x_). not not [Greek: ph] (_x_, _t_), + +where "[Greek: ph] (_x_, _t_)" denotes the unasserted propositional +function "the doing wrong to the person _x_ at the instant _t_." By +means of the principle of double negation we can at once simplify the +above expression into: + + (_t_). not (_x_). [Greek: ph] (_x_, _t_); + +which can be thus read: "If at every instant of his life there was at +least one person _x_ to whom he did no wrong (at that instant)." It is +difficult to imagine any one so sunk in iniquity that he would not +satisfy this hypothesis. We are forced, then, unless our imagination for +evil is to be distrusted, to conclude that any one might have been there +to have heard that song. Now this conclusion is plainly false, possibly +on physical grounds, and certainly on æsthetic grounds. It may be added, +by the way, that it is quite possible that De Morgan was mistaken in his +interpretation of the above proposition owing to the fact that he was +unacquainted with Frege's work. In fact, if he had not noticed the fact +that _any_ two of the "not's" cannot be cancelled against one another he +would have concluded that the interpretation was: "If he had never done +any wrong to anybody." + +According as the symbol for "not" comes before the (_x_) or between the +(_x_) and the [Greek: ph], we have an expression of what Frege called +respectively the denial of generality, and the generality of denial. The +denial of the generality of a denial is the form of all existential +propositions, while the assertion of or denial of generality is the +general form of all non-existential or universal propositions. + +FOOTNOTES: + +[48] To which De Morgan drew attention in a letter; see (Mrs.) S. E. De +Morgan, _Memoir of Augustus De Morgan_, London, 1882, p. 324. + +[49] _Pa. Ma._, p. 16. + +[50] However, here, for the printer's convenience, we depart from Mr. +Russell's usage so far as to write "not" for a curly minus sign. + + + + +CHAPTER XIX + +IMPLICATION + + +A good illustration of the fact that what is called "implication" in +logic is such that a false proposition implies any other proposition, +true or false, is given by Lewis Carroll's puzzle of the three +barbers.[51] + +Allen, Brown, and Carr keep a barber's shop together; so that one of +them must be in during working hours. Allen has lately had an illness of +such a nature that, if Allen is out, Brown must be accompanying him. +Further, if Carr is out, then, if Allen is out, Brown must be in for +obvious business reasons. The problem is, may Carr ever go out? + +Putting _p_ for "Carr is out," _q_ for "Allen is out" and _r_ for "Brown +is out," we have: + + (1) _q_ implies _r_, + (2) _p_ implies that _q_ implies not-_r_. + +Lewis Carroll supposed that "_q_ implies _r_" and "_q_ implies not-_r_" +are inconsistent, and hence that _p_ must be false. But these +propositions are not inconsistent, and are, in fact, both true if _q_ is +false. The contradictory of "_q_ implies _r_" is "_q_ does not imply +_r_" which is not a consequence of "_q_ implies not-_r_." It seems to be +true theoretically that, if Mr. X is a Christian, he is not an Atheist, +but we cannot conclude from this alone that his being a Christian does +not imply that he is an Atheist, unless we assume that the class of +Christians is not null. Thus, if _p_ is true, _q_ is false; or, if Carr +is out, Allen is in. The odd part of this conclusion is that it is the +one which common-sense would have drawn in that particular case. + +A distinguished philosopher (M) once thought that the logical use of the +word "implication"--any false proposition being said to "imply" any +proposition true or false--is absurd, on the grounds that it is +ridiculous to suppose that the proposition "2 and 2 make 5" implies the +proposition "M is the Pope." This is a most unfortunate instance, +because it so happens that the false proposition that 2 and 2 make 5 can +rigorously be proved to imply that M, or anybody else other than the +Pope, is the Pope. For if 2 and 2 make 5, since they also make 4, we +would conclude that 5 is equal to 4. Consequently, subtracting 3 from +both sides, we conclude that 2 would be equal to 1. But if this were +true, since M and the Pope are two, they would be one, and obviously +then M would be the Pope. + +The principle that the false implies the true has very important +applications in political arguments. In fact, it is hard to find a +single principle of politics of which false propositions are not the +main support. + +If _p_ and _q_ are two propositions, and _p_ implies _q_; then, if, and +only if, _q_ and _p_ are both false or both true, we also have: _q_ +implies _p_. The most important applications of this invertibility were +made by the late Samuel Butler[52] and Mr. G. B. Shaw. A political +application may be made as follows: In a country where only those with +middling-sized incomes are taxed, conservative and _bourgeois_ +politicians would still maintain that the proposition "the rich are +taxed" implies the proposition "the poor are taxed," and this +implication, which is true because both premiss and conclusion are +false, would be quite unnecessarily supported by many false practical +arguments. It is equally true that "the poor are taxed" implies that +"the rich are taxed." And this can be proved, in certain cases, on other +grounds. For the taxation of the poor would imply, ultimately, that the +poor could not afford to pay a little more for the necessities of life +than, in strict justice, they ought; and this would mean the cessation +of one of the chief means of production of individual wealth. + +We also see why a valuable means for the discovery of truth is given by +the inversion of platitudinous implications. It may happen that another +platitude is the result of inversion; but it is the fate of any true +remark, especially if it is easy to remember by reason of a paradoxical +form, to become a platitude in course of time. There are rare cases of a +platitude remaining unrepeated for so long that, by a converse process, +it has become paradoxical. Such, for example, is Plato's remark that a +lie is less important than an error in thought. + +Of late years, a method of disguising platitudes as paradoxes has been +too extensively used by Mr. G. K. Chesterton. The method is as follows. +Take any proposition _p_ which holds of an entity _a_; choose _p_ so +that it seems plausible that _p_ also holds of at least two other +entities _b_ and _c_; call _a_, _b_, _c_, and any others for which _p_ +holds or seems to hold, the class A, and _p_ the "A-ness" or "A-ity" of +A; let _d_ be an entity for which _p_ does not hold; and put _d_ among +the A's when you think that nobody is looking. Then state your paradox: +"Some A's do not have A-ness." By further manipulation you can get the +proposition "No A's have A-ness." But it is possible to make a very +successful _coup_ if A is the null-class, which has the advantage that +manipulation is unnecessary. Thus, Mr. Chesterton, in his _Orthodoxy_ +put A for the class of doubters who doubt the possibility of logic, and +proved that such agnostics refuted themselves--a conclusion which seems +to have pleased many clergymen. + +In this way, Mr. Chesterton has been enabled readily to write many books +and to maintain, on almost every page, such theses as that simplicity is +not simple, heterodoxy is not heterodox, poets are not poetical, and so +on; thereby building up the gigantic platitude that Mr. Chesterton is +Chestertonian. + +In the chapter on Identity we have illustrated the use of a case of the +principle that any proposition implies any true proposition. This +important principle may be called _the principle of the irrelevant +premiss_;[53] and is of great service in oratory, because it does not +matter what the premiss is, true or false. There is a _principle of the +irrelevant conclusion_, but, except in law-courts, interruptions of +meetings, and family life, this is seldom used, partly because of the +limitation involved in the logical impossibility for the conclusion to +be false if the premiss be true, but chiefly because the conclusion is +more important than the premiss, being usually a matter of prejudice. + +Certain modern logicians, such as Frege, have found it necessary so to +extend the meaning of implication of _q_ by _p_ that it holds when _p_ +is not a proposition at all. Hitherto, politicians, finding that either +identical or false propositions are sufficient for their needs, have +made no use of this principle; but it is obvious that their stock of +arguments would be vastly increased thereby. + +Logical implication is often an enemy of dignity and eloquence. De +Morgan[54] relates "a tradition of a Cambridge professor who was once +asked in a mathematical discussion, 'I suppose you will admit that the +whole is greater than its part?' and who answered, 'Not I, until I see +what use you are going to make of it.'" And the care displayed by +cautious mathematicians like Poincaré, Schoenflies, Borel, Hobson, and +Baire in abstaining from pushing their arguments to their logical +conclusions is probably founded on the unconscious--but no less +well-grounded--fear of appearing ridiculous if they dealt with such +extreme cases as "the series of all ordinal numbers."[55] They are, +probably, as unconscious of implication as Gibbon, when he remarked that +he always had a copy of Horace in his pocket, and often in his hand, was +of the necessary implication of these propositions that his hand was +sometimes in his pocket. + +FOOTNOTES: + +[51] _Md._, N. S., vol. iii., 1894, pp. 436-8. Cf. the discussions by W. +E. Johnson (_ibid._, p. 583) and Russell (_P. M._, p. 18, note, and +_Md._, N. S., vol. xiv., 1905, pp. 400-1). + +[52] The inhabitants of "Erewhon" punished invalids more severely than +criminals. In modern times, one frequently hears the statement that +crime is a disease; and if so, it is surely false that criminals ought +to be punished. + +[53] _Irrelevant_ in a popular sense; one would not say, speaking +loosely, that the fact that Brutus killed Cæsar implies that the sea is +salt; and yet this conclusion is implied both by the above premiss, and +the premiss that Cæsar killed Brutus. Cf. on such questions Venn, +_S. L._, 2nd ed., pp. 240-4. + +[54] _F. L._, p. 264. + +[55] Cf. Chapters XXIX and XXXVII. + + + + +CHAPTER XX + +DIGNITY + + +We have seen, at the end of the preceding chapter, that logical +implication is often an enemy of dignity. The subject of dignity is not +usually considered in treatises on logic, but, as we have remarked, many +mathematicians implicitly or explicitly seem to fear either that the +dignity of mathematics will be impaired if she follows out conclusions +logically, or that only an act of faith can save us from the belief +that, if we followed out conclusions logically, we should find out +something alarming about the past, present, or future of mathematics. + +Thus it seems necessary to inquire rather more closely into the nature +of dignity, with a view to the discovery of whether it is, as is +commonly supposed, a merit in life and logic. + +The chief use of dignity is to veil ignorance. Thus, it is well known +that the most dignified people, as a rule, are schoolmasters, and +schoolmasters are usually so occupied with teaching that they have no +time to learn anything. And because dignity is used to hide ignorance, +it is plain that impudence is not always the opposite of dignity, but +that dignity is sometimes impudence. Dignity is said to inspire respect; +and this may be in part why respect for others is an error of judgment +and self-respect is ridiculous. + +Self-respect is, of course, self-esteem. William James has remarked that +self-esteem depends, not simply upon our success, but upon the ratio of +our success to our pretensions, and can therefore be increased by +diminishing our pretensions. Thus if a man is successful, but only then, +can he be both ambitious and dignified. James also implies that +happiness increases with self-esteem. Likeness of thought with one's +friends, then, does not make one happy, for otherwise a man who esteemed +himself little would be indeed happy. Also if a man is unhappy he could +not, from our premisses, by the principles of the syllogism and of +contraposition, be dignified--a conclusion which should be fatal to many +novelists' heroes. + +A reflection on pessimism to which this discussion gives rise is the +following: It would appear that a man's self-esteem would be increased +by a conviction of the unworthiness of his neighbours. A man, therefore, +who thinks that the world and all its inhabitants, except himself, are +very bad, should be extremely happy. In fact, the effects would hardly +be distinguishable from those of optimism. And optimism, as everybody +knows, is a state of mind induced by stupidity. + + + + +CHAPTER XXI + +THE SYNTHETIC NATURE OF DEDUCTION + + +Doubt has often been expressed as to whether a syllogism can add to +our knowledge in any way. John Stuart Mill and Henri Poincaré, in +particular, held the opinion that the conclusion of a syllogism is an +"analytic" judgment in the sense of Kant, and therefore could be +obtained by the mere dissection of the premisses. Any one, then, who +maintains that mathematics is founded solely on logical principles would +appear to maintain that mathematics, in the last instance, reduces to a +huge tautology. + +Mill, in Chapter III of Book II of his _System of Logic_, said that "it +must be granted that in every syllogism, considered as an argument to +prove the conclusion, there is a _petitio principii_. When we say + + All men are mortal, + Socrates is a man, + +therefore + + Socrates is mortal, + +it is unanswerably urged by the adversaries of the syllogistic theory, +that the proposition, Socrates is mortal, is presupposed in the more +general assumption, All men are mortal; that we cannot be assured of the +mortality of all men unless we are already certain of the mortality of +every individual man; that if it be still doubtful whether Socrates, or +any other individual we choose to name, be mortal or not, the same +degree of uncertainty must hang over the assertion, All men are mortal; +that the general principle, instead of being given as evidence of the +particular case, cannot itself be taken for true without exception until +every shadow of doubt which could affect any case comprised with it is +dispelled by evidence _aliunde_; and then what remains for the syllogism +to prove? That, in short, no reasoning from general to particular can, +as such, prove anything, since from a general principle we cannot infer +any particulars but those which the principle itself assumes as known. +This doctrine appears to me irrefragable...." + +But it is not difficult to see that in certain cases at least deduction +gives us _new_ knowledge.[56] If we already know that two and two always +make four, and that Asquith and Lloyd George are two and so are the +German Emperor and the Crown Prince, we can deduce that Asquith and +Lloyd George and the German Emperor and the Crown Prince are four. This +is new knowledge, not contained in our premisses, because the general +proposition, "two and two are four," never told us there were such +people as Asquith and Lloyd George and the German Emperor and the Crown +Prince, and the particular premisses did not tell us that there were +four of them, whereas the particular proposition deduced does tell us +both these things. But the newness of the knowledge is much less certain +if we take the stock instance of deduction that is always given in books +on logic, namely "All men are mortal; Socrates is a man, therefore +Socrates is mortal." In this case what we really know beyond reasonable +doubt is that certain men, A, B, C, were mortal, since, in fact, they +have died. If Socrates is one of these men, it is foolish to go the +roundabout way through "all men are mortal" to arrive at the conclusion +that _probably_ Socrates is mortal. If Socrates is not one of the men on +whom our induction is based, we shall still do better to argue straight +from our A, B, C, to Socrates, than to go round by the general +proposition, "all men are mortal." For the probability that Socrates is +mortal is greater, on our data, than the probability that all men are +mortal. This is obvious, because if all men are mortal, so is Socrates; +but if Socrates is mortal, it does not follow that all men are mortal. +Hence we shall reach the conclusion that Socrates is mortal, with a +greater approach to certainty if we make our argument purely inductive +than if we go by way of "all men are mortal" and then use deduction. + +Many years ago there appeared, principally owing to the initiative of +Dr. F. C. S. Schiller of Oxford, a comic number of _Mind_. The idea was +extraordinarily good, not so the execution. A German friend of Dr. +Schiller was puzzled by the appearance of the advertisements, which were +doubtfully humorous. However, by a syllogistic process, he acquired +information which was new and useful to him, and thus incidentally +refuted Mill. Presumably he started from the title of the magazine +(_Mind!_), for a mark of exclamation seems nearly always in German to be +a sign of an intended joke (including of course the mark after the +politeness expressed in the first sentence of a private letter or a +public address). There would be, then, the following syllogism: + + This is a book of would-be jokes (i.e. everything in this book is a + would-be joke); + This advertisement is in this book; + Therefore, this advertisement is a would-be joke. + +Thus the syllogism may be almost as powerful an agent in the detection +of humour as M. Bergson's criterion, to be described in a future +chapter.[57] + +FOOTNOTES: + +[56] [The following passage is almost word for word the same as a +passage on pp. 123-5 of Mr. Russell's _Problems of Philosophy_, first +published in 1912, a year after Mr. R*ss*ll's death. It is easy hastily +to conclude that Mr. Russell was indebted to Mr. R*ss*ll to a greater +degree than is usually supposed. But an examination of the internal +evidence leads us to another conclusion. The two texts, it will be +found, differ only in the names of the German Emperor, the Crown Prince +and the other personages being replaced, in the book of 1912, by those +of Messrs. Brown, Jones, Smith, and Robinson. Now, Mr. Russell, in a new +edition of his _Problems_ issued near the beginning of the European war +and before the Russian revolution, substituted "the Emperor of Russia" +for "the Emperor of China" of the first edition. Hence it seems quite +likely that Mr. Russell, who has always shown a tendency to substitute +existents for nonentities, wrote Mr. R*ss*ll's notes.--<sc>Ed.</sc>] + +[57] [See Chapter XLII.--ED.] + + + + +CHAPTER XXII + +THE MORTALITY OF SOCRATES + + +The mortality of Socrates is so often asserted in books on logic that it +may be as well briefly to consider what it means. The phrase "Socrates +is mortal" may be thus defined: "There is at least one instant _t_ such +that _t_ has not to Socrates the one-many relation R which is the +converse of the relation 'exists at,' and all instants following _t_ +have not the relation R to Socrates, and there is at least one instant +_t´_ such that neither _t´_ nor any instant preceding _t´_ has the +relation R to Socrates." + +This definition has many merits. In the first place, no assumption is +made that Socrates ever lived at all. In the second place, no assumption +is made that the instants of time form a continuous series. In the third +place, no assumption is made as to whether Socrates had a first or last +moment of his existence. If time be indeed a continuous series, then we +can easily deduce[58] that there must have been _either_ a first moment +of his non-existence _or_ a last one of his existence, but not both; +just as there seems to be either a greatest weight that a man can lift +or a least weight that he cannot lift, but not both.[59] This may be set +forth as follows: for the present we will not concern ourselves with +evidence for or against human immortality; I will merely try to present +some logical questions which persistently arise whenever we think of +eternal life. One of the greatest merits of modern logic is that it has +allowed us to give precision to such problems, while definitely +abandoning any pretensions of solving them; and I will now apply the +logico-analytical method to one of the problems of our knowledge of the +eternal world.[60] + +We will start from the generally accepted proposition that all men are +mortal. Clearly, if we could know each individual man, and know that he +was mortal, that would not enable us to know that all men are mortal, +unless we knew, in addition, that those were all the men there are. But +we need not here assume any such knowledge of general propositions; and, +though most of us will admit that the proposition in question has great +intrinsic plausibility, it is not strictly necessary for our present +purpose to assume anything more than the still more probable proposition +"Socrates is mortal." This last proposition, quite apart from the fact +that we have a large amount of historical evidence for its truth, has +been repeated so often in books on logic that it has taken on the +respectable air of a platitude while preserving the character of an +exceedingly probable truth. The truth also results from the fact that it +is used as the conclusion of a syllogism. For it is a well-known fact +that syllogisms can only be regarded as forming part of a sound +education if the conclusions are obviously true. The use of a syllogism +of the form "All cats are ducks and all ducks are mice, therefore all +cats are mice," would introduce grave doubts into the University of +Oxford as to whether logic could any longer be considered as a valuable +mental training for what are amusingly called the "learned professions." + +If, then, we divide all the instants of time, whether past, present, or +future, into two series--those instants at which Socrates was alive, and +those instants at which he was not alive--and leave out of +consideration, for the sake of greater simplicity, all those instants +before he lived, we see at once, by the simple application of Dedekind's +Axiom, that, if Socrates entered into eternal life after his death, +there must have been either a last moment of his earthly life _or_ a +first moment of his eternal life, but not both. + +Logic alone can give us no information as to which of these cases +actually occurred, and we are thrown back on to a discussion of +empirical evidence. It is no unusual thing to read of people who thought +"that every moment would be their last." In this case it is quite +obvious that they consequently thought that eternity would have no +beginning. + +Now here we must consider two things: (1) It is plainly unsafe to +conclude from what people think will happen to what will happen; (2) +even if we could so conclude, it would be unsafe to deduce that there +was a last moment in the life of Socrates: we could only make the guess +plausible, as we should be using the inductive method. + +There are two other pieces of evidence that there is a last moment of +any earthly existence, which we may now briefly consider. That this was +so was held by Carlo Michaelstaedter; but since he apparently only +believed this because he wanted, by attributing a supposed ethical value +to that moment, to give support to his theory of suicide, we ought not +to give great weight to this evidence. Secondly, Thomas Hobbes objected +to the principle "that a quantity may grow less and less eternally, so +as at last to be equal to another quantity; or, which is all one, that +there is a last in eternity" as "void of sense." Now, the principle +meant is true, so that, although the other proposition mentioned by +Hobbes does not follow logically from the first, there is some evidence +that this other is true. In fact, that Hobbes thought that such-and-such +a proposition followed from another proposition which he wrongly +believed to be false, is far better evidence for the truth of +such-and-such a proposition than any we have for the truth of most of +our most cherished beliefs. + +Thirdly, Leibniz, in a dialogue[61] written on his journey of 1676 to +visit Spinoza, raised the question whether the moment at which a man +dies may be regarded as both the last moment at which he is alive and +the first at which he is dead, as it must be by Aristotle's theory of +continuity. Agreement with this view violates the law of contradiction; +denial of it implies that two moments can be immediately adjacent. By +the denial, then, we are led to regard space and time as made up of +indivisible points and moments, and thus, since we can draw one and only +one parallel from any point in the diagonal of a square to a given side, +the diagonal will contain the same (infinite) number of points as that +side, and will therefore be equal to it. In this Leibniz repeated an +argument used by the ancient Arabs, Roger Bacon, and William of Occam. +This Leibniz considered to be a proof that a line cannot be an aggregate +of points. Indeed, their number would be "the number of all numbers" of +the greatest possible integer, which _is_ not. + +It does not seem, further, that any light is thrown on the logical +question of human mortality or immortality by legal decisions. It would +appear that one can, legally speaking, be alive for any period less than +twenty-four hours after one is dead and be dead for any period less than +twenty-four hours before one's death. At least, according to _Salkeld_, +i. 44, it was "adjudged that if one be born the first of February at +eleven at night, and the last of January in the twenty-first year of his +age, at one of the clock in the morning, he makes his will of lands, and +dies, it is a good will, for he was then of age." In Sir Robert Howard's +case (_ibid._, ii. 625) it was held by Chief Justice Holt that "if A be +born on the third day of September; and on the second day of September +twenty-one years afterwards he make his will, this is a good will; for +the law will make no fraction of a day, and by consequence he was of +age." But it is hardly necessary to remark that in this way the problem +with which we are concerned is merely shifted and not solved. For the +question as to whether there is or is not a last moment of a man's life +is not answered by the decision that he dies legally twenty-four hours +before or after he dies in the usual sense of the word, and the problem +arises as to whether there is or is not a last moment of his legal +age.[62] + +So assuming that there was a last moment of Socrates's earthly life, and +consequently no first moment of his eternal life, we see, further, that, +unless the possibility of infinite numbers is granted, it would be quite +possible for us logically to doubt the possibility of an eternal life +for Socrates on the same grounds as those which led Zeno to assert that +motion was impossible and that Achilles could never overtake the +Tortoise. If, on the other hand, it be admitted that eternity, at least +in the case of Socrates, had a beginning, these same arguments of Zeno +would lead any one who denies the possibility of infinite number to +conclude that Socrates, like the worm, can never die. Thus is it quite +plain that the difficulties about immortality which meet us at the very +outset of our inquiry can partly be solved only by the help of the +theory of infinite numbers and partly, it would seem, not at all. + +There is another difficulty about immortality which is quite distinct +from this and is analogous to another argument of Zeno. If, indeed, all +the instants of time be divided, as before, into the two series of +instants at which Socrates was alive and instants at which he was not +alive, it follows at once that no instant of time is not accounted for. +At none of these instants, however, does Socrates die; obviously he +cannot die either when he is alive or when he is dead. Thus it would +appear that Socrates never died, and that we ought to re-define the term +"mortal" to mean "a human being who is alive at some moments and dead at +some." Consequently we must avoid the very tempting conclusion that, +because Socrates never died, he was therefore immortal. + +It is very important carefully to distinguish between the two arguments +I have just set forth. The second argument proves quite rigidly that +Socrates and, indeed, anybody else, never dies, whether there is or is +not a last moment of his life on earth. The first argument proves that, +if there is a first moment of Socrates's eternal life, his life on earth +never ends. But we have seen that we cannot conclude that this unending +life proves that he never is or will be in a state of eternity. + +FOOTNOTES: + +[58] By "Dedekind's Axiom," _E. N._, p. 11. + +[59] _M._, vol. xx., 1910, pp. 134-5. + +[60] [Here, again, Mr. R*ss*ll's work seems to anticipate some of Mr. +Russell's later work, e.g. in _Our Knowledge of the External World as a +Field for Scientific Method in Philosophy_, Chicago and London, 1914, +pp. 3-4, 55-6, _et passim._--ED.] + +[61] "Pacidius Philalethi" in Louis Couturat, _Opuscules et Fragments +inédits de Leibniz_, Paris, 1903, pp. 594-627, especially pp. 599, 601, +608, 611. Cf. [A. E. Taylor, Hastings' _Encyclopædia of Religion and +Ethics_, vol. iv., Part 2, Edinburgh, 1912, p. 96.--ED.]; Robert Latta, +_Leibniz: The Monadology and other Philosophical Writings_, Oxford, +1898, pp. 21 ff, 29 (note); Couturat, _La Logique de Leibniz d'après des +documents inédits_, Paris, 1901, pp. 130, 132; and Russell, _Ph. L._, +pp. 108-16, 243-9. + +[62] [It may be remarked that, according to _The Times_ of December 20, +1917, Mr. Justice Sargant, in the Chancery Division, also held that "the +law did not recognize fractions of a day," and that Lord Blackburn, in +his decision (9 _App. Cas._, 371, 373) that a man born on the thirteenth +of May 1853 attained the age of twenty-one on the thirteenth of May 1874 +"was not speaking strictly."--ED.] + + + + +CHAPTER XXIII + +DENOTING + + +A concept _denotes_ when, if it occurs in a proposition, the proposition +is not about the concept, but _about_ a term connected in a certain +peculiar way with the concept. Some people often assert that man is +mortal, and yet we never see announced in _The Times_ that Man died on a +certain day at his villa residence "Camelot" at Upper Tooting,[63] nor +do we hear that Procrastination was again the butt of Mr. Plowden's +jokes at Marylebone Police Court last week. + +That two phrases may have different _meanings_ and the same _denotation_ +was discovered by Alice and Frege. Alice[64] observed that the road +which led to Tweedledum's house was that which led to the house of +Tweedledee; and Frege pointed out that the phrases "the house to which +the road that leads to Tweedledum's house leads" and "the house to which +the road that leads to Tweedledee's house leads" have different _Sinn_, +but the same _Bedeutung_. + +FOOTNOTES: + +[63] Cf. _P. M._, pp. 53-4. + +[64] See Appendix M. + + + + +CHAPTER XXIV + +THE + + +The word "the" implies existence and uniqueness; it is a mistake to talk +of "the son of So-and-So" if So-and-So has a fine family of ten +sons.[65] People who refer to "the Oxford Movement" imply that Oxford +only moved once; and those quaint people who say that "A is quite the +gentleman" imply both the doubtful proposition that there is only one +gentleman in the world, and the indubitably false proposition that he is +that man. Probably A is one of those persons who add to the confusion in +the use of the definite article by speaking of his wife as "the wife." + +In a certain Children's Hymn Book one reads: + + The river vast and small. + +Few would deny that there is not more than one such river, but +unfortunately it is doubtful if there is such a river at all. The case +is exactly the same with the ontological proof of the existence of the +most perfect being.[66] + +According to the _Daily Mail_ of October 9, 1906, Judge Russell decided +against a claim brought by an agent against his company for appointing +another agent, the claim being on the ground that he was appointed as +"the" agent. + +Most people admit that the number 2 can be added to the number 2 to give +the number 4, but this is a mistake. They concede, when they use _the_, +that there is only one number 2, and yet they imagine that, when they +consider it apart as the first term of our above sum, they can find +another to add to it, and thereby form the third term. The truth is that +"2 + 2 = 4" is a very misleading equation, and what we really mean by +that faultily abbreviated statement is more precisely: If _x_ and _y_ +denote any things which form a class B, and _x´_ and _y´_ any other +things that form a class (A) which, like that of _x_ and _y_, is a +member of the class (which we call "2") of those classes which have a +one-one correspondence with B (so that any member of A corresponds to +one, and only one, member of B, and conversely), the class of all the +terms of A and B together is a member of that class of classes which, +analogously, we call "4." In this, for the sake of shortness, we have +introduced abbreviations which should not be used in a rigorous logical +statement. + +FOOTNOTES: + +[65] Cf. _Md._, N. S., vol. xiv., 1905, pp. 481, 484. + +[66] Cf. _ibid._, p. 491, note. + + + + +CHAPTER XXV + +NON-ENTITY + + +When people say that such-and-such a thing "is non-existent" they +usually mean that there is not any "thing" of the kind spoken of. Venn +meant this when he described[67] his encounter with what he imagined to +be a very ingenious tradesman: "I once had some strawberry plants +furnished me which the vendor admitted would not bear many berries. But +he assured me that this did not matter, since they made up in their size +what they lost in their number. (He gave me, in fact, the hyperbolic +formula, _xy = c_, to connect the number and magnitude.) When summer +came, _no_ fruit whatever appeared. I saw that it would be no use to +complain, because the man would urge that the size of the non-existent +berry was infinite, which I could not see my way to disprove. I had +forgotten to bar zero values of either variable." + +It is to be regretted that this useful note was omitted in the second +edition of Venn's book; one can imagine that it might have protected Mr. +MacColl and Herr Meinong (who believed, unlike Alice in what may be +called her first theory,[68] in round squares and fabulous monsters) +against the dishonest practices of traders who were too ready with +promises. For the death-blow to this kind of trade was not given until +1905, when Mr. Russell published his article "On Denoting,"[69] and took +up the position of the White King in opposition to Alice's later +assertions.[70] + +Venn's experience illustrates another characteristic of mathematical +logic. It is necessary, in order to make our arguments conclusive, to +devote great care to the elimination of difficulties which rarely occur. +The White Knight--who was like Boole in being a pioneer of mathematical +logic in this way, and yet seems to have held, like Boole, those +philosophical opinions which would base logic on psychology--recognized +the necessity of taking precautions against any unusual appearance of +mice on a horse's back.[71] + +FOOTNOTES: + +[67] _S. L._, 1881, p. 339, note. + +[68] See Appendix N. + +[69] _Md._, N. S., vol. xiv., October 1905, pp. 479-93. + +[70] See Appendix N. + +[71] See Appendix O. + + + + +CHAPTER XXVI + +IS + + +_Is_ has four perfectly distinct meanings in English, besides misuses of +the word. Among the misuses, perhaps the most important are those +referred to by De Morgan:[72] "... We say 'murder _is_ death to the +perpetrator' where the copula is _brings_; 'two and two _are_ four,' the +copula being 'have the value of,' etc." + +Schröder[73] quite satisfactorily pointed out the well-known distinction +between an _is_ where subject and predicate can be interchanged (such +as: "the class whose members are Shem, Ham and Japhet is the class of +the sons of Noah") and an _is_ or _are_ where they cannot (such as: +Englishmen are Britons), but failed to see[74] the more important +distinction (made by Peano) of is in the sense of "is a member of." If +Englishmen are Britons, and Britons are civilized people, it follows +that Englishmen are civilized people; but, though the _Harmsworth +Encyclopædia_ is a member of the class Book (of one or more volumes), +and this class is the member of a class A of which it is the only +member, yet the _Harmsworth Encyclopædia_ is not a member of A, for it +is not true that it is the whole class of books; and such a statement +would not even be made except possibly in the form of an advertisement. + +The fourth meaning of _is_ is _exists_; it is in certain rare moods a +matter for regret that there are difficulties in the way of using one +word to denote four different things. For, if there were not, we might +prove the existence of any thing we please by making it the subject of a +proposition, and thereby earn the gratitude of theologians. + +FOOTNOTES: + +[72] _F. L._, p. 268. + +[73] _A. d. L._, i. pp. 127 sqq. + +[74] _Ibid._, vol. ii. pp. 461, 597. + + + + +CHAPTER XXVII + +_AND_ AND _OR_ + + +When, with Boole, alternatives (A, B) are considered as mutually +exclusive, logical addition may be described as the process of taking A +_and_ B or A _or_ B. It is a great and rare convenience to have two +terms for denoting the same thing: commonly, people denote several +things by the same term, and only the Germans have the privilege of +referring to, say, _continuity_ as _Stetigkeit_ or _Kontinuierlichkeit_. +But Jevons[75] quoted Milton, Shakespeare, and Darwin to prove that +alternatives are not exclusive, and so attained first to recognized +views by arguments which were plainly irrelevant. + +Of course, _and_ is often used as the sign of logical addition: thus one +may speak of one's brothers _and_ sisters, without being understood to +mean the null-class (as should be the case), or pray for one's +"relations and friends," without being sure that one's prayer would be +answered,--as it certainly would if one meant to pray for the +null-class, this being the class indicated. And a word like _while_ is +often used for a logical addition, when exclusiveness of the +alternatives is almost implied. Thus, a reviewer in _Mind_,[76] noticing +the translation of Mach's _Popular Scientific Lectures_ into American, +said of the lectures that: "Most of them will be familiar ... to +epistemologists and experimental psychologists: while the remainder, +which deal with physical questions, are well worth reading." The reader +has the impression, probably given unintentionally, that Professor +Mach's epistemological and psychological lectures are not, in the +reviewer's opinion, worth reading. + +FOOTNOTES: + +[75] _Pure Logic_ ..., London, 1864, pp. 76-9. Cf. Venn, _S. L._, 2nd +ed., pp. 40-8. + +[76] N. S., vol. iv. p. 261. + + + + +CHAPTER XXVIII + +THE CONVERSION OF RELATIONS + + +The "Conversion of Relations" does not mean what it might be supposed to +mean; it has nothing to do with what Kant called "the wholesome art of +persuasion." What concerns us here is the convertibility of a logical +relation. If A has a certain relation R to B, the relation of B to A, +which may be denoted by [vR], is called the _converse_ of R. As De +Morgan[77] remarked, this conversion may sometimes present difficulties. +The following is De Morgan's example: + +"Teacher: 'Now, boys, Shem, Ham and Japheth were Noah's sons; who was +the father of Shem, Ham and Japheth?' No answer. + +"Teacher: 'Boys, you know Mr. Smith, the carpenter, opposite; has he any +sons?' + +"Boys: 'Oh! yes, sir! there's Bill and Ben.' + +"Teacher: 'And who is the father of Bill and Ben Smith?' + +"Boys: 'Why, Mr. Smith, to be sure.' + +"Teacher: 'Well, then, once more, Shem, Ham and Japheth were _Noah's_ +sons; who was the father of Shem, Ham and Japheth?' + +"A long pause; at last a boy, indignant at what he thought the attempted +trick, cried out: 'It _couldn't_ have been Mr. Smith.' These boys had +never converted the relation of father and son...." + +FOOTNOTES: + +[77] _Trans. Camb. Phil. Soc._, vol. x., 1864, part ii., note on page +334. + + + + +CHAPTER XXIX + +PREVIOUS PHILOSOPHICAL THEORIES OF MATHEMATICS + + +Mathematicians usually try to found mathematics on two principles:[78] +one is the principle of confusion between the sign and the thing +signified (they call this principle the foundation-stone of the formal +theory), and the other is the Principle of the Identity of Discernibles +(which they call the principle of the permanence of equivalent forms). + +But the truth is that if we set sail on a voyage of discovery with Logic +alone at the helm, we must either throw such principles as "the identity +of those conceptions which have in common the properties that interest +us" and "the principle of permanence" overboard, or, if we do not like +to act in such a way to old companions with whom we are so familiar that +we can hardly feel contempt for them, at least recognize them clearly as +having no logical validity and merely as psychological principles, and +reduce them to the humble rank of stewards, to minister to our human +weaknesses on the voyage. And then, if we adopt the wise policy of +keeping our axioms down to the minimum number, we must refrain from +creating or thinking that we are creating new numbers to fill up gaps +among the older ones, and thence recognize that our rational numbers are +not particular cases of "real" numbers, and so on. + +We thus get a world of conceptions which looks, and is, very different +from that which ordinary mathematicians think they see; and perhaps this +is the reason why some mathematicians of great eminence, such as Hilbert +and Poincaré, have produced such absurd discussions on the fundamental +principles of mathematics,[79] showing once more the truth of the not +quite original remark of Aunt Jane, who + + ... observed, the second time + She tumbled off a 'bus: + "The step is short from the sublime + To the ridiculous." + +In their readiness to consider many different things as one thing--to +consider, for example, the ratio 2:1 as the same thing as the cardinal +number 2--such mathematicians as Peacock, Hankel, and Schubert were +forestalled by the Pigeon, who thought that Alice and the Serpent were +the same creature, because both had long necks and ate eggs.[80] It is, +however, doubtful whether the Pigeon would have followed the example of +the mathematicians just mentioned so far as to embrace the creed of +nominalism and so to feel no difficulty in subtracting from zero--a +difficulty which was pointed out with great acuteness by the Hatter[81] +and modern mathematical logicians. + +FOOTNOTES: + +[78] These principles, after many attempts to state them by Peacock, the +Red and the White Queen (see Appendix P), Hankel, Schröder, and Schubert +had been made, were first precisely formulated by Frege in _Z. S._; cf. +also Chapter VII. + +[79] See Couturat, _R. M. M._, vol. xiv., March, 1906, pp. 208-50, and +Russell, _ibid._, September, 1906, pp. 627-34. + +[80] See Appendix P. + +[81] See _ibid._ + + + + +CHAPTER XXX + +FINITE AND INFINITE + + +I was once shown a statement made by an eminent mathematician of +Cambridge from which one would conclude that this mathematician thought +that finite distances became infinite when they were great enough. In +one of those splendidly printed books, bound in blue, published by the +University Press, and sold at about a guinea as a guide to some advanced +branch of pure mathematics, one may read, even in the second edition +published in 1900, the words: "Representation [of a complex variable] on +a plane is obviously more effective for points at a finite distance from +the origin than for points at a very great distance." + +Plainly some of the points at a very great distance are at a _finite_ +distance, for the same author mentions that Neumann's sphere for +representing the positions of points on a plane "has the advantage ... +of exhibiting the uniqueness of _z_ = [infinity symbol] as a value of +the variable." + + + + +CHAPTER XXXI + +THE MATHEMATICAL ATTAINMENTS OF TRISTRAM SHANDY + + +Tristram Shandy[82] said that his father was sometimes a gainer by +misfortune; for if the pleasure of haranguing about it was as ten, and +the misfortune itself only as five, he gained "half in half," and was +well off again as if the misfortune had never happened. + +Suppose that the unit (arbitrary) of pleasure is denoted by A, Tristram +Shandy, by neglecting, in this ethical discussion, to introduce negative +quantities (Kant's pamphlet advocating this introduction into philosophy +was made subsequently[83]), apparently made 15A to result, and this can +hardly be maintained to be the half of 10A. It is possible, however, +that Tristram Shandy succeeded in proving the apparently paradoxical +equation + + 15A = 5A + +by remarking that the axiom "the whole is greater than the part" does +not always hold. This remark follows at once from what Mr. Russell[84] +has called "The Paradox of Tristram Shandy." This paradox is described +by Mr. Russell as follows: + +"Tristram Shandy, as we know, took two years writing the history of the +first two days of his life, and lamented that, at this rate, material +would accumulate faster than he could deal with it, so that he could +never come to an end. Now I maintain that, if he had lived for ever, +and not wearied of his task, then, even if his life had continued as +eventfully as it began, no part of his biography would have remained +unwritten." + +This paradox is strictly correlative to the well-known paradox of Zeno +about Achilles and the Tortoise.[85] "The Achilles proves that two +variables in a continuous series, which approach equality from the same +side, cannot ever have a common limit: the Tristram Shandy proves that +two variables which start from a common term, and proceed in the same +direction, but diverge more and more, may yet determine the same +limiting class (which, however, is not necessarily a segment, because +segments were defined as having terms beyond them). The Achilles assumes +that whole and part cannot be similar, and deduces a paradox; the other, +starting from a platitude, deduces that whole and part may be similar. +For common-sense, it must be confessed that it is a most unfortunate +state of things." And Mr. Russell considers that, in the face of proofs, +it ought to commit suicide in despair. + +Now, I suggest the extremely unlikely possibility that Tristram Shandy, +by reflection on his own life and literary labours, was led to the +correct course of accepting the paradox which resulted from this +reflection and rejecting the Achilles. Thus, he concluded that an +infinite whole may be similar (or, in Cantor's terminology, +"equivalent") to a proper part of itself, and hence, by a confusion of +similarity with identity (or equivalence with equality) which he shares +with some subsequent philosophers,[86] that a whole may be equal to a +proper part of itself. If A is an infinite class, it is not difficult to +see that we can have + + 10A = 5A. + +In this way many have avoided an opinion which rests on no better +foundation than that formerly entertained by the inductive philosophers +of Central Africa, that all men are black.[87] + +FOOTNOTES: + +[82] Cf. a letter of De Morgan in Mrs. De Morgan's _Memoir of Augustus +De Morgan_, p. 324. + +[83] Kant's tract was published in 1763, while _Tristram Shandy_ was +published in 1760. + +[84] _P. M._, pp. 358-9 [Cf. _M._, vol. xxii., January 1912, p. +187.--ED.] + +[85] Cf. _P. M._, pp. 350, 358-9; _M._, vol. xxii., 1912, p. 157. + +[86] [Cf. for example, Cosmo Guastella, _Dell' infinito_, Palermo, +1912.--ED.] + +[87] Cf. Russell, _P. M._, p. 360. + + + + +CHAPTER XXXII + +THE HARDSHIPS OF A MAN WITH AN UNLIMITED INCOME + + +I once heard a man refer to his income as limited, in order to +illustrate the hardship of a class of men, of which he of course was +one, in having to pay a somewhat high income-tax. It is obvious that +this man spoke enviously, and consequently admitted the existence of +more fortunately placed individuals who had unlimited incomes. A little +reflection would have shown the man that he was not taking up a +paradoxical attitude. A "paradoxical attitude" is of course the +assertion of one or more propositions of which the truth cannot be +perceived by a philosopher--and particularly an idealist--and can be +perceived by a logician and occasionally, but not always, by a man of +common-sense. Such propositions are: "The cat is hungry," "Columbus +discovered America," and "A thing which is always at rest may move from +the position A to the different position B." + +Now, if a man had an unlimited income, it is an immediate inference +that, however low income-tax might be, he would have to pay annually to +the Exchequer of his nation a sum equal in value to his whole income. +Further, if his income was derived from a capital invested at a finite +rate of interest (as is usual), the annual payments of income-tax would +each be equal in value to the man's whole capital. If, then, the man +with an unlimited income chose to be discontented, he would be sure of a +sympathetic audience among philosophers and business acquaintances; but +discontent could not last long, for the thought of the difficulties he +was putting in the way of the Chancellor of the Exchequer, who would +find the drawing up of his budget most puzzling, would be amusing. +Again, the discovery that, after paying an infinite income-tax, the +income would be quite undiminished, would obviously afford satisfaction, +though perhaps the satisfaction might be mixed with a slight uneasiness +as to any action the Commissioners of Income-Tax might take in view of +this fact. + +A problem of a wholly different nature is connected with the possible +purchase by the man with an unlimited income of an enumerable infinity +of pairs of boots. If he wished to prove that he had an even number of +boots, it would be easy if right boots were distinguishable from left +ones, but if the man were a faddist of such a kind that he insisted that +his left boots should not be made in any way differently from his right +ones, it would not be possible for him to prove the theorem mentioned +unless he assumed what is known as "the multiplicative axiom." In fact +this axiom shows that it is legitimate to pick out an infinite +succession of members of an infinite class in an arbitrary way. In the +case of the pairs of boots, each pair contains two members, and if there +is no means of distinguishing between them, when we wish to pick out one +of them for each of the infinity of pairs, we cannot say which ones we +mean to pick out unless we assume, by means of the above axiom, that a +particularized member can always be found even with things of each of +which it can be said that, like Private James in the _Bab Ballads_, + + No characteristic trait had he + Of any distinctive kind. + +However, a solution of the puzzle was given by Dr. Dénes König of +Budapest. You first prove that there are points in space such that, if P +is one of them, not more than a finite number of pairs of boots are such +that each centre of mass of the two members of a pair is equidistant +from P. Taking a point P of this sort, select from each pair the boot +whose centre of mass is nearest P. (There may be a finite number of +pairs left over, but they can be dealt with arbitrarily.) + +Another form of the problem is as follows. Every time the man bought a +pair of boots he also bought a pair of socks to go with it; he had an +enumerable infinity of pairs of each, and the problem is to prove that +he had as many boots as he had socks. In this case the boots, we will +suppose, can be divided into right and left, but the socks cannot. Thus +there are an enumerable infinity of boots, but the number of the socks +cannot be determined without admitting the axiom mentioned above. A +further difficulty might arise if the owner of the boots and socks lost +one leg in some accident, and told his butler to give away half his +socks. Naturally the butler would find great logical difficulties in so +doing, and it would seem to be an interesting ethical problem whether he +should be dismissed from his situation for failing to prove the +multiplicative axiom. Again, if the butler stole a pair of boots, the +millionaire would have as many pairs as before, but might have fewer +boots. There is as yet no evidence that the number of his boots is equal +to or greater than the number of pairs. + + + + +CHAPTER XXXIII + +THE RELATIONS OF MAGNITUDE OF CARDINAL NUMBERS + + +The theorems of cardinal arithmetic are frequently used in ordinary +conversation. What is known as the Schröder-Bernstein theorem was used, +long before Bernstein or Schröder, by Edward Thurlow, afterward the +law-lord Lord Thurlow, when an undergraduate of Caius College, +Cambridge. Thurlow was rebuked for idleness by the Master, who said to +him: "Whenever I look out of the window, Mr. Thurlow, I see you crossing +the Court." The provost thus asserted a one-one correspondence between +the class A of his acts of looking out of the window and a part of the +class B of Thurlow's acts of crossing the Court. Thurlow asserted in +reply a one-one correspondence between B and a part of A: "Whenever +I cross the Court I see you looking out of the window." The +Schröder-Bernstein theorem, then, allows us to conclude that there is a +one-one correspondence between the classes A and B. That A and B were +finite classes is not the fault of the Master or Thurlow; nor is it +relevant logically. + + + + +CHAPTER XXXIV + +THE UNKNOWABLE + + +According to Mr. S. N. Gupta,[88] the first thing that every student of +Hindu logic has to learn when he is said to begin the study of inference +is that "all H is S" is not always equivalent to "No H is not S." "The +latter proposition is an absurdity when S is _Kebalánvayi_, i.e. covers +the whole sphere of thought and existence.... 'Knowable' and 'Nameable' +are among the examples of _Kebalánvayi_ terms. If you say there is a +thing not-knowable, how do you know it? If you say there is a thing +not-nameable, you must point that out, i.e. somehow name it. Thus you +contradict yourself." + +Mr. Herbert Spencer's doctrine of the "Unknowable" gives rise to some +amusing thoughts. To state that all knowledge of such and such a thing +is above a certain person's intelligence is not self-contradictory, but +merely rude: to state that all knowledge of a certain thing is above all +possible human intelligence is nonsense, in spite of its modest, +platitudinous appearance. For the statement seems to show that we do +know something of it, viz. that it is unknowable. + +To the last (1900) edition of _First Principles_ was added a "Postscript +to Part I," in which the justice of this simple and well-known criticism +as to the contradiction involved in speaking of an "Unknowable," which +had been often made during the forty odd years in which the various +editions had been on the market, was grudgingly acknowledged as +follows:[89] + +"It is doubtless true that saying what a thing is not, is, in some +measure, saying what it is;... Hence it cannot be denied that to affirm +of the Ultimate Reality that it is unknowable is, in a remote way, to +assert some knowledge of it, and therefore involves a contradiction." + +The "Postscript" reminds one of the postscript to a certain Irishman's +letter. This Irishman, missing his razors after his return from a visit +to a friend, wrote to his friend, giving precise directions where to +look for the missing razors; but, before posting the letter, added a +postscript to the effect that he had found the razors. + +One is tempted to inquire, analogously, what might be, in view of the +Postscript, the point of much of Spencer's Part I. It is, to use De +Morgan's[90] description of the arguments of some who maintain that we +can know nothing about infinity, of the same force as that of the man +who answered the question how long he had been deaf and dumb. + +But the best part of the joke against Mr. Spencer is that he, as we +shall see in Chapter XXXVIII, was refuted by a fallacious argument, and +thus mistakenly asserted the validity of the refutation of remarks which +happen to be unsound. + +The analogy of the contradiction of Burali-Forti with the contradiction +involved in the notion of an "unknowable" may be set forth as follows. +If A should say to B: "I know things which you never by any possibility +can know," he may be speaking the truth. In the same way, [Greek: ô] may +be said, without contradiction, to transcend all the _finite_ integers. +But if some one else, C, should say: "There are some things which no +human being can ever know anything about," he is talking nonsense.[91] +And in the same way if we succeeded in imagining a number which +transcends _all_ numbers, we have succeeded in imagining the absurdity +of a number which transcends itself. + +All the paradoxes of logic (or "the theory of aggregates") are +analogous to the difficulty arising from a man's statement: "I am +lying."[92] In fact, if this is true, it is false, and _vice versa_. If +such a statement is spread out a little, it becomes an amusing hoax or +an epigram. Thus, one may present to a friend a card bearing on both +sides the words: "The statement on the other side of this card is +false"; while the first of the epigrams derived from this principle +seems to have been written by a Greek satirist:[93] + + Lerians are bad; not _some_ bad and some _not_; + But all; there's not a Lerian in the lot, + Save Procles, that you could a good man call;-- + And Procles--is a Lerian after all. + +This is the original of a well-known epigram by Porson, who remarked +that all Germans are ignorant of Greek metres, + + All, save only Hermann;-- + And Hermann's a German. + +FOOTNOTES: + +[88] _Md._, N. S., vol. iv., 1895, p. 168. + +[89] _First Principles_, 6th ed., 1900, pp. 107-10. The first edition +was published in 1862. + +[90] Note on p. 6 of his paper: "On Infinity; and on the Sign of +Equality," _Trans. Camb. Phil. Soc._, vol. xi., part i., pp. 1-45 (read +May 16, 1864). + +[91] The assertion of the finitude of a man's mind appears to be +nonsense; both because, if we say that the mind of man is limited we +tacitly postulate an "unknowable," and because, even if the human mind +were finite, there is no more reason against its conceiving the infinite +than there is for a mind to be blue in order to conceive a pair of blue +eyes (cf. De Morgan, _loc. cit._). + +[92] Russell, _R. M. M._, vol. xiv., September 1906, pp. 632-3, 640-4. + +[93] _The Greek Anthology_, by Lord Neaves (Ancient Classics for English +Readers), Edinburgh and London, 1897, p. 194. + + + + +CHAPTER XXXV + +MR. SPENCER, THE ATHANASIAN CREED AND THE ARTICLES + + +When, in what I believe is misleadingly known as "The Athanasian Creed," +people say "The Father incomprehensible," and so on, they are not +falling into the same error as Mr. Spencer, for the Latin equivalent for +"incomprehensible" is merely "_immensus_," and Bishop Hilsey translated +it more correctly as "immeasurable."[94] It is a regrettable fact that +Dr. Blunt,[95] in his mistaken modesty, has added a note to this passage +that: "Yet it is true that a meaning not intended in the Creed has +developed itself through this change of language, for the Nature of God +is as far beyond the grasp of the mind as it is beyond the possibility +of being contained within local bounds." + +Mr. Spencer seems no happier when we compare his statements with those +in the Anglican Articles of Religion. There God is never referred to as +infinite. It is true that His power and goodness are so referred to; but +this deficiency was presumably brought about intentionally, so that +faith might gain in meaning as time went on. + +FOOTNOTES: + +[94] _A. C. P._, p. 217. + +[95] _Ibid._, p. 218. + + + + +CHAPTER XXXVI + +THE HUMOUR OF MATHEMATICIANS + + +Brahmagupta's problem[96] appears to be the earliest instance of a kind +of joke which has been much used by mathematicians. For the sake of +giving a certain picturesqueness to the data of problems, and so to +excite that sort of interest which is partly expressed by a smile, +mathematicians have got into the habit of talking, for example, of +monkeys in the form of geometrical points climbing up massless ropes. +Professor P. Stäckel[97] truly remarked that physiological +mechanics--the mechanics of bones, muscles, and so on--is wholly +different from this. There was once a lecturer on mathematics at +Cambridge who used yearly to propound to his pupils a problem in rigid +dynamics which related to the motion of a garden roller supposed to be +without mass or friction, when a heavy and perfectly rough insect walked +round the interior of it in the direction of normal rolling. + +Hitherto this has been the only mathematical outlet for the humour of +mathematicians; and those who really had the interests of mathematics at +heart saw with alarm the growing tendency towards scholasticism in +mathematical jokes. Fortunately the discovery of logic by some +mathematicians has removed this danger. Still to many mathematicians +logic is still unknown, and to them--to Professor A. Schoenflies for +example--modern mathematics, owing to its alliance with logic, appears +to be sinking into scholasticism. It is true that the word +"scholasticism" is not used by Professor Schoenflies in any +intentionally precise signification, but merely as a vague epithet of +disapproval, as the word "socialism" is used by the ordinary philistine, +and this would certainly serve as a sufficient excuse. But no excuse is +needed: these opinions are themselves a source of mathematical jokes. + +FOOTNOTES: + +[96] See Chapter XII. + +[97] _Encykl. der math. Wiss._, vol. iv., part i., p. 474. + + + + +CHAPTER XXXVII + +THE PARADOXES OF LOGIC + + +We have already[98] referred to the contempt shown by some +mathematicians for exact thought, which they condemn under the name +of "scholasticism." An example of this is given by Schoenflies in +the second part of his publication usually known as the _Bericht +über Mengenlehre_.[99] Here[100] a battle-cry in italics-- + + "_Against all resignation, but also against all scholasticism!_"-- + +found utterance. Later on, Schoenflies[101] became bolder and adopted a +more personal battle-cry, also in italics, and with a whole line to +itself: + + "_For Cantorism but against Russellism!_" + +"Cantorism" means the theory of transfinite aggregates and numbers +erected for the most part by Georg Cantor. Shortly speaking, the great +sin of "Russellism" is to have gone too far in the chain of logical +deduction for many mathematicians, who were perhaps, like +Schoenflies,[102] blinded by their rather uncritical love of +mathematics. Thus it comes about that Schoenflies[103] denounces +Russellism as "scholastic and unhealthy." This queer blend of qualities +would surely arouse the curiosity of the most _blasé_ as to what strange +thing Russellism must be.[104] + +Schoenflies[105] said that some mathematicians attributed to the logical +paradoxes which have given Russell so much trouble to clear up, +"especially to those that are artificially constructed, a significance +that they do not have." Yet no grounds were given for this assertion, +from which it might be concluded that the rigid examination of any +concept was unimportant. The paradoxes are simply the necessary results +of certain logical views which are currently held, which views do not, +except when they are examined rather closely, appear to contain any +difficulty. The contradiction is not felt, as it happens, by people who +confine their attention to the first few number-classes of Cantor, and +this seems to have given rise to the opinion, which it is a little +surprising to find that some still hold, that cases not usually met +with, though falling under the same concept as those usually met with, +are of little importance. One might just as well maintain that +continuous but not differentiable functions are unimportant because they +are artificially constructed--a term which I suppose means that they do +not present themselves when unasked for. Rather should we say that it is +by the discovery and investigation of such cases that the concept in +question can alone be judged, and the validity of certain theorems--if +they are valid--conclusively proved. That this has been done, chiefly by +the work of Russell, is simply a fact; that this work has been and is +misunderstood by many[106] is regrettable for this reason, among others, +that it proves that, at the present time, as in the days in which +_Gulliver's Travels_ were written, some mathematicians are bad +reasoners.[107] + +Nearly all mathematicians agreed that the way to solve these paradoxes +was simply not to mention them; but there was some divergence of opinion +as to how they were to be unmentioned. It was clearly unsatisfactory +merely not to mention them. Thus Poincaré was apparently of opinion that +the best way of avoiding such awkward subjects was to mention that they +were not to be mentioned. But[108] "one might as well, in talking to a +man with a long nose, say: 'When I speak of noses, I except such as are +inordinately long,' which would not be a very successful effort to avoid +a painful topic." + +Schoenflies, in his paper of 1911 mentioned above, adopted the +convenient plan of referring these logical difficulties at the root of +mathematics to a department of knowledge which he called "philosophy." +He said[109] of the theory of aggregates that though "born of the +acuteness of the mathematical spirit, it has gradually fallen into +philosophical ways, and has lost to some extent the compelling force +which dwells in the mathematical process of conclusion." + +The majority of mathematicians have followed Schoenflies rather than +Poincaré, and have thus adopted tactics rather like those of the March +Hare and the Gryphon,[110] who promptly changed the subject when Alice +raised awkward questions. Indeed, the process of the first of these +creatures of a child's dream is rather preferable to that of +Schoenflies. The March Hare refused to discuss the subject because he +was bored when difficulties arose. Schoenflies would not say that he was +bored--he professed interest in philosophical matters, but simply called +the logical continuation of a subject by another name when he did not +wish to discuss the continuation, and thus implied that he had discussed +the whole subject. Further, Schoenflies would not apparently admit that +the one method of logic could be applied to the solution of both +mathematical and philosophical problems, in so far as these problems are +soluble at all; but the March Hare, shortly before the remark we have +just quoted, rightly showed great astonishment that butter did not help +to cure both hunger and watches that would not go.[111] The judgment of +Schoenflies by which certain apparently mathematical questions were +condemned as "philosophical," rested on grounds as flimsy as those in +the Dreyfus Case, or the Trial in _Wonderland_.[112] + +FOOTNOTES: + +[98] Chapters VII and XXXVI. + +[99] _Die Entwickelung der Lehre von den Punktmannigfaltigkeiten._ +Bericht, erstattet der deutschen Mathematiker-Vereinigung, Leipzig, +1908. + +[100] _Ibid._, p. 7. The battle-cry is: "_Gegen jede Resignation, aber +auch gegen jede Scholastik!_" + +[101] "Ueber die Stellung der Definition in der Axiomatik," _Jahresber, +der deutsch. Math.-Ver._, vol. xx., 1911, pp. 222-5. The battle-cry is +on p. 256 and is: "Für den Cantorismus aber gegen den Russellismus!" + +[102] _Ibid._, p. 251. "Es ist also," he exclaims with the eloquence of +emotion and the emotion of eloquence, "nicht die Geringschätzung der +Philosophie, die mich dabei treibt, sondern die Liebe zur +Mathematik;..." + +[103] "Ueber die Stellung der Definition in der Axiomatik," _Jahresber, +der deutsch. Math.-Ver._, vol. xx., 1911, p. 251. + +[104] [Cf. for this, _M._, vol. xxii., January 1912, pp. 149-58.--ED.] + +[105] _Bericht_, 1908, p. 76, note; cf. p. 72. + +[106] E.g. in F. Hausdorff's review of Russell's _Principles_ of 1903 in +the _Vierteljahrsschr. für wiss. Philos. und Soziologie_. + +[107] [Cf. _M._, vol. xxv., 1915, pp. 333-8.--ED.] + +[108] Russell, _A. J. M._, vol. xxx., 1908, p. 226. + +[109] _Loc. cit._, p. 222. + +[110] See Appendix Q. + +[111] See Appendix R. + +[112] See Appendix S. + + + + +CHAPTER XXXVIII + +MODERN LOGIC AND SOME PHILOSOPHICAL ARGUMENTS + + +The most noteworthy reformation of recent years in logic is the +discovery and development by Mr. Bertrand Russell of the fact that the +paradoxes--of Burali-Forti, Russell, König, Richard, and others--which +have appeared of late years in the mathematical theory of aggregates and +have just been referred to, are of an entirely _logical_ nature, and +that their avoidance requires us to take account of a principle which +has been hitherto unrecognized, and which renders invalid several +well-known arguments in refutation of scepticism, agnosticism, and the +statement of a man that he asserts nothing. + +Dr. Whitehead and Mr. Russell say:[113] "The principle which enables us +to avoid illegitimate totalities may be stated as follows: 'Whatever +involves _all_ of a collection must not be one of the collection,' or +conversely: 'If, provided a certain collection had a total, it would +have members only definable in terms of that total, then the said +collection has no total.' We shall call this the 'vicious-circle +principle,' because it enables us to avoid the vicious circles involved +in the assumption of illegitimate totalities. Arguments which are +condemned by the vicious-circle principle will be called 'vicious-circle +fallacies.' Such arguments, in certain circumstances, may lead to +contradictions, but it often happens that the conclusions to which they +lead are in fact true, though the arguments are fallacious. Take, for +example, the law of excluded middle in the form 'all propositions are +true or false.' If from this law we argue that, because the law of +excluded middle is a proposition, therefore the law of excluded middle +is true or false, we incur a vicious-circle fallacy. 'All propositions' +must be in some way limited before it becomes a legitimate totality, and +any limitation which makes it legitimate must make any statement about +the totality fall outside the totality. Similarly the imaginary sceptic +who asserts that he knows nothing and is refuted by being asked if he +knows that he knows nothing, has asserted nonsense, and has been +fallaciously refuted by an argument which involves a vicious-circle +fallacy. In order that the sceptic's assertion may become significant it +is necessary to place some limitation upon the things of which he is +asserting his ignorance; the proposition that he is ignorant of every +member of this collection must not itself be one of the collection. +Hence any significant scepticism is not open to the above form of +refutation." + +In fact, the world of things falls into various sets of things of the +same "type." For every propositional function [Greek: ph](_x_) there is +a range of values of _x_ for which [Greek: ph](_x_) has a signification +as a true or a false proposition. Until this theory was brought forward, +there were occasionally discussions as to whether an object which did +not belong to the range of a certain propositional function possessed +the corresponding property or not. Thus, Jevons, in early days,[114] was +of opinion that virtue is neither black nor not-black because it is not +coloured, but rather later[115] he admitted that virtue is not +triangular.[116] + +FOOTNOTES: + +[113] _Pa. Ma._, p. 40. + +[114] _S. o. S._ pp. 36-7. + +[115] _E. L. L._, pp. 120-1. + +[116] [It may perhaps be added that, some years after Mr. R*ss*ll's +death, Dr. Whitehead stated, in an address delivered in 1916 and +reprinted in his book on _The Organisation of Thought_ (London, 1917, p. +120), that "the specific heat of virtue is 0.003 is, I should imagine, +not a proposition at all, so that it is neither true nor +false...."--ED.] + + + + +CHAPTER XXXIX + +THE HIERARCHY OF JOKES + + +Jokes may be divided into various types. Thus a joke or class of jokes +can only be the subject of a joke of higher order. Otherwise we would +get the same vicious-circle fallacy which gives rise to so many +paradoxes in logic and mathematics. A certain Oxford scholar succeeded, +to his own satisfaction, in reducing all jokes to primitive types, +consisting of thirty-seven proto-Aryan jokes. When any proposition was +propounded to him, he would reflect and afterwards pronounce on the +question as to whether the proposition was a joke or not. If he decided, +by his theory, that it was a joke, he would solemnly say: "There _is_ +that joke." If this narration is accepted as a joke, since it cannot be +reduced to one of the proto-Aryan jokes under pain of leading us to +commit a vicious-circle fallacy, we must conclude that there is at least +one joke which is not proto-Aryan; and, in fact, is of a higher type. +There is no great difficulty in forming a hierarchy of jokes of various +types. Thus a joke of the fourth type (or order) is as follows: A joke +of the first order was told to a Scotchman, who, as we would expect, was +unable to see it.[117] The person (A) who told this joke told the story +of how the joke was received to another Scotchman thereby making a joke +about a joke of the first order, and thus making a joke of the second +order. A remarked on this joke that no joke could penetrate the head of +the Scotchman to whom the joke of the first order was told, even if it +were fired into his head with a gun. The Scotchman, after severe +thought, replied: "But ye couldn't do that, ye know!" A repeated the +whole story, which constituted a joke of the third order, to a third +Scotchman. This last Scotchman again, after prolonged thought, replied: +"He had ye there!" This whole story is a joke of the fourth order. + +Most known jokes are of the first order, for the simple reason that the +majority of people find that the slightest mental effort effectually +destroys any perception of humour. It seems to me that a joke becomes +more pleasurable in proportion as logical faculties are brought into +play by it; and hence that logical power is allied, or possibly +identical, with the power of grasping more subtle jokes. The jokes which +amuse the frequenters of music-halls, Conservatives, and M. Bergson--and +which usually deal with accidents, physical defects, mothers-in-law, +foreigners, or over-ripe cheese--are usually jokes of the first order. +Jokes of the second, and even of the third, order appeal to ordinary +well-educated people; jokes of higher order require either special +ability or a sound logical training on the part of the hearer if the +joke is to be appreciated; while jokes of transfinite order presumably +only excite the inaudible laughter of the gods. + +FOOTNOTES: + +[117] [It may be that, like certain remarks about cheese and +mothers-in-law (see below), the statement that Scotchmen cannot see +jokes is a joke of the first order.--ED.] + + + + +CHAPTER XL + +THE EVIDENCE OF GEOMETRICAL PROPOSITIONS + + +It has often been maintained that the twentieth proposition of the first +book of Euclid--that two sides of a triangle are together greater than +the third side--is evident even to asses. This does not, however, seem +to me generally true. I once asked a coastguardsman the distance from A +to B; he replied: "Eight miles." On further inquiry I elicited the fact +that the distance from A to C was two miles and the distance from C to B +was twenty-two miles. Now the paths from A to B and from C to B were by +sea; while the path from A to C was by land. Hence if the path by land +was rugged and the distance along the road was two miles, it would +appear that the coastguardsman believed that not only could one side of +a triangle be greater than the other two, but that one straight side of +a triangle might be greater than one straight side and any curvilinear +side of the same triangle. The only escape from part of this astonishing +creed would be by assuming that the distance of two miles from A to C +was measured "as the crow flies," while the road A to C was so hilly +that a pedestrian would traverse more than fourteen miles when +proceeding from A to C. Then indeed the coastguardsman could maintain +the true proposition that there is at least one triangle ABC, with the +side AC curvilinear, such that the sum of the lengths of AB and AC is +greater than the length of BC, and only deny the twentieth proposition +of the first book of Euclid. + +Reasoning with the coastguardsman only had the effect of his adducing +the authority of one Captain Jones in support of the accuracy of his +data. Possibly Captain Jones held strange views as to the influence of +temperature or other physical circumstances, or even the nature of space +itself, on the lengths of lines in the neighbourhood of the triangle +ABC. + + + + +CHAPTER XLI + +ABSOLUTE AND RELATIVE POSITION + + +Some people maintain that position in space or time must be relative +because, if we try to determine the position of a body A, if bodies B, +C, D with respect to which the position of A could be determined were +not present, we should be trying to determine something about A without +having our senses affected by other things. These people seem to me to +be like the cautious guest who refused to say anything about his host's +port-wine until he had tasted red ink. + +"Wherein, then," says Mr. Russell,[118] "lies the plausibility of the +notion that all points are exactly alike? This notion is, I believe, a +psychological illusion, due to the fact that we cannot remember a point +so as to know it when we meet again. Among simultaneously presented +points it is easy to distinguish; but though we are perpetually moving, +and thus being brought among new points, we are quite unable to detect +this fact by our senses, and we recognize places only by the objects +they contain. But this seems to be a mere blindness on our part--there +is no difficulty, so far as I can see, in supposing an immediate +difference between points, as between colours, but a difference which +our senses are not constructed to be aware of. Let us take an analogy: +Suppose a man with a very bad memory for faces; he would be able to +know, at any moment, whether he saw one face or many, but he would not +be aware whether he had seen any of the faces before. Thus he might be +led to define people by the rooms in which he saw them, and to suppose +it self-contradictory that new people should come to his lectures, or +that old people should cease to do so. In the latter point at least it +will be admitted by lecturers that he would be mistaken. And as with +faces, so with points--inability to recognize them must be attributed, +not to the absence of individuality, but merely to our incapacity." + +Another form of this tendency is shown by Kronecker, Borel, Poincaré, +and many other mathematicians, who refuse mere logical determination of +a conception and require that it be actually described in a finite +number of terms. These eminent mathematicians were anticipated by the +empirical philosopher who would not pronounce that the "law of thought" +that A is either in the place B or not is true until he had looked to +make sure. This philosopher was of the same school as J. S. Mill and +Buckle, who seem to have maintained implicitly not only that, in view of +the fact that the breadth of a geometrical line depends upon the +material out of which it is constructed, or upon which it is drawn, that +there ought to be a paste-board geometry, a stone geometry, and so +on;[119] but also that the foundations of logic are inductive in their +nature.[120] "We cannot," says Mill,[121] "conceive a round square, not +merely because no such object has ever presented itself in our +experience, for that would not be enough. Neither, for anything we know, +are the two ideas in themselves incompatible. To conceive a body all +black and yet white would only be to conceive two different sensations +as produced in us simultaneously by the same object--a conception +familiar to our experience--and we should probably be as well able to +conceive a round square as a hard square, or a heavy square, if it were +not that in our uniform experience, at the instant when a thing begins +to be round, it ceases to be square, so that the beginning of the one +impression is inseparably associated with the departure or cessation of +the other. Thus our inability to form a conception always arises from +our being compelled to form another contradictory to it." + +FOOTNOTES: + +[118] _Md._, N. S., vol. x., July, 1901, pp. 313-14. + +[119] J. B. Stallo, _The Concepts and Theories of Modern Physics_, 4th +ed., London, 1900, pp. 217-27. + +[120] _Ibid._, pp. 140-4. + +[121] _Examination of the Philosophy of Sir William Hamilton_, vol. i. +p. 88, Amer. ed. + + + + +CHAPTER XLII + +LAUGHTER + + +[It seemed advisable to give here[122] some views on laughter, most of +which were also held by Mr. R*ss*ll, though no written expression of his +views has yet been found. In a review[123] of M. Bergson's book on +_Laughter_,[124] Mr. Russell has remarked: + +"It has long been recognized by publishers that everybody desires to be +a perfect lady or gentleman (as the case may be); to this fact we owe +the constant stream of etiquette-books. But if there is one thing which +people desire even more, it is to have a faultless sense of humour. Yet +so far as I know, there is no book called 'Jokes without Tears, by Mr. +McQuedy.' This extraordinary lacuna has now been filled. Those to whom +laughter has hitherto been an unintelligible vagary, in which one must +join, though one could never tell when it would break out, need only +study M. Bergson's book to acquire the finest flower of Parisian wit. By +observing a very simple formula they will know infallibly what is funny +and what is not; if they sometimes surprise their unlearned friends, +they have only to mention their authority in order to silence doubt. +'The attitudes, gestures and movements of the human body,' says M. +Bergson, 'are laughable in exact proportion as that body reminds us of a +mere machine.' When an elderly gentleman slips on a piece of orange-peel +and falls, we laugh, because his body follows the laws of dynamics +instead of a human purpose. When a man falls from a scaffolding and +breaks his neck on the pavement, we presumably laugh even more, since +the movement is even more completely mechanical. When the clown makes a +bad joke for the first time, we keep our countenance, but at the fifth +repetition we smile, and at the tenth we roar with laughter, because we +begin to feel him a mere automaton. We laugh at Molière's misers, +misanthropists and hypocrites, because they are mere types mechanically +dominated by a master impulse. Presumably we laugh at Balzac's +characters for the same reason; and presumably we never smile at +Falstaff, because he is individual throughout." + +The review concludes with the reflection that "it would seem to be +impossible to find any such formula as M. Bergson seeks. Every formula +treats what is living as if it were mechanical, and is therefore by his +own rule a fitting object of laughter." Now, this undoubtedly true +conclusion has been obtained, as is readily seen, by a vicious-circle +fallacy which Mr. R*ss*ll would hardly have committed.--ED.] + +FOOTNOTES: + +[122] From a remark on p. 47 above, it is evident that Mr. R*ss*ll +intended to write some such chapter as this. + +[123] _The Professor's Guide to Laughter, The Cambridge Review_, vol. +xxxii., 1912, pp. 193-4. + +[124] _Laughter, an Essay on the Meaning of the Comic_, English +translation by C. Brereton and F. Rothwell, London, 1911. + + + + +CHAPTER XLIII + +"GEDANKENEXPERIMENTE" AND EVOLUTIONARY ETHICS + + +The "Gedankenexperimente," upon which so much weight has been laid by +Mach[125] and Heymans,[126] had already been investigated by the White +Queen,[127] who, however, seems to have perceived that the results of +such experiments are not always logically valid. The psychological +founding of logic appears to be not without analogy with the surprising +method of advocates of evolutionary ethics, who expect to discover what +_is_ good by inquiring what cannibals have _thought_ good. I sometimes +feel inclined to apply the historical method to the multiplication +table. I should make a statistical inquiry among school-children, before +their pristine wisdom had been biassed by teachers. I should put down +their answers as to what 6 times 9 amounts to, I should work out the +average of their answers to six places of decimals, and should then +decide that, at the present stage of human development, this average is +the value of 6 times 9. + +FOOTNOTES: + +[125] See, e.g., _E. u. I._, pp. 183-200. + +[126] _G. u. E._, vol. i. + +[127] See Appendix T. + + + + +APPENDIXES + + +A. LOGIC AND THE PRINCIPLE OF IDENTITY. + +_T. L. G._, p. 45: "'Contrariwise," continued Tweedledee, "if it was so, +it might be; and if it were so, it would be: but as it isn't, it ain't. +That's logic." + + * * * * * + +_S. B._, p. 159: The Professor said: "The day is the same length as +anything that is the same length as _it_." + + * * * * * + +_S. B._, p. 161: Bruno observed that, when the Other Professor lost +himself, he should shout: "He'd be sure to hear hisself, 'cause he +couldn't be far off." + + +B. SYNTHESIS OF CONTRADICTORIES. + +_T. L. G._, p. 71: "'What a beautiful belt you've got on!' Alice +suddenly remarked.... 'At least,' she corrected herself on second +thoughts, 'a beautiful cravat, I should have said--no, a belt, I mean--I +beg your pardon!' she added in dismay, for Humpty-Dumpty looked +thoroughly offended, and she began to wish she hadn't chosen that +subject. 'If only I knew,' she thought to herself, 'which was neck and +which was waist!'" + + +C. EMPIRICAL PHILOSOPHERS AND MATHEMATICS. + +_T. L. G._, p. 79: "'... Now if you had the two eyes on the same side of +the nose, for instance--or the mouth at the top--that would be _some_ +help.' + +"'It wouldn't look nice,' Alice objected. But Humpty-Dumpty only shut +his eyes and said: 'Wait till you've tried.'" + + * * * * * + +_T. L. G._, p. 72: "'And if you take one from three hundred and +sixty-five, what remains?' + +"'Three hundred and sixty-four, of course.' + +"Humpty-Dumpty looked doubtful. 'I'd rather see that done on paper,' +he said." + + +D. NOMINAL DEFINITION. + +_T. L. G._, p. 73: "'When _I_ used a word,' Humpty-Dumpty said in rather +a scornful tone, 'it means just what I choose it to mean--neither more +nor less.' + +"'The question is,' said Alice, 'whether you _can_ make words mean +different things.' + +"'The question is,' said Humpty-Dumpty, 'which is to be master--that's +all.'" + + +E. CONFORMITY OF A PARADOXICAL LOGIC WITH COMMON-SENSE. + +_T. L. G._, p. 100: + + "But I was thinking of a plan + To dye one's whiskers green, + And always use so large a fan + That they could not be seen." + (Verse from White Knight's song.) + + +F. IDEALISTS AND THE LAWS OF LOGIC. + +_T. L. G._, p. 52-3: Tweedledee exclaimed: "'... if he [the Red King] +left off dreaming about you [Alice], where do you suppose you'd be?' + +"'Where I am now, of course,' said Alice. + +"'Not you!' Tweedledee retorted contemptuously. 'You'd be nowhere. Why, +you're only a sort of thing in his dream!' + +"'If that there King was to wake,' added Tweedledum, 'you'd go +out--bang!--just like a candle!' + +"'I shouldn't!' Alice exclaimed indignantly. 'Besides, if _I'm_ only a +sort of thing in his dream, what are _you_, I should like to know?' + +"'Ditto,' said Tweedledum...; 'you know very well you're not real.' + +"'I _am_ real!' said Alice, and began to cry." + + * * * * * + +_T. L. G._, p. 97: "'How _can_ you go on talking so quietly, head +downwards?' Alice asked, as she dragged him out by the feet, and laid +him in a heap on the bank. + +"The Knight looked surprised at the question. 'What does it matter where +my body happens to be?' he said. 'My mind goes on working all the same. +In fact, the more head downwards I am, the more I keep inventing new +things.'" + + * * * * * + +_T. L. G._, p. 98: "'... Everybody that hears me sing--either it brings +the _tears_ into their eyes, or else----' + +"'Or else what?' said Alice, for the Knight had made a sudden pause. + +"'Or else it doesn't, you know.'" + + +G. DISTINCTION BETWEEN SIGN AND SIGNIFICATION. + +_T. L. G._, pp. 98-9: "'The name of the song is called "_Haddocks' +Eyes_."' + +"'Oh, that's the name of the song, is it?' Alice said, trying to feel +interested. + +"'No, you don't understand,' the Knight said looking a little vexed. +'That's what the name is _called_. The name really _is_ "_The Aged Aged +Man_."' + +"'Then I ought to have said "That's what the _song_ is called"?' Alice +corrected herself. + +"'No, you oughtn't: that's another thing. The _song_ is called "_Ways +and Means_": but that's only what it's _called_, you know!' + +"'Well, what _is_ the song, then?' said Alice, who was by this time +completely bewildered. + +"'I was coming to that,' the Knight said. 'The song really _is +"A-sitting on a Gate_"....'" + + +H. NOMINALISM. + +_A. A. W._, p. 70: "'Then you should say what you mean,' the March Hare +went on. + +"'I do,' Alice hastily replied; 'at least--at least I mean what I +say--that's the same thing, you know.' + +"'Not the same thing a bit!' said the Hatter. 'Why, you might just as +well say that "I see what I eat" is the same thing as "I eat what I +see."' + +"'You might just as well say,' added the March Hare, 'that "I like what +I get" is the same thing as "I get what I like"!' + +"'You might just as well say,' added the Dormouse, which seemed to be +talking in its sleep, 'that "I breathe when I sleep" is the same as "I +sleep when I breathe"!' + +"'It _is_ the same thing with you,' said the Hatter; and here the +conversation dropped,..." + + +I. UTILITY OF SYMBOLIC LOGIC. + +_A. A. W._, p. 92: "'I quite agree with you,' said the Duchess, 'and the +moral of that is--"Be what you would seem to be"--or if you'd like it +put more simply--"Never imagine yourself not to be otherwise than what +it might appear to others that what you were or might have been was not +otherwise than what you had been would have appeared to them to be +otherwise."' + +"'I think I should understand that better,' Alice said very politely, +'if I had it written down: but I can't quite follow it as you say it.' + +"'That's nothing to what I could say if I chose,' the Duchess replied, +in a pleased tone." + + +J. MISTAKE AS TO THE NATURE OF CRITICISM. + +_T. L. G._, p. 105: "'She's in that state of mind,' said the White +Queen, 'that she wants to deny _something_--only she doesn't know what +to deny.' + +"'A nasty, vicious temper,' the White Queen remarked; and then there was +an uncomfortable silence for a minute or two." + + +K. A CRITERION OF TRUTH. + +_H. S._, p. 3: + + "Just the place for a Snark! I have said it twice: + That alone should encourage the crew. + Just the place for a Snark! I have said it thrice: + What I tell you three times is true." + + * * * * * + +_H. S._, p. 50: + + "'Tis the note of the Jubjub! Keep count. I entreat; + You will find I have told it you twice. + 'Tis the song of the Jubjub! The proof is complete, + If only I've stated it thrice." + + +L. UNIVERSAL AND PARTICULAR PROPOSITIONS. + +_T. L. G._, p. 40: The Gnat had told Alice that the Bread-and-butterfly +lives on weak tea with cream in it; so: + +"'Supposing it couldn't find any?' she suggested. + +"'Then it would die, of course.' + +"'But that must happen very often,' Alice remarked thoughtfully. + +"'It always happens,' said the Gnat." + + +M. DENOTING. + +_T. L. G._, p. 43: Tweedledum and Tweedledee were, in many respects, +indistinguishable, and Alice, walking along the road, noticed that +"whenever the road divided there were sure to be two finger-posts +pointing the same way, one marked 'TO TWEEDLEDUM'S HOUSE' and the other +'TO THE HOUSE OF TWEEDLEDEE.' + +"'I do believe,' said Alice at last, 'that they live in the same +house!...'" + + +N. NON-ENTITY. + +_T. L. G._, p. 87: "'I always thought they [human children] were +fabulous monsters!' said the Unicorn.... + +"'Do you know [said Alice], I always thought Unicorns were fabulous +monsters, too! I never saw one alive before!' + +"'Well, now that we _have_ seen each other,' said the Unicorn, 'if +you'll believe in me, I'll believe in you. Is that a bargain?'" + + * * * * * + +_T. L. G._, pp. 80-1: "'I see nobody on the road,' said Alice. + +"'I only wish _I_ had such eyes,' the [White] King remarked in a fretful +tone. 'To be able to see Nobody! And at that distance, too! Why, it's as +much as _I_ can do to see real people by this light!'" + + * * * * * + +_A. A. W._, p. 17: "And she [Alice] tried to fancy what the flame of a +candle looks like after the candle is blown out, for she could not +remember ever having seen such a thing." + + * * * * * + +_A. A. W._, p. 68: "... This time it [the Cheshire Cat] vanished quite +slowly, beginning with the end of the tail, and ending with the grin, +which remained some time after the rest of it had gone. + +"'Well! I've often seen a cat without a grin,' thought Alice; 'but a +grin without a cat! It's the most curious thing I ever saw in all my +life!'" + + * * * * * + +_A. A. W._, p. 77: "... The Dormouse went on,...; 'and they drew all +manner of things--everything that begins with an M.' + +"'Why with an M?' said Alice. + +"'Why not?' said the March Hare. + +"Alice was silent. + +"... [The Dormouse] went on: '--that begins with an M, such as +mouse-traps, and the moon, and memory, and muchness, you know you say +things are "much of a muchness"--did you ever see such a thing as a +drawing of a muchness?' + +"'Really, now you ask me,' said Alice, very much confused, 'I don't +think----' + +"'Then you shouldn't talk,' said the Hatter." + + +O. OBJECTS OF MATHEMATICAL LOGIC. + +_T. L. G._, p. 93: "'I was wondering what the mouse-trap [fastened to +the White Knight's saddle] was for,' said Alice. 'It isn't very likely +there would be any mice on the horse's back.' + +"'Not very likely, perhaps,' said the Knight, 'but, if they _do_ come, I +don't choose to have them running all about.' + +"'You see,' he went on after a pause, 'it's as well to be provided for +_everything_. That's the reason the horse has all these anklets round +his feet.' + +"'But what are they for?' Alice asked in a tone of great curiosity. + +"'To guard against the bites of sharks,' the Knight replied." + + +P. THE PRINCIPLE OF PERMANENCE. + +_T. L. G._, p. 106: "'Can you do Subtraction? [said the Red Queen] Take +nine from eight.' + +"'Nine from eight I can't, you know,' Alice replied very readily 'but--' + +"'She can't do Substraction,' said the White Queen." + + * * * * * + +_A. A. W._, p. 56: [Said the Pigeon to Alice]: "'... No, no! You're a +serpent; and there's no use denying it. I suppose you'll be telling me +next that you never tasted an egg!' + +"'I _have_ tasted eggs certainly,' said Alice, who was a very truthful +child; 'but little girls eat eggs quite as much as serpents do, you +know.' + +"'I don't believe it,' said the Pigeon; 'but if they do, why then +they're a kind of serpent, that's all I can say.' + +"This was such a new idea to Alice, that she was quite silent for a +minute or two, which gave the Pigeon the opportunity of adding, 'You're +looking for eggs, I know _that_ well enough; and what does it matter to +me whether you're a little girl or a serpent?' + +"'It matters a good deal to _me_,' said Alice hastily;..." + + * * * * * + +_A. A. W._, p. 75: "'But why [asked Alice] did they live at the bottom +of a well?' + +"'Take some more tea,' the March Hare said to Alice, very earnestly. + +"'I've had nothing yet,' Alice replied in an offended tone, 'so I can't +take more.' + +"'You mean you can't take _less_,' said the Hatter: 'it's very easy to +take _more_ than nothing.'" + + +Q. MATHEMATICIANS' TREATMENT OF LOGIC. + +_A. A. W._, p. 74: The Hatter had told of his quarrel with Time, and of +Time's refusal now to do anything he asked: "'... It's always six +o'clock now!' + +"A bright idea came into Alice's head. 'Is that the reason so many tea +things are put out here?' she asked. + +"'Yes, that's it,' said the Hatter, with a sigh: 'it's always tea time, +and we've no time to wash the things between whiles.' + +"'Then you keep moving round, I suppose?' said Alice. + +"'Exactly so,' said the Hatter: 'as the things get used up.' + +"'But what happens when you come to the beginning again?' Alice ventured +to ask. + +"'Suppose we change the subject,' the March Hare interrupted, yawning. +'I'm getting tired of this.'" + + * * * * * + +_A. A. W._, p. 99: "'And how many hours a day did you do lessons?' said +Alice, in a hurry to change the subject. + +"'Ten hours the first day,' said the Mock Turtle, 'nine the next, and so +on.' + +"'What a curious plan!' exclaimed Alice. + +"'That's the reason they're called lessons,' the Gryphon remarked, +'because they lessen from day to day.' + +"This was quite a new idea to Alice, and she thought it over a little +before she made her next remark. 'Then the eleventh day must have been a +holiday.' + +"'Of course it was,' said the Mock Turtle. + +"'And how did you manage on the twelfth?' Alice went on eagerly. + +"'That's enough about lessons,' the Gryphon interrupted in a very +decided tone...." + + +R. METHOD IN MATHEMATICS AND LOGIC. + +_A. A. W._, p. 71: "'Two days wrong!' sighed the Hatter. 'I told you +butter wouldn't suit the works!' he added, looking angrily at the March +Hare. + +"'It was the _best_ butter,' the March Hare meekly replied. + +"'Yes, but some crumbs must have got in as well,' the Hatter grumbled; +'you shouldn't have put it in with the bread-knife.' + +"The March Hare took the watch and looked at it gloomily: then he dipped +it into his cup of tea, and looked at it again: but he could think of +nothing better to say than his first remark, 'It was the _best_ butter, +you know.'" + + +S. VERDICT THAT LOGIC IS PHILOSOPHY. + +_A. A. W._, pp. 119-23: "... 'Consider your verdict,' he [the King] said +to the jury, in a low trembling voice. + +"'There's more evidence to come yet, please your Majesty,' said the +White Rabbit, jumping up in a great hurry: 'this paper has just been +picked up.' + +"'What's in it?' said the Queen. + +"'I haven't opened it yet,' said the White Rabbit, 'but it seems to be a +letter written by the prisoner to--to somebody.' + +"'It must have been that,' said the King, 'unless it was written to +nobody, which isn't usual, you know.' + +"'Who is it directed to?' said one of the jurymen. + +"'It isn't directed at all,' said the White Rabbit, 'in fact there's +nothing written on the _outside_.' He unfolded the paper as he spoke, +and added, 'It isn't a letter, after all: it's a set of verses.' + +"'Are they in the prisoner's handwriting?' asked another of the jurymen. + +"'No they're not,' said the White Rabbit, 'and that's the queerest thing +about it.' (The jury all looked puzzled). + +"'He must have imitated somebody else's hand,' said the King. (The jury +brightened up again.) + +"'Please your Majesty,' said the Knave, 'I didn't write it, and they +can't prove that I did: there's no name signed at the end.' + +"'If you didn't sign it, said the King, that only makes the matter +worse. You _must_ have meant some mischief, or else you'd have signed +your name like an honest man.' + +"There was a general clapping of hands at this: it was the first really +clever thing the King had said that day. + +"'That _proves_ his guilt, of course,' said the Queen, 'so, off +with----' + +"'It doesn't prove anything of the sort!' said Alice. 'Why, you don't +even know what they're about!' + +"'Read them,' said the King. + +"The White Rabbit put on his spectacles. 'Where shall I begin, please +your Majesty?' he asked. + +"'Begin at the beginning,' the King said very gravely, 'and go on till +you come to the end: then stop.' + +"There was dead silence in the court, whilst the White Rabbit read out +these verses: + + "'_They told me you had been to her, + And mentioned me to him; + She gave me a good character, + But said I could not swim._ + + _He sent them word I had not gone + (We know it to be true): + If she should push the matter on, + What would become of you?_ + + _I gave her one, they gave him two, + You gave us three or more; + They all returned from him to you, + Though they were mine before._ + + _If I or she should chance to be + Involved in this affair, + He trusts to you to set them free + Exactly as they were._ + + _My notion was that you had been + (Before she had this fit) + An obstacle that came between + Him, and ourselves, and it._ + + _Don't let him know she liked them best, + For this must ever be + A secret kept from all the rest, + Between yourself and me._' + +"'That's the most important piece of evidence we've heard yet,' said the +King, rubbing his hands, 'so now let the jury----' + +"'If any one of them can explain it,' said Alice (she had grown so large +in the last few minutes that she wasn't a bit afraid of interrupting +him), 'I'll give him sixpence. _I_ don't believe there's an atom of +meaning in it.' + +"The jury all wrote down on their slates, 'She doesn't believe there's +an atom of meaning in it,' but none of them attempted to explain the +paper. + +"'If there's no meaning in it,' said the King, 'that saves a world of +trouble, you know, as we needn't try to find any. And yet I don't know,' +he went on, spreading out the verses on his knee and looking at them +with one eye; 'I seem to see some meaning in them after all. "_-- said +I could not swim_"; you can't swim, can you?' he added, turning to the +Knave. + +"The Knave shook his head sadly. 'Do I look like it?' he said. (Which he +certainly did _not_, being made entirely of cardboard.) + +"'All right, so far,' said the King; and he went on muttering over the +verses to himself: ''_We know it to be true_'--that's the jury, of +course--'_If she should push the matter on_'--that must be the +Queen--'_What would become of you?_' What indeed!--'_I gave her one, +they gave him two!_' why, that must be what he did with the tarts, you +know----' + +"'But it goes on, '_They all returned from him to you_,'' said Alice. + +"'Why, there they are!' said the King, triumphantly pointing to the +tarts on the table. 'Nothing can be clearer than that. Then +again--'_Before she had this fit_'--you never had fits, my dear, I +think?' he said to the Queen. + +"'Never!' said the Queen furiously, throwing an inkstand at the Lizard +as she spoke. (The unfortunate little Bill had left off writing on his +slate with one finger, as he found it made no mark; but he now hastily +began again, using the ink that was trickling down his face, as long as +it lasted.) + +"'Then the words don't _fit_ you,' said the King, looking round the +court with a smile. There was a dead silence. + +"'It's a pun!' the King added in an angry tone, and everybody laughed. + +"'Let the jury consider their verdict,' the King said, for about the +twentieth time that day. + +"'No, no!' said the Queen. 'Sentence first--verdict afterwards.' + +"'Stuff and nonsense!' said Alice loudly. 'The idea of having the +sentence first!' + +"'Hold your tongue!' said the Queen, turning purple...." + + +T. "GEDANKENEXPERIMENTE." + +_T. L. G._, p. 61: "Alice laughed. 'There's no use trying,' she said: +'one _can't_ believe impossible things.' + +"'I daresay you haven't had much practice,' said the [White] Queen. +'When I was your age, I always did it for half-an-hour a day. Why, +sometimes I've believed as many as six impossible things before +breakfast.'" + + +_Printed in Great Britain by_ + +UNWIN BROTHERS, LIMITED, THE GRESHAM PRESS, WOKING AND LONDON + + + + + +End of Project Gutenberg's The philosophy of B*rtr*nd R*ss*ll, by Various + +*** END OF THIS PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF B*RTR*ND R*SS*LL *** + +***** This file should be named 38430-8.txt or 38430-8.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/8/4/3/38430/ + +Produced by Adrian Mastronardi and the Online Distributed +Proofreading Team at http://www.pgdp.net (This file was +produced from images generously made available by The +Internet Archive/Canadian Libraries) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: The philosophy of B*rtr*nd R*ss*ll + +Author: Various + +Editor: Philip E. B. Jordain + +Release Date: December 28, 2011 [EBook #38430] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF B*RTR*ND R*SS*LL *** + + + + +Produced by Adrian Mastronardi and the Online Distributed +Proofreading Team at http://www.pgdp.net (This file was +produced from images generously made available by The +Internet Archive/Canadian Libraries) + + + + + + +</pre> + + +<h1>THE PHILOSOPHY OF<br /> +MR. B*RTR*ND R*SS*LL</h1> + +<h4>WITH AN APPENDIX OF LEADING<br /> +PASSAGES FROM CERTAIN OTHER WORKS</h4> + +<h4><small>EDITED BY</small><br /> + +<big>PHILIP E. B. JOURDAIN</big></h4> + +<div class="figcenter" style="width: 120px;"> +<img src="images/printdevice.jpg" width="120" height="120" alt="" title="" /> +</div> + +<h4>LONDON: GEORGE ALLEN & UNWIN LTD.<br /> +<small>RUSKIN HOUSE 40 MUSEUM STREET, W.C. 1</small><br /> +CHICAGO: THE OPEN COURT PUBLISHING CO. +</h4> +<hr /> + +<p class="center"><i>First published in 1918</i></p> + +<p class="center">(<i>All rights reserved</i>)</p> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_3" id="Page_3">[Pg 3]</a></span></p> +<h2><a name="EDITORS_NOTE" id="EDITORrsquoS_NOTE"></a>EDITOR’S NOTE</h2> + + +<p>When Mr. B*rtr*nd R*ss*ll, following the advice of Mr. +W*ll**m J*m*s, again “got into touch with reality” and +in July 1911 was torn to pieces by Anti-Suffragists, many +of whom were political opponents of Mr. R*ss*ll and held +strong views on the Necessity of Protection of Trade and +person, a manuscript which was almost ready for the press +was fortunately saved from the flames on the occasion when +a body of eager champions of the Sacredness of Personal +Property burnt the late Mr. R*ss*ll’s house. This manuscript, +together with some further fragments found in the +late Mr. R*ss*ll’s own interleaved copy of his <i>Prayer-Book +of Free Man’s Worship</i>, which was fortunately rescued with +a few of the great author’s other belongings, was first given +to the world in the <i>Monist</i> for October 1911 and January +1916, and has here been arranged and completed by some +other hitherto undecipherable manuscripts. The title of the +above-mentioned <i>Prayer-Book</i>, it may perhaps be mentioned, +was apparently suggested to Mr. R*ss*ll by that of the +Essay on “The Free Man’s Worship” in the <i>Philosophical +Essays</i> (London, 1910, pp. 59-70<a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a>) of Mr. R*ss*ll’s distinguished +contemporary, Mr. Bertrand Russell, from whom +much of Mr. R*ss*ll’s philosophy was derived. And, indeed, +the influence of Mr. Russell extended even beyond philosophical +views to arrangement and literary style. The +method of arrangement of the present work seems to have +been borrowed from Mr. Russell’s <i>Philosophy of Leibniz</i> of +1900; in the selection of subjects dealt with, Mr. R*ss*ll +seems to have been guided by Mr. Russell’s <i>Principles of +Mathematics</i> of 1903; while Mr. R*ss*ll’s literary style fortunately +<span class='pagenum'><a name="Page_4" id="Page_4">[Pg 4]</a></span>reminds us more of Mr. Russell’s later clear and +charming subtleties than his earlier brilliant and no less +subtle obscurities. But, on the other hand, some important +points of Mr. Russell’s doctrine, which first appeared in +books published after Mr. R*ss*ll’s death, were anticipated +in Mr. R*ss*ll’s notes, and these anticipations, so interesting +for future historians of philosophy, have been provided by +the editor with references to the later works of Mr. Russell. +All editorial notes are enclosed in square brackets, to indicate +that they were not written by the late Mr. R*ss*ll.</p> + +<p>At the present time we have come to take a calm view +of the question so much debated seven years ago as to the +legitimacy of logical arguments in political discussions. +No longer, fortunately, can that intense feeling be roused +which then found expression in the famous cry, “Justice—right +or wrong,” and which played such a large part in +the politics of that time. Thus it will not be out of place +in this unimpassioned record of some of the truths and errors +in the world to refer briefly to Mr. R*ss*ll’s short and stormy +career. Before he was torn to pieces, he had been forbidden +to lecture on philosophy or mathematics by some well-intentioned +advocates of freedom in speech who thought +that the cause of freedom might be endangered by allowing +Mr. R*ss*ll to speak freely on points of logic, on the grounds, +apparently, that logic is both harmful and unnecessary +and might be applied to politics unless strong measures +were taken for its suppression. On much the same grounds, +his liberty was taken from him by those who remarked +that, if necessary, they would die in defence of the sacred +principle of liberty; and it was in prison that the greater +part of the present work was written. Shortly after his +liberation, which, like all actions of public bodies, was brought +about by the combined honour and interests of those in +authority, occurred his lamentable death to which we have +referred above.</p> + +<p>Mr. R*ss*ll maintained that the chief use of “implication” +in politics is to draw conclusions, which are thought to be +true, and which are consequently false, from identical propositions, +and we can see these views expressed in +Chapters III and XIX of the present work. These +chapters were apparently written before the Government,<span class='pagenum'><a name="Page_5" id="Page_5">[Pg 5]</a></span> +in the spring of 1910, arrived at the famous secret decision +that only “certain implications” are permitted in discussion. +Naturally the secret decision gave rise to much +speculation among logicians as to which kinds of implication +were barred, and Mr. R*ss*ll and Mr. Bertrand +Russell had many arguments on the subject, which naturally +could not be published at the time. However, after Mr. +R*ss*ll’s death, successive prosecutions which were made by +the Government at last made it quite clear that the opinion +held by Mr. R*ss*ll was the correct one. There had been +numerous prosecutions of people who, from true but not +identical premisses, had deduced true conclusions, so that +the possible legitimate forms of “implication” were reduced. +Further, the other doubtful cases were cleared up in course +of time by the prosecution of (1) members of the Aristotelian +Society for deducing true conclusions from false premisses; +(2) members of the <i>Mind</i> Association for deducing false +conclusions from false premisses; and also by the attempted +prosecution of an eminent lady for deducing true conclusions +from identities. Fortunately this lady was able to defend +herself successfully by pleading that one eminent philosopher +believed them to be true—which, of course, means that the +conclusions are false. Thus appeared the true nature of +legitimate political arguments.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a> [This Essay is also reprinted in Mr. Russell’s <i>Mysticism and Logic</i>, +London and New York, 1918, pp. 46-57.—<span class="smcap">Ed.</span>]</p></div> + + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_6" id="Page_6">[Pg 6]</a></span></p> + +<div class="blockquot"> +<p class="center">“Even a joke should have some meaning....”</p> + +<p style='text-align: right'>(The Red Queen, <i>T. L. G.</i>, p. 105).</p> +</div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_7" id="Page_7">[Pg 7]</a></span></p> +<h2>CONTENTS</h2> + + +<div class='center'> +<table border="0" cellpadding="4" cellspacing="0" summary=""> +<tr><td align='right'></td><td align='right'></td><td align='right'><small>PAGE</small></td></tr> +<tr><td align='right'></td><td class='smcap1'>Editor’s Note</td><td align='right'><a href="#Page_3">3</a></td></tr> +<tr><td align='right'></td><td class='smcap1'>Abbreviations</td><td align='right'><a href="#Page_9">9</a></td></tr> +<tr><td align='right'><small>CHAPTER</small></td></tr> +<tr><td align='right'>I.</td><td class='smcap1'>The Indefinables of Logic</td><td align='right'><a href="#Page_11">11</a></td></tr> +<tr><td align='right'>II.</td><td class='smcap1'>Objective Validity of the “Laws of Thought”</td><td align='right'><a href="#Page_15">15</a></td></tr> +<tr><td align='right'>III.</td><td class='smcap1'>Identity</td><td align='right'><a href="#Page_16">16</a></td></tr> +<tr><td align='right'>IV.</td><td class='smcap1'>Identity of Classes</td><td align='right'><a href="#Page_18">18</a></td></tr> +<tr><td align='right'>V.</td><td class='smcap1'>Ethical Applications of the Law of Identity</td><td align='right'><a href="#Page_19">19</a></td></tr> +<tr><td align='right'>VI.</td><td class='smcap1'>The Law of Contradiction in Modern Logic</td><td align='right'><a href="#Page_21">21</a></td></tr> +<tr><td align='right'>VII.</td><td class='smcap1'>Symbolism and Meaning</td><td align='right'><a href="#Page_22">22</a></td></tr> +<tr><td align='right'>VIII.</td><td class='smcap1'>Nominalism</td><td align='right'><a href="#Page_24">24</a></td></tr> +<tr><td align='right'>IX.</td><td class='smcap1'>Ambiguity and Symbolic Logic</td><td align='right'><a href="#Page_26">26</a></td></tr> +<tr><td align='right'>X.</td><td class='smcap1'>Logical Addition and the Utility of Symbolism</td><td align='right'><a href="#Page_27">27</a></td></tr> +<tr><td align='right'>XI.</td><td class='smcap1'>Criticism</td><td align='right'><a href="#Page_29">29</a></td></tr> +<tr><td align='right'>XII.</td><td class='smcap1'>Historical Criticism</td><td align='right'><a href="#Page_30">30</a></td></tr> +<tr><td align='right'>XIII.</td><td class='smcap1'>Is the Mind in the Head?</td><td align='right'><a href="#Page_31">31</a></td></tr> +<tr><td align='right'>XIV.</td><td class='smcap1'>The Pragmatist Theory of Truth</td><td align='right'><a href="#Page_32">32</a></td></tr> +<tr><td align='right'>XV.</td><td class='smcap1'>Assertion</td><td align='right'><a href="#Page_34">34</a></td></tr> +<tr><td align='right'>XVI.</td><td class='smcap1'>The Commutative Law</td><td align='right'><a href="#Page_35">35</a></td></tr> +<tr><td align='right'>XVII.</td><td class='smcap1'>Universal and Particular Propositions</td><td align='right'><a href="#Page_36">36</a></td></tr> +<tr><td align='right'>XVIII.</td><td class='smcap1'>Denial of Generality and Generality of Denial</td><td align='right'><a href="#Page_37">37</a></td></tr> +<tr><td align='right'>XIX.</td><td class='smcap1'>Implication</td><td align='right'><a href="#Page_39">39</a></td></tr> +<tr><td align='right'>XX.</td><td class='smcap1'>Dignity</td><td align='right'><a href="#Page_43">43</a></td></tr> +<tr><td align='right'>XXI.</td><td class='smcap1'>The Synthetic Nature of Deduction</td><td align='right'><a href="#Page_45">45</a></td></tr> +<tr><td align='right'>XXII.</td><td class='smcap1'>The Mortality of Socrates</td><td align='right'><a href="#Page_48">48</a><span class='pagenum'><a name="Page_8" id="Page_8">[Pg 8]</a></span></td></tr> +<tr><td align='right'>XXIII.</td><td class='smcap1'>Denoting</td><td align='right'><a href="#Page_53">53</a></td></tr> +<tr><td align='right'>XXIV.</td><td class='smcap1'>The</td><td align='right'><a href="#Page_54">54</a></td></tr> +<tr><td align='right'>XXV.</td><td class='smcap1'>Non-Entity</td><td align='right'><a href="#Page_56">56</a></td></tr> +<tr><td align='right'>XXVI.</td><td class='smcap1'>Is</td><td align='right'><a href="#Page_58">58</a></td></tr> +<tr><td align='right'>XXVII.</td><td class='smcap1'>And and Or</td><td align='right'><a href="#Page_59">59</a></td></tr> +<tr><td align='right'>XXVIII.</td><td class='smcap1'>The Conversion of Relations</td><td align='right'><a href="#Page_60">60</a></td></tr> +<tr><td align='right'>XXIX.</td><td class='smcap1'>Previous Philosophical Theories of Mathematics</td><td align='right'><a href="#Page_61">61</a></td></tr> +<tr><td align='right'>XXX.</td><td class='smcap1'>Finite and Infinite</td><td align='right'><a href="#Page_63">63</a></td></tr> +<tr><td align='right'>XXXI.</td><td class='smcap1'>The Mathematical Attainments of Tristram Shandy</td><td align='right'><a href="#Page_64">64</a></td></tr> +<tr><td align='right'>XXXII.</td><td class='smcap1'>The Hardships of a Man with an Unlimited Income</td><td align='right'><a href="#Page_66">66</a></td></tr> +<tr><td align='right'>XXXIII.</td><td class='smcap1'>The Relations of Magnitude of Cardinal Numbers</td><td align='right'><a href="#Page_69">69</a></td></tr> +<tr><td align='right'>XXXIV.</td><td class='smcap1'>The Unknowable</td><td align='right'><a href="#Page_70">70</a></td></tr> +<tr><td align='right'>XXXV.</td><td class='smcap1'>Mr. Spencer, the Athanasian Creed, and the Articles</td><td align='right'><a href="#Page_73">73</a></td></tr> +<tr><td align='right'>XXXVI.</td><td class='smcap1'>The Humour of Mathematicians</td><td align='right'><a href="#Page_74">74</a></td></tr> +<tr><td align='right'>XXXVII.</td><td class='smcap1'>The Paradoxes of Logic</td><td align='right'><a href="#Page_75">75</a></td></tr> +<tr><td align='right'>XXXVIII.</td><td class='smcap1'>Modern Logic and some Philosophical Arguments</td><td align='right'><a href="#Page_79">79</a></td></tr> +<tr><td align='right'>XXXIX.</td><td class='smcap1'>The Hierarchy of Jokes</td><td align='right'><a href="#Page_81">81</a></td></tr> +<tr><td align='right'>XL.</td><td class='smcap1'>The Evidence of Geometrical Propositions</td><td align='right'><a href="#Page_83">83</a></td></tr> +<tr><td align='right'>XLI.</td><td class='smcap1'>Absolute and Relative Position</td><td align='right'><a href="#Page_84">84</a></td></tr> +<tr><td align='right'>XLII.</td><td class='smcap1'>Laughter</td><td align='right'><a href="#Page_86">86</a></td></tr> +<tr><td align='right'>XLIII.</td><td class='smcap1'>“Gedankenexperimente” and Evolutionary Ethics</td><td align='right'><a href="#Page_88">88</a></td></tr> +<tr><td align='right'></td><td class='smcap1'>Appendixes</td><td align='right'><a href="#Page_89">89</a></td></tr> +</table></div> + + +<hr class="full" /> + +<p><span class='pagenum'><a name="Page_9" id="Page_9">[Pg 9]</a></span></p> + + +<h2>ABBREVIATIONS</h2> + + + +<div class='center'> +<table border="0" cellpadding="6" cellspacing="4" summary=""> +<colgroup><col width="10%" /><col width="90%" /></colgroup> +<tr><td align='left'><i>A. A. W.</i></td><td align='left'>Lewis Carroll: <i>Alice’s Adventures in Wonderland</i>, London, 1908. [This book was first published much earlier, but this was the edition used by Mr. R*ss*ll. The same applies to <i>H. S.</i> and <i>T. L. G.</i>]</td></tr> +<tr><td align='left'><i>A. C. P.</i></td><td align='left'>John Henry Blunt (ed. by): <i>The Annotated Book of Common Prayer</i>, London, new edition, 1888.</td></tr> +<tr><td align='left'><i>A. d. L.</i></td><td align='left'>Ernst Schröder: <i>Vorlesungen über die Algebra der Logik, Leipzig</i>, vol. i., 1890; vol. ii. (two parts), 1891 and 1905; vol. iii.: <i>Algebra und Logik der Relative</i>, 1895.</td></tr> +<tr><td align='left'><i>E. N.</i></td><td align='left'>Richard Dedekind: <i>Essays on the Theory of Numbers</i>, Chicago and London, 1901.</td></tr> +<tr><td align='left'><i>E. L. L.</i></td><td align='left'>William Stanley Jevons: <i>Elementary Lessons in Logic, Deductive and Inductive. With copious Questions and Examples, and a Vocabulary of Logical Terms</i>, London, 24th ed., 1907 [first published in 1870].</td></tr> +<tr><td align='left'><i>E. u. I.</i></td><td align='left'>Ernst Mach: <i>Erkenntnis und Irrtum: Skizzen zur Psychologie der Forschung</i>, Leipzig, 1906.</td></tr> +<tr><td align='left'><i>F. L.</i></td><td align='left'>Augustus De Morgan: <i>Formal Logic: or The Calculus of Inference, Necessary and Probable</i>, London, 1847.</td></tr> +<tr><td align='left'><i>Fm. L.</i></td><td align='left'>John Neville Keynes: <i>Studies and Exercises in Formal Logic</i>, 4th ed., London, 1906.</td></tr> +<tr><td align='left'><i>Gg.</i></td><td align='left'>Gottlob Frege: <i>Grundgesetze der Arithmetik begriffschriftlich abgeleitet</i>, Jena, vol. i., 1893; vol. ii., 1903.</td></tr> +<tr><td align='left'><i>Gl.</i></td><td align='left'>Gottlob Frege: <i>Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl</i>, Breslau, 1884.</td></tr> +<tr><td align='left'><i>G. u. E.</i></td><td align='left'>G. Heymans: <i>Die Gesetze und Elemente des wisenschaftlichen Denkens</i>, Leiden, vol. i., 1890; vol. ii., 1894.</td></tr> +<tr><td align='left'><i>H. J.</i></td><td align='left'><i>The Hibbert Journal: a Quarterly Review of Religion, Theology and Philosophy</i>, London and New York.<span class='pagenum'><a name="Page_10" id="Page_10">[Pg 10]</a></span></td></tr> +<tr><td align='left'><i>H. S.</i></td><td align='left'>Lewis Carroll: <i>The Hunting of the Snark: an Agony in Eight Fits</i>, London, 1911.</td></tr> +<tr><td align='left'><i>M.</i></td><td align='left'><i>The Monist: a Quarterly Magazine Devoted to Science and Philosophy</i>, Chicago and London.</td></tr> +<tr><td align='left'><i>Md.</i></td><td align='left'><i>Mind: a Quarterly Review of Psychology and Philosophy</i>, London and New York.</td></tr> +<tr><td align='left'><i>Pa. Ma.</i></td><td align='left'>Alfred North Whitehead and Bertrand Russell: <i>Principia Mathematica</i>, vol. i., Cambridge, 1910. [Other volumes were published in 1912 and 1913.]</td></tr> +<tr><td align='left'><i>P. E.</i></td><td align='left'>Bertrand Russell: <i>Philosophical Essays</i>, London and New York, 1910.</td></tr> +<tr><td align='left'><i>Ph. L.</i></td><td align='left'>Bertrand Russell: <i>A Critical Exposition of the Philosophy of Leibniz, with an Appendix of Leading Passages</i>, Cambridge, 1900.</td></tr> +<tr><td align='left'><i>P. M.</i></td><td align='left'>Bertrand Russell: <i>The Principles of Mathematics</i>, vol. i., Cambridge, 1903.</td></tr> +<tr><td align='left'><i>R. M. M.</i></td><td align='left'><i>Revue de Métaphysique et de Morale</i>, Paris.</td></tr> +<tr><td align='left'><i>S. B.</i></td><td align='left'>Lewis Carroll: <i>Sylvie and Bruno</i>, London, 1889.</td></tr> +<tr><td align='left'><i>S. L.</i></td><td align='left'>John Venn: <i>Symbolic Logic</i>, London, 1881; 2nd ed., 1894.</td></tr> +<tr><td align='left'><i>S. o. S.</i></td><td align='left'>William Stanley Jevons: <i>The Substitution of Similars, the True Principle of Reasoning derived from a Modification of Aristotle’s Dictum</i>, London, 1869.</td></tr> +<tr><td align='left'><i>T. L. G.</i></td><td align='left'>Lewis Carroll: <i>Through the Looking-Glass, and what Alice found there</i>, London, 1911.</td></tr> +<tr><td align='left'><i>Z. S.</i></td><td align='left'>Gottlob Frege: <i>Ueber die Zahlen des Herrn H. Schubert</i>, Jena, 1899.</td></tr> +</table></div> + + + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_11" id="Page_11">[Pg 11]</a></span></p> +<h3><a name="CHAPTER_I" id="CHAPTER_I"></a>CHAPTER I</h3> + +<h2>THE INDEFINABLES OF LOGIC</h2> + + +<p>The view that the fundamental principles of logic consist +solely of the law of identity was held by Leibniz,<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">[2]</a> Drobisch, +Uberweg,<a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">[3]</a> and Tweedledee. Tweedledee, it may be remembered,<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">[4]</a> +remarked that certain identities “are” logic. +Now, there is some doubt as to whether he, like Jevons,<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">[5]</a> +understood “are” to mean what mathematicians mean by +“=,” or, like Schröder<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">[6]</a> and most logicians, to have the +same meaning as the relation of subsumption. The first +alternative alone would justify our contention; and we may, +I think, conclude from an opposition to authority that may +have been indicated by Tweedledee’s frequent use of the +word “contrariwise” that he did not follow the majority +of logicians, but held, like Jevons,<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">[7]</a> the mistaken<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">[8]</a> view +that the quantification of the predicate is relevant to symbolic +logic.</p> + +<p>It may be mentioned, by the way, that it is probable that +Humpty-Dumpty’s “is” is the “is” of identity. In fact, +it is not unlikely that Humpty-Dumpty was a Hegelian; +for, although his ability for clear explanation may seem to +militate against this, yet his inability to understand mathematics,<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">[9]</a> +together with his synthesis of a cravat and a belt, +<span class='pagenum'><a name="Page_12" id="Page_12">[Pg 12]</a></span>which usually serve different purposes,<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">[10]</a> and his proclivity +towards riddles seem to make out a good case for those who +hold that he was in fact a Hegelian. Indeed, riddles are +very closely allied to puns, and it was upon a pun, consisting +of the confusion of the “is” of predication with the “is” +of identity—so that, for example, “Socrates” was identified +with “mortal” and more generally the particular with the +universal—that Hegel’s system of philosophy was founded.<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">[11]</a> +But the question of Humpty-Dumpty’s philosophical opinions +must be left for final verification to the historians of philosophy: +here I am only concerned with an <i>a priori</i> logical +construction of what his views might have been if they +formed a consistent whole.<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">[12]</a></p> + +<p>If the principle of identity were indeed the sole principle +of logic, the principles of logic could hardly be said to be, +as in fact they are, a body of propositions whose consistency +it is impossible to prove.<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">[13]</a> This characteristic is important +and one of the marks of the greatest possible security. For +example, while a great achievement of late years has been +to prove the consistency of the principles of arithmetic, a +science which is unreservedly accepted except by some +empiricists,<a name="FNanchor_14_14" id="FNanchor_14_14"></a><a href="#Footnote_14_14" class="fnanchor">[14]</a> it can be proved formally that one foundation +of arithmetic is shattered.<a name="FNanchor_15_15" id="FNanchor_15_15"></a><a href="#Footnote_15_15" class="fnanchor">[15]</a> It is true that, quite lately, +it has been shown that this conclusion may be avoided, and, +by a re-moulding of logic, we can draw instead the paradoxical +conclusion that the opinions held by common-sense for so +many years are, in part, justified. But it is quite certain +that, with the principles of logic, no such proof of consistency, +and no such paradoxical result of further investigations +is to be feared.<span class='pagenum'><a name="Page_13" id="Page_13">[Pg 13]</a></span></p> + +<p>Still, this re-moulding has had the result of bringing logic +into a fuller agreement with common-sense than might be +expected. There were only two alternatives: if we chose +principles in accordance with common-sense, we arrived at +conclusions which shocked common-sense; by starting with +paradoxical principles, we arrived at ordinary conclusions. +Like the White Knight, we have dyed our whiskers an +unusual colour and then hidden them.<a name="FNanchor_16_16" id="FNanchor_16_16"></a><a href="#Footnote_16_16" class="fnanchor">[16]</a></p> + +<p>The quaint name of “Laws of Thought,” which is often +applied to the principles of Logic, has given rise to confusion +in two ways: in the first place, the “Laws,” unlike other +laws, cannot be broken, even in thought; and, in the second +place, people think that the “Laws” have something to +do with holding for the operations of their minds, just as +laws of nature hold for events in the world around us.<a name="FNanchor_17_17" id="FNanchor_17_17"></a><a href="#Footnote_17_17" class="fnanchor">[17]</a> +But that the laws are not psychological laws follows from +the facts that a thing may be true even if nobody believes +it, and something else may be false if everybody believes +it. Such, it may be remarked, is usually the case.</p> + +<p>Perhaps the most frequent instance of the assumption +that the laws of logic are mental is the treatment of an +identity as if its validity were an affair of our permission. +Some people suggest to others that they should “let bygones +be bygones.” Another important piece of evidence that +the truth of propositions has nothing to do with mind is +given by the phrase “it is morally certain that such-and-such +a proposition is true.” Now, in the first place, morality, +curiously enough, seems to be closely associated with mental +acts: we have professorships and lectureships of, and +examinations in, “mental and moral philosophy.” In the +second place, it is plain that a “morally certain” proposition +is a highly doubtful one. Thus it is as vain to expect +any information about our minds from a study of the “Laws +of Thought” as it would be to expect a description of a +certain social event from Miss E. E. C. Jones’s book <i>An +Introduction to General Logic</i>.</p> + +<p>Fortunately, the principles or laws of Logic are not a +matter of philosophical discussion. Idealists like Tweedle<span class='pagenum'><a name="Page_14" id="Page_14">[Pg 14]</a></span>dum +and Tweedledee, and even practical idealists like the +White Knight, explicitly accept laws like the law of identity +and the excluded middle.<a name="FNanchor_18_18" id="FNanchor_18_18"></a><a href="#Footnote_18_18" class="fnanchor">[18]</a> In fact, throughout all logic +and mathematics, the existence of the human or any other +mind is totally irrelevant; mental processes are studied by +means of logic, but the subject-matter of logic does not +presuppose mental processes, and would be equally true +if there were no mental processes. It is true that, in that +case, we should not know logic; but our knowledge must +not be confounded with the truths which we know.<a name="FNanchor_19_19" id="FNanchor_19_19"></a><a href="#Footnote_19_19" class="fnanchor">[19]</a> An +apple is not confused with the eating of it except by savages, +idealists, and people who are too hungry to think.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">[2]</span></a> Russell, <i>Ph. L.</i>, pp. 17, 19, 207-8.</p></div> + +<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">[3]</span></a> Schröder, <i>A. d. L.</i>, i. p. 4.</p></div> + +<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">[4]</span></a> See <a href="#App_A">Appendix A</a>. This Appendix also illustrates the importance +attached to the Principle of Identity by the Professor and Bruno.</p></div> + +<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">[5]</span></a> <i>S. o. S.</i>, pp. 9-15.</p></div> + +<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">[6]</span></a> <i>A. d. L.</i>, i. p. 132.</p></div> + +<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">[7]</span></a> Cf., besides the reference in the last note but one, <i>E. L. L.</i>, +pp. 183, 191. “Contrariwise,” it may be remarked, is not a term +used in traditional logic.</p></div> + +<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">[8]</span></a> <i>S. L.</i>, 1881, pp. 173-5, 324-5; 1894, pp. 194-6.</p></div> + +<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">[9]</span></a> Cf. <a href="#App_C">Appendix C</a>, and William Robertson Smith, “Hegel and the +Metaphysics of the Fluxional Calculus,” <i>Trans. Roy. Soc., Edinb.</i>, +vol. xxv., 1869, pp. 491-511.</p></div> + +<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">[10]</span></a> See <a href="#App_B">Appendix B</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">[11]</span></a> [This is a remarkable anticipation of the note on pp. 39-40 of +Mr. Russell’s book, published about three years after the death of Mr. +R*ss*ll, and entitled <i>Our Knowledge of the External World as a Field +for Scientific Method in Philosophy</i>, Chicago and London, 1914.—<span class="smcap">Ed.</span>]</p></div> + +<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">[12]</span></a> Cf. <i>Ph. L.</i>, pp. v.-vi. 3.</p></div> + +<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">[13]</span></a> Cf. Pieri, <i>R. M. M.</i>, March 1906, p. 199.</p></div> + +<div class="footnote"><p><a name="Footnote_14_14" id="Footnote_14_14"></a><a href="#FNanchor_14_14"><span class="label">[14]</span></a> As a type of these, Humpty-Dumpty, with his inability to admit +anything not empirically given and his lack of comprehension of pure +mathematics, may be taken (see <a href="#App_C">Appendix C</a>). In his (correct) thesis +that definitions are nominal, too, Humpty-Dumpty reminds one of +J. S. Mill (see <a href="#App_D">Appendix D</a>).</p></div> + +<div class="footnote"><p><a name="Footnote_15_15" id="Footnote_15_15"></a><a href="#FNanchor_15_15"><span class="label">[15]</span></a> See Frege, <i>Gg.</i>, ii. p. 253.</p></div> + +<div class="footnote"><p><a name="Footnote_16_16" id="Footnote_16_16"></a><a href="#FNanchor_16_16"><span class="label">[16]</span></a> See <a href="#App_E">Appendix E</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_17_17" id="Footnote_17_17"></a><a href="#FNanchor_17_17"><span class="label">[17]</span></a> See Frege, <i>Gg.</i>, i. p. 15.</p></div> + +<div class="footnote"><p><a name="Footnote_18_18" id="Footnote_18_18"></a><a href="#FNanchor_18_18"><span class="label">[18]</span></a> See the above references and also <a href="#App_F">Appendix F</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_19_19" id="Footnote_19_19"></a><a href="#FNanchor_19_19"><span class="label">[19]</span></a> Cf. B. Russell, <i>H. J.</i>, July 1904, p. 812.</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_15" id="Page_15">[Pg 15]</a></span></p> +<h3><a name="CHAPTER_II" id="CHAPTER_II"></a>CHAPTER II</h3> + +<h2>OBJECTIVE VALIDITY OF THE “LAWS OF +THOUGHT”</h2> + + +<p>I once inquired of a maid-servant whether her mistress +was at home. She replied, in a doubtful fashion, that she +<i>thought</i> that her mistress was in unless she was out. I concluded +that the maid was uncertain as to the objective +validity of the law of excluded middle, and remarked that +to her mistress. But since I used the phrase “laws of +thought,” the mistress perhaps supposed that a “law of +thought” has something to do with thinking and seemed +to imagine that I wished to impute to the maid some moral +defect of an unimportant nature. Thus she remonstrated +with me in an amused way, since she probably imagined +that I meant to find fault with the maid’s capacity for +thinking.</p> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_16" id="Page_16">[Pg 16]</a></span></p> +<h3><a name="CHAPTER_III" id="CHAPTER_III"></a>CHAPTER III</h3> + +<h2>IDENTITY</h2> + + +<p>In the first chapter we have noticed the opinion that +identities are fundamental to all logic. We will now consider +some other views of the value of identities.</p> + +<p>Identities are frequently used in common life by people +who seem to imagine that they can draw important conclusions +respecting conduct or matters of fact from them. +I have heard of a man who gained the double reputation +of being a philosopher and a fatalist by the repeated enunciation +of the identity “Whatever will be, will be”; and +the Italian equivalent of this makes up an appreciable part +of one of Mr. Robert Hichens’ novels. Further, the identity +“Life is Life” has not only been often accepted as an explanation +for a particular way of living but has even been +considered by an authoress who calls herself “Zack” to be +an appropriate title for a novel; while “Business is Business” +is frequently thought to provide an excuse for dishonesty +in trading, for which purpose it is plainly inadequate.</p> + +<p>Another example is given by a poem of Mr. Kipling, where +he seems to assert that “East is East” and “West is West” +imply that “never the twain shall meet.” The conclusion, +now, is false; for, since the world is round—as geography +books still maintain by arguments which strike every intelligent +child as invalid<a name="FNanchor_20_20" id="FNanchor_20_20"></a><a href="#Footnote_20_20" class="fnanchor">[20]</a>—what is called the “West” does, +in fact, merge into the “East.” Even if we are to take<span class='pagenum'><a name="Page_17" id="Page_17">[Pg 17]</a></span> +the statement metaphorically, it is still untrue, as the +Japanese nation has shown.</p> + +<p>The law-courts are often rightly blamed for being strenuous +opponents of the spread of modern logic: the frequent +misuse of <i>and</i>, <i>or</i>, <i>the</i>, and <i>provided that</i> in them is notorious. +But the fault seems partly to lie in the uncomplicated nature +of the logical problems which are dealt with in them. Thus +it is no uncommon thing for somebody to appear there who +is unable to establish his own identity, or for A to assert +that B was “not himself” when he made a will leaving +his money to C.</p> + +<p>The chief use of identities is in implication. Since, in +logic, we so understand <i>implication</i><a name="FNanchor_21_21" id="FNanchor_21_21"></a><a href="#Footnote_21_21" class="fnanchor">[21]</a> that any true proposition +implies and is implied by any other true proposition; +if one is convinced of the truth of the proposition Q, it is +advisable to choose one or more identities P, whose truth +is undoubted, and say that P implies Q. Thus, Mr. Austen +Chamberlain, according to <i>The Times</i> of March 27, 1909, +professed to deduce the conclusion that it is not right that +women should have votes from the premisses that “man +is man” and “woman is woman.” This method requires +that one should have made up one’s mind about the conclusion +before discovering the premisses—by what, no doubt, +Jevons would call an “inverse or inductive method.” Thus +the method is of use only in speeches and in giving good +advice.</p> + +<p>Mr. Austen Chamberlain afterwards rather destroyed one’s +belief in the truth of his premisses by putting limits to the +validity of the principle of identity. In the course of the +Debate on the Budget of 1909, he maintained, against Mr. +Lloyd George, that a joke was a joke except when it was +an untruth: Mr. Lloyd George, apparently, being of the +plausible opinion that a joke is a joke under all circumstances.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_20_20" id="Footnote_20_20"></a><a href="#FNanchor_20_20"><span class="label">[20]</span></a> The argument about the hull of a ship disappearing first is not +convincing, since it would equally well prove that the surface of the +earth was, for example, corrugated on a large scale. If the common-sense +of the reader were supposed to dismiss the possibility of water +clinging to such corrugations, it might equally be supposed to dismiss +the possibility of water clinging to a spherical earth. Traditional +geography books, no doubt, gave rise to the opinions held by Lady +Blount and the Zetetic Society.</p></div> + +<div class="footnote"><p><a name="Footnote_21_21" id="Footnote_21_21"></a><a href="#FNanchor_21_21"><span class="label">[21]</span></a> The subject of Implication will be further considered in Chapter +XIX.</p></div> + + + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_18" id="Page_18">[Pg 18]</a></span></p> +<h3><a name="CHAPTER_IV" id="CHAPTER_IV"></a>CHAPTER IV</h3> + +<h2>IDENTITY OF CLASSES</h2> + + +<p>I once heard of a meritorious lady who was extremely +conventional; on the slender grounds of carefully acquired +habits of preferring the word “woman” to the +word “lady” and of going to the post-office without a hat, +imagined that she was unconventional and altogether a +remarkable person; and who once remarked with great satisfaction +that she was a “very queer person,” and that nothing +shocked her “except, of course, bad form.”</p> + +<p>Thus, she asserted that all the things which shocked her +were actions in bad form; and she would undoubtedly agree, +though she did not actually state it, that all the things which +were done in bad form would shock her. Consequently +she asserted that the class of things which shocked her was +the class of actions in bad form. Consequently the statement +of this lady that some or all of the actions done in bad +form shocked her is an identical proposition of the form +“nothing shocks me, except, of course, the things which do, +in fact, shock me”; and this statement the lady certainly +did not intend to make.</p> + +<p>This excellent lady, had she but known it, was logically +justified in making any statement whatever about her unconventionality. +For the class of her unconventional actions +was the null class. Thus she might logically have made +inconsistent statements about this class of actions. As a +matter of fact she did make inconsistent statements, but +unfortunately she justified them by stating that, “It is the +privilege of woman to be inconsistent.” She was one of +those persons who say things like that.</p> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_19" id="Page_19">[Pg 19]</a></span></p> +<h3><a name="CHAPTER_V" id="CHAPTER_V"></a>CHAPTER V</h3> + +<h2>ETHICAL APPLICATIONS OF THE LAW OF +IDENTITY</h2> + + +<p>It may be remembered that Mr. Podsnap remarked, with +sadness tempered by satisfaction, that he regretted to say +that “Foreign nations do as they do do.” Besides aiding +the comforting expression of moral disapproval, the law of +identity has yet another useful purpose in practical ethics: +It serves the welcome purpose of providing an excuse for +infractions of the moral law. There was once a man who +treated his wife badly, was unfaithful to her, was dishonest +in business, and was not particular in his use of language; +and yet his life on earth was described in the lines:</p> + +<p class="poem"> +This man maintained a wife’s a wife,<br /> +Men are as they are made,<br /> +Business is business, life is life;<br /> +And called a spade a spade.<br /> +</p> + +<p>One of the objects of Dr. G. E. Moore’s <i>Principia Ethica</i><a name="FNanchor_22_22" id="FNanchor_22_22"></a><a href="#Footnote_22_22" class="fnanchor">[22]</a> +was to argue that the word “good” means simply good, +and not pleasant or anything else. Appropriately enough, +this book bore on its title-page the quotation from the preface +to the <i>Sermons</i>, published in 1726, of Bishop Joseph Butler, +the author of the <i>Analogy</i>: “Everything is what it is and +not another thing.”</p> + +<p>But another famous Butler—Samuel Butler, the author +of <i>Hudibras</i>—went farther than this, and maintained that +identities were the highest attainment of metaphysics itself. +At the beginning of the first Canto of <i>Hudibras</i>, in the description +of Hudibras himself, Butler wrote:</p> + +<p class="poem"> +He knew what’s what, and that’s as high<br /> +As metaphysic wit can fly.<br /> +</p> + +<p><span class='pagenum'><a name="Page_20" id="Page_20">[Pg 20]</a></span>I once conducted what I imagined to be an æsthetic +investigation for the purpose of discovery, by the continual +use of the word “Why?”<a name="FNanchor_23_23" id="FNanchor_23_23"></a><a href="#Footnote_23_23" class="fnanchor">[23]</a> the grounds upon which certain +people choose to put milk into a tea-cup before the tea. +I was surprised to discover that it was an ethical, and not +an æsthetic problem; for I soon elicited the fact that it +was done because it was “right.” A continuance of my +patient questioning elicited further evidence of the fundamental +character of the principle of identity in ethics; for +it was right, I learned, because “right is right.”</p> + +<p>It appears that some people unconsciously think that the +principle of identity is the foundation, in certain religions, +of the reasons which can be alleged for moral conduct, and +are surprised when this fact is pointed out to them. The +late Sir Leslie Stephen, when travelling by railway, fell +into conversation with an officer of the Salvation Army, +who tried hard to convert him. Failing in this laudable +endeavour, the Salvationist at last remarked: “But if you +aren’t saved, you can’t go to heaven!” “That, my friend,” +replied Stephen, “is an identical proposition.”</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_22_22" id="Footnote_22_22"></a><a href="#FNanchor_22_22"><span class="label">[22]</span></a> Cambridge, 1903.</p></div> + +<div class="footnote"><p><a name="Footnote_23_23" id="Footnote_23_23"></a><a href="#FNanchor_23_23"><span class="label">[23]</span></a> Cf. <i>P. E.</i>, p. 2.</p></div> + + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_21" id="Page_21">[Pg 21]</a></span></p> +<h3><a name="CHAPTER_VI" id="CHAPTER_VI"></a>CHAPTER VI</h3> + +<h2>THE LAW OF CONTRADICTION IN MODERN LOGIC</h2> + + +<p>Considering the important place assigned by philosophers +and logicians to the law of contradiction, the remark will +naturally be resented by many of the older schools of philosophy, +and especially by Kantians, that “in spite of its fame +we have found few occasions for its use.”<a name="FNanchor_24_24" id="FNanchor_24_24"></a><a href="#Footnote_24_24" class="fnanchor">[24]</a> Also in modern +times, Benedetto Croce, an opponent of both traditional +logic and mathematical logic, began the preface of the book +of 1908 on Logic<a name="FNanchor_25_25" id="FNanchor_25_25"></a><a href="#Footnote_25_25" class="fnanchor">[25]</a> by saying that that volume “is and is +not” a certain memoir of his which had been published in +1905.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_24_24" id="Footnote_24_24"></a><a href="#FNanchor_24_24"><span class="label">[24]</span></a> <i>Pa. Ma.</i>, p. 116.</p></div> + +<div class="footnote"><p><a name="Footnote_25_25" id="Footnote_25_25"></a><a href="#FNanchor_25_25"><span class="label">[25]</span></a> [English translation of the third Italian edition by Douglas Ainslie, +under the title: <i>Logic as the Science of the Pure Concept</i>, London +1917.—<span class="smcap">Ed.</span>]</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_22" id="Page_22">[Pg 22]</a></span></p> +<h3><a name="CHAPTER_VII" id="CHAPTER_VII"></a>CHAPTER VII</h3> + +<h2>SYMBOLISM AND MEANING</h2> + + +<p>When people write down any statement such as “The curfew +tolls the knell of parting day,”<a name="FNanchor_26_26" id="FNanchor_26_26"></a><a href="#Footnote_26_26" class="fnanchor">[26]</a> which we will call “C” +for shortness, what they mean is not “C” but the <i>meaning</i> +of “C”; and not “the meaning of ‘C’” but the <i>meaning</i> of +“the meaning of ‘C’.” And so on, <i>ad infinitum</i>. Thus, in +writing or in speech, we always fail to state the meaning of +any proposition whatever. Sometimes, indeed, we succeed +in <i>conveying</i> it; but there is danger in too great a disregard +of statement and preoccupation with conveyance of meaning. +Thus many mathematicians have been so anxious to convey +to us a perfectly distinct and unmetaphysical concept of +number that they have stripped away from it everything +that they considered unessential (like its logical nature) +and have finally delivered it to us as a mere <i>sign</i>. By the +labours of Helmholtz, Kronecker, Heine, Stolz, Thomae, +Pringsheim, and Schubert, many people were persuaded +that, when they said “‘2’ is a number” they were speaking +the truth, and hold that “Paris” is a town containing +the letter “P.” When Frege pointed out<a name="FNanchor_27_27" id="FNanchor_27_27"></a><a href="#Footnote_27_27" class="fnanchor">[27]</a> this difficulty +he was almost universally denounced in Germany as “<i>spitzfindig</i>.” +In fact, Germans seem to have been influenced +perhaps by that great contemner of “<i>Spitzfindigkeit</i>,” Kant, +to reject the White Knight’s<a name="FNanchor_28_28" id="FNanchor_28_28"></a><a href="#Footnote_28_28" class="fnanchor">[28]</a> distinctions between words +and their denotations and to regard subtlety with disfavour +to such a degree that their only mathematical logician except +Frege, namely Schröder—the least subtle of mortals, by +the way—seems to have been filled with such fear of being<span class='pagenum'><a name="Page_23" id="Page_23">[Pg 23]</a></span> +thought subtle, that he made his books so prolix that nobody +has read them.</p> + +<p>Another term which, as we shall see when discussing +the paradoxes of logic, mathematicians are accustomed to +apply to thought which is more exact than any to which +they are accustomed is “scholastic.”<a name="FNanchor_29_29" id="FNanchor_29_29"></a><a href="#Footnote_29_29" class="fnanchor">[29]</a> By this, I suppose, +they mean that the pursuits of certain acute people of the +Middle Ages are unimportant in contrast with the great +achievements of modern thought, as exemplified by a +method of making plausible guesses known as induction,<a name="FNanchor_30_30" id="FNanchor_30_30"></a><a href="#Footnote_30_30" class="fnanchor">[30]</a> +the bicycle, and the gramophone—all of them instruments +of doubtful merit.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_26_26" id="Footnote_26_26"></a><a href="#FNanchor_26_26"><span class="label">[26]</span></a> Cf. <i>Md</i>, N. S., vol. xiv., October 1905, p. 486.</p></div> + +<div class="footnote"><p><a name="Footnote_27_27" id="Footnote_27_27"></a><a href="#FNanchor_27_27"><span class="label">[27]</span></a> In <i>Z. S.</i>, for example.</p></div> + +<div class="footnote"><p><a name="Footnote_28_28" id="Footnote_28_28"></a><a href="#FNanchor_28_28"><span class="label">[28]</span></a> See <a href="#App_G">Appendix G</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_29_29" id="Footnote_29_29"></a><a href="#FNanchor_29_29"><span class="label">[29]</span></a> Cf. Chapter XXXVII below.</p></div> + +<div class="footnote"><p><a name="Footnote_30_30" id="Footnote_30_30"></a><a href="#FNanchor_30_30"><span class="label">[30]</span></a> Cf. <i>P. M.</i>, p. 11, note.</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_24" id="Page_24">[Pg 24]</a></span></p> +<h3><a name="CHAPTER_VIII" id="CHAPTER_VIII"></a>CHAPTER VIII</h3> + +<h2>NOMINALISM</h2> + + +<p>De Morgan<a name="FNanchor_31_31" id="FNanchor_31_31"></a><a href="#Footnote_31_31" class="fnanchor">[31]</a> said that, “if all mankind had spoken one +language, we cannot doubt that there would have been a +powerful, perhaps universal, school of philosophers who +would have believed in the inherent connexion between +names and things; who would have taken the sound <i>man</i> +to be the mode of agitating the air which is essentially communicative +of the ideas of reason, cookery, bipedality, etc.... +‘The French,’ said the sailor, ‘call a cabbage a <i>shoe</i>; the +fools! Why can’t they call it a cabbage, when they must +know it is one?’”</p> + +<p>One of the chief differences between logicians and men +of letters is that the latter mean many different things by +one word, whereas the former do not—at least nowadays. +Most mathematicians belong to the class of men of letters.</p> + +<p>I once had a manservant who told me on a certain occasion +that he “never thought a word about it.” I was doubtful +whether to class him with such eminent mathematicians +as are mentioned in the last chapter, or as a supporter of +Max Müller’s theory of the identity of thought and language. +However, since the man was very untruthful, and he told +me that he meant what he said and said what he meant,<a name="FNanchor_32_32" id="FNanchor_32_32"></a><a href="#Footnote_32_32" class="fnanchor">[32]</a> +the conclusion is probably correct that he really believed +that the meanings of his words were not the words themselves. +Thus I think it most probable that my manservant had been +a mathematician but had escaped by the aid of logic.</p> + +<p>As regards his remark that he meant what he said and<span class='pagenum'><a name="Page_25" id="Page_25">[Pg 25]</a></span> +said what he meant, he plainly wished to pride himself on +certain virtues which he did not possess, and was not indifferent +to applause, which, however, was never evoked. +The virtues, if so they be, and the applause were withheld +for other reasons than that the above statements are either +nonsensical or false. Suppose that “I say what I mean” +expresses a truth. What I say (or write) is always a symbol—words +(or marks); and what I mean by the symbol is +the meaning of the symbol and not the symbol itself. So +the remark cannot express a truth, any more than the name +“Wellington” won the battle of Waterloo.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_31_31" id="Footnote_31_31"></a><a href="#FNanchor_31_31"><span class="label">[31]</span></a> <i>F. L.</i>, pp. 246-7.</p></div> + +<div class="footnote"><p><a name="Footnote_32_32" id="Footnote_32_32"></a><a href="#FNanchor_32_32"><span class="label">[32]</span></a> The Hatter (see <a href="#App_H">Appendix H</a>) pointed out that there is a difference +between these two assertions. Thus, he clearly showed that he was +a nominalist, and philosophically opposed to the March Hare who had +recommended Alice to say what she meant.</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_26" id="Page_26">[Pg 26]</a></span></p> +<h3><a name="CHAPTER_IX" id="CHAPTER_IX"></a>CHAPTER IX</h3> + +<h2>AMBIGUITY AND SYMBOLIC LOGIC</h2> + + +<p>The universal use of some system of Symbolic Logic would +not only enable everybody easily to deal with exceedingly complicated +arguments, but would prevent ambiguous arguments. +In denying the indispensability of Symbolic Logic in the +former state of things, Keynes<a name="FNanchor_33_33" id="FNanchor_33_33"></a><a href="#Footnote_33_33" class="fnanchor">[33]</a> is probably alone, against +the need strongly felt by Alice when speaking to the Duchess,<a name="FNanchor_34_34" id="FNanchor_34_34"></a><a href="#Footnote_34_34" class="fnanchor">[34]</a> +and most modern logicians. It may be noticed that the +Duchess is more consistent than Keynes, for Keynes really +uses the signs for logical multiplication and addition of Boole +and Venn under the different shapes of the words “and” +and “or.”</p> + +<p>As regards ambiguity, a translation of <i>Hymns Ancient +and Modern</i> into, say, Peanesque, would prevent the puzzle +of childhood as to whether the “his” in</p> + +<p class="poem"> +And Satan trembles when he sees<br /> +The weakest saint upon his knees<br /> +</p> + +<p class="noidt">refers to the saint’s knees or Satan’s.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_33_33" id="Footnote_33_33"></a><a href="#FNanchor_33_33"><span class="label">[33]</span></a> In his <i>Fm. L.</i></p></div> + +<div class="footnote"><p><a name="Footnote_34_34" id="Footnote_34_34"></a><a href="#FNanchor_34_34"><span class="label">[34]</span></a> See <a href="#App_I">Appendix I</a>.</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_27" id="Page_27">[Pg 27]</a></span></p> +<h3><a name="CHAPTER_X" id="CHAPTER_X"></a>CHAPTER X</h3> + +<h2>LOGICAL ADDITION AND THE UTILITY OF +SYMBOLISM</h2> + + +<p>Frequently ordinary language contains subtle psychological +implications which cannot be translated into symbolic +logic except at great length. Thus if a man (say Mr. Jones) +wishes to speak collectively of himself and his wife, the +order of mentioning the terms in the class considered and +the names applied to these terms are, logically speaking, +irrelevant. And yet more or less definite information is +given about Mr. Jones according as he talks to his friends of:</p> + + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="1" summary=""> +<tr><td align='left'></td><td align='left'>(1) Mrs. Jones and I,</td></tr> +<tr><td align='left'></td><td align='left'>(2) I (or me) and my wife (or missus),</td></tr> +<tr><td align='left'></td><td align='left'>(3) My wife and I,</td></tr> +<tr><td align='left'>or </td><td align='left'>(4) I (or me) and Mrs. Jones.</td></tr> +</table></div> + +<p>In case (1) one is probably correct in placing Mr. Jones +among the clergy or the small professional men who make +up the bulk of the middle-class; in case (2) one would conclude +that Mr. Jones belonged to the lower middle-class; +the form (3) would be used by Mr. Jones if he were a member +of the upper, upper middle, or lower class; while form (4) is +only used by retired shopkeepers of the lower middle-class, of +which a male member usually combines belief in the supremacy +of man with belief in the dignity of his wife as well +as himself. A further complication is introduced if a wife +is referred to as “the wife.”<a name="FNanchor_35_35" id="FNanchor_35_35"></a><a href="#Footnote_35_35" class="fnanchor">[35]</a> Cases (2) and (3) then each +give rise to one more case. Cases (1) and (4) do not, since +nobody has hitherto referred to his wife as “the Mrs. +Jones”—at least without a qualifying adjective before +the “Mrs.<span class='pagenum'><a name="Page_28" id="Page_28">[Pg 28]</a></span>”</p> + +<p>On the other hand, certain descriptive phrases and certain +propositions can be expressed more shortly and more accurately +by means of symbolic logic. Let us consider the +proposition “No man marries his deceased wife’s sister.” +If we assume, as a first approximation, that all marriages +are fertile and that all children are legitimate, then, with +only four primitive ideas: the relation of parent to child (P) +and the three classes of males, females, and dead people, we +can define “wife” (a female who has the relation formed +by taking the relative product of P and P̌<a name="FNanchor_36_36" id="FNanchor_36_36"></a><a href="#Footnote_36_36" class="fnanchor">[36]</a> to a male), +“sister,” “deceased wife,” and “deceased wife’s sister” in +terms of these ideas and of the fundamental notions of logic. +Then the proposition “No man marries his deceased wife’s +sister” can be expressed unambiguously by about twenty-nine +simple signs on paper, whereas, in words, the unasserted +statement consists of no less than thirty-four letters. +Although, legally speaking, we should have to adopt somewhat +different definitions and possibly increase the complications +of our proposition, it must be remembered that, +on the other hand, we always reduce the number of symbols +in any proposition by increasing the number of definitions +in the preliminaries to it.</p> + +<p>But the utility of symbolic logic should not be estimated +by the brevity with which propositions may sometimes +be expressed by its means. Logical simplicity, in fact, +can very often only be obtained by apparently complicated +statements. For example, the logical interpretation of +“The father of Charles II was executed” is, “It is not always +false of <i>x</i> that <i>x</i> begat Charles II, and that <i>x</i> was executed +and that ‘if <i>y</i> begat Charles II, <i>y</i> is identical with <i>x</i>’ is always +true of <i>y</i>.”<a name="FNanchor_37_37" id="FNanchor_37_37"></a><a href="#Footnote_37_37" class="fnanchor">[37]</a> From the point of view of logic, we may say +that the apparently simple is most often very complicated, +and, even if it is not so, symbolism will make it seem so,<a name="FNanchor_38_38" id="FNanchor_38_38"></a><a href="#Footnote_38_38" class="fnanchor">[38]</a> +and thus draw attention to what might otherwise easily +be overlooked.</p> + +<hr /> + +<div class="footnote"><p><a name="Footnote_35_35" id="Footnote_35_35"></a><a href="#FNanchor_35_35"><span class="label">[35]</span></a> Cf. Chapter XXIV below.</p></div> + +<div class="footnote"><p><a name="Footnote_36_36" id="Footnote_36_36"></a><a href="#FNanchor_36_36"><span class="label">[36]</span></a> C. S. Peirce’s notation for the relation “converse of P.”</p></div> + +<div class="footnote"><p><a name="Footnote_37_37" id="Footnote_37_37"></a><a href="#FNanchor_37_37"><span class="label">[37]</span></a> Russell, <i>Md.</i>, N. S., vol. xiv., October 1905, p. 482.</p></div> + +<div class="footnote"><p><a name="Footnote_38_38" id="Footnote_38_38"></a><a href="#FNanchor_38_38"><span class="label">[38]</span></a> Russell, <i>International Monthly</i>, vol. iv., 1901, pp. 85-6; cf. <i>M.</i>, +vol. xxii., 1912, p. 153. [This essay is reprinted in <i>Mysticism and +Logic</i>, London and New York, 1918, pp. 74-96.—<span class="smcap">Ed.</span>]</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_29" id="Page_29">[Pg 29]</a></span></p> +<h3><a name="CHAPTER_XI" id="CHAPTER_XI"></a>CHAPTER XI</h3> + +<h2>CRITICISM</h2> + + +<p>Those people who think that it is more godlike to seem to +turn water into wine than to seem to turn wine into water +surprise me. I cannot imagine an intolerable critic. It +seems to me that, if A resents B’s criticism in trying to put +his (A’s) discovery in the right or wrong place, A acts as +if he thought he had some private property in truth. The +White Queen seems to have shared the popular misconception +as to the nature of criticism.<a name="FNanchor_39_39" id="FNanchor_39_39"></a><a href="#Footnote_39_39" class="fnanchor">[39]</a></p> + +<hr /> + +<div class="footnote"><p><a name="Footnote_39_39" id="Footnote_39_39"></a><a href="#FNanchor_39_39"><span class="label">[39]</span></a> See <a href="#App_J">Appendix J</a>.</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_30" id="Page_30">[Pg 30]</a></span></p> +<h3><a name="CHAPTER_XII" id="CHAPTER_XII"></a>CHAPTER XII</h3> + +<h2>HISTORICAL CRITICISM</h2> + + +<p>From a problem in Diophantus’s <i>Arithmetic</i> about the price +of some wine it would seem that the wine was of poor quality, +and Paul Tannery has suggested that the prices mentioned +for such a wine are higher than were usual until after the +end of the second century. He therefore rejected the view +which was formerly held that Diophantus lived in that +century.<a name="FNanchor_40_40" id="FNanchor_40_40"></a><a href="#Footnote_40_40" class="fnanchor">[40]</a></p> + +<p>The same method applied to a problem given by the ancient +Hindu algebraist Brahmagupta, who lived in the seventh +century after Christ, might result in placing Brahmagupta +in prehistoric times. This is the problem:<a name="FNanchor_41_41" id="FNanchor_41_41"></a><a href="#Footnote_41_41" class="fnanchor">[41]</a> “Two apes +lived at the top of a cliff of height <i>h</i>, whose base was distant +<i>mh</i> from a neighbouring village. One descended the cliff +and walked to the village, the other flew up a height <i>x</i> and +then flew in a straight line to the village. The distance +traversed by each was the same. Find <i>x</i>.”</p> + +<hr /> + +<div class="footnote"><p><a name="Footnote_40_40" id="Footnote_40_40"></a><a href="#FNanchor_40_40"><span class="label">[40]</span></a> W. W. Rouse Ball, <i>A Short Account of the History of Mathematics</i>, +4th edition, London, 1908, p. 109.</p></div> + +<div class="footnote"><p><a name="Footnote_41_41" id="Footnote_41_41"></a><a href="#FNanchor_41_41"><span class="label">[41]</span></a> <i>Ibid.</i>, pp. 148-9.</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_31" id="Page_31">[Pg 31]</a></span></p> +<h3><a name="CHAPTER_XIII" id="CHAPTER_XIII"></a>CHAPTER XIII</h3> + +<h2>IS THE MIND IN THE HEAD?</h2> + + +<p>The contrary opinion has been maintained by idealists and +a certain election agent with whom I once had to deal and +who remarked that something slipped his mind and then +went out of his head altogether. At some period, then, +a remembrance was in his head and out of his mind; his +mind was not, then, wholly within his head. Also, one +is sometimes assured that with certain people “out of sight +is out of mind.” What is in their minds is therefore in +sight, and cannot therefore be inside their heads.</p> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_32" id="Page_32">[Pg 32]</a></span></p> +<h3><a name="CHAPTER_XIV" id="CHAPTER_XIV"></a>CHAPTER XIV</h3> + +<h2>THE PRAGMATIST THEORY OF TRUTH</h2> + + +<p>The pragmatist theory that “truth” is a belief which works +well sometimes conflicts with common-sense and not with +logic. It is commonly supposed that it is always better +to be sometimes right than to be never right. But this is +by no means true. For example, consider the case of a +watch which has stopped; it is exactly right twice every day. +A watch, on the other hand, which is always five minutes +slow is never exactly right. And yet there can be no question +but that a belief in the accuracy of the watch which was +never right would, on the whole, produce better results than +such a belief in the one which had altogether stopped. The +pragmatist would, then, conclude that the watch which +was always inaccurate gave truer results than the one which +was sometimes accurate. In this conclusion the pragmatist +would seem to be correct, and this is an instance of how +the false premisses of pragmatism may give rise to true +conclusions.</p> + +<p>From the text written above the church clock in a certain +English village, “Be ye ready, for ye know not the time,” +it would be concluded that the clock never stopped for a +period as long as twelve hours. For the text is rather a +vague symbolical expression of a propositional function +which is asserted to be true at all instants. The proposition +that a presumably not illiterate and credulous observer of +the clock at any definite instant does not know the time +implies, then, that the clock is always wrong. Now, if the +clock stopped for twelve hours, it would be absolutely right +at least once. It must be right twice if it were right at the +first instant it stopped or the last instant at which it went;<a name="FNanchor_42_42" id="FNanchor_42_42"></a><a href="#Footnote_42_42" class="fnanchor">[42]</a><span class='pagenum'><a name="Page_33" id="Page_33">[Pg 33]</a></span> +but the second possibility is excluded by hypothesis, and +the occurrence of the first possibility—or of the analogous +possibility of the stopped clock being right three times in +twenty-four hours—does not affect the present question. +Hence the clock can never stop for twelve hours.</p> + +<p>The pragmatist’s criterion of truth appears to be far more +difficult to apply than the Bellman’s,<a name="FNanchor_43_43" id="FNanchor_43_43"></a><a href="#Footnote_43_43" class="fnanchor">[43]</a> that what he said +three times is true, and to give results just as insecure.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_42_42" id="Footnote_42_42"></a><a href="#FNanchor_42_42"><span class="label">[42]</span></a> Both cases cannot occur; the question is similar to that arising +in the discussion of the mortality of Socrates (see Chapter XXII).</p></div> + +<div class="footnote"><p><a name="Footnote_43_43" id="Footnote_43_43"></a><a href="#FNanchor_43_43"><span class="label">[43]</span></a> See <a href="#App_K">Appendix K</a>.</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_34" id="Page_34">[Pg 34]</a></span></p> +<h3><a name="CHAPTER_XV" id="CHAPTER_XV"></a>CHAPTER XV</h3> + +<h2>ASSERTION</h2> + + +<p>The subject of the present chapter must not be confused +with the assertion of ordinary life. Commonly, an unasserted +proposition is synonymous with a probably false statement, +while an asserted proposition is synonymous with one that +is certainly false. But in logic we apply assertion also to +true propositions, and, as Lewis Carroll showed in his version +of “What the Tortoise said to Achilles,”<a name="FNanchor_44_44" id="FNanchor_44_44"></a><a href="#Footnote_44_44" class="fnanchor">[44]</a> usually pass +over unconsciously an infinite series of implications in so +doing. If <i>p</i> and <i>q</i> are propositions, <i>p</i> is true, and <i>p</i> implies <i>q</i>, +then, at first sight, one would think that one might assert <i>q</i>. +But, from (A) <i>p</i> is true, and (B) <i>p</i> implies <i>q</i>, we must, in order +to deduce (Z) <i>q</i> is true, accept the hypothetical: (C) If A and +B are true, Z must be true. And then, in order to deduce +Z from A, B, and C, we must accept another hypothetical: +(D) If A, B, and C are true, Z must be true; and so on <i>ad +infinitum</i>. Thus, in deducing Z, we pass over an infinite +series of hypotheticals which increase in complexity. Thus +we need a new principle to be able to assert <i>q</i>.</p> + +<p>Frege was the first logician sharply to distinguish between +an asserted proposition, like “A is greater than B,” and +one which is merely considered, like “A’s being greater than +B,” although an analogous distinction had been made in +our common discourse on certain psychological grounds, +for long previously. In fact, soon after the invention of +speech, the necessity of distinguishing between a considered +proposition and an asserted one became evident, on account +of the state of things referred to at the beginning of this +chapter.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_44_44" id="Footnote_44_44"></a><a href="#FNanchor_44_44"><span class="label">[44]</span></a> <i>Md.</i> N. S., vol. iv., 1895, pp. 278-80. Cf. Russell, <i>P. M.</i>, p. 35.</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_35" id="Page_35">[Pg 35]</a></span></p> +<h3><a name="CHAPTER_XVI" id="CHAPTER_XVI"></a>CHAPTER XVI</h3> + +<h2>THE COMMUTATIVE LAW</h2> + + +<p>Often the meaning of a sentence tacitly implies that the +commutative law does not hold. We are all familiar with +the passage in which Macaulay pointed out that, by using +the commutative law because of exigencies of metre, Robert +Montgomery unintentionally made Creation tremble at the +Atheist’s nod instead of the Almighty’s. This use of the +commutative law by writers of verse renders it doubtful +whether, in the hymn-line:</p> + +<p class="center"> +The humble poor believe,<br /> +</p> + +<p class="noidt">we are to understand a statement about the humble poor, +or a doubtful maxim as to the attitude of our minds to +statements made by the humble poor.</p> + +<p>The non-commutativity of English titles offers difficulties +to some novelists and Americans who refer to Mary Lady +So-and-So as Lady Mary So-and-So, and <i>vice versa</i>.</p> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_36" id="Page_36">[Pg 36]</a></span></p> +<h3><a name="CHAPTER_XVII" id="CHAPTER_XVII"></a>CHAPTER XVII</h3> + +<h2>UNIVERSAL AND PARTICULAR PROPOSITIONS</h2> + + +<p>People who are cynical as to the morality of the English +are often unpleasantly surprised to learn that “All trespassers +will be prosecuted” does not necessarily imply that +“some trespassers will be prosecuted.” The view that +universal propositions are non-existential is now generally +held: Bradley and Venn seem to have been the first to hold +this, while older logicians, such as De Morgan,<a name="FNanchor_45_45" id="FNanchor_45_45"></a><a href="#Footnote_45_45" class="fnanchor">[45]</a> considered +universal propositions to be existential, like particular ones.</p> + +<p>If the Gnat<a name="FNanchor_46_46" id="FNanchor_46_46"></a><a href="#Footnote_46_46" class="fnanchor">[46]</a> had been content to affirm his proposition +about the means of subsistence of Bread-and-Butter flies, +in consequence of their lack of which such flies always die, +without pointing out such an insect and thereby proving +that the class of them is not null, Alice’s doubt as to the +existence of the class in question, even if it were proved to +be well founded, would not have affected the validity of +the proposition.</p> + +<p>This brings us to a great convenience in treating universal +propositions as non-existential: we can maintain that all +<i>x</i>’s are <i>y</i>’s at the same time as that no <i>x</i>’s are <i>y</i>’s, if only +<i>x</i> is the null-class. Thus, when Mr. MacColl<a name="FNanchor_47_47" id="FNanchor_47_47"></a><a href="#Footnote_47_47" class="fnanchor">[47]</a> objected to +other symbolic logicians that their premisses imply that all +Centaurs are flower-pots, they could reply that their premisses +also imply the more usual view that Centaurs are +not flower-pots.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_45_45" id="Footnote_45_45"></a><a href="#FNanchor_45_45"><span class="label">[45]</span></a> Cf., e.g., <i>F. L.</i>, p. 4.</p></div> + +<div class="footnote"><p><a name="Footnote_46_46" id="Footnote_46_46"></a><a href="#FNanchor_46_46"><span class="label">[46]</span></a> See <a href="#App_L">Appendix L</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_47_47" id="Footnote_47_47"></a><a href="#FNanchor_47_47"><span class="label">[47]</span></a> Cf., e.g., <i>Md.</i>, N. S., vol. xiv., July, 1905, pp. 399-400.</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_37" id="Page_37">[Pg 37]</a></span></p> +<h3><a name="CHAPTER_XVIII" id="CHAPTER_XVIII"></a>CHAPTER XVIII</h3> + +<h2>DENIAL OF GENERALITY AND GENERALITY +OF DENIAL</h2> + + +<p>The conclusion of a certain song<a name="FNanchor_48_48" id="FNanchor_48_48"></a><a href="#Footnote_48_48" class="fnanchor">[48]</a> about a young man who +poisoned his sweetheart with sheep’s-head broth, and was +frightened to death by a voice exclaiming:</p> + +<p class="poem"> +“Where’s that young maid<br /> +What you did poison with my head?”<br /> +</p> + +<p class="noidt">at his bedside, gives rise to difficulties which are readily +solved by a symbolism that brings into relief the principle +that the denial of a universal and non-existential proposition +is a particular and existential one. The conclusion +of the song is:</p> + +<p class="poem"> +Now all young men, both high and low,<br /> +Take warning by this dismal go!<br /> +For if he’d never done nobody no wrong,<br /> +He might have been here to have heard this song.<br /> +</p> + +<p>It is an obvious error, say Whitehead and Russell,<a name="FNanchor_49_49" id="FNanchor_49_49"></a><a href="#Footnote_49_49" class="fnanchor">[49]</a> though +one easy to commit, to assume that the cases: (1) all the +propositions of a certain class are true; and (2) no proposition +of the class is true; are each other’s contradictories. +However, in the modification<a name="FNanchor_50_50" id="FNanchor_50_50"></a><a href="#Footnote_50_50" class="fnanchor">[50]</a> of Frege’s symbolism which +was used by Russell</p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="1" summary=""> +<tr><td align='left'></td><td align='left'>(1) is (<i>x</i>). <i>x</i>,</td></tr> +<tr><td align='left'>and </td><td align='left'>(2) is (<i>x</i>). not <i>x</i>;</td></tr> +</table></div> +<p><span class='pagenum'><a name="Page_38" id="Page_38">[Pg 38]</a></span></p> + +<p class='noidt'>while the contradictory of (1) is:</p> + +<p class='center'> +not (<i>x</i>). <i>x</i>.<br /> +</p> + +<p>The last line but one of the above verse may, then, be +written:</p> + +<p class='center'> +(<i>t</i>). not (<i>x</i>). not not ϕ(<i>x</i>, <i>t</i>),<br /> +</p> + +<p class='noidt'>where “ϕ(<i>x</i>, <i>t</i>)” denotes the unasserted propositional function +“the doing wrong to the person <i>x</i> at the instant <i>t</i>.” By +means of the principle of double negation we can at once +simplify the above expression into:</p> + +<p class='center'> +(<i>t</i>). not (<i>x</i>). ϕ(<i>x</i>, <i>t</i>);<br /> +</p> + +<p class='noidt'>which can be thus read: “If at every instant of his life +there was at least one person <i>x</i> to whom he did no wrong +(at that instant).” It is difficult to imagine any one so +sunk in iniquity that he would not satisfy this hypothesis. +We are forced, then, unless our imagination for evil is to +be distrusted, to conclude that any one might have been +there to have heard that song. Now this conclusion is +plainly false, possibly on physical grounds, and certainly +on æsthetic grounds. It may be added, by the way, that +it is quite possible that De Morgan was mistaken in his +interpretation of the above proposition owing to the fact +that he was unacquainted with Frege’s work. In fact, if +he had not noticed the fact that <i>any</i> two of the “not’s” +cannot be cancelled against one another he would have +concluded that the interpretation was: “If he had never +done any wrong to anybody.”</p> + +<p>According as the symbol for “not” comes before the +(<i>x</i>) or between the (<i>x</i>) and the ϕ, we have an expression of +what Frege called respectively the denial of generality, +and the generality of denial. The denial of the generality +of a denial is the form of all existential propositions, while +the assertion of or denial of generality is the general form +of all non-existential or universal propositions.</p> + +<hr /> + +<div class="footnote"><p><a name="Footnote_48_48" id="Footnote_48_48"></a><a href="#FNanchor_48_48"><span class="label">[48]</span></a> To which De Morgan drew attention in a letter; see (Mrs.) S. E. +De Morgan, <i>Memoir of Augustus De Morgan</i>, London, 1882, p. 324.</p></div> + +<div class="footnote"><p><a name="Footnote_49_49" id="Footnote_49_49"></a><a href="#FNanchor_49_49"><span class="label">[49]</span></a> <i>Pa. Ma.</i>, p. 16.</p></div> + +<div class="footnote"><p><a name="Footnote_50_50" id="Footnote_50_50"></a><a href="#FNanchor_50_50"><span class="label">[50]</span></a> However, here, for the printer’s convenience, we depart from +Mr. Russell’s usage so far as to write “not” for a curly minus sign.</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_39" id="Page_39">[Pg 39]</a></span></p> +<h3><a name="CHAPTER_XIX" id="CHAPTER_XIX"></a>CHAPTER XIX</h3> + +<h2>IMPLICATION</h2> + + +<p>A good illustration of the fact that what is called “implication” +in logic is such that a false proposition implies any +other proposition, true or false, is given by Lewis Carroll’s +puzzle of the three barbers.<a name="FNanchor_51_51" id="FNanchor_51_51"></a><a href="#Footnote_51_51" class="fnanchor">[51]</a></p> + +<p>Allen, Brown, and Carr keep a barber’s shop together; so +that one of them must be in during working hours. Allen +has lately had an illness of such a nature that, if Allen is +out, Brown must be accompanying him. Further, if Carr +is out, then, if Allen is out, Brown must be in for obvious +business reasons. The problem is, may Carr ever go out?</p> + +<p>Putting <i>p</i> for “Carr is out,” <i>q</i> for “Allen is out” and <i>r</i> +for “Brown is out,” we have:</p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="1" summary=""> +<tr><td align='left'>(1) <i>q</i> implies <i>r</i>,</td></tr> +<tr><td align='left'>(2) <i>p</i> implies that <i>q</i> implies not-<i>r</i>.</td></tr> +</table></div> + +<p>Lewis Carroll supposed that “<i>q</i> implies <i>r</i>” and “<i>q</i> implies +not-<i>r</i>” are inconsistent, and hence that <i>p</i> must be false. +But these propositions are not inconsistent, and are, in +fact, both true if <i>q</i> is false. The contradictory of “<i>q</i> implies +<i>r</i>” is “<i>q</i> does not imply <i>r</i>” which is not a consequence of +“<i>q</i> implies not-<i>r</i>.” It seems to be true theoretically that, +if Mr. X is a Christian, he is not an Atheist, but we cannot +conclude from this alone that his being a Christian does not +imply that he is an Atheist, unless we assume that the class +of Christians is not null. Thus, if <i>p</i> is true, <i>q</i> is false; or, +if Carr is out, Allen is in. The odd part of this conclusion +is that it is the one which common-sense would have drawn +in that particular case.<span class='pagenum'><a name="Page_40" id="Page_40">[Pg 40]</a></span></p> + +<p>A distinguished philosopher (M) once thought that the +logical use of the word “implication”—any false proposition +being said to “imply” any proposition true or false—is +absurd, on the grounds that it is ridiculous to suppose that +the proposition “2 and 2 make 5” implies the proposition +“M is the Pope.” This is a most unfortunate instance, +because it so happens that the false proposition that 2 and 2 +make 5 can rigorously be proved to imply that M, or anybody +else other than the Pope, is the Pope. For if 2 and 2 +make 5, since they also make 4, we would conclude that +5 is equal to 4. Consequently, subtracting 3 from both +sides, we conclude that 2 would be equal to 1. But if this +were true, since M and the Pope are two, they would be one, +and obviously then M would be the Pope.</p> + +<p>The principle that the false implies the true has very +important applications in political arguments. In fact, it +is hard to find a single principle of politics of which false +propositions are not the main support.</p> + +<p>If <i>p</i> and <i>q</i> are two propositions, and <i>p</i> implies <i>q</i>; then, +if, and only if, <i>q</i> and <i>p</i> are both false or both true, we also +have: <i>q</i> implies <i>p</i>. The most important applications of this +invertibility were made by the late Samuel Butler<a name="FNanchor_52_52" id="FNanchor_52_52"></a><a href="#Footnote_52_52" class="fnanchor">[52]</a> and +Mr. G. B. Shaw. A political application may be made as +follows: In a country where only those with middling-sized +incomes are taxed, conservative and <i>bourgeois</i> politicians +would still maintain that the proposition “the rich are +taxed” implies the proposition “the poor are taxed,” and +this implication, which is true because both premiss and +conclusion are false, would be quite unnecessarily supported +by many false practical arguments. It is equally true that +“the poor are taxed” implies that “the rich are taxed.” +And this can be proved, in certain cases, on other grounds. +For the taxation of the poor would imply, ultimately, that +the poor could not afford to pay a little more for the necessities +of life than, in strict justice, they ought; and this +would mean the cessation of one of the chief means of +production of individual wealth.<span class='pagenum'><a name="Page_41" id="Page_41">[Pg 41]</a></span></p> + +<p>We also see why a valuable means for the discovery of +truth is given by the inversion of platitudinous implications. +It may happen that another platitude is the result of inversion; +but it is the fate of any true remark, especially +if it is easy to remember by reason of a paradoxical form, +to become a platitude in course of time. There are rare +cases of a platitude remaining unrepeated for so long that, +by a converse process, it has become paradoxical. Such, +for example, is Plato’s remark that a lie is less important +than an error in thought.</p> + +<p>Of late years, a method of disguising platitudes as paradoxes +has been too extensively used by Mr. G. K. Chesterton. +The method is as follows. Take any proposition <i>p</i> which +holds of an entity <i>a</i>; choose <i>p</i> so that it seems plausible that +<i>p</i> also holds of at least two other entities <i>b</i> and <i>c</i>; call +<i>a</i>, <i>b</i>, <i>c</i>, and any others for which <i>p</i> holds or seems to hold, +the class A, and <i>p</i> the “A-ness” or “A-ity” of A; let <i>d</i> +be an entity for which <i>p</i> does not hold; and put <i>d</i> among the +A’s when you think that nobody is looking. Then state +your paradox: “Some A’s do not have A-ness.” By further +manipulation you can get the proposition “No A’s have +A-ness.” But it is possible to make a very successful <i>coup</i> +if A is the null-class, which has the advantage that manipulation +is unnecessary. Thus, Mr. Chesterton, in his <i>Orthodoxy</i> +put A for the class of doubters who doubt the possibility of +logic, and proved that such agnostics refuted themselves—a +conclusion which seems to have pleased many clergymen.</p> + +<p>In this way, Mr. Chesterton has been enabled readily to +write many books and to maintain, on almost every page, +such theses as that simplicity is not simple, heterodoxy is +not heterodox, poets are not poetical, and so on; thereby +building up the gigantic platitude that Mr. Chesterton is +Chestertonian.</p> + +<p>In the chapter on Identity we have illustrated the use +of a case of the principle that any proposition implies any +true proposition. This important principle may be called +<i>the principle of the irrelevant premiss</i>;<a name="FNanchor_53_53" id="FNanchor_53_53"></a><a href="#Footnote_53_53" class="fnanchor">[53]</a> and is of great service<span class='pagenum'><a name="Page_42" id="Page_42">[Pg 42]</a></span> +in oratory, because it does not matter what the premiss is, +true or false. There is a <i>principle of the irrelevant conclusion</i>, +but, except in law-courts, interruptions of meetings, and +family life, this is seldom used, partly because of the limitation +involved in the logical impossibility for the conclusion +to be false if the premiss be true, but chiefly because the +conclusion is more important than the premiss, being usually +a matter of prejudice.</p> + +<p>Certain modern logicians, such as Frege, have found it +necessary so to extend the meaning of implication of <i>q</i> by <i>p</i> +that it holds when <i>p</i> is not a proposition at all. Hitherto, +politicians, finding that either identical or false propositions +are sufficient for their needs, have made no use of this principle; +but it is obvious that their stock of arguments would +be vastly increased thereby.</p> + +<p>Logical implication is often an enemy of dignity and +eloquence. De Morgan<a name="FNanchor_54_54" id="FNanchor_54_54"></a><a href="#Footnote_54_54" class="fnanchor">[54]</a> relates “a tradition of a Cambridge +professor who was once asked in a mathematical discussion, +‘I suppose you will admit that the whole is greater than +its part?’ and who answered, ‘Not I, until I see what use +you are going to make of it.’” And the care displayed by +cautious mathematicians like Poincaré, Schoenflies, Borel, +Hobson, and Baire in abstaining from pushing their arguments +to their logical conclusions is probably founded on the +unconscious—but no less well-grounded—fear of appearing +ridiculous if they dealt with such extreme cases as “the +series of all ordinal numbers.”<a name="FNanchor_55_55" id="FNanchor_55_55"></a><a href="#Footnote_55_55" class="fnanchor">[55]</a> They are, probably, as +unconscious of implication as Gibbon, when he remarked +that he always had a copy of Horace in his pocket, and +often in his hand, was of the necessary implication of +these propositions that his hand was sometimes in his +pocket.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_51_51" id="Footnote_51_51"></a><a href="#FNanchor_51_51"><span class="label">[51]</span></a> <i>Md.</i>, N. S., vol. iii., 1894, pp. 436-8. Cf. the discussions by +W. E. Johnson (<i>ibid.</i>, p. 583) and Russell (<i>P. M.</i>, p. 18, note, and +<i>Md.</i>, N. S., vol. xiv., 1905, pp. 400-1).</p></div> + +<div class="footnote"><p><a name="Footnote_52_52" id="Footnote_52_52"></a><a href="#FNanchor_52_52"><span class="label">[52]</span></a> The inhabitants of “Erewhon” punished invalids more severely +than criminals. In modern times, one frequently hears the statement +that crime is a disease; and if so, it is surely false that criminals +ought to be punished.</p></div> + +<div class="footnote"><p><a name="Footnote_53_53" id="Footnote_53_53"></a><a href="#FNanchor_53_53"><span class="label">[53]</span></a> <i>Irrelevant</i> in a popular sense; one would not say, speaking loosely, +that the fact that Brutus killed Cæsar implies that the sea is salt; +and yet this conclusion is implied both by the above premiss, and +the premiss that Cæsar killed Brutus. Cf. on such questions Venn, +<i>S. L.</i>, 2nd ed., pp. 240-4.</p></div> + +<div class="footnote"><p><a name="Footnote_54_54" id="Footnote_54_54"></a><a href="#FNanchor_54_54"><span class="label">[54]</span></a> <i>F. L.</i>, p. 264.</p></div> + +<div class="footnote"><p><a name="Footnote_55_55" id="Footnote_55_55"></a><a href="#FNanchor_55_55"><span class="label">[55]</span></a> Cf. Chapters XXIX and XXXVII.</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_43" id="Page_43">[Pg 43]</a></span></p> +<h3><a name="CHAPTER_XX" id="CHAPTER_XX"></a>CHAPTER XX</h3> + +<h2>DIGNITY</h2> + + +<p>We have seen, at the end of the preceding chapter, that +logical implication is often an enemy of dignity. The subject +of dignity is not usually considered in treatises on logic, +but, as we have remarked, many mathematicians implicitly +or explicitly seem to fear either that the dignity of mathematics +will be impaired if she follows out conclusions logically, +or that only an act of faith can save us from the belief that, +if we followed out conclusions logically, we should find out +something alarming about the past, present, or future of +mathematics.</p> + +<p>Thus it seems necessary to inquire rather more closely +into the nature of dignity, with a view to the discovery of +whether it is, as is commonly supposed, a merit in life and +logic.</p> + +<p>The chief use of dignity is to veil ignorance. Thus, it is +well known that the most dignified people, as a rule, are +schoolmasters, and schoolmasters are usually so occupied with +teaching that they have no time to learn anything. And +because dignity is used to hide ignorance, it is plain that +impudence is not always the opposite of dignity, but that +dignity is sometimes impudence. Dignity is said to inspire +respect; and this may be in part why respect for others is +an error of judgment and self-respect is ridiculous.</p> + +<p>Self-respect is, of course, self-esteem. William James has +remarked that self-esteem depends, not simply upon our +success, but upon the ratio of our success to our pretensions, +and can therefore be increased by diminishing our pretensions. +Thus if a man is successful, but only then, can he be both +ambitious and dignified. James also implies that happiness +increases with self-esteem. Likeness of thought with one’s<span class='pagenum'><a name="Page_44" id="Page_44">[Pg 44]</a></span> +friends, then, does not make one happy, for otherwise a man +who esteemed himself little would be indeed happy. Also if +a man is unhappy he could not, from our premisses, by the +principles of the syllogism and of contraposition, be dignified—a +conclusion which should be fatal to many novelists’ +heroes.</p> + +<p>A reflection on pessimism to which this discussion gives +rise is the following: It would appear that a man’s self-esteem +would be increased by a conviction of the unworthiness +of his neighbours. A man, therefore, who thinks that the +world and all its inhabitants, except himself, are very bad, +should be extremely happy. In fact, the effects would +hardly be distinguishable from those of optimism. And +optimism, as everybody knows, is a state of mind induced +by stupidity.</p> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_45" id="Page_45">[Pg 45]</a></span></p> +<h3><a name="CHAPTER_XXI" id="CHAPTER_XXI"></a>CHAPTER XXI</h3> + +<h2>THE SYNTHETIC NATURE OF DEDUCTION</h2> + + +<p>Doubt has often been expressed as to whether a syllogism +can add to our knowledge in any way. John Stuart Mill +and Henri Poincaré, in particular, held the opinion that +the conclusion of a syllogism is an “analytic” judgment +in the sense of Kant, and therefore could be obtained by +the mere dissection of the premisses. Any one, then, who +maintains that mathematics is founded solely on logical +principles would appear to maintain that mathematics, +in the last instance, reduces to a huge tautology.</p> + +<p>Mill, in Chapter III of Book II of his <i>System of Logic</i>, +said that “it must be granted that in every syllogism, considered +as an argument to prove the conclusion, there is +a <i>petitio principii</i>. When we say</p> + + +<div class='center'> +<table border="0" cellpadding="4" cellspacing="0" summary=""> +<tr><td align='left'></td><td align='left'>All men are mortal,<br />Socrates is a man,</td></tr> +<tr><td align='left'>therefore</td></tr> +<tr><td align='left'></td><td align='left'>Socrates is mortal,</td></tr> +</table></div> + +<p class='noidt'>it is unanswerably urged by the adversaries of the syllogistic +theory, that the proposition, Socrates is mortal, is presupposed +in the more general assumption, All men are mortal; +that we cannot be assured of the mortality of all men unless +we are already certain of the mortality of every individual +man; that if it be still doubtful whether Socrates, or any +other individual we choose to name, be mortal or not, the +same degree of uncertainty must hang over the assertion, +All men are mortal; that the general principle, instead of +being given as evidence of the particular case, cannot itself +be taken for true without exception until every shadow of<span class='pagenum'><a name="Page_46" id="Page_46">[Pg 46]</a></span> +doubt which could affect any case comprised with it is dispelled +by evidence <i>aliunde</i>; and then what remains for the +syllogism to prove? That, in short, no reasoning from +general to particular can, as such, prove anything, since +from a general principle we cannot infer any particulars +but those which the principle itself assumes as known. This +doctrine appears to me irrefragable....”</p> + +<p>But it is not difficult to see that in certain cases at least +deduction gives us <i>new</i> knowledge.<a name="FNanchor_56_56" id="FNanchor_56_56"></a><a href="#Footnote_56_56" class="fnanchor">[56]</a> If we already know +that two and two always make four, and that Asquith and +Lloyd George are two and so are the German Emperor and +the Crown Prince, we can deduce that Asquith and Lloyd +George and the German Emperor and the Crown Prince +are four. This is new knowledge, not contained in our +premisses, because the general proposition, “two and two +are four,” never told us there were such people as Asquith +and Lloyd George and the German Emperor and the Crown +Prince, and the particular premisses did not tell us that there +were four of them, whereas the particular proposition deduced +does tell us both these things. But the newness of the +knowledge is much less certain if we take the stock instance +of deduction that is always given in books on logic, namely +“All men are mortal; Socrates is a man, therefore Socrates +is mortal.” In this case what we really know beyond reasonable +doubt is that certain men, A, B, C, were mortal, since, +in fact, they have died. If Socrates is one of these men, it +is foolish to go the roundabout way through “all men are +mortal” to arrive at the conclusion that <i>probably</i> Socrates +is mortal. If Socrates is not one of the men on whom our<span class='pagenum'><a name="Page_47" id="Page_47">[Pg 47]</a></span> +induction is based, we shall still do better to argue straight +from our A, B, C, to Socrates, than to go round by the general +proposition, “all men are mortal.” For the probability that +Socrates is mortal is greater, on our data, than the probability +that all men are mortal. This is obvious, because if +all men are mortal, so is Socrates; but if Socrates is mortal, +it does not follow that all men are mortal. Hence we shall +reach the conclusion that Socrates is mortal, with a greater +approach to certainty if we make our argument purely +inductive than if we go by way of “all men are mortal” +and then use deduction.</p> + +<p>Many years ago there appeared, principally owing to the +initiative of Dr. F. C. S. Schiller of Oxford, a comic number +of <i>Mind</i>. The idea was extraordinarily good, not so the +execution. A German friend of Dr. Schiller was puzzled by +the appearance of the advertisements, which were doubtfully +humorous. However, by a syllogistic process, he +acquired information which was new and useful to him, and +thus incidentally refuted Mill. Presumably he started +from the title of the magazine (<i>Mind!</i>), for a mark of +exclamation seems nearly always in German to be a sign +of an intended joke (including of course the mark after the +politeness expressed in the first sentence of a private letter +or a public address). There would be, then, the following +syllogism:</p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="1" summary=""> +<tr><td align='left'>This is a book of would-be jokes (i.e. everything in this book is a would-be joke);</td></tr> +<tr><td align='left'>This advertisement is in this book;</td></tr> +<tr><td align='left'>Therefore, this advertisement is a would-be joke.</td></tr> +</table></div> + +<p>Thus the syllogism may be almost as powerful an agent +in the detection of humour as M. Bergson’s criterion, to +be described in a future chapter.<a name="FNanchor_57_57" id="FNanchor_57_57"></a><a href="#Footnote_57_57" class="fnanchor">[57]</a></p> + +<hr /> +<div class="footnote"><p><a name="Footnote_56_56" id="Footnote_56_56"></a><a href="#FNanchor_56_56"><span class="label">[56]</span></a> [The following passage is almost word for word the same as a +passage on pp. 123-5 of Mr. Russell’s <i>Problems of Philosophy</i>, first +published in 1912, a year after Mr. R*ss*ll’s death. It is easy hastily +to conclude that Mr. Russell was indebted to Mr. R*ss*ll to a greater +degree than is usually supposed. But an examination of the internal +evidence leads us to another conclusion. The two texts, it will be +found, differ only in the names of the German Emperor, the Crown +Prince and the other personages being replaced, in the book of 1912, +by those of Messrs. Brown, Jones, Smith, and Robinson. Now, Mr. +Russell, in a new edition of his <i>Problems</i> issued near the beginning of +the European war and before the Russian revolution, substituted “the +Emperor of Russia” for “the Emperor of China” of the first edition. +Hence it seems quite likely that Mr. Russell, who has always shown a +tendency to substitute existents for nonentities, wrote Mr. R*ss*ll’s +notes.—<span class="smcap">Ed.</span>]</p></div> + +<div class="footnote"><p><a name="Footnote_57_57" id="Footnote_57_57"></a><a href="#FNanchor_57_57"><span class="label">[57]</span></a> [See Chapter XLII.—<span class="smcap">Ed.</span>]</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_48" id="Page_48">[Pg 48]</a></span></p> +<h3><a name="CHAPTER_XXII" id="CHAPTER_XXII"></a>CHAPTER XXII</h3> + +<h2>THE MORTALITY OF SOCRATES</h2> + + +<p>The mortality of Socrates is so often asserted in books on +logic that it may be as well briefly to consider what it means. +The phrase “Socrates is mortal” may be thus defined: +“There is at least one instant <i>t</i> such that <i>t</i> has not to Socrates +the one-many relation R which is the converse of the relation +‘exists at,’ and all instants following <i>t</i> have not the relation +R to Socrates, and there is at least one instant <i>t´</i> such that +neither <i>t´</i> nor any instant preceding <i>t´</i> has the relation R +to Socrates.”</p> + +<p>This definition has many merits. In the first place, no +assumption is made that Socrates ever lived at all. In the +second place, no assumption is made that the instants of +time form a continuous series. In the third place, no assumption +is made as to whether Socrates had a first or last moment +of his existence. If time be indeed a continuous series, +then we can easily deduce<a name="FNanchor_58_58" id="FNanchor_58_58"></a><a href="#Footnote_58_58" class="fnanchor">[58]</a> that there must have been <i>either</i> +a first moment of his non-existence <i>or</i> a last one of his existence, +but not both; just as there seems to be either a greatest +weight that a man can lift or a least weight that he cannot +lift, but not both.<a name="FNanchor_59_59" id="FNanchor_59_59"></a><a href="#Footnote_59_59" class="fnanchor">[59]</a> This may be set forth as follows: for +the present we will not concern ourselves with evidence for +or against human immortality; I will merely try to present +some logical questions which persistently arise whenever +we think of eternal life. One of the greatest merits of modern +logic is that it has allowed us to give precision to such +problems, while definitely abandoning any pretensions of +solving them; and I will now apply the logico-analytical<span class='pagenum'><a name="Page_49" id="Page_49">[Pg 49]</a></span> +method to one of the problems of our knowledge of the +eternal world.<a name="FNanchor_60_60" id="FNanchor_60_60"></a><a href="#Footnote_60_60" class="fnanchor">[60]</a></p> + +<p>We will start from the generally accepted proposition that +all men are mortal. Clearly, if we could know each individual +man, and know that he was mortal, that would not +enable us to know that all men are mortal, unless we knew, +in addition, that those were all the men there are. But +we need not here assume any such knowledge of general +propositions; and, though most of us will admit that the +proposition in question has great intrinsic plausibility, it +is not strictly necessary for our present purpose to assume +anything more than the still more probable proposition +“Socrates is mortal.” This last proposition, quite apart +from the fact that we have a large amount of historical +evidence for its truth, has been repeated so often in books +on logic that it has taken on the respectable air of a platitude +while preserving the character of an exceedingly probable +truth. The truth also results from the fact that it is used +as the conclusion of a syllogism. For it is a well-known +fact that syllogisms can only be regarded as forming part +of a sound education if the conclusions are obviously true. +The use of a syllogism of the form “All cats are ducks and +all ducks are mice, therefore all cats are mice,” would introduce +grave doubts into the University of Oxford as to whether +logic could any longer be considered as a valuable mental +training for what are amusingly called the “learned professions.”</p> + +<p>If, then, we divide all the instants of time, whether past, +present, or future, into two series—those instants at which +Socrates was alive, and those instants at which he was not +alive—and leave out of consideration, for the sake of greater +simplicity, all those instants before he lived, we see at once, +by the simple application of Dedekind’s Axiom, that, if +Socrates entered into eternal life after his death, there must +have been either a last moment of his earthly life <i>or</i> a first +moment of his eternal life, but not both.</p> + +<p>Logic alone can give us no information as to which of<span class='pagenum'><a name="Page_50" id="Page_50">[Pg 50]</a></span> +these cases actually occurred, and we are thrown back on +to a discussion of empirical evidence. It is no unusual +thing to read of people who thought “that every moment +would be their last.” In this case it is quite obvious that they +consequently thought that eternity would have no beginning.</p> + +<p>Now here we must consider two things: (1) It is plainly +unsafe to conclude from what people think will happen to +what will happen; (2) even if we could so conclude, it would +be unsafe to deduce that there was a last moment in the +life of Socrates: we could only make the guess plausible, +as we should be using the inductive method.</p> + +<p>There are two other pieces of evidence that there is a +last moment of any earthly existence, which we may now +briefly consider. That this was so was held by Carlo +Michaelstaedter; but since he apparently only believed +this because he wanted, by attributing a supposed ethical +value to that moment, to give support to his theory of +suicide, we ought not to give great weight to this evidence. +Secondly, Thomas Hobbes objected to the principle “that +a quantity may grow less and less eternally, so as at last +to be equal to another quantity; or, which is all one, that +there is a last in eternity” as “void of sense.” Now, the +principle meant is true, so that, although the other proposition +mentioned by Hobbes does not follow logically from +the first, there is some evidence that this other is true. In +fact, that Hobbes thought that such-and-such a proposition +followed from another proposition which he wrongly believed +to be false, is far better evidence for the truth of such-and-such +a proposition than any we have for the truth of most +of our most cherished beliefs.</p> + +<p>Thirdly, Leibniz, in a dialogue<a name="FNanchor_61_61" id="FNanchor_61_61"></a><a href="#Footnote_61_61" class="fnanchor">[61]</a> written on his journey of +1676 to visit Spinoza, raised the question whether the moment +at which a man dies may be regarded as both the last moment +at which he is alive and the first at which he is dead, as it<span class='pagenum'><a name="Page_51" id="Page_51">[Pg 51]</a></span> +must be by Aristotle’s theory of continuity. Agreement +with this view violates the law of contradiction; denial of +it implies that two moments can be immediately adjacent. +By the denial, then, we are led to regard space and time +as made up of indivisible points and moments, and thus, +since we can draw one and only one parallel from any point +in the diagonal of a square to a given side, the diagonal will +contain the same (infinite) number of points as that side, +and will therefore be equal to it. In this Leibniz repeated +an argument used by the ancient Arabs, Roger Bacon, and +William of Occam. This Leibniz considered to be a proof +that a line cannot be an aggregate of points. Indeed, their +number would be “the number of all numbers” of the +greatest possible integer, which <i>is</i> not.</p> + +<p>It does not seem, further, that any light is thrown on the +logical question of human mortality or immortality by legal +decisions. It would appear that one can, legally speaking, +be alive for any period less than twenty-four hours after +one is dead and be dead for any period less than twenty-four +hours before one’s death. At least, according to <i>Salkeld</i>, i. 44, +it was “adjudged that if one be born the first of February +at eleven at night, and the last of January in the twenty-first +year of his age, at one of the clock in the morning, he +makes his will of lands, and dies, it is a good will, for he +was then of age.” In Sir Robert Howard’s case (<i>ibid.</i>, ii. 625) +it was held by Chief Justice Holt that “if A be born on +the third day of September; and on the second day of September +twenty-one years afterwards he make his will, this +is a good will; for the law will make no fraction of a day, +and by consequence he was of age.” But it is hardly +necessary to remark that in this way the problem with which +we are concerned is merely shifted and not solved. For the +question as to whether there is or is not a last moment of +a man’s life is not answered by the decision that he dies +legally twenty-four hours before or after he dies in the usual +sense of the word, and the problem arises as to whether +there is or is not a last moment of his legal age.<a name="FNanchor_62_62" id="FNanchor_62_62"></a><a href="#Footnote_62_62" class="fnanchor">[62]</a></p> +<p><span class='pagenum'><a name="Page_52" id="Page_52">[Pg 52]</a></span></p> +<p>So assuming that there was a last moment of Socrates’s +earthly life, and consequently no first moment of his eternal +life, we see, further, that, unless the possibility of infinite +numbers is granted, it would be quite possible for us logically +to doubt the possibility of an eternal life for Socrates on the +same grounds as those which led Zeno to assert that motion +was impossible and that Achilles could never overtake the +Tortoise. If, on the other hand, it be admitted that eternity, +at least in the case of Socrates, had a beginning, these same +arguments of Zeno would lead any one who denies the possibility +of infinite number to conclude that Socrates, like the +worm, can never die. Thus is it quite plain that the difficulties +about immortality which meet us at the very outset +of our inquiry can partly be solved only by the help of the +theory of infinite numbers and partly, it would seem, not at all.</p> + +<p>There is another difficulty about immortality which is +quite distinct from this and is analogous to another argument +of Zeno. If, indeed, all the instants of time be +divided, as before, into the two series of instants at which +Socrates was alive and instants at which he was not alive, +it follows at once that no instant of time is not accounted +for. At none of these instants, however, does Socrates die; +obviously he cannot die either when he is alive or when he +is dead. Thus it would appear that Socrates never died, +and that we ought to re-define the term “mortal” to mean +“a human being who is alive at some moments and dead +at some.” Consequently we must avoid the very tempting +conclusion that, because Socrates never died, he was therefore +immortal.</p> + +<p>It is very important carefully to distinguish between the +two arguments I have just set forth. The second argument +proves quite rigidly that Socrates and, indeed, anybody +else, never dies, whether there is or is not a last moment of +his life on earth. The first argument proves that, if there +is a first moment of Socrates’s eternal life, his life on earth +never ends. But we have seen that we cannot conclude +that this unending life proves that he never is or will be in +a state of eternity.</p> + + +<hr /> +<div class="footnote"><p><a name="Footnote_58_58" id="Footnote_58_58"></a><a href="#FNanchor_58_58"><span class="label">[58]</span></a> By “Dedekind’s Axiom,” <i>E. N.</i>, p. 11.</p></div> + +<div class="footnote"><p><a name="Footnote_59_59" id="Footnote_59_59"></a><a href="#FNanchor_59_59"><span class="label">[59]</span></a> <i>M.</i>, vol. xx., 1910, pp. 134-5.</p></div> + +<div class="footnote"><p><a name="Footnote_60_60" id="Footnote_60_60"></a><a href="#FNanchor_60_60"><span class="label">[60]</span></a> [Here, again, Mr. R*ss*ll’s work seems to anticipate some of Mr. +Russell’s later work, e.g. in <i>Our Knowledge of the External World as +a Field for Scientific Method in Philosophy</i>, Chicago and London, 1914, +pp. 3-4, 55-6, <i>et passim.</i>—<span class="smcap">Ed.</span>]</p></div> + +<div class="footnote"><p><a name="Footnote_61_61" id="Footnote_61_61"></a><a href="#FNanchor_61_61"><span class="label">[61]</span></a> “Pacidius Philalethi” in Louis Couturat, <i>Opuscules et Fragments +inédits de Leibniz</i>, Paris, 1903, pp. 594-627, especially pp. 599, 601, 608, +611. Cf. [A. E. Taylor, Hastings’ <i>Encyclopædia of Religion and Ethics</i>, +vol. iv., Part 2, Edinburgh, 1912, p. 96.—<span class="smcap">Ed.</span>]; Robert Latta, <i>Leibniz: +The Monadology and other Philosophical Writings</i>, Oxford, 1898, pp. 21 ff, +29 (note); Couturat, <i>La Logique de Leibniz d’après des documents inédits</i>, +Paris, 1901, pp. 130, 132; and Russell, <i>Ph. L.</i>, pp. 108-16, 243-9.</p></div> + +<div class="footnote"><p><a name="Footnote_62_62" id="Footnote_62_62"></a><a href="#FNanchor_62_62"><span class="label">[62]</span></a> [It may be remarked that, according to <i>The Times</i> of December +20, 1917, Mr. Justice Sargant, in the Chancery Division, also held that +“the law did not recognize fractions of a day,” and that Lord Blackburn, +in his decision (9 <i>App. Cas.</i>, 371, 373) that a man born on +the thirteenth of May 1853 attained the age of twenty-one on the +thirteenth of May 1874 “was not speaking strictly.”—<span class="smcap">Ed.</span>]</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_53" id="Page_53">[Pg 53]</a></span></p> +<h3><a name="CHAPTER_XXIII" id="CHAPTER_XXIII"></a>CHAPTER XXIII</h3> + +<h2>DENOTING</h2> + + +<p>A concept <i>denotes</i> when, if it occurs in a proposition, the +proposition is not about the concept, but <i>about</i> a term connected +in a certain peculiar way with the concept. Some +people often assert that man is mortal, and yet we never +see announced in <i>The Times</i> that Man died on a certain day +at his villa residence “Camelot” at Upper Tooting,<a name="FNanchor_63_63" id="FNanchor_63_63"></a><a href="#Footnote_63_63" class="fnanchor">[63]</a> nor do +we hear that Procrastination was again the butt of Mr. +Plowden’s jokes at Marylebone Police Court last week.</p> + +<p>That two phrases may have different <i>meanings</i> and the +same <i>denotation</i> was discovered by Alice and Frege. Alice<a name="FNanchor_64_64" id="FNanchor_64_64"></a><a href="#Footnote_64_64" class="fnanchor">[64]</a> +observed that the road which led to Tweedledum’s house +was that which led to the house of Tweedledee; and Frege +pointed out that the phrases “the house to which the road +that leads to Tweedledum’s house leads” and “the house to +which the road that leads to Tweedledee’s house leads” have +different <i>Sinn</i>, but the same <i>Bedeutung</i>.</p> + + +<hr /> +<div class="footnote"><p><a name="Footnote_63_63" id="Footnote_63_63"></a><a href="#FNanchor_63_63"><span class="label">[63]</span></a> Cf. <i>P. M.</i>, pp. 53-4.</p></div> + +<div class="footnote"><p><a name="Footnote_64_64" id="Footnote_64_64"></a><a href="#FNanchor_64_64"><span class="label">[64]</span></a> See <a href="#App_M">Appendix M</a>.</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_54" id="Page_54">[Pg 54]</a></span></p> +<h3><a name="CHAPTER_XXIV" id="CHAPTER_XXIV"></a>CHAPTER XXIV</h3> + +<h2>THE</h2> + + +<p>The word “the” implies existence and uniqueness; it is +a mistake to talk of “the son of So-and-So” if So-and-So +has a fine family of ten sons.<a name="FNanchor_65_65" id="FNanchor_65_65"></a><a href="#Footnote_65_65" class="fnanchor">[65]</a> People who refer to “the +Oxford Movement” imply that Oxford only moved once; +and those quaint people who say that “A is quite the gentleman” +imply both the doubtful proposition that there is +only one gentleman in the world, and the indubitably false +proposition that he is that man. Probably A is one of those +persons who add to the confusion in the use of the definite +article by speaking of his wife as “the wife.”</p> + +<p>In a certain Children’s Hymn Book one reads:</p> + +<p class='center'> +The river vast and small.<br /> +</p> + +<p class='noidt'>Few would deny that there is not more than one such river, +but unfortunately it is doubtful if there is such a river at +all. The case is exactly the same with the ontological proof +of the existence of the most perfect being.<a name="FNanchor_66_66" id="FNanchor_66_66"></a><a href="#Footnote_66_66" class="fnanchor">[66]</a></p> + +<p>According to the <i>Daily Mail</i> of October 9, 1906, Judge +Russell decided against a claim brought by an agent against +his company for appointing another agent, the claim being +on the ground that he was appointed as “the” agent.</p> + +<p>Most people admit that the number 2 can be added to +the number 2 to give the number 4, but this is a mistake. +They concede, when they use <i>the</i>, that there is only one +number 2, and yet they imagine that, when they consider +it apart as the first term of our above sum, they can find +another to add to it, and thereby form the third term. The +truth is that “2 + 2 = 4” is a very misleading equation,<span class='pagenum'><a name="Page_55" id="Page_55">[Pg 55]</a></span> +and what we really mean by that faultily abbreviated statement +is more precisely: If <i>x</i> and <i>y</i> denote any things which +form a class B, and <i>x´</i> and <i>y´</i> any other things that form a +class (A) which, like that of <i>x</i> and <i>y</i>, is a member of the +class (which we call “2”) of those classes which have a +one-one correspondence with B (so that any member of A +corresponds to one, and only one, member of B, and conversely), +the class of all the terms of A and B together is +a member of that class of classes which, analogously, we +call “4.” In this, for the sake of shortness, we have +introduced abbreviations which should not be used in a +rigorous logical statement.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_65_65" id="Footnote_65_65"></a><a href="#FNanchor_65_65"><span class="label">[65]</span></a> Cf. <i>Md.</i>, N. S., vol. xiv., 1905, pp. 481, 484.</p></div> + +<div class="footnote"><p><a name="Footnote_66_66" id="Footnote_66_66"></a><a href="#FNanchor_66_66"><span class="label">[66]</span></a> Cf. <i>ibid.</i>, p. 491, note.</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_56" id="Page_56">[Pg 56]</a></span></p> +<h3><a name="CHAPTER_XXV" id="CHAPTER_XXV"></a>CHAPTER XXV</h3> + +<h2>NON-ENTITY</h2> + + +<p>When people say that such-and-such a thing “is non-existent” +they usually mean that there is not any “thing” +of the kind spoken of. Venn meant this when he described<a name="FNanchor_67_67" id="FNanchor_67_67"></a><a href="#Footnote_67_67" class="fnanchor">[67]</a> +his encounter with what he imagined to be a very ingenious +tradesman: “I once had some strawberry plants furnished +me which the vendor admitted would not bear many berries. +But he assured me that this did not matter, since they made +up in their size what they lost in their number. (He gave +me, in fact, the hyperbolic formula, <i>xy = c</i>, to connect the +number and magnitude.) When summer came, <i>no</i> fruit +whatever appeared. I saw that it would be no use to complain, +because the man would urge that the size of the +non-existent berry was infinite, which I could not see my +way to disprove. I had forgotten to bar zero values of +either variable.”</p> + +<p>It is to be regretted that this useful note was omitted +in the second edition of Venn’s book; one can imagine that +it might have protected Mr. MacColl and Herr Meinong (who +believed, unlike Alice in what may be called her first theory,<a name="FNanchor_68_68" id="FNanchor_68_68"></a><a href="#Footnote_68_68" class="fnanchor">[68]</a> +in round squares and fabulous monsters) against the dishonest +practices of traders who were too ready with promises. +For the death-blow to this kind of trade was not given until +1905, when Mr. Russell published his article “On Denoting,”<a name="FNanchor_69_69" id="FNanchor_69_69"></a><a href="#Footnote_69_69" class="fnanchor">[69]</a> +and took up the position of the White King in opposition +to Alice’s later assertions.<a name="FNanchor_70_70" id="FNanchor_70_70"></a><a href="#Footnote_70_70" class="fnanchor">[70]</a></p> + +<p>Venn’s experience illustrates another characteristic of +mathematical logic. It is necessary, in order to make our<span class='pagenum'><a name="Page_57" id="Page_57">[Pg 57]</a></span> +arguments conclusive, to devote great care to the elimination +of difficulties which rarely occur. The White Knight—who +was like Boole in being a pioneer of mathematical logic in +this way, and yet seems to have held, like Boole, those +philosophical opinions which would base logic on psychology—recognized +the necessity of taking precautions against any +unusual appearance of mice on a horse’s back.<a name="FNanchor_71_71" id="FNanchor_71_71"></a><a href="#Footnote_71_71" class="fnanchor">[71]</a></p> + +<hr /> + +<div class="footnote"><p><a name="Footnote_67_67" id="Footnote_67_67"></a><a href="#FNanchor_67_67"><span class="label">[67]</span></a> <i>S. L.</i>, 1881, p. 339, note.</p></div> + +<div class="footnote"><p><a name="Footnote_68_68" id="Footnote_68_68"></a><a href="#FNanchor_68_68"><span class="label">[68]</span></a> See <a href="#App_N">Appendix N</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_69_69" id="Footnote_69_69"></a><a href="#FNanchor_69_69"><span class="label">[69]</span></a> <i>Md.</i>, N. S., vol. xiv., October 1905, pp. 479-93.</p></div> + +<div class="footnote"><p><a name="Footnote_70_70" id="Footnote_70_70"></a><a href="#FNanchor_70_70"><span class="label">[70]</span></a> See <a href="#App_N">Appendix N</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_71_71" id="Footnote_71_71"></a><a href="#FNanchor_71_71"><span class="label">[71]</span></a> See <a href="#App_O">Appendix O</a>.</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_58" id="Page_58">[Pg 58]</a></span></p> +<h3><a name="CHAPTER_XXVI" id="CHAPTER_XXVI"></a>CHAPTER XXVI</h3> + +<h2>IS</h2> + + +<p><i>Is</i> has four perfectly distinct meanings in English, besides +misuses of the word. Among the misuses, perhaps the most +important are those referred to by De Morgan:<a name="FNanchor_72_72" id="FNanchor_72_72"></a><a href="#Footnote_72_72" class="fnanchor">[72]</a> “... We +say ‘murder <i>is</i> death to the perpetrator’ where the copula +is <i>brings</i>; ‘two and two <i>are</i> four,’ the copula being ‘have +the value of,’ etc.”</p> + +<p>Schröder<a name="FNanchor_73_73" id="FNanchor_73_73"></a><a href="#Footnote_73_73" class="fnanchor">[73]</a> quite satisfactorily pointed out the well-known +distinction between an <i>is</i> where subject and predicate can +be interchanged (such as: “the class whose members are +Shem, Ham and Japhet is the class of the sons of Noah”) +and an <i>is</i> or <i>are</i> where they cannot (such as: Englishmen +are Britons), but failed to see<a name="FNanchor_74_74" id="FNanchor_74_74"></a><a href="#Footnote_74_74" class="fnanchor">[74]</a> the more important distinction +(made by Peano) of is in the sense of “is a member of.” +If Englishmen are Britons, and Britons are civilized people, +it follows that Englishmen are civilized people; but, though +the <i>Harmsworth Encyclopædia</i> is a member of the class Book +(of one or more volumes), and this class is the member of +a class A of which it is the only member, yet the <i>Harmsworth +Encyclopædia</i> is not a member of A, for it is not true that +it is the whole class of books; and such a statement +would not even be made except possibly in the form of an +advertisement.</p> + +<p>The fourth meaning of <i>is</i> is <i>exists</i>; it is in certain rare +moods a matter for regret that there are difficulties in the +way of using one word to denote four different things. For, +if there were not, we might prove the existence of any thing +we please by making it the subject of a proposition, and +thereby earn the gratitude of theologians.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_72_72" id="Footnote_72_72"></a><a href="#FNanchor_72_72"><span class="label">[72]</span></a> <i>F. L.</i>, p. 268.</p></div> + +<div class="footnote"><p><a name="Footnote_73_73" id="Footnote_73_73"></a><a href="#FNanchor_73_73"><span class="label">[73]</span></a> <i>A. d. L.</i>, i. pp. 127 sqq.</p></div> + +<div class="footnote"><p><a name="Footnote_74_74" id="Footnote_74_74"></a><a href="#FNanchor_74_74"><span class="label">[74]</span></a> <i>Ibid.</i>, vol. ii. pp. 461, 597.</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_59" id="Page_59">[Pg 59]</a></span></p> +<h3><a name="CHAPTER_XXVII" id="CHAPTER_XXVII"></a>CHAPTER XXVII</h3> + +<h2><i>AND</i> AND <i>OR</i></h2> + + +<p>When, with Boole, alternatives (A, B) are considered as +mutually exclusive, logical addition may be described as +the process of taking A <i>and</i> B or A <i>or</i> B. It is a great and +rare convenience to have two terms for denoting the same +thing: commonly, people denote several things by the same +term, and only the Germans have the privilege of referring +to, say, <i>continuity</i> as <i>Stetigkeit</i> or <i>Kontinuierlichkeit</i>. But +Jevons<a name="FNanchor_75_75" id="FNanchor_75_75"></a><a href="#Footnote_75_75" class="fnanchor">[75]</a> quoted Milton, Shakespeare, and Darwin to prove +that alternatives are not exclusive, and so attained first to +recognized views by arguments which were plainly irrelevant.</p> + +<p>Of course, <i>and</i> is often used as the sign of logical addition: +thus one may speak of one’s brothers <i>and</i> sisters, without +being understood to mean the null-class (as should be the +case), or pray for one’s “relations and friends,” without +being sure that one’s prayer would be answered,—as it +certainly would if one meant to pray for the null-class, this +being the class indicated. And a word like <i>while</i> is often +used for a logical addition, when exclusiveness of the alternatives +is almost implied. Thus, a reviewer in <i>Mind</i>,<a name="FNanchor_76_76" id="FNanchor_76_76"></a><a href="#Footnote_76_76" class="fnanchor">[76]</a> +noticing the translation of Mach’s <i>Popular Scientific Lectures</i> +into American, said of the lectures that: “Most of them will +be familiar ... to epistemologists and experimental psychologists: +while the remainder, which deal with physical +questions, are well worth reading.” The reader has the +impression, probably given unintentionally, that Professor +Mach’s epistemological and psychological lectures are not, +in the reviewer’s opinion, worth reading.</p> +<hr /> +<div class="footnote"><p><a name="Footnote_75_75" id="Footnote_75_75"></a><a href="#FNanchor_75_75"><span class="label">[75]</span></a> <i>Pure Logic</i> ..., London, 1864, pp. 76-9. Cf. Venn, <i>S. L.</i>, 2nd ed., +pp. 40-8.</p></div> + +<div class="footnote"><p><a name="Footnote_76_76" id="Footnote_76_76"></a><a href="#FNanchor_76_76"><span class="label">[76]</span></a> N. S., vol. iv. p. 261.</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_60" id="Page_60">[Pg 60]</a></span></p> +<h3><a name="CHAPTER_XXVIII" id="CHAPTER_XXVIII"></a>CHAPTER XXVIII</h3> + +<h2>THE CONVERSION OF RELATIONS</h2> + + +<p>The “Conversion of Relations” does not mean what it +might be supposed to mean; it has nothing to do with what +Kant called “the wholesome art of persuasion.” What +concerns us here is the convertibility of a logical relation. +If A has a certain relation R to B, the relation of B to A, +which may be denoted by Ř, is called the <i>converse</i> of R. +As De Morgan<a name="FNanchor_77_77" id="FNanchor_77_77"></a><a href="#Footnote_77_77" class="fnanchor">[77]</a> remarked, this conversion may sometimes +present difficulties. The following is De Morgan’s example:</p> + +<p>“Teacher: ‘Now, boys, Shem, Ham and Japheth were +Noah’s sons; who was the father of Shem, Ham and Japheth?’ +No answer.</p> + +<p>“Teacher: ‘Boys, you know Mr. Smith, the carpenter, +opposite; has he any sons?’</p> + +<p>“Boys: ‘Oh! yes, sir! there’s Bill and Ben.’</p> + +<p>“Teacher: ‘And who is the father of Bill and Ben Smith?’</p> + +<p>“Boys: ‘Why, Mr. Smith, to be sure.’</p> + +<p>“Teacher: ‘Well, then, once more, Shem, Ham and +Japheth were <i>Noah’s</i> sons; who was the father of Shem, +Ham and Japheth?’</p> + +<p>“A long pause; at last a boy, indignant at what he thought +the attempted trick, cried out: ‘It <i>couldn’t</i> have been Mr. +Smith.’ These boys had never converted the relation of +father and son....”</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_77_77" id="Footnote_77_77"></a><a href="#FNanchor_77_77"><span class="label">[77]</span></a> <i>Trans. Camb. Phil. Soc.</i>, vol. x., 1864, part ii., note on page 334.</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_61" id="Page_61">[Pg 61]</a></span></p> +<h3><a name="CHAPTER_XXIX" id="CHAPTER_XXIX"></a>CHAPTER XXIX</h3> + +<h2>PREVIOUS PHILOSOPHICAL THEORIES OF +MATHEMATICS</h2> + + +<p>Mathematicians usually try to found mathematics on two +principles:<a name="FNanchor_78_78" id="FNanchor_78_78"></a><a href="#Footnote_78_78" class="fnanchor">[78]</a> one is the principle of confusion between the +sign and the thing signified (they call this principle the +foundation-stone of the formal theory), and the other is the +Principle of the Identity of Discernibles (which they call +the principle of the permanence of equivalent forms).</p> + +<p>But the truth is that if we set sail on a voyage of discovery +with Logic alone at the helm, we must either throw such +principles as “the identity of those conceptions which have +in common the properties that interest us” and “the principle +of permanence” overboard, or, if we do not like to +act in such a way to old companions with whom we are so +familiar that we can hardly feel contempt for them, at least +recognize them clearly as having no logical validity and +merely as psychological principles, and reduce them to the +humble rank of stewards, to minister to our human weaknesses +on the voyage. And then, if we adopt the wise +policy of keeping our axioms down to the minimum number, +we must refrain from creating or thinking that we are creating +new numbers to fill up gaps among the older ones, and +thence recognize that our rational numbers are not particular +cases of “real” numbers, and so on.</p> + +<p>We thus get a world of conceptions which looks, and is, +very different from that which ordinary mathematicians +think they see; and perhaps this is the reason why some +mathematicians of great eminence, such as Hilbert and<span class='pagenum'><a name="Page_62" id="Page_62">[Pg 62]</a></span> +Poincaré, have produced such absurd discussions on the +fundamental principles of mathematics,<a name="FNanchor_79_79" id="FNanchor_79_79"></a><a href="#Footnote_79_79" class="fnanchor">[79]</a> showing once more +the truth of the not quite original remark of Aunt Jane, who</p> + +<p class="poem"> +... observed, the second time<br /> +She tumbled off a ’bus:<br /> +“The step is short from the sublime<br /> +To the ridiculous.”<br /> +</p> + +<p>In their readiness to consider many different things as +one thing—to consider, for example, the ratio 2:1 as the +same thing as the cardinal number 2—such mathematicians +as Peacock, Hankel, and Schubert were forestalled by the +Pigeon, who thought that Alice and the Serpent were the +same creature, because both had long necks and ate eggs.<a name="FNanchor_80_80" id="FNanchor_80_80"></a><a href="#Footnote_80_80" class="fnanchor">[80]</a> +It is, however, doubtful whether the Pigeon would have +followed the example of the mathematicians just mentioned +so far as to embrace the creed of nominalism and so to feel +no difficulty in subtracting from zero—a difficulty which +was pointed out with great acuteness by the Hatter<a name="FNanchor_81_81" id="FNanchor_81_81"></a><a href="#Footnote_81_81" class="fnanchor">[81]</a> and +modern mathematical logicians.</p> + +<hr /> + +<div class="footnote"><p><a name="Footnote_78_78" id="Footnote_78_78"></a><a href="#FNanchor_78_78"><span class="label">[78]</span></a> These principles, after many attempts to state them by Peacock, +the Red and the White Queen (see <a href="#App_P">Appendix P</a>), Hankel, Schröder, +and Schubert had been made, were first precisely formulated by Frege +in <i>Z. S.</i>; cf. also Chapter VII.</p></div> + +<div class="footnote"><p><a name="Footnote_79_79" id="Footnote_79_79"></a><a href="#FNanchor_79_79"><span class="label">[79]</span></a> See Couturat, <i>R. M. M.</i>, vol. xiv., March, 1906, pp. 208-50, and +Russell, <i>ibid.</i>, September, 1906, pp. 627-34.</p></div> + +<div class="footnote"><p><a name="Footnote_80_80" id="Footnote_80_80"></a><a href="#FNanchor_80_80"><span class="label">[80]</span></a> See <a href="#App_P">Appendix P</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_81_81" id="Footnote_81_81"></a><a href="#FNanchor_81_81"><span class="label">[81]</span></a> See <i>ibid.</i></p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_63" id="Page_63">[Pg 63]</a></span></p> +<h3><a name="CHAPTER_XXX" id="CHAPTER_XXX"></a>CHAPTER XXX</h3> + +<h2>FINITE AND INFINITE</h2> + + +<p>I was once shown a statement made by an eminent mathematician +of Cambridge from which one would conclude +that this mathematician thought that finite distances became +infinite when they were great enough. In one of those +splendidly printed books, bound in blue, published by the +University Press, and sold at about a guinea as a guide to +some advanced branch of pure mathematics, one may read, +even in the second edition published in 1900, the words: +“Representation [of a complex variable] on a plane is +obviously more effective for points at a finite distance from +the origin than for points at a very great distance.”</p> + +<p>Plainly some of the points at a very great distance are +at a <i>finite</i> distance, for the same author mentions that +Neumann’s sphere for representing the positions of points +on a plane “has the advantage ... of exhibiting the +uniqueness of <i>z</i> = ∞ as a value of the variable.”</p> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_64" id="Page_64">[Pg 64]</a></span></p> +<h3><a name="CHAPTER_XXXI" id="CHAPTER_XXXI"></a>CHAPTER XXXI</h3> + +<h2>THE MATHEMATICAL ATTAINMENTS OF +TRISTRAM SHANDY</h2> + + +<p>Tristram Shandy<a name="FNanchor_82_82" id="FNanchor_82_82"></a><a href="#Footnote_82_82" class="fnanchor">[82]</a> said that his father was sometimes a +gainer by misfortune; for if the pleasure of haranguing +about it was as ten, and the misfortune itself only as five, +he gained “half in half,” and was well off again as if the +misfortune had never happened.</p> + +<p>Suppose that the unit (arbitrary) of pleasure is denoted +by A, Tristram Shandy, by neglecting, in this ethical +discussion, to introduce negative quantities (Kant’s pamphlet +advocating this introduction into philosophy was made +subsequently<a name="FNanchor_83_83" id="FNanchor_83_83"></a><a href="#Footnote_83_83" class="fnanchor">[83]</a>), apparently made 15A to result, and this +can hardly be maintained to be the half of 10A. It is +possible, however, that Tristram Shandy succeeded in proving +the apparently paradoxical equation</p> + +<p class="center"> +15A = 5A<br /> +</p> + +<p class="noidt">by remarking that the axiom “the whole is greater than +the part” does not always hold. This remark follows at +once from what Mr. Russell<a name="FNanchor_84_84" id="FNanchor_84_84"></a><a href="#Footnote_84_84" class="fnanchor">[84]</a> has called “The Paradox of +Tristram Shandy.” This paradox is described by Mr. Russell +as follows:</p> + +<p>“Tristram Shandy, as we know, took two years writing +the history of the first two days of his life, and lamented +that, at this rate, material would accumulate faster than +he could deal with it, so that he could never come to an +end. Now I maintain that, if he had lived for ever, and<span class='pagenum'><a name="Page_65" id="Page_65">[Pg 65]</a></span> +not wearied of his task, then, even if his life had continued +as eventfully as it began, no part of his biography would +have remained unwritten.”</p> + +<p>This paradox is strictly correlative to the well-known +paradox of Zeno about Achilles and the Tortoise.<a name="FNanchor_85_85" id="FNanchor_85_85"></a><a href="#Footnote_85_85" class="fnanchor">[85]</a> “The +Achilles proves that two variables in a continuous series, +which approach equality from the same side, cannot ever +have a common limit: the Tristram Shandy proves that +two variables which start from a common term, and proceed +in the same direction, but diverge more and more, may yet +determine the same limiting class (which, however, is not +necessarily a segment, because segments were defined as +having terms beyond them). The Achilles assumes that +whole and part cannot be similar, and deduces a paradox; +the other, starting from a platitude, deduces that whole +and part may be similar. For common-sense, it must be +confessed that it is a most unfortunate state of things.” +And Mr. Russell considers that, in the face of proofs, it ought +to commit suicide in despair.</p> + +<p>Now, I suggest the extremely unlikely possibility that +Tristram Shandy, by reflection on his own life and literary +labours, was led to the correct course of accepting the paradox +which resulted from this reflection and rejecting the Achilles. +Thus, he concluded that an infinite whole may be similar +(or, in Cantor’s terminology, “equivalent”) to a proper +part of itself, and hence, by a confusion of similarity with +identity (or equivalence with equality) which he shares with +some subsequent philosophers,<a name="FNanchor_86_86" id="FNanchor_86_86"></a><a href="#Footnote_86_86" class="fnanchor">[86]</a> that a whole may be equal +to a proper part of itself. If A is an infinite class, it is not +difficult to see that we can have</p> + +<p class="center"> +10A = 5A.<br /> +</p> + +<p>In this way many have avoided an opinion which rests +on no better foundation than that formerly entertained by +the inductive philosophers of Central Africa, that all men +are black.<a name="FNanchor_87_87" id="FNanchor_87_87"></a><a href="#Footnote_87_87" class="fnanchor">[87]</a></p> + +<hr /> + +<div class="footnote"><p><a name="Footnote_82_82" id="Footnote_82_82"></a><a href="#FNanchor_82_82"><span class="label">[82]</span></a> Cf. a letter of De Morgan in Mrs. De Morgan’s <i>Memoir of Augustus +De Morgan</i>, p. 324.</p></div> + +<div class="footnote"><p><a name="Footnote_83_83" id="Footnote_83_83"></a><a href="#FNanchor_83_83"><span class="label">[83]</span></a> Kant’s tract was published in 1763, while <i>Tristram Shandy</i> was +published in 1760.</p></div> + +<div class="footnote"><p><a name="Footnote_84_84" id="Footnote_84_84"></a><a href="#FNanchor_84_84"><span class="label">[84]</span></a> <i>P. M.</i>, pp. 358-9 [Cf. <i>M.</i>, vol. xxii., January 1912, p. 187.—<span class="smcap">Ed.</span>]</p></div> + +<div class="footnote"><p><a name="Footnote_85_85" id="Footnote_85_85"></a><a href="#FNanchor_85_85"><span class="label">[85]</span></a> Cf. <i>P. M.</i>, pp. 350, 358-9; <i>M.</i>, vol. xxii., 1912, p. 157.</p></div> + +<div class="footnote"><p><a name="Footnote_86_86" id="Footnote_86_86"></a><a href="#FNanchor_86_86"><span class="label">[86]</span></a> [Cf. for example, Cosmo Guastella, <i>Dell’ infinito</i>, Palermo, 1912.—<span class="smcap">Ed.</span>]</p></div> + +<div class="footnote"><p><a name="Footnote_87_87" id="Footnote_87_87"></a><a href="#FNanchor_87_87"><span class="label">[87]</span></a> Cf. Russell, <i>P. M.</i>, p. 360.</p></div> + + + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_66" id="Page_66">[Pg 66]</a></span></p> +<h3><a name="CHAPTER_XXXII" id="CHAPTER_XXXII"></a>CHAPTER XXXII</h3> + +<h2>THE HARDSHIPS OF A MAN WITH AN +UNLIMITED INCOME</h2> + + +<p>I once heard a man refer to his income as limited, in order +to illustrate the hardship of a class of men, of which he +of course was one, in having to pay a somewhat high +income-tax. It is obvious that this man spoke enviously, +and consequently admitted the existence of more fortunately +placed individuals who had unlimited incomes. A +little reflection would have shown the man that he was not +taking up a paradoxical attitude. A “paradoxical attitude” +is of course the assertion of one or more propositions of +which the truth cannot be perceived by a philosopher—and +particularly an idealist—and can be perceived by a logician +and occasionally, but not always, by a man of common-sense. +Such propositions are: “The cat is hungry,” +“Columbus discovered America,” and “A thing which is +always at rest may move from the position A to the different +position B.”</p> + +<p>Now, if a man had an unlimited income, it is an immediate +inference that, however low income-tax might be, he would +have to pay annually to the Exchequer of his nation a sum +equal in value to his whole income. Further, if his income +was derived from a capital invested at a finite rate of interest +(as is usual), the annual payments of income-tax would each +be equal in value to the man’s whole capital. If, then, +the man with an unlimited income chose to be discontented, +he would be sure of a sympathetic audience among philosophers +and business acquaintances; but discontent could +not last long, for the thought of the difficulties he was putting +in the way of the Chancellor of the Exchequer, who would +find the drawing up of his budget most puzzling, would be<span class='pagenum'><a name="Page_67" id="Page_67">[Pg 67]</a></span> +amusing. Again, the discovery that, after paying an infinite +income-tax, the income would be quite undiminished, would +obviously afford satisfaction, though perhaps the satisfaction +might be mixed with a slight uneasiness as to any action +the Commissioners of Income-Tax might take in view of +this fact.</p> + +<p>A problem of a wholly different nature is connected with +the possible purchase by the man with an unlimited income +of an enumerable infinity of pairs of boots. If he wished +to prove that he had an even number of boots, it would be +easy if right boots were distinguishable from left ones, but +if the man were a faddist of such a kind that he insisted that +his left boots should not be made in any way differently +from his right ones, it would not be possible for him to prove +the theorem mentioned unless he assumed what is known as +“the multiplicative axiom.” In fact this axiom shows that +it is legitimate to pick out an infinite succession of members +of an infinite class in an arbitrary way. In the case of the +pairs of boots, each pair contains two members, and if there +is no means of distinguishing between them, when we wish +to pick out one of them for each of the infinity of pairs, we +cannot say which ones we mean to pick out unless we assume, +by means of the above axiom, that a particularized member +can always be found even with things of each of which it +can be said that, like Private James in the <i>Bab Ballads</i>,</p> + +<p class="poem"> +No characteristic trait had he<br /> +Of any distinctive kind.<br /> +</p> + +<p>However, a solution of the puzzle was given by Dr. +Dénes König of Budapest. You first prove that there are +points in space such that, if P is one of them, not more +than a finite number of pairs of boots are such that each +centre of mass of the two members of a pair is equidistant +from P. Taking a point P of this sort, select from each pair +the boot whose centre of mass is nearest P. (There may be +a finite number of pairs left over, but they can be dealt with +arbitrarily.)</p> + +<p>Another form of the problem is as follows. Every time +the man bought a pair of boots he also bought a pair of socks +to go with it; he had an enumerable infinity of pairs<span class='pagenum'><a name="Page_68" id="Page_68">[Pg 68]</a></span> +of each, and the problem is to prove that he had as many +boots as he had socks. In this case the boots, we will suppose, +can be divided into right and left, but the socks cannot. +Thus there are an enumerable infinity of boots, but the +number of the socks cannot be determined without admitting +the axiom mentioned above. A further difficulty might +arise if the owner of the boots and socks lost one leg in some +accident, and told his butler to give away half his socks. +Naturally the butler would find great logical difficulties in +so doing, and it would seem to be an interesting ethical +problem whether he should be dismissed from his situation +for failing to prove the multiplicative axiom. Again, if the +butler stole a pair of boots, the millionaire would have as +many pairs as before, but might have fewer boots. There +is as yet no evidence that the number of his boots is equal +to or greater than the number of pairs.</p> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_69" id="Page_69">[Pg 69]</a></span></p> +<h3><a name="CHAPTER_XXXIII" id="CHAPTER_XXXIII"></a>CHAPTER XXXIII</h3> + +<h2>THE RELATIONS OF MAGNITUDE OF +CARDINAL NUMBERS</h2> + + +<p>The theorems of cardinal arithmetic are frequently used +in ordinary conversation. What is known as the Schröder-Bernstein +theorem was used, long before Bernstein or +Schröder, by Edward Thurlow, afterward the law-lord Lord +Thurlow, when an undergraduate of Caius College, Cambridge. +Thurlow was rebuked for idleness by the Master, +who said to him: “Whenever I look out of the window, +Mr. Thurlow, I see you crossing the Court.” The provost +thus asserted a one-one correspondence between the class A +of his acts of looking out of the window and a part of the +class B of Thurlow’s acts of crossing the Court. Thurlow +asserted in reply a one-one correspondence between B and +a part of A: “Whenever I cross the Court I see you looking +out of the window.” The Schröder-Bernstein theorem, then, +allows us to conclude that there is a one-one correspondence +between the classes A and B. That A and B were finite +classes is not the fault of the Master or Thurlow; nor is +it relevant logically.</p> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_70" id="Page_70">[Pg 70]</a></span></p> +<h3><a name="CHAPTER_XXXIV" id="CHAPTER_XXXIV"></a>CHAPTER XXXIV</h3> + +<h2>THE UNKNOWABLE</h2> + + +<p>According to Mr. S. N. Gupta,<a name="FNanchor_88_88" id="FNanchor_88_88"></a><a href="#Footnote_88_88" class="fnanchor">[88]</a> the first thing that every +student of Hindu logic has to learn when he is said to begin +the study of inference is that “all H is S” is not always +equivalent to “No H is not S.” “The latter proposition +is an absurdity when S is <i>Kebalánvayi</i>, i.e. covers the whole +sphere of thought and existence.... ‘Knowable’ and +‘Nameable’ are among the examples of <i>Kebalánvayi</i> terms. +If you say there is a thing not-knowable, how do you know +it? If you say there is a thing not-nameable, you must +point that out, i.e. somehow name it. Thus you contradict +yourself.”</p> + +<p>Mr. Herbert Spencer’s doctrine of the “Unknowable” +gives rise to some amusing thoughts. To state that all +knowledge of such and such a thing is above a certain +person’s intelligence is not self-contradictory, but merely +rude: to state that all knowledge of a certain thing is above +all possible human intelligence is nonsense, in spite of its +modest, platitudinous appearance. For the statement seems +to show that we do know something of it, viz. that it is +unknowable.</p> + +<p>To the last (1900) edition of <i>First Principles</i> was added a +“Postscript to Part I,” in which the justice of this simple and +well-known criticism as to the contradiction involved in speaking +of an “Unknowable,” which had been often made during +the forty odd years in which the various editions had been on +the market, was grudgingly acknowledged as follows:<a name="FNanchor_89_89" id="FNanchor_89_89"></a><a href="#Footnote_89_89" class="fnanchor">[89]</a></p> + +<p>“It is doubtless true that saying what a thing is not, is, +in some measure, saying what it is;... Hence it cannot<span class='pagenum'><a name="Page_71" id="Page_71">[Pg 71]</a></span> +be denied that to affirm of the Ultimate Reality that it is +unknowable is, in a remote way, to assert some knowledge +of it, and therefore involves a contradiction.”</p> + +<p>The “Postscript” reminds one of the postscript to a +certain Irishman’s letter. This Irishman, missing his razors +after his return from a visit to a friend, wrote to his friend, +giving precise directions where to look for the missing razors; +but, before posting the letter, added a postscript to the +effect that he had found the razors.</p> + +<p>One is tempted to inquire, analogously, what might be, +in view of the Postscript, the point of much of Spencer’s +Part I. It is, to use De Morgan’s<a name="FNanchor_90_90" id="FNanchor_90_90"></a><a href="#Footnote_90_90" class="fnanchor">[90]</a> description of the arguments +of some who maintain that we can know nothing +about infinity, of the same force as that of the man who +answered the question how long he had been deaf and dumb.</p> + +<p>But the best part of the joke against Mr. Spencer is that +he, as we shall see in Chapter XXXVIII, was refuted +by a fallacious argument, and thus mistakenly asserted the +validity of the refutation of remarks which happen to be +unsound.</p> + +<p>The analogy of the contradiction of Burali-Forti with the +contradiction involved in the notion of an “unknowable” +may be set forth as follows. If A should say to B: “I +know things which you never by any possibility can know,” +he may be speaking the truth. In the same way, ω may +be said, without contradiction, to transcend all the <i>finite</i> +integers. But if some one else, C, should say: “There are +some things which no human being can ever know anything +about,” he is talking nonsense.<a name="FNanchor_91_91" id="FNanchor_91_91"></a><a href="#Footnote_91_91" class="fnanchor">[91]</a> And in the same way if +we succeeded in imagining a number which transcends <i>all</i> +numbers, we have succeeded in imagining the absurdity of +a number which transcends itself.</p> + +<p>All the paradoxes of logic (or “the theory of aggregates”)<span class='pagenum'><a name="Page_72" id="Page_72">[Pg 72]</a></span> +are analogous to the difficulty arising from a man’s statement: +“I am lying.”<a name="FNanchor_92_92" id="FNanchor_92_92"></a><a href="#Footnote_92_92" class="fnanchor">[92]</a> In fact, if this is true, it is false, +and <i>vice versa</i>. If such a statement is spread out a little, +it becomes an amusing hoax or an epigram. Thus, one may +present to a friend a card bearing on both sides the words: +“The statement on the other side of this card is false”; +while the first of the epigrams derived from this principle +seems to have been written by a Greek satirist:<a name="FNanchor_93_93" id="FNanchor_93_93"></a><a href="#Footnote_93_93" class="fnanchor">[93]</a></p> + +<p class="poem"> +Lerians are bad; not <i>some</i> bad and some <i>not</i>;<br /> +But all; there’s not a Lerian in the lot,<br /> +Save Procles, that you could a good man call;—<br /> +And Procles—is a Lerian after all.<br /> +</p> + +<p>This is the original of a well-known epigram by Porson, +who remarked that all Germans are ignorant of Greek metres,</p> + +<p class="poem"> +All, save only Hermann;—<br /> +And Hermann’s a German.<br /> +</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_88_88" id="Footnote_88_88"></a><a href="#FNanchor_88_88"><span class="label">[88]</span></a> <i>Md.</i>, N. S., vol. iv., 1895, p. 168.</p></div> + +<div class="footnote"><p><a name="Footnote_89_89" id="Footnote_89_89"></a><a href="#FNanchor_89_89"><span class="label">[89]</span></a> <i>First Principles</i>, 6th ed., 1900, pp. 107-10. The first edition was +published in 1862.</p></div> + +<div class="footnote"><p><a name="Footnote_90_90" id="Footnote_90_90"></a><a href="#FNanchor_90_90"><span class="label">[90]</span></a> Note on p. 6 of his paper: “On Infinity; and on the Sign of +Equality,” <i>Trans. Camb. Phil. Soc.</i>, vol. xi., part i., pp. 1-45 (read +May 16, 1864).</p></div> + +<div class="footnote"><p><a name="Footnote_91_91" id="Footnote_91_91"></a><a href="#FNanchor_91_91"><span class="label">[91]</span></a> The assertion of the finitude of a man’s mind appears to be nonsense; +both because, if we say that the mind of man is limited we +tacitly postulate an “unknowable,” and because, even if the human +mind were finite, there is no more reason against its conceiving the +infinite than there is for a mind to be blue in order to conceive a pair +of blue eyes (cf. De Morgan, <i>loc. cit.</i>).</p></div> + +<div class="footnote"><p><a name="Footnote_92_92" id="Footnote_92_92"></a><a href="#FNanchor_92_92"><span class="label">[92]</span></a> Russell, <i>R. M. M.</i>, vol. xiv., September 1906, pp. 632-3, 640-4.</p></div> + +<div class="footnote"><p><a name="Footnote_93_93" id="Footnote_93_93"></a><a href="#FNanchor_93_93"><span class="label">[93]</span></a> <i>The Greek Anthology</i>, by Lord Neaves (Ancient Classics for English +Readers), Edinburgh and London, 1897, p. 194.</p></div> + + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_73" id="Page_73">[Pg 73]</a></span></p> +<h3><a name="CHAPTER_XXXV" id="CHAPTER_XXXV"></a>CHAPTER XXXV</h3> + +<h2>MR. SPENCER, THE ATHANASIAN CREED AND +THE ARTICLES</h2> + + +<p>When, in what I believe is misleadingly known as “The +Athanasian Creed,” people say “The Father incomprehensible,” +and so on, they are not falling into the same error +as Mr. Spencer, for the Latin equivalent for “incomprehensible” +is merely “<i>immensus</i>,” and Bishop Hilsey translated +it more correctly as “immeasurable.”<a name="FNanchor_94_94" id="FNanchor_94_94"></a><a href="#Footnote_94_94" class="fnanchor">[94]</a> It is a +regrettable fact that Dr. Blunt,<a name="FNanchor_95_95" id="FNanchor_95_95"></a><a href="#Footnote_95_95" class="fnanchor">[95]</a> in his mistaken modesty, +has added a note to this passage that: “Yet it is true that +a meaning not intended in the Creed has developed itself +through this change of language, for the Nature of God is +as far beyond the grasp of the mind as it is beyond the +possibility of being contained within local bounds.”</p> + +<p>Mr. Spencer seems no happier when we compare his statements +with those in the Anglican Articles of Religion. There +God is never referred to as infinite. It is true that His power +and goodness are so referred to; but this deficiency was +presumably brought about intentionally, so that faith might +gain in meaning as time went on.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_94_94" id="Footnote_94_94"></a><a href="#FNanchor_94_94"><span class="label">[94]</span></a> <i>A. C. P.</i>, p. 217.</p></div> + +<div class="footnote"><p><a name="Footnote_95_95" id="Footnote_95_95"></a><a href="#FNanchor_95_95"><span class="label">[95]</span></a> <i>Ibid.</i>, p. 218.</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_74" id="Page_74">[Pg 74]</a></span></p> +<h3><a name="CHAPTER_XXXVI" id="CHAPTER_XXXVI"></a>CHAPTER XXXVI</h3> + +<h2>THE HUMOUR OF MATHEMATICIANS</h2> + + +<p>Brahmagupta’s problem<a name="FNanchor_96_96" id="FNanchor_96_96"></a><a href="#Footnote_96_96" class="fnanchor">[96]</a> appears to be the earliest instance +of a kind of joke which has been much used by mathematicians. +For the sake of giving a certain picturesqueness +to the data of problems, and so to excite that sort of interest +which is partly expressed by a smile, mathematicians have +got into the habit of talking, for example, of monkeys in +the form of geometrical points climbing up massless ropes. +Professor P. Stäckel<a name="FNanchor_97_97" id="FNanchor_97_97"></a><a href="#Footnote_97_97" class="fnanchor">[97]</a> truly remarked that physiological +mechanics—the mechanics of bones, muscles, and so on—is +wholly different from this. There was once a lecturer +on mathematics at Cambridge who used yearly to propound +to his pupils a problem in rigid dynamics which related to +the motion of a garden roller supposed to be without mass +or friction, when a heavy and perfectly rough insect walked +round the interior of it in the direction of normal rolling.</p> + +<p>Hitherto this has been the only mathematical outlet for +the humour of mathematicians; and those who really had +the interests of mathematics at heart saw with alarm the +growing tendency towards scholasticism in mathematical +jokes. Fortunately the discovery of logic by some mathematicians +has removed this danger. Still to many mathematicians +logic is still unknown, and to them—to Professor +A. Schoenflies for example—modern mathematics, owing to +its alliance with logic, appears to be sinking into scholasticism. +It is true that the word “scholasticism” is not used by +Professor Schoenflies in any intentionally precise signification, +but merely as a vague epithet of disapproval, as the word +“socialism” is used by the ordinary philistine, and this +would certainly serve as a sufficient excuse. But no excuse +is needed: these opinions are themselves a source of +mathematical jokes.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_96_96" id="Footnote_96_96"></a><a href="#FNanchor_96_96"><span class="label">[96]</span></a> See Chapter XII.</p></div> + +<div class="footnote"><p><a name="Footnote_97_97" id="Footnote_97_97"></a><a href="#FNanchor_97_97"><span class="label">[97]</span></a> <i>Encykl. der math. Wiss.</i>, vol. iv., part i., p. 474.</p></div> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_75" id="Page_75">[Pg 75]</a></span></p> +<h3><a name="CHAPTER_XXXVII" id="CHAPTER_XXXVII"></a>CHAPTER XXXVII</h3> + +<h2>THE PARADOXES OF LOGIC</h2> + + +<p>We have already<a name="FNanchor_98_98" id="FNanchor_98_98"></a><a href="#Footnote_98_98" class="fnanchor">[98]</a> referred to the contempt shown by some +mathematicians for exact thought, which they condemn +under the name of “scholasticism.” An example of this +is given by Schoenflies in the second part of his publication +usually known as the <i>Bericht über Mengenlehre</i>.<a name="FNanchor_99_99" id="FNanchor_99_99"></a><a href="#Footnote_99_99" class="fnanchor">[99]</a> Here<a name="FNanchor_100_100" id="FNanchor_100_100"></a><a href="#Footnote_100_100" class="fnanchor">[100]</a> a +battle-cry in italics—</p> + +<p class="poem"> +“<i>Against all resignation, but also against all scholasticism!</i>”—<br /> +</p> + +<p class="noidt">found utterance. Later on, Schoenflies<a name="FNanchor_101_101" id="FNanchor_101_101"></a><a href="#Footnote_101_101" class="fnanchor">[101]</a> became bolder and +adopted a more personal battle-cry, also in italics, and +with a whole line to itself:</p> + +<p class="poem"> +“<i>For Cantorism but against Russellism!</i>”<br /> +</p> + +<p>“Cantorism” means the theory of transfinite aggregates +and numbers erected for the most part by Georg Cantor. +Shortly speaking, the great sin of “Russellism” is to have +gone too far in the chain of logical deduction for many +mathematicians, who were perhaps, like Schoenflies,<a name="FNanchor_102_102" id="FNanchor_102_102"></a><a href="#Footnote_102_102" class="fnanchor">[102]</a> blinded<span class='pagenum'><a name="Page_76" id="Page_76">[Pg 76]</a></span> +by their rather uncritical love of mathematics. Thus it +comes about that Schoenflies<a name="FNanchor_103_103" id="FNanchor_103_103"></a><a href="#Footnote_103_103" class="fnanchor">[103]</a> denounces Russellism as +“scholastic and unhealthy.” This queer blend of qualities +would surely arouse the curiosity of the most <i>blasé</i> as to what +strange thing Russellism must be.<a name="FNanchor_104_104" id="FNanchor_104_104"></a><a href="#Footnote_104_104" class="fnanchor">[104]</a></p> + +<p>Schoenflies<a name="FNanchor_105_105" id="FNanchor_105_105"></a><a href="#Footnote_105_105" class="fnanchor">[105]</a> said that some mathematicians attributed to +the logical paradoxes which have given Russell so much +trouble to clear up, “especially to those that are artificially +constructed, a significance that they do not have.” Yet +no grounds were given for this assertion, from which it might +be concluded that the rigid examination of any concept was +unimportant. The paradoxes are simply the necessary +results of certain logical views which are currently held, +which views do not, except when they are examined rather +closely, appear to contain any difficulty. The contradiction +is not felt, as it happens, by people who confine their attention +to the first few number-classes of Cantor, and this seems +to have given rise to the opinion, which it is a little surprising +to find that some still hold, that cases not usually met with, +though falling under the same concept as those usually met +with, are of little importance. One might just as well maintain +that continuous but not differentiable functions are +unimportant because they are artificially constructed—a term +which I suppose means that they do not present themselves +when unasked for. Rather should we say that it is by the +discovery and investigation of such cases that the concept +in question can alone be judged, and the validity of certain +theorems—if they are valid—conclusively proved. That +this has been done, chiefly by the work of Russell, is simply +a fact; that this work has been and is misunderstood by +many<a name="FNanchor_106_106" id="FNanchor_106_106"></a><a href="#Footnote_106_106" class="fnanchor">[106]</a> is regrettable for this reason, among others, that it +proves that, at the present time, as in the days in which +<i>Gulliver’s Travels</i> were written, some mathematicians are +bad reasoners.<a name="FNanchor_107_107" id="FNanchor_107_107"></a><a href="#Footnote_107_107" class="fnanchor">[107]</a><span class='pagenum'><a name="Page_77" id="Page_77">[Pg 77]</a></span></p> + +<p>Nearly all mathematicians agreed that the way to solve +these paradoxes was simply not to mention them; but there +was some divergence of opinion as to how they were to be +unmentioned. It was clearly unsatisfactory merely not to +mention them. Thus Poincaré was apparently of opinion +that the best way of avoiding such awkward subjects was +to mention that they were not to be mentioned. But<a name="FNanchor_108_108" id="FNanchor_108_108"></a><a href="#Footnote_108_108" class="fnanchor">[108]</a> +“one might as well, in talking to a man with a long nose, +say: ‘When I speak of noses, I except such as are inordinately +long,’ which would not be a very successful effort to +avoid a painful topic.”</p> + +<p>Schoenflies, in his paper of 1911 mentioned above, adopted +the convenient plan of referring these logical difficulties at +the root of mathematics to a department of knowledge which +he called “philosophy.” He said<a name="FNanchor_109_109" id="FNanchor_109_109"></a><a href="#Footnote_109_109" class="fnanchor">[109]</a> of the theory of aggregates +that though “born of the acuteness of the mathematical +spirit, it has gradually fallen into philosophical ways, and +has lost to some extent the compelling force which dwells +in the mathematical process of conclusion.”</p> + +<p>The majority of mathematicians have followed Schoenflies +rather than Poincaré, and have thus adopted tactics rather +like those of the March Hare and the Gryphon,<a name="FNanchor_110_110" id="FNanchor_110_110"></a><a href="#Footnote_110_110" class="fnanchor">[110]</a> who promptly +changed the subject when Alice raised awkward questions. +Indeed, the process of the first of these creatures of a child’s +dream is rather preferable to that of Schoenflies. The March +Hare refused to discuss the subject because he was bored +when difficulties arose. Schoenflies would not say that he +was bored—he professed interest in philosophical matters, +but simply called the logical continuation of a subject by +another name when he did not wish to discuss the continuation, +and thus implied that he had discussed the whole +subject. Further, Schoenflies would not apparently admit +that the one method of logic could be applied to the solution +of both mathematical and philosophical problems, in so far +as these problems are soluble at all; but the March Hare, +shortly before the remark we have just quoted, rightly +showed great astonishment that butter did not help to cure<span class='pagenum'><a name="Page_78" id="Page_78">[Pg 78]</a></span> +both hunger and watches that would not go.<a name="FNanchor_111_111" id="FNanchor_111_111"></a><a href="#Footnote_111_111" class="fnanchor">[111]</a> The judgment +of Schoenflies by which certain apparently mathematical +questions were condemned as “philosophical,” rested on +grounds as flimsy as those in the Dreyfus Case, or the Trial +in <i>Wonderland</i>.<a name="FNanchor_112_112" id="FNanchor_112_112"></a><a href="#Footnote_112_112" class="fnanchor">[112]</a></p> + +<hr /> +<div class="footnote"><p><a name="Footnote_98_98" id="Footnote_98_98"></a><a href="#FNanchor_98_98"><span class="label">[98]</span></a> Chapters VII and XXXVI.</p></div> + +<div class="footnote"><p><a name="Footnote_99_99" id="Footnote_99_99"></a><a href="#FNanchor_99_99"><span class="label">[99]</span></a> <i>Die Entwickelung der Lehre von den Punktmannigfaltigkeiten.</i> +Bericht, erstattet der deutschen Mathematiker-Vereinigung, Leipzig, +1908.</p></div> + +<div class="footnote"><p><a name="Footnote_100_100" id="Footnote_100_100"></a><a href="#FNanchor_100_100"><span class="label">[100]</span></a> <i>Ibid.</i>, p. 7. The battle-cry is: “<i>Gegen jede Resignation, aber +auch gegen jede Scholastik!</i>”</p></div> + +<div class="footnote"><p><a name="Footnote_101_101" id="Footnote_101_101"></a><a href="#FNanchor_101_101"><span class="label">[101]</span></a> “Ueber die Stellung der Definition in der Axiomatik,” <i>Jahresber, +der deutsch. Math.-Ver.</i>, vol. xx., 1911, pp. 222-5. The battle-cry is +on p. 256 and is: “Für den Cantorismus aber gegen den Russellismus!”</p></div> + +<div class="footnote"><p><a name="Footnote_102_102" id="Footnote_102_102"></a><a href="#FNanchor_102_102"><span class="label">[102]</span></a> <i>Ibid.</i>, p. 251. “Es ist also,” he exclaims with the eloquence of +emotion and the emotion of eloquence, “nicht die Geringschätzung +der Philosophie, die mich dabei treibt, sondern die Liebe zur Mathematik;...”</p></div> + +<div class="footnote"><p><a name="Footnote_103_103" id="Footnote_103_103"></a><a href="#FNanchor_103_103"><span class="label">[103]</span></a> “Ueber die Stellung der Definition in der Axiomatik,” <i>Jahresber, +der deutsch. Math.-Ver.</i>, vol. xx., 1911, p. 251.</p></div> + +<div class="footnote"><p><a name="Footnote_104_104" id="Footnote_104_104"></a><a href="#FNanchor_104_104"><span class="label">[104]</span></a> [Cf. for this, <i>M.</i>, vol. xxii., January 1912, pp. 149-58.—<span class="smcap">Ed.</span>]</p></div> + +<div class="footnote"><p><a name="Footnote_105_105" id="Footnote_105_105"></a><a href="#FNanchor_105_105"><span class="label">[105]</span></a> <i>Bericht</i>, 1908, p. 76, note; cf. p. 72.</p></div> + +<div class="footnote"><p><a name="Footnote_106_106" id="Footnote_106_106"></a><a href="#FNanchor_106_106"><span class="label">[106]</span></a> E.g. in F. Hausdorff’s review of Russell’s <i>Principles</i> of 1903 in +the <i>Vierteljahrsschr. für wiss. Philos. und Soziologie</i>.</p></div> + +<div class="footnote"><p><a name="Footnote_107_107" id="Footnote_107_107"></a><a href="#FNanchor_107_107"><span class="label">[107]</span></a> [Cf. <i>M.</i>, vol. xxv., 1915, pp. 333-8.—<span class="smcap">Ed.</span>]</p></div> + +<div class="footnote"><p><a name="Footnote_108_108" id="Footnote_108_108"></a><a href="#FNanchor_108_108"><span class="label">[108]</span></a> Russell, <i>A. J. M.</i>, vol. xxx., 1908, p. 226.</p></div> + +<div class="footnote"><p><a name="Footnote_109_109" id="Footnote_109_109"></a><a href="#FNanchor_109_109"><span class="label">[109]</span></a> <i>Loc. cit.</i>, p. 222.</p></div> + +<div class="footnote"><p><a name="Footnote_110_110" id="Footnote_110_110"></a><a href="#FNanchor_110_110"><span class="label">[110]</span></a> See <a href="#App_Q">Appendix Q</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_111_111" id="Footnote_111_111"></a><a href="#FNanchor_111_111"><span class="label">[111]</span></a> See <a href="#App_R">Appendix R</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_112_112" id="Footnote_112_112"></a><a href="#FNanchor_112_112"><span class="label">[112]</span></a> See <a href="#App_S">Appendix S</a>.</p></div> + + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_79" id="Page_79">[Pg 79]</a></span></p> +<h3><a name="CHAPTER_XXXVIII" id="CHAPTER_XXXVIII"></a>CHAPTER XXXVIII</h3> + +<h2>MODERN LOGIC AND SOME PHILOSOPHICAL +ARGUMENTS</h2> + + +<p>The most noteworthy reformation of recent years in logic +is the discovery and development by Mr. Bertrand Russell +of the fact that the paradoxes—of Burali-Forti, Russell, +König, Richard, and others—which have appeared of late +years in the mathematical theory of aggregates and have +just been referred to, are of an entirely <i>logical</i> nature, and +that their avoidance requires us to take account of a principle +which has been hitherto unrecognized, and which +renders invalid several well-known arguments in refutation +of scepticism, agnosticism, and the statement of a man that +he asserts nothing.</p> + +<p>Dr. Whitehead and Mr. Russell say:<a name="FNanchor_113_113" id="FNanchor_113_113"></a><a href="#Footnote_113_113" class="fnanchor">[113]</a> “The principle +which enables us to avoid illegitimate totalities may be +stated as follows: ‘Whatever involves <i>all</i> of a collection +must not be one of the collection,’ or conversely: ‘If, +provided a certain collection had a total, it would have +members only definable in terms of that total, then the said +collection has no total.’ We shall call this the ‘vicious-circle +principle,’ because it enables us to avoid the vicious +circles involved in the assumption of illegitimate totalities. +Arguments which are condemned by the vicious-circle principle +will be called ‘vicious-circle fallacies.’ Such arguments, +in certain circumstances, may lead to contradictions, but it +often happens that the conclusions to which they lead are +in fact true, though the arguments are fallacious. Take, +for example, the law of excluded middle in the form ‘all +propositions are true or false.’ If from this law we argue +that, because the law of excluded middle is a proposition,<span class='pagenum'><a name="Page_80" id="Page_80">[Pg 80]</a></span> +therefore the law of excluded middle is true or false, we +incur a vicious-circle fallacy. ‘All propositions’ must be +in some way limited before it becomes a legitimate totality, +and any limitation which makes it legitimate must make +any statement about the totality fall outside the totality. +Similarly the imaginary sceptic who asserts that he knows +nothing and is refuted by being asked if he knows that he +knows nothing, has asserted nonsense, and has been fallaciously +refuted by an argument which involves a vicious-circle +fallacy. In order that the sceptic’s assertion may +become significant it is necessary to place some limitation +upon the things of which he is asserting his ignorance; the +proposition that he is ignorant of every member of this +collection must not itself be one of the collection. Hence +any significant scepticism is not open to the above form of +refutation.”</p> + +<p>In fact, the world of things falls into various sets of things +of the same “type.” For every propositional function ϕ(<i>x</i>) +there is a range of values of <i>x</i> for which ϕ(<i>x</i>) has a signification +as a true or a false proposition. Until this theory +was brought forward, there were occasionally discussions +as to whether an object which did not belong to the range +of a certain propositional function possessed the corresponding +property or not. Thus, Jevons, in early days,<a name="FNanchor_114_114" id="FNanchor_114_114"></a><a href="#Footnote_114_114" class="fnanchor">[114]</a> +was of opinion that virtue is neither black nor not-black +because it is not coloured, but rather later<a name="FNanchor_115_115" id="FNanchor_115_115"></a><a href="#Footnote_115_115" class="fnanchor">[115]</a> he admitted that +virtue is not triangular.<a name="FNanchor_116_116" id="FNanchor_116_116"></a><a href="#Footnote_116_116" class="fnanchor">[116]</a></p> + +<hr /> +<div class="footnote"><p><a name="Footnote_113_113" id="Footnote_113_113"></a><a href="#FNanchor_113_113"><span class="label">[113]</span></a> <i>Pa. Ma.</i>, p. 40.</p></div> + +<div class="footnote"><p><a name="Footnote_114_114" id="Footnote_114_114"></a><a href="#FNanchor_114_114"><span class="label">[114]</span></a> <i>S. o. S.</i> pp. 36-7.</p></div> + +<div class="footnote"><p><a name="Footnote_115_115" id="Footnote_115_115"></a><a href="#FNanchor_115_115"><span class="label">[115]</span></a> <i>E. L. L.</i>, pp. 120-1.</p></div> + +<div class="footnote"><p><a name="Footnote_116_116" id="Footnote_116_116"></a><a href="#FNanchor_116_116"><span class="label">[116]</span></a> [It may perhaps be added that, some years after Mr. R*ss*ll’s +death, Dr. Whitehead stated, in an address delivered in 1916 and +reprinted in his book on <i>The Organisation of Thought</i> (London, 1917, +p. 120), that “the specific heat of virtue is 0.003 is, I should imagine, +not a proposition at all, so that it is neither true nor false....”—<span class="smcap">Ed.</span>]</p></div> + + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_81" id="Page_81">[Pg 81]</a></span></p> +<h3><a name="CHAPTER_XXXIX" id="CHAPTER_XXXIX"></a>CHAPTER XXXIX</h3> + +<h2>THE HIERARCHY OF JOKES</h2> + + +<p>Jokes may be divided into various types. Thus a joke or +class of jokes can only be the subject of a joke of +higher order. Otherwise we would get the same vicious-circle +fallacy which gives rise to so many paradoxes in +logic and mathematics. A certain Oxford scholar succeeded, +to his own satisfaction, in reducing all jokes to +primitive types, consisting of thirty-seven proto-Aryan +jokes. When any proposition was propounded to him, +he would reflect and afterwards pronounce on the question +as to whether the proposition was a joke or not. If +he decided, by his theory, that it was a joke, he would +solemnly say: “There <i>is</i> that joke.” If this narration is +accepted as a joke, since it cannot be reduced to one of the +proto-Aryan jokes under pain of leading us to commit a +vicious-circle fallacy, we must conclude that there is at +least one joke which is not proto-Aryan; and, in fact, is +of a higher type. There is no great difficulty in forming +a hierarchy of jokes of various types. Thus a joke of the +fourth type (or order) is as follows: A joke of the first order +was told to a Scotchman, who, as we would expect, was +unable to see it.<a name="FNanchor_117_117" id="FNanchor_117_117"></a><a href="#Footnote_117_117" class="fnanchor">[117]</a> The person (A) who told this joke told +the story of how the joke was received to another Scotchman +thereby making a joke about a joke of the first order, and +thus making a joke of the second order. A remarked on +this joke that no joke could penetrate the head of the +Scotchman to whom the joke of the first order was told, +even if it were fired into his head with a gun. The Scotch<span class='pagenum'><a name="Page_82" id="Page_82">[Pg 82]</a></span>man, +after severe thought, replied: “But ye couldn’t do +that, ye know!” A repeated the whole story, which +constituted a joke of the third order, to a third Scotchman. +This last Scotchman again, after prolonged thought, replied: +“He had ye there!” This whole story is a joke of the +fourth order.</p> + +<p>Most known jokes are of the first order, for the simple +reason that the majority of people find that the slightest +mental effort effectually destroys any perception of humour. +It seems to me that a joke becomes more pleasurable in +proportion as logical faculties are brought into play by it; +and hence that logical power is allied, or possibly identical, +with the power of grasping more subtle jokes. The jokes +which amuse the frequenters of music-halls, Conservatives, +and M. Bergson—and which usually deal with accidents, +physical defects, mothers-in-law, foreigners, or over-ripe +cheese—are usually jokes of the first order. Jokes of the +second, and even of the third, order appeal to ordinary +well-educated people; jokes of higher order require either +special ability or a sound logical training on the part of the +hearer if the joke is to be appreciated; while jokes of transfinite +order presumably only excite the inaudible laughter +of the gods.</p> +<hr /> +<div class="footnote"><p><a name="Footnote_117_117" id="Footnote_117_117"></a><a href="#FNanchor_117_117"><span class="label">[117]</span></a> [It may be that, like certain remarks about cheese and mothers-in-law +(see below), the statement that Scotchmen cannot see jokes +is a joke of the first order.—<span class="smcap">Ed.</span>]</p></div> + + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_83" id="Page_83">[Pg 83]</a></span></p> +<h3><a name="CHAPTER_XL" id="CHAPTER_XL"></a>CHAPTER XL</h3> + +<h2>THE EVIDENCE OF GEOMETRICAL PROPOSITIONS</h2> + + +<p>It has often been maintained that the twentieth proposition +of the first book of Euclid—that two sides of a triangle are +together greater than the third side—is evident even to asses. +This does not, however, seem to me generally true. I once +asked a coastguardsman the distance from A to B; he +replied: “Eight miles.” On further inquiry I elicited the +fact that the distance from A to C was two miles and the +distance from C to B was twenty-two miles. Now the paths +from A to B and from C to B were by sea; while the path +from A to C was by land. Hence if the path by land was +rugged and the distance along the road was two miles, it +would appear that the coastguardsman believed that not +only could one side of a triangle be greater than the other +two, but that one straight side of a triangle might be greater +than one straight side and any curvilinear side of the same +triangle. The only escape from part of this astonishing +creed would be by assuming that the distance of two miles +from A to C was measured “as the crow flies,” while the +road A to C was so hilly that a pedestrian would traverse +more than fourteen miles when proceeding from A to C. +Then indeed the coastguardsman could maintain the true +proposition that there is at least one triangle ABC, with +the side AC curvilinear, such that the sum of the lengths +of AB and AC is greater than the length of BC, and only +deny the twentieth proposition of the first book of Euclid.</p> + +<p>Reasoning with the coastguardsman only had the effect +of his adducing the authority of one Captain Jones in support +of the accuracy of his data. Possibly Captain Jones held +strange views as to the influence of temperature or other +physical circumstances, or even the nature of space itself, +on the lengths of lines in the neighbourhood of the +triangle ABC.</p> + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_84" id="Page_84">[Pg 84]</a></span></p> +<h3><a name="CHAPTER_XLI" id="CHAPTER_XLI"></a>CHAPTER XLI</h3> + +<h2>ABSOLUTE AND RELATIVE POSITION</h2> + + +<p>Some people maintain that position in space or time must +be relative because, if we try to determine the position of +a body A, if bodies B, C, D with respect to which the position +of A could be determined were not present, we should be +trying to determine something about A without having our +senses affected by other things. These people seem to me +to be like the cautious guest who refused to say anything +about his host’s port-wine until he had tasted red ink.</p> + +<p>“Wherein, then,” says Mr. Russell,<a name="FNanchor_118_118" id="FNanchor_118_118"></a><a href="#Footnote_118_118" class="fnanchor">[118]</a> “lies the plausibility +of the notion that all points are exactly alike? This notion +is, I believe, a psychological illusion, due to the fact that we +cannot remember a point so as to know it when we meet +again. Among simultaneously presented points it is easy +to distinguish; but though we are perpetually moving, and +thus being brought among new points, we are quite unable +to detect this fact by our senses, and we recognize places +only by the objects they contain. But this seems to be a +mere blindness on our part—there is no difficulty, so far as +I can see, in supposing an immediate difference between +points, as between colours, but a difference which our senses +are not constructed to be aware of. Let us take an analogy: +Suppose a man with a very bad memory for faces; he would +be able to know, at any moment, whether he saw one face +or many, but he would not be aware whether he had seen +any of the faces before. Thus he might be led to define +people by the rooms in which he saw them, and to suppose +it self-contradictory that new people should come to his +lectures, or that old people should cease to do so. In the +latter point at least it will be admitted by lecturers that<span class='pagenum'><a name="Page_85" id="Page_85">[Pg 85]</a></span> +he would be mistaken. And as with faces, so with points—inability +to recognize them must be attributed, not to +the absence of individuality, but merely to our incapacity.”</p> + +<p>Another form of this tendency is shown by Kronecker, +Borel, Poincaré, and many other mathematicians, who +refuse mere logical determination of a conception and require +that it be actually described in a finite number of terms. +These eminent mathematicians were anticipated by the +empirical philosopher who would not pronounce that the +“law of thought” that A is either in the place B or not +is true until he had looked to make sure. This philosopher +was of the same school as J. S. Mill and Buckle, who seem +to have maintained implicitly not only that, in view of the +fact that the breadth of a geometrical line depends upon +the material out of which it is constructed, or upon which +it is drawn, that there ought to be a paste-board geometry, +a stone geometry, and so on;<a name="FNanchor_119_119" id="FNanchor_119_119"></a><a href="#Footnote_119_119" class="fnanchor">[119]</a> but also that the foundations +of logic are inductive in their nature.<a name="FNanchor_120_120" id="FNanchor_120_120"></a><a href="#Footnote_120_120" class="fnanchor">[120]</a> “We cannot,” says +Mill,<a name="FNanchor_121_121" id="FNanchor_121_121"></a><a href="#Footnote_121_121" class="fnanchor">[121]</a> “conceive a round square, not merely because no such +object has ever presented itself in our experience, for that +would not be enough. Neither, for anything we know, are +the two ideas in themselves incompatible. To conceive a +body all black and yet white would only be to conceive +two different sensations as produced in us simultaneously +by the same object—a conception familiar to our experience—and +we should probably be as well able to conceive a round +square as a hard square, or a heavy square, if it were not +that in our uniform experience, at the instant when a thing +begins to be round, it ceases to be square, so that the beginning +of the one impression is inseparably associated with the +departure or cessation of the other. Thus our inability to +form a conception always arises from our being compelled +to form another contradictory to it.”</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_118_118" id="Footnote_118_118"></a><a href="#FNanchor_118_118"><span class="label">[118]</span></a> <i>Md.</i>, N. S., vol. x., July, 1901, pp. 313-14.</p></div> + +<div class="footnote"><p><a name="Footnote_119_119" id="Footnote_119_119"></a><a href="#FNanchor_119_119"><span class="label">[119]</span></a> J. B. Stallo, <i>The Concepts and Theories of Modern Physics</i>, 4th ed., +London, 1900, pp. 217-27.</p></div> + +<div class="footnote"><p><a name="Footnote_120_120" id="Footnote_120_120"></a><a href="#FNanchor_120_120"><span class="label">[120]</span></a> <i>Ibid.</i>, pp. 140-4.</p></div> + +<div class="footnote"><p><a name="Footnote_121_121" id="Footnote_121_121"></a><a href="#FNanchor_121_121"><span class="label">[121]</span></a> <i>Examination of the Philosophy of Sir William Hamilton</i>, vol. i. +p. 88, Amer. ed.</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_86" id="Page_86">[Pg 86]</a></span></p> +<h3><a name="CHAPTER_XLII" id="CHAPTER_XLII"></a>CHAPTER XLII</h3> + +<h2>LAUGHTER</h2> + + +<p>[It seemed advisable to give here<a name="FNanchor_122_122" id="FNanchor_122_122"></a><a href="#Footnote_122_122" class="fnanchor">[122]</a> some views on laughter, most +of which were also held by Mr. R*ss*ll, though no written +expression of his views has yet been found. In a review<a name="FNanchor_123_123" id="FNanchor_123_123"></a><a href="#Footnote_123_123" class="fnanchor">[123]</a> of +M. Bergson’s book on <i>Laughter</i>,<a name="FNanchor_124_124" id="FNanchor_124_124"></a><a href="#Footnote_124_124" class="fnanchor">[124]</a> Mr. Russell has remarked:</p> + +<p>“It has long been recognized by publishers that everybody +desires to be a perfect lady or gentleman (as the case may +be); to this fact we owe the constant stream of etiquette-books. +But if there is one thing which people desire even +more, it is to have a faultless sense of humour. Yet so +far as I know, there is no book called ‘Jokes without Tears, +by Mr. McQuedy.’ This extraordinary lacuna has now been +filled. Those to whom laughter has hitherto been an unintelligible +vagary, in which one must join, though one could +never tell when it would break out, need only study +M. Bergson’s book to acquire the finest flower of Parisian wit. +By observing a very simple formula they will know infallibly +what is funny and what is not; if they sometimes surprise +their unlearned friends, they have only to mention their +authority in order to silence doubt. ‘The attitudes, gestures +and movements of the human body,’ says M. Bergson, ‘are +laughable in exact proportion as that body reminds us of +a mere machine.’ When an elderly gentleman slips on a +piece of orange-peel and falls, we laugh, because his body +follows the laws of dynamics instead of a human purpose. +When a man falls from a scaffolding and breaks his neck on +the pavement, we presumably laugh even more, since the<span class='pagenum'><a name="Page_87" id="Page_87">[Pg 87]</a></span> +movement is even more completely mechanical. When the +clown makes a bad joke for the first time, we keep our countenance, +but at the fifth repetition we smile, and at the tenth +we roar with laughter, because we begin to feel him a mere +automaton. We laugh at Molière’s misers, misanthropists +and hypocrites, because they are mere types mechanically +dominated by a master impulse. Presumably we laugh at +Balzac’s characters for the same reason; and presumably we +never smile at Falstaff, because he is individual throughout.”</p> + +<p>The review concludes with the reflection that “it would +seem to be impossible to find any such formula as M. Bergson +seeks. Every formula treats what is living as if it were +mechanical, and is therefore by his own rule a fitting object +of laughter.” Now, this undoubtedly true conclusion has +been obtained, as is readily seen, by a vicious-circle fallacy +which Mr. R*ss*ll would hardly have committed.—<span class="smcap">Ed.</span>]</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_122_122" id="Footnote_122_122"></a><a href="#FNanchor_122_122"><span class="label">[122]</span></a> From a remark on p. 47 above, it is evident that Mr. R*ss*ll +intended to write some such chapter as this.</p></div> + +<div class="footnote"><p><a name="Footnote_123_123" id="Footnote_123_123"></a><a href="#FNanchor_123_123"><span class="label">[123]</span></a> <i>The Professor’s Guide to Laughter, The Cambridge Review</i>, vol. +xxxii., 1912, pp. 193-4.</p></div> + +<div class="footnote"><p><a name="Footnote_124_124" id="Footnote_124_124"></a><a href="#FNanchor_124_124"><span class="label">[124]</span></a> <i>Laughter, an Essay on the Meaning of the Comic</i>, English translation +by C. Brereton and F. Rothwell, London, 1911.</p></div> + + + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_88" id="Page_88">[Pg 88]</a></span></p> + +<h3><a name="CHAPTER_XLIII" id="CHAPTER_XLIII"></a>CHAPTER XLIII</h3> + +<h2>“GEDANKENEXPERIMENTE” AND EVOLUTIONARY +ETHICS</h2> + + +<p>The “Gedankenexperimente,” upon which so much weight +has been laid by Mach<a name="FNanchor_125_125" id="FNanchor_125_125"></a><a href="#Footnote_125_125" class="fnanchor">[125]</a> and Heymans,<a name="FNanchor_126_126" id="FNanchor_126_126"></a><a href="#Footnote_126_126" class="fnanchor">[126]</a> had already been +investigated by the White Queen,<a name="FNanchor_127_127" id="FNanchor_127_127"></a><a href="#Footnote_127_127" class="fnanchor">[127]</a> who, however, seems to +have perceived that the results of such experiments are not +always logically valid. The psychological founding of logic +appears to be not without analogy with the surprising +method of advocates of evolutionary ethics, who expect +to discover what <i>is</i> good by inquiring what cannibals have +<i>thought</i> good. I sometimes feel inclined to apply the historical +method to the multiplication table. I should make +a statistical inquiry among school-children, before their +pristine wisdom had been biassed by teachers. I should +put down their answers as to what 6 times 9 amounts to, +I should work out the average of their answers to six places +of decimals, and should then decide that, at the present +stage of human development, this average is the value of +6 times 9.</p> + +<hr /> +<div class="footnote"><p><a name="Footnote_125_125" id="Footnote_125_125"></a><a href="#FNanchor_125_125"><span class="label">[125]</span></a> See, e.g., <i>E. u. I.</i>, pp. 183-200.</p></div> + +<div class="footnote"><p><a name="Footnote_126_126" id="Footnote_126_126"></a><a href="#FNanchor_126_126"><span class="label">[126]</span></a> <i>G. u. E.</i>, vol. i.</p></div> + +<div class="footnote"><p><a name="Footnote_127_127" id="Footnote_127_127"></a><a href="#FNanchor_127_127"><span class="label">[127]</span></a> See <a href="#App_T">Appendix T</a>.</p></div> + + +<hr class="full" /> +<p><span class='pagenum'><a name="Page_89" id="Page_89">[Pg 89]</a></span></p> +<h2><a name="APPENDIXES" id="APPENDIXES"></a>APPENDIXES</h2> + + +<h4><a name="App_A" id="App_A"></a>A. LOGIC AND THE PRINCIPLE OF IDENTITY.</h4> + +<p><i>T. L. G.</i>, p. 45: “‘Contrariwise,” continued Tweedledee, “if it +was so, it might be; and if it were so, it would be: but as it isn’t, +it ain’t. That’s logic.”</p> + +<hr style='width: 15%;' /> + +<p><i>S. B.</i>, p. 159: The Professor said: “The day is the same length +as anything that is the same length as <i>it</i>.”</p> + +<hr style='width: 15%;' /> + +<p><i>S. B.</i>, p. 161: Bruno observed that, when the Other Professor lost +himself, he should shout: “He’d be sure to hear hisself, ‘cause he +couldn’t be far off.”</p> + + +<h4><a name="App_B" id="App_B"></a>B. SYNTHESIS OF CONTRADICTORIES.</h4> + +<p><i>T. L. G.</i>, p. 71: “‘What a beautiful belt you’ve got on!’ Alice +suddenly remarked.... ‘At least,’ she corrected herself on second +thoughts, ‘a beautiful cravat, I should have said—no, a belt, I mean—I +beg your pardon!’ she added in dismay, for Humpty-Dumpty +looked thoroughly offended, and she began to wish she hadn’t chosen +that subject. ‘If only I knew,’ she thought to herself, ‘which was +neck and which was waist!’”</p> + + +<h4><a name="App_C" id="App_C"></a>C. EMPIRICAL PHILOSOPHERS AND MATHEMATICS.</h4> + +<p><i>T. L. G.</i>, p. 79: “‘... Now if you had the two eyes on the same +side of the nose, for instance—or the mouth at the top—that would +be <i>some</i> help.’</p> + +<p>“‘It wouldn’t look nice,’ Alice objected. But Humpty-Dumpty only +shut his eyes and said: ‘Wait till you’ve tried.’”</p> + +<hr style='width: 15%;' /> + +<p><i>T. L. G.</i>, p. 72: “‘And if you take one from three hundred and +sixty-five, what remains?’</p> + +<p>“‘Three hundred and sixty-four, of course.’</p> + +<p>“Humpty-Dumpty looked doubtful. ‘I’d rather see that done on +paper,’ he said.”</p> + + +<h4><a name="App_D" id="App_D"></a>D. NOMINAL DEFINITION.</h4> + +<p><i>T. L. G.</i>, p. 73: “‘When <i>I</i> used a word,’ Humpty-Dumpty said +in rather a scornful tone, ‘it means just what I choose it to mean—neither +more nor less.’</p> + +<p>“‘The question is,’ said Alice, ‘whether you <i>can</i> make words mean +different things.’</p> + +<p>“‘The question is,’ said Humpty-Dumpty, ‘which is to be master—that’s +all.<span class='pagenum'><a name="Page_90" id="Page_90">[Pg 90]</a></span>’”</p> + + +<h4><a name="App_E" id="App_E"></a>E. CONFORMITY OF A PARADOXICAL LOGIC WITH +COMMON-SENSE.</h4> + +<p><i>T. L. G.</i>, p. 100:</p> + +<p class="poem"> +“But I was thinking of a plan<br /> +To dye one’s whiskers green,<br /> +And always use so large a fan<br /> +That they could not be seen.”<br /> +<span style="margin-left: 16em;">(Verse from White Knight’s song.)</span><br /> +</p> + + +<h4><a name="App_F" id="App_F"></a>F. IDEALISTS AND THE LAWS OF LOGIC.</h4> + +<p><i>T. L. G.</i>, p. 52-3: Tweedledee exclaimed: “‘... if he [the Red +King] left off dreaming about you [Alice], where do you suppose you’d +be?’</p> + +<p>“‘Where I am now, of course,’ said Alice.</p> + +<p>“‘Not you!’ Tweedledee retorted contemptuously. ‘You’d be +nowhere. Why, you’re only a sort of thing in his dream!’</p> + +<p>“‘If that there King was to wake,’ added Tweedledum, ‘you’d +go out—bang!—just like a candle!’</p> + +<p>“‘I shouldn’t!’ Alice exclaimed indignantly. ‘Besides, if <i>I’m</i> +only a sort of thing in his dream, what are <i>you</i>, I should like to know?’</p> + +<p>“‘Ditto,’ said Tweedledum...; ‘you know very well you’re not +real.’</p> + +<p>“‘I <i>am</i> real!’ said Alice, and began to cry.”</p> + +<hr style='width: 15%;' /> + +<p><i>T. L. G.</i>, p. 97: “‘How <i>can</i> you go on talking so quietly, head +downwards?’ Alice asked, as she dragged him out by the feet, and +laid him in a heap on the bank.</p> + +<p>“The Knight looked surprised at the question. ‘What does it +matter where my body happens to be?’ he said. ‘My mind goes +on working all the same. In fact, the more head downwards I am, +the more I keep inventing new things.’”</p> + +<hr style='width: 15%;' /> + +<p><i>T. L. G.</i>, p. 98: “‘... Everybody that hears me sing—either it +brings the <i>tears</i> into their eyes, or else——’</p> + +<p>“‘Or else what?’ said Alice, for the Knight had made a sudden +pause.</p> + +<p>“‘Or else it doesn’t, you know.’”</p> + + +<h4><a name="App_G" id="App_G"></a>G. DISTINCTION BETWEEN SIGN AND SIGNIFICATION.</h4> + +<p><i>T. L. G.</i>, pp. 98-9: “‘The name of the song is called “<i>Haddocks’ +Eyes</i>.”’</p> + +<p>“‘Oh, that’s the name of the song, is it?’ Alice said, trying to +feel interested.</p> + +<p>“‘No, you don’t understand,’ the Knight said looking a little +vexed. ‘That’s what the name is <i>called</i>. The name really <i>is</i> “<i>The +Aged Aged Man</i>.”’</p> + +<p>“‘Then I ought to have said “That’s what the <i>song</i> is called”?’ +Alice corrected herself.</p> + +<p>“‘No, you oughtn’t: that’s another thing. The <i>song</i> is called +“<i>Ways and Means</i>”: but that’s only what it’s <i>called</i>, you know!’</p> + +<p>“‘Well, what <i>is</i> the song, then?’ said Alice, who was by this +time completely bewildered.</p> + +<p>“‘I was coming to that,’ the Knight said. ‘The song really <i>is +“A-sitting on a Gate</i>”....<span class='pagenum'><a name="Page_91" id="Page_91">[Pg 91]</a></span>’”</p> + + +<h4><a name="App_H" id="App_H"></a>H. NOMINALISM.</h4> + +<p><i>A. A. W.</i>, p. 70: “‘Then you should say what you mean,’ the +March Hare went on.</p> + +<p>“‘I do,’ Alice hastily replied; ‘at least—at least I mean what +I say—that’s the same thing, you know.’</p> + +<p>“‘Not the same thing a bit!’ said the Hatter. ‘Why, you might +just as well say that “I see what I eat” is the same thing as “I eat +what I see.”’</p> + +<p>“‘You might just as well say,’ added the March Hare, ‘that “I +like what I get” is the same thing as “I get what I like”!’</p> + +<p>“‘You might just as well say,’ added the Dormouse, which seemed +to be talking in its sleep, ‘that “I breathe when I sleep” is the same +as “I sleep when I breathe”!’</p> + +<p>“‘It <i>is</i> the same thing with you,’ said the Hatter; and here the +conversation dropped,...”</p> + + +<h4><a name="App_I" id="App_I"></a>I. UTILITY OF SYMBOLIC LOGIC.</h4> + +<p><i>A. A. W.</i>, p. 92: “‘I quite agree with you,’ said the Duchess, +‘and the moral of that is—“Be what you would seem to be”—or +if you’d like it put more simply—“Never imagine yourself not to be +otherwise than what it might appear to others that what you were +or might have been was not otherwise than what you had been would +have appeared to them to be otherwise.”’</p> + +<p>“‘I think I should understand that better,’ Alice said very politely, +‘if I had it written down: but I can’t quite follow it as you say it.’</p> + +<p>“‘That’s nothing to what I could say if I chose,’ the Duchess +replied, in a pleased tone.”</p> + + +<h4><a name="App_J" id="App_J"></a>J. MISTAKE AS TO THE NATURE OF CRITICISM.</h4> + +<p><i>T. L. G.</i>, p. 105: “‘She’s in that state of mind,’ said the White +Queen, ‘that she wants to deny <i>something</i>—only she doesn’t know +what to deny.’</p> + +<p>“‘A nasty, vicious temper,’ the White Queen remarked; and +then there was an uncomfortable silence for a minute or two.”</p> + + +<h4><a name="App_K" id="App_K"></a>K. A CRITERION OF TRUTH.</h4> + +<p><i>H. S.</i>, p. 3:</p> + +<p class="poem"> +“Just the place for a Snark! I have said it twice:<br /> +That alone should encourage the crew.<br /> +Just the place for a Snark! I have said it thrice:<br /> +What I tell you three times is true.”<br /> +</p> + +<hr style='width: 15%;' /> +<p><i>H. S.</i>, p. 50:</p> + +<p class="poem"> +“’Tis the note of the Jubjub! Keep count. I entreat;<br /> +You will find I have told it you twice.<br /> +’Tis the song of the Jubjub! The proof is complete,<br /> +If only I’ve stated it thrice.”<br /> +</p> +<p><span class='pagenum'><a name="Page_92" id="Page_92">[Pg 92]</a></span></p> + +<h4><a name="App_L" id="App_L"></a>L. UNIVERSAL AND PARTICULAR PROPOSITIONS.</h4> + +<p><i>T. L. G.</i>, p. 40: The Gnat had told Alice that the Bread-and-butterfly +lives on weak tea with cream in it; so:</p> + +<p>“‘Supposing it couldn’t find any?’ she suggested.</p> + +<p>“‘Then it would die, of course.’</p> + +<p>“‘But that must happen very often,’ Alice remarked thoughtfully.</p> + +<p>“‘It always happens,’ said the Gnat.”</p> + + +<h4><a name="App_M" id="App_M"></a>M. DENOTING.</h4> + +<p><i>T. L. G.</i>, p. 43: Tweedledum and Tweedledee were, in many respects, +indistinguishable, and Alice, walking along the road, noticed that +“whenever the road divided there were sure to be two finger-posts +pointing the same way, one marked ‘<span class="smcap">To Tweedledum’s House</span>’ +and the other ‘<span class="smcap">To the House of Tweedledee</span>.’</p> + +<p>“‘I do believe,’ said Alice at last, ‘that they live in the same +house!...’”</p> + + +<h4><a name="App_N" id="App_N"></a>N. NON-ENTITY.</h4> + +<p><i>T. L. G.</i>, p. 87: “‘I always thought they [human children] were +fabulous monsters!’ said the Unicorn....</p> + +<p>“‘Do you know [said Alice], I always thought Unicorns were +fabulous monsters, too! I never saw one alive before!’</p> + +<p>“‘Well, now that we <i>have</i> seen each other,’ said the Unicorn, +‘if you’ll believe in me, I’ll believe in you. Is that a bargain?’”</p> + +<hr style='width: 15%;' /> + +<p><i>T. L. G.</i>, pp. 80-1: “‘I see nobody on the road,’ said Alice.</p> + +<p>“‘I only wish <i>I</i> had such eyes,’ the [White] King remarked in a +fretful tone. ‘To be able to see Nobody! And at that distance, +too! Why, it’s as much as <i>I</i> can do to see real people by this light!’”</p> + +<hr style='width: 15%;' /> + +<p><i>A. A. W.</i>, p. 17: “And she [Alice] tried to fancy what the flame of +a candle looks like after the candle is blown out, for she could not +remember ever having seen such a thing.”</p> + +<hr style='width: 15%;' /> + +<p><i>A. A. W.</i>, p. 68: “... This time it [the Cheshire Cat] vanished +quite slowly, beginning with the end of the tail, and ending with the +grin, which remained some time after the rest of it had gone.</p> + +<p>“‘Well! I’ve often seen a cat without a grin,’ thought Alice; +‘but a grin without a cat! It’s the most curious thing I ever saw +in all my life!’”</p> + +<hr style='width: 15%;' /> + +<p><i>A. A. W.</i>, p. 77: “... The Dormouse went on,...; ‘and they +drew all manner of things—everything that begins with an M.’</p> + +<p>“‘Why with an M?’ said Alice.</p> + +<p>“‘Why not?’ said the March Hare.</p> + +<p>“Alice was silent.</p> + +<p>“... [The Dormouse] went on: ‘—that begins with an M, such +as mouse-traps, and the moon, and memory, and muchness, you know +you say things are “much of a muchness”—did you ever see such +a thing as a drawing of a muchness?’</p> + +<p>“‘Really, now you ask me,’ said Alice, very much confused, ‘I +don’t think——’</p> + +<p>“‘Then you shouldn’t talk,’ said the Hatter.<span class='pagenum'><a name="Page_93" id="Page_93">[Pg 93]</a></span>”</p> + + +<h4><a name="App_O" id="App_O"></a>O. OBJECTS OF MATHEMATICAL LOGIC.</h4> + +<p><i>T. L. G.</i>, p. 93: “‘I was wondering what the mouse-trap [fastened +to the White Knight’s saddle] was for,’ said Alice. ‘It isn’t very +likely there would be any mice on the horse’s back.’</p> + +<p>“‘Not very likely, perhaps,’ said the Knight, ‘but, if they <i>do</i> +come, I don’t choose to have them running all about.’</p> + +<p>“‘You see,’ he went on after a pause, ‘it’s as well to be provided +for <i>everything</i>. That’s the reason the horse has all these anklets round +his feet.’</p> + +<p>“‘But what are they for?’ Alice asked in a tone of great curiosity.</p> + +<p>“‘To guard against the bites of sharks,’ the Knight replied.”</p> + + +<h4><a name="App_P" id="App_P"></a>P. THE PRINCIPLE OF PERMANENCE.</h4> + +<p><i>T. L. G.</i>, p. 106: “‘Can you do Subtraction? [said the Red Queen] +Take nine from eight.’</p> + +<p>“‘Nine from eight I can’t, you know,’ Alice replied very readily +‘but—’</p> + +<p>“‘She can’t do Substraction,’ said the White Queen.”</p> + +<hr style='width: 15%;' /> + +<p><i>A. A. W.</i>, p. 56: [Said the Pigeon to Alice]: “‘... No, no! +You’re a serpent; and there’s no use denying it. I suppose you’ll +be telling me next that you never tasted an egg!’</p> + +<p>“‘I <i>have</i> tasted eggs certainly,’ said Alice, who was a very truthful +child; ‘but little girls eat eggs quite as much as serpents do, you +know.’</p> + +<p>“‘I don’t believe it,’ said the Pigeon; ‘but if they do, why then +they’re a kind of serpent, that’s all I can say.’</p> + +<p>“This was such a new idea to Alice, that she was quite silent for +a minute or two, which gave the Pigeon the opportunity of adding, +‘You’re looking for eggs, I know <i>that</i> well enough; and what does it +matter to me whether you’re a little girl or a serpent?’</p> + +<p>“‘It matters a good deal to <i>me</i>,’ said Alice hastily;...”</p> + +<hr style='width: 15%;' /> + +<p><i>A. A. W.</i>, p. 75: “‘But why [asked Alice] did they live at the +bottom of a well?’</p> + +<p>“‘Take some more tea,’ the March Hare said to Alice, very +earnestly.</p> + +<p>“‘I’ve had nothing yet,’ Alice replied in an offended tone, ‘so I +can’t take more.’</p> + +<p>“‘You mean you can’t take <i>less</i>,’ said the Hatter: ‘it’s very easy +to take <i>more</i> than nothing.’”</p> + + +<h4><a name="App_Q" id="App_Q"></a>Q. MATHEMATICIANS’ TREATMENT OF LOGIC.</h4> + +<p><i>A. A. W.</i>, p. 74: The Hatter had told of his quarrel with Time, +and of Time’s refusal now to do anything he asked: “‘... It’s always +six o’clock now!’</p> + +<p>“A bright idea came into Alice’s head. ‘Is that the reason so +many tea things are put out here?’ she asked.</p> + +<p>“‘Yes, that’s it,’ said the Hatter, with a sigh: ‘it’s always tea +time, and we’ve no time to wash the things between whiles.’</p> + +<p>“‘Then you keep moving round, I suppose?’ said Alice.</p> + +<p>“‘Exactly so,’ said the Hatter: ‘as the things get used up.<span class='pagenum'><a name="Page_94" id="Page_94">[Pg 94]</a></span>’</p> + +<p>“‘But what happens when you come to the beginning again?’ +Alice ventured to ask.</p> + +<p>“‘Suppose we change the subject,’ the March Hare interrupted, +yawning. ‘I’m getting tired of this.’”</p> + +<hr style='width: 15%;' /> + +<p><i>A. A. W.</i>, p. 99: “‘And how many hours a day did you do lessons?’ +said Alice, in a hurry to change the subject.</p> + +<p>“‘Ten hours the first day,’ said the Mock Turtle, ‘nine the next, +and so on.’</p> + +<p>“‘What a curious plan!’ exclaimed Alice.</p> + +<p>“‘That’s the reason they’re called lessons,’ the Gryphon remarked, +‘because they lessen from day to day.’</p> + +<p>“This was quite a new idea to Alice, and she thought it over a +little before she made her next remark. ‘Then the eleventh day +must have been a holiday.’</p> + +<p>“‘Of course it was,’ said the Mock Turtle.</p> + +<p>“‘And how did you manage on the twelfth?’ Alice went on +eagerly.</p> + +<p>“‘That’s enough about lessons,’ the Gryphon interrupted in a +very decided tone....”</p> + + +<h4><a name="App_R" id="App_R"></a>R. METHOD IN MATHEMATICS AND LOGIC.</h4> + +<p><i>A. A. W.</i>, p. 71: “‘Two days wrong!’ sighed the Hatter. ‘I +told you butter wouldn’t suit the works!’ he added, looking angrily +at the March Hare.</p> + +<p>“‘It was the <i>best</i> butter,’ the March Hare meekly replied.</p> + +<p>“‘Yes, but some crumbs must have got in as well,’ the Hatter +grumbled; ‘you shouldn’t have put it in with the bread-knife.’</p> + +<p>“The March Hare took the watch and looked at it gloomily: then +he dipped it into his cup of tea, and looked at it again: but he could +think of nothing better to say than his first remark, ‘It was the <i>best</i> +butter, you know.’”</p> + + +<h4><a name="App_S" id="App_S"></a>S. VERDICT THAT LOGIC IS PHILOSOPHY.</h4> + +<p><i>A. A. W.</i>, pp. 119-23: “... ‘Consider your verdict,’ he [the +King] said to the jury, in a low trembling voice.</p> + +<p>“‘There’s more evidence to come yet, please your Majesty,’ said +the White Rabbit, jumping up in a great hurry: ‘this paper has just +been picked up.’</p> + +<p>“‘What’s in it?’ said the Queen.</p> + +<p>“‘I haven’t opened it yet,’ said the White Rabbit, ‘but it seems +to be a letter written by the prisoner to—to somebody.’</p> + +<p>“‘It must have been that,’ said the King, ‘unless it was written +to nobody, which isn’t usual, you know.’</p> + +<p>“‘Who is it directed to?’ said one of the jurymen.</p> + +<p>“‘It isn’t directed at all,’ said the White Rabbit, ‘in fact there’s +nothing written on the <i>outside</i>.’ He unfolded the paper as he spoke, +and added, ‘It isn’t a letter, after all: it’s a set of verses.’</p> + +<p>“‘Are they in the prisoner’s handwriting?’ asked another of the +jurymen.</p> + +<p>“‘No they’re not,’ said the White Rabbit, ‘and that’s the queerest +thing about it.’ (The jury all looked puzzled).<span class='pagenum'><a name="Page_95" id="Page_95">[Pg 95]</a></span></p> + +<p>“‘He must have imitated somebody else’s hand,’ said the King. +(The jury brightened up again.)</p> + +<p>“‘Please your Majesty,’ said the Knave, ‘I didn’t write it, and they +can’t prove that I did: there’s no name signed at the end.’</p> + +<p>“‘If you didn’t sign it, said the King, that only makes the matter +worse. You <i>must</i> have meant some mischief, or else you’d have +signed your name like an honest man.’</p> + +<p>“There was a general clapping of hands at this: it was the first +really clever thing the King had said that day.</p> + +<p>“‘That <i>proves</i> his guilt, of course,’ said the Queen, ‘so, off +with——’</p> + +<p>“‘It doesn’t prove anything of the sort!’ said Alice. ‘Why, you +don’t even know what they’re about!’</p> + +<p>“‘Read them,’ said the King.</p> + +<p>“The White Rabbit put on his spectacles. ‘Where shall I begin, +please your Majesty?’ he asked.</p> + +<p>“‘Begin at the beginning,’ the King said very gravely, ‘and go +on till you come to the end: then stop.’</p> + +<p>“There was dead silence in the court, whilst the White Rabbit +read out these verses:</p> + +<p class="poem"> +“‘<i>They told me you had been to her,<br /> +And mentioned me to him;<br /> +She gave me a good character,<br /> +But said I could not swim.</i><br /> +<br /> +<i>He sent them word I had not gone<br /> +(We know it to be true):<br /> +If she should push the matter on,<br /> +What would become of you?</i><br /> +<br /> +<i>I gave her one, they gave him two,<br /> +You gave us three or more;<br /> +They all returned from him to you,<br /> +Though they were mine before.</i><br /> +<br /> +<i>If I or she should chance to be<br /> +Involved in this affair,<br /> +He trusts to you to set them free<br /> +Exactly as they were.</i><br /> +<br /> +<i>My notion was that you had been<br /> +(Before she had this fit)<br /> +An obstacle that came between<br /> +Him, and ourselves, and it.</i><br /> +<br /> +<i>Don’t let him know she liked them best,<br /> +For this must ever be<br /> +A secret kept from all the rest,<br /> +Between yourself and me.</i>’<br /> +</p> + +<p>“‘That’s the most important piece of evidence we’ve heard yet,’ +said the King, rubbing his hands, ‘so now let the jury——’</p> + +<p>“‘If any one of them can explain it,’ said Alice (she had grown +so large in the last few minutes that she wasn’t a bit afraid of interrupting +him), ‘I’ll give him sixpence. <i>I</i> don’t believe there’s an atom +of meaning in it.<span class='pagenum'><a name="Page_96" id="Page_96">[Pg 96]</a></span>’</p> + +<p>“The jury all wrote down on their slates, ‘She doesn’t believe +there’s an atom of meaning in it,’ but none of them attempted to +explain the paper.</p> + +<p>“‘If there’s no meaning in it,’ said the King, ‘that saves a world +of trouble, you know, as we needn’t try to find any. And yet I don’t +know,’ he went on, spreading out the verses on his knee and looking at +them with one eye; ‘I seem to see some meaning in them after all. +“<i>— said I could not swim</i>”; you can’t swim, can you?’ he added, +turning to the Knave.</p> + +<p>“The Knave shook his head sadly. ‘Do I look like it?’ he said. +(Which he certainly did <i>not</i>, being made entirely of cardboard.)</p> + +<p>“‘All right, so far,’ said the King; and he went on muttering +over the verses to himself: ‘‘<i>We know it to be true</i>’—that’s the jury, +of course—‘<i>If she should push the matter on</i>’—that must be the +Queen—‘<i>What would become of you?</i>’ What indeed!—‘<i>I gave her +one, they gave him two!</i>’ why, that must be what he did with the tarts, +you know——’</p> + +<p>“‘But it goes on, ‘<i>They all returned from him to you</i>,’’ said Alice.</p> + +<p>“‘Why, there they are!’ said the King, triumphantly pointing +to the tarts on the table. ‘Nothing can be clearer than that. Then +again—‘<i>Before she had this fit</i>’—you never had fits, my dear, I think?’ +he said to the Queen.</p> + +<p>“‘Never!’ said the Queen furiously, throwing an inkstand at the +Lizard as she spoke. (The unfortunate little Bill had left off writing +on his slate with one finger, as he found it made no mark; but he +now hastily began again, using the ink that was trickling down his +face, as long as it lasted.)</p> + +<p>“‘Then the words don’t <i>fit</i> you,’ said the King, looking round the +court with a smile. There was a dead silence.</p> + +<p>“‘It’s a pun!’ the King added in an angry tone, and everybody +laughed.</p> + +<p>“‘Let the jury consider their verdict,’ the King said, for about +the twentieth time that day.</p> + +<p>“‘No, no!’ said the Queen. ‘Sentence first—verdict afterwards.’</p> + +<p>“‘Stuff and nonsense!’ said Alice loudly. ‘The idea of having +the sentence first!’</p> + +<p>“‘Hold your tongue!’ said the Queen, turning purple....”</p> + + +<h4><a name="App_T" id="App_T"></a>T. “GEDANKENEXPERIMENTE.”</h4> + +<p><i>T. L. G.</i>, p. 61: “Alice laughed. ‘There’s no use trying,’ she +said: ‘one <i>can’t</i> believe impossible things.’</p> + +<p>“‘I daresay you haven’t had much practice,’ said the [White] +Queen. ‘When I was your age, I always did it for half-an-hour a +day. Why, sometimes I’ve believed as many as six impossible things +before breakfast.’”</p> + +<p> </p> + +<h5><i>Printed in Great Britain by</i><br /> +UNWIN BROTHERS, LIMITED, THE GRESHAM PRESS, WOKING AND LONDON</h5> + + + + + + + + + +<pre> + + + + + +End of Project Gutenberg's The philosophy of B*rtr*nd R*ss*ll, by Various + +*** END OF THIS PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF B*RTR*ND R*SS*LL *** + +***** This file should be named 38430-h.htm or 38430-h.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/8/4/3/38430/ + +Produced by Adrian Mastronardi and the Online Distributed +Proofreading Team at http://www.pgdp.net (This file was +produced from images generously made available by The +Internet Archive/Canadian Libraries) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: The philosophy of B*rtr*nd R*ss*ll + +Author: Various + +Editor: Philip E. B. Jordain + +Release Date: December 28, 2011 [EBook #38430] + +Language: English + +Character set encoding: ASCII + +*** START OF THIS PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF B*RTR*ND R*SS*LL *** + + + + +Produced by Adrian Mastronardi and the Online Distributed +Proofreading Team at http://www.pgdp.net (This file was +produced from images generously made available by The +Internet Archive/Canadian Libraries) + + + + + + + + + + THE PHILOSOPHY OF + MR. B*RTR*ND R*SS*LL + + WITH AN APPENDIX OF LEADING + PASSAGES FROM CERTAIN OTHER WORKS + + EDITED BY + + PHILIP E. B. JOURDAIN + + [Illustration] + + LONDON: GEORGE ALLEN & UNWIN LTD. + RUSKIN HOUSE 40 MUSEUM STREET, W.C. 1 + CHICAGO: THE OPEN COURT PUBLISHING CO. + + + + _First published in 1918_ + + (_All rights reserved_) + + + + +EDITOR'S NOTE + + +When Mr. B*rtr*nd R*ss*ll, following the advice of Mr. W*ll**m J*m*s, +again "got into touch with reality" and in July 1911 was torn to pieces +by Anti-Suffragists, many of whom were political opponents of Mr. +R*ss*ll and held strong views on the Necessity of Protection of Trade +and person, a manuscript which was almost ready for the press was +fortunately saved from the flames on the occasion when a body of eager +champions of the Sacredness of Personal Property burnt the late Mr. +R*ss*ll's house. This manuscript, together with some further fragments +found in the late Mr. R*ss*ll's own interleaved copy of his _Prayer-Book +of Free Man's Worship_, which was fortunately rescued with a few of the +great author's other belongings, was first given to the world in the +_Monist_ for October 1911 and January 1916, and has here been arranged +and completed by some other hitherto undecipherable manuscripts. The +title of the above-mentioned _Prayer-Book_, it may perhaps be mentioned, +was apparently suggested to Mr. R*ss*ll by that of the Essay on +"The Free Man's Worship" in the _Philosophical Essays_ (London, 1910, +pp. 59-70[1]) of Mr. R*ss*ll's distinguished contemporary, Mr. Bertrand +Russell, from whom much of Mr. R*ss*ll's philosophy was derived. And, +indeed, the influence of Mr. Russell extended even beyond philosophical +views to arrangement and literary style. The method of arrangement of +the present work seems to have been borrowed from Mr. Russell's +_Philosophy of Leibniz_ of 1900; in the selection of subjects dealt +with, Mr. R*ss*ll seems to have been guided by Mr. Russell's _Principles +of Mathematics_ of 1903; while Mr. R*ss*ll's literary style fortunately +reminds us more of Mr. Russell's later clear and charming subtleties +than his earlier brilliant and no less subtle obscurities. But, on the +other hand, some important points of Mr. Russell's doctrine, which first +appeared in books published after Mr. R*ss*ll's death, were anticipated +in Mr. R*ss*ll's notes, and these anticipations, so interesting for +future historians of philosophy, have been provided by the editor with +references to the later works of Mr. Russell. All editorial notes are +enclosed in square brackets, to indicate that they were not written by +the late Mr. R*ss*ll. + +At the present time we have come to take a calm view of the question so +much debated seven years ago as to the legitimacy of logical arguments +in political discussions. No longer, fortunately, can that intense +feeling be roused which then found expression in the famous cry, +"Justice--right or wrong," and which played such a large part in the +politics of that time. Thus it will not be out of place in this +unimpassioned record of some of the truths and errors in the world to +refer briefly to Mr. R*ss*ll's short and stormy career. Before he was +torn to pieces, he had been forbidden to lecture on philosophy or +mathematics by some well-intentioned advocates of freedom in speech who +thought that the cause of freedom might be endangered by allowing Mr. +R*ss*ll to speak freely on points of logic, on the grounds, apparently, +that logic is both harmful and unnecessary and might be applied to +politics unless strong measures were taken for its suppression. On much +the same grounds, his liberty was taken from him by those who remarked +that, if necessary, they would die in defence of the sacred principle of +liberty; and it was in prison that the greater part of the present work +was written. Shortly after his liberation, which, like all actions of +public bodies, was brought about by the combined honour and interests of +those in authority, occurred his lamentable death to which we have +referred above. + +Mr. R*ss*ll maintained that the chief use of "implication" in politics +is to draw conclusions, which are thought to be true, and which are +consequently false, from identical propositions, and we can see these +views expressed in Chapters III and XIX of the present work. These +chapters were apparently written before the Government, in the spring +of 1910, arrived at the famous secret decision that only "certain +implications" are permitted in discussion. Naturally the secret decision +gave rise to much speculation among logicians as to which kinds of +implication were barred, and Mr. R*ss*ll and Mr. Bertrand Russell had +many arguments on the subject, which naturally could not be published at +the time. However, after Mr. R*ss*ll's death, successive prosecutions +which were made by the Government at last made it quite clear that the +opinion held by Mr. R*ss*ll was the correct one. There had been numerous +prosecutions of people who, from true but not identical premisses, had +deduced true conclusions, so that the possible legitimate forms of +"implication" were reduced. Further, the other doubtful cases were +cleared up in course of time by the prosecution of (1) members of the +Aristotelian Society for deducing true conclusions from false premisses; +(2) members of the _Mind_ Association for deducing false conclusions +from false premisses; and also by the attempted prosecution of an +eminent lady for deducing true conclusions from identities. Fortunately +this lady was able to defend herself successfully by pleading that one +eminent philosopher believed them to be true--which, of course, means +that the conclusions are false. Thus appeared the true nature of +legitimate political arguments. + +FOOTNOTES: + +[1] [This Essay is also reprinted in Mr. Russell's _Mysticism and +Logic_, London and New York, 1918, pp. 46-57.--ED.] + + + + + "Even a joke should have some meaning...." + + (The Red Queen, _T. L. G._, p. 105). + + + + +CONTENTS + + + PAGE + EDITOR'S NOTE 3 + ABBREVIATIONS 9 + CHAPTER + I. THE INDEFINABLES OF LOGIC 11 + II. OBJECTIVE VALIDITY OF THE "LAWS OF THOUGHT" 15 + III. IDENTITY 16 + IV. IDENTITY OF CLASSES 18 + V. ETHICAL APPLICATIONS OF THE LAW OF IDENTITY 19 + VI. THE LAW OF CONTRADICTION IN MODERN LOGIC 21 + VII. SYMBOLISM AND MEANING 22 + VIII. NOMINALISM 24 + IX. AMBIGUITY AND SYMBOLIC LOGIC 26 + X. LOGICAL ADDITION AND THE UTILITY OF SYMBOLISM 27 + XI. CRITICISM 29 + XII. HISTORICAL CRITICISM 30 + XIII. IS THE MIND IN THE HEAD? 31 + XIV. THE PRAGMATIST THEORY OF TRUTH 32 + XV. ASSERTION 34 + XVI. THE COMMUTATIVE LAW 35 + XVII. UNIVERSAL AND PARTICULAR PROPOSITIONS 36 + XVIII. DENIAL OF GENERALITY AND GENERALITY OF DENIAL 37 + XIX. IMPLICATION 39 + XX. DIGNITY 43 + XXI. THE SYNTHETIC NATURE OF DEDUCTION 45 + XXII. THE MORTALITY OF SOCRATES 48 + XXIII. DENOTING 53 + XXIV. THE 54 + XXV. NON-ENTITY 56 + XXVI. IS 58 + XXVII. AND AND OR 59 + XXVIII. THE CONVERSION OF RELATIONS 60 + XXIX. PREVIOUS PHILOSOPHICAL THEORIES OF MATHEMATICS 61 + XXX. FINITE AND INFINITE 63 + XXXI. THE MATHEMATICAL ATTAINMENTS OF TRISTRAM SHANDY 64 + XXXII. THE HARDSHIPS OF A MAN WITH AN UNLIMITED INCOME 66 + XXXIII. THE RELATIONS OF MAGNITUDE OF CARDINAL NUMBERS 69 + XXXIV. THE UNKNOWABLE 70 + XXXV. MR. SPENCER, THE ATHANASIAN CREED, AND THE ARTICLES 73 + XXXVI. THE HUMOUR OF MATHEMATICIANS 74 + XXXVII. THE PARADOXES OF LOGIC 75 + XXXVIII. MODERN LOGIC AND SOME PHILOSOPHICAL ARGUMENTS 79 + XXXIX. THE HIERARCHY OF JOKES 81 + XL. THE EVIDENCE OF GEOMETRICAL PROPOSITIONS 83 + XLI. ABSOLUTE AND RELATIVE POSITION 84 + XLII. LAUGHTER 86 + XLIII. "GEDANKENEXPERIMENTE" AND EVOLUTIONARY ETHICS 88 + APPENDIXES 89 + + + + +ABBREVIATIONS + + + _A. A. W._ Lewis Carroll: _Alice's Adventures in Wonderland_, London, + 1908. [This book was first published much earlier, but + this was the edition used by Mr. R*ss*ll. The same applies + to _H. S._ and _T. L. G._] + + _A. C. P._ John Henry Blunt (ed. by): _The Annotated Book of Common + Prayer_, London, new edition, 1888. + + _A. d. L._ Ernst Schroeder: _Vorlesungen ueber die Algebra der Logik, + Leipzig_, vol. i., 1890; vol. ii. (two parts), 1891 and + 1905; vol. iii.: _Algebra und Logik der Relative_, 1895. + + _E. N._ Richard Dedekind: _Essays on the Theory of Numbers_, + Chicago and London, 1901. + + _E. L. L._ William Stanley Jevons: _Elementary Lessons in Logic, + Deductive and Inductive. With copious Questions and + Examples, and a Vocabulary of Logical Terms_, London, + 24th ed., 1907 [first published in 1870]. + + _E. u. I._ Ernst Mach: _Erkenntnis und Irrtum: Skizzen zur + Psychologie der Forschung_, Leipzig, 1906. + + _F. L._ Augustus De Morgan: _Formal Logic: or The Calculus of + Inference, Necessary and Probable_, London, 1847. + + _Fm. L._ John Neville Keynes: _Studies and Exercises in Formal + Logic_, 4th ed., London, 1906. + + _Gg._ Gottlob Frege: _Grundgesetze der Arithmetik + begriffschriftlich abgeleitet_, Jena, vol. i., 1893; + vol. ii., 1903. + + _Gl._ Gottlob Frege: _Die Grundlagen der Arithmetik: eine + logisch-mathematische Untersuchung ueber den Begriff der + Zahl_, Breslau, 1884. + + _G. u. E._ G. Heymans: _Die Gesetze und Elemente des + wisenschaftlichen Denkens_, Leiden, vol. i., 1890; + vol. ii., 1894. + + _H. J._ _The Hibbert Journal: a Quarterly Review of Religion, + Theology and Philosophy_, London and New York. + + _H. S._ Lewis Carroll: _The Hunting of the Snark: an Agony in + Eight Fits_, London, 1911. + + _M._ _The Monist: a Quarterly Magazine Devoted to Science and + Philosophy_, Chicago and London. + + _Md._ _Mind: a Quarterly Review of Psychology and Philosophy_, + London and New York. + + _Pa. Ma._ Alfred North Whitehead and Bertrand Russell: _Principia + Mathematica_, vol. i., Cambridge, 1910. [Other volumes + were published in 1912 and 1913.] + + _P. E._ Bertrand Russell: _Philosophical Essays_, London and New + York, 1910. + + _Ph. L._ Bertrand Russell: _A Critical Exposition of the Philosophy + of Leibniz, with an Appendix of Leading Passages_, + Cambridge, 1900. + + _P. M._ Bertrand Russell: _The Principles of Mathematics_, + vol. i., Cambridge, 1903. + + _R. M. M._ _Revue de Metaphysique et de Morale_, Paris. + + _S. B._ Lewis Carroll: _Sylvie and Bruno_, London, 1889. + + _S. L._ John Venn: _Symbolic Logic_, London, 1881; 2nd ed., 1894. + + _S. o. S._ William Stanley Jevons: _The Substitution of Similars, the + True Principle of Reasoning derived from a Modification of + Aristotle's Dictum_, London, 1869. + + _T. L. G._ Lewis Carroll: _Through the Looking-Glass, and what Alice + found there_, London, 1911. + + _Z. S._ Gottlob Frege: _Ueber die Zahlen des Herrn H. Schubert_, + Jena, 1899. + + + + +CHAPTER I + +THE INDEFINABLES OF LOGIC + + +The view that the fundamental principles of logic consist solely of the +law of identity was held by Leibniz,[2] Drobisch, Uberweg,[3] and +Tweedledee. Tweedledee, it may be remembered,[4] remarked that certain +identities "are" logic. Now, there is some doubt as to whether he, like +Jevons,[5] understood "are" to mean what mathematicians mean by "=," or, +like Schroeder[6] and most logicians, to have the same meaning as the +relation of subsumption. The first alternative alone would justify our +contention; and we may, I think, conclude from an opposition to +authority that may have been indicated by Tweedledee's frequent use of +the word "contrariwise" that he did not follow the majority of +logicians, but held, like Jevons,[7] the mistaken[8] view that the +quantification of the predicate is relevant to symbolic logic. + +It may be mentioned, by the way, that it is probable that +Humpty-Dumpty's "is" is the "is" of identity. In fact, it is not +unlikely that Humpty-Dumpty was a Hegelian; for, although his ability +for clear explanation may seem to militate against this, yet his +inability to understand mathematics,[9] together with his synthesis of a +cravat and a belt, which usually serve different purposes,[10] and his +proclivity towards riddles seem to make out a good case for those who +hold that he was in fact a Hegelian. Indeed, riddles are very closely +allied to puns, and it was upon a pun, consisting of the confusion of +the "is" of predication with the "is" of identity--so that, for example, +"Socrates" was identified with "mortal" and more generally the +particular with the universal--that Hegel's system of philosophy was +founded.[11] But the question of Humpty-Dumpty's philosophical opinions +must be left for final verification to the historians of philosophy: +here I am only concerned with an _a priori_ logical construction of what +his views might have been if they formed a consistent whole.[12] + +If the principle of identity were indeed the sole principle of logic, +the principles of logic could hardly be said to be, as in fact they are, +a body of propositions whose consistency it is impossible to prove.[13] +This characteristic is important and one of the marks of the greatest +possible security. For example, while a great achievement of late years +has been to prove the consistency of the principles of arithmetic, a +science which is unreservedly accepted except by some empiricists,[14] +it can be proved formally that one foundation of arithmetic is +shattered.[15] It is true that, quite lately, it has been shown that +this conclusion may be avoided, and, by a re-moulding of logic, we can +draw instead the paradoxical conclusion that the opinions held by +common-sense for so many years are, in part, justified. But it is quite +certain that, with the principles of logic, no such proof of +consistency, and no such paradoxical result of further investigations is +to be feared. + +Still, this re-moulding has had the result of bringing logic into a +fuller agreement with common-sense than might be expected. There were +only two alternatives: if we chose principles in accordance with +common-sense, we arrived at conclusions which shocked common-sense; by +starting with paradoxical principles, we arrived at ordinary +conclusions. Like the White Knight, we have dyed our whiskers an unusual +colour and then hidden them.[16] + +The quaint name of "Laws of Thought," which is often applied to the +principles of Logic, has given rise to confusion in two ways: in the +first place, the "Laws," unlike other laws, cannot be broken, even in +thought; and, in the second place, people think that the "Laws" have +something to do with holding for the operations of their minds, just as +laws of nature hold for events in the world around us.[17] But that the +laws are not psychological laws follows from the facts that a thing may +be true even if nobody believes it, and something else may be false if +everybody believes it. Such, it may be remarked, is usually the case. + +Perhaps the most frequent instance of the assumption that the laws of +logic are mental is the treatment of an identity as if its validity were +an affair of our permission. Some people suggest to others that they +should "let bygones be bygones." Another important piece of evidence +that the truth of propositions has nothing to do with mind is given by +the phrase "it is morally certain that such-and-such a proposition is +true." Now, in the first place, morality, curiously enough, seems to be +closely associated with mental acts: we have professorships and +lectureships of, and examinations in, "mental and moral philosophy." In +the second place, it is plain that a "morally certain" proposition is a +highly doubtful one. Thus it is as vain to expect any information about +our minds from a study of the "Laws of Thought" as it would be to expect +a description of a certain social event from Miss E. E. C. Jones's book +_An Introduction to General Logic_. + +Fortunately, the principles or laws of Logic are not a matter of +philosophical discussion. Idealists like Tweedledum and Tweedledee, and +even practical idealists like the White Knight, explicitly accept laws +like the law of identity and the excluded middle.[18] In fact, +throughout all logic and mathematics, the existence of the human or any +other mind is totally irrelevant; mental processes are studied by means +of logic, but the subject-matter of logic does not presuppose mental +processes, and would be equally true if there were no mental processes. +It is true that, in that case, we should not know logic; but our +knowledge must not be confounded with the truths which we know.[19] An +apple is not confused with the eating of it except by savages, +idealists, and people who are too hungry to think. + +FOOTNOTES: + +[2] Russell, _Ph. L._, pp. 17, 19, 207-8. + +[3] Schroeder, _A. d. L._, i. p. 4. + +[4] See Appendix A. This Appendix also illustrates the importance +attached to the Principle of Identity by the Professor and Bruno. + +[5] _S. o. S._, pp. 9-15. + +[6] _A. d. L._, i. p. 132. + +[7] Cf., besides the reference in the last note but one, _E. L. L._, pp. +183, 191. "Contrariwise," it may be remarked, is not a term used in +traditional logic. + +[8] _S. L._, 1881, pp. 173-5, 324-5; 1894, pp. 194-6. + +[9] Cf. Appendix C, and William Robertson Smith, "Hegel and the +Metaphysics of the Fluxional Calculus," _Trans. Roy. Soc., Edinb._, vol. +xxv., 1869, pp. 491-511. + +[10] See Appendix B. + +[11] [This is a remarkable anticipation of the note on pp. 39-40 of Mr. +Russell's book, published about three years after the death of Mr. +R*ss*ll, and entitled _Our Knowledge of the External World as a Field +for Scientific Method in Philosophy_, Chicago and London, 1914.--ED.] + +[12] Cf. _Ph. L._, pp. v.-vi. 3. + +[13] Cf. Pieri, _R. M. M._, March 1906, p. 199. + +[14] As a type of these, Humpty-Dumpty, with his inability to admit +anything not empirically given and his lack of comprehension of pure +mathematics, may be taken (see Appendix C). In his (correct) thesis that +definitions are nominal, too, Humpty-Dumpty reminds one of J. S. Mill +(see Appendix D). + +[15] See Frege, _Gg._, ii. p. 253. + +[16] See Appendix E. + +[17] See Frege, _Gg._, i. p. 15. + +[18] See the above references and also Appendix F. + +[19] Cf. B. Russell, _H. J._, July 1904, p. 812. + + + + +CHAPTER II + +OBJECTIVE VALIDITY OF THE "LAWS OF THOUGHT" + + +I once inquired of a maid-servant whether her mistress was at home. She +replied, in a doubtful fashion, that she _thought_ that her mistress was +in unless she was out. I concluded that the maid was uncertain as to the +objective validity of the law of excluded middle, and remarked that to +her mistress. But since I used the phrase "laws of thought," the +mistress perhaps supposed that a "law of thought" has something to do +with thinking and seemed to imagine that I wished to impute to the maid +some moral defect of an unimportant nature. Thus she remonstrated with +me in an amused way, since she probably imagined that I meant to find +fault with the maid's capacity for thinking. + + + + +CHAPTER III + +IDENTITY + + +In the first chapter we have noticed the opinion that identities are +fundamental to all logic. We will now consider some other views of the +value of identities. + +Identities are frequently used in common life by people who seem to +imagine that they can draw important conclusions respecting conduct or +matters of fact from them. I have heard of a man who gained the double +reputation of being a philosopher and a fatalist by the repeated +enunciation of the identity "Whatever will be, will be"; and the Italian +equivalent of this makes up an appreciable part of one of Mr. Robert +Hichens' novels. Further, the identity "Life is Life" has not only been +often accepted as an explanation for a particular way of living but has +even been considered by an authoress who calls herself "Zack" to be an +appropriate title for a novel; while "Business is Business" is +frequently thought to provide an excuse for dishonesty in trading, for +which purpose it is plainly inadequate. + +Another example is given by a poem of Mr. Kipling, where he seems to +assert that "East is East" and "West is West" imply that "never the +twain shall meet." The conclusion, now, is false; for, since the world +is round--as geography books still maintain by arguments which strike +every intelligent child as invalid[20]--what is called the "West" does, +in fact, merge into the "East." Even if we are to take the statement +metaphorically, it is still untrue, as the Japanese nation has shown. + +The law-courts are often rightly blamed for being strenuous opponents of +the spread of modern logic: the frequent misuse of _and_, _or_, _the_, +and _provided that_ in them is notorious. But the fault seems partly to +lie in the uncomplicated nature of the logical problems which are dealt +with in them. Thus it is no uncommon thing for somebody to appear there +who is unable to establish his own identity, or for A to assert that B +was "not himself" when he made a will leaving his money to C. + +The chief use of identities is in implication. Since, in logic, we so +understand _implication_[21] that any true proposition implies and is +implied by any other true proposition; if one is convinced of the truth +of the proposition Q, it is advisable to choose one or more identities +P, whose truth is undoubted, and say that P implies Q. Thus, Mr. Austen +Chamberlain, according to _The Times_ of March 27, 1909, professed to +deduce the conclusion that it is not right that women should have votes +from the premisses that "man is man" and "woman is woman." This method +requires that one should have made up one's mind about the conclusion +before discovering the premisses--by what, no doubt, Jevons would call +an "inverse or inductive method." Thus the method is of use only in +speeches and in giving good advice. + +Mr. Austen Chamberlain afterwards rather destroyed one's belief in the +truth of his premisses by putting limits to the validity of the +principle of identity. In the course of the Debate on the Budget of +1909, he maintained, against Mr. Lloyd George, that a joke was a joke +except when it was an untruth: Mr. Lloyd George, apparently, being of +the plausible opinion that a joke is a joke under all circumstances. + +FOOTNOTES: + +[20] The argument about the hull of a ship disappearing first is not +convincing, since it would equally well prove that the surface of the +earth was, for example, corrugated on a large scale. If the common-sense +of the reader were supposed to dismiss the possibility of water clinging +to such corrugations, it might equally be supposed to dismiss the +possibility of water clinging to a spherical earth. Traditional +geography books, no doubt, gave rise to the opinions held by Lady Blount +and the Zetetic Society. + +[21] The subject of Implication will be further considered in Chapter +XIX. + + + + +CHAPTER IV + +IDENTITY OF CLASSES + + +I once heard of a meritorious lady who was extremely conventional; on +the slender grounds of carefully acquired habits of preferring the word +"woman" to the word "lady" and of going to the post-office without a +hat, imagined that she was unconventional and altogether a remarkable +person; and who once remarked with great satisfaction that she was a +"very queer person," and that nothing shocked her "except, of course, +bad form." + +Thus, she asserted that all the things which shocked her were actions in +bad form; and she would undoubtedly agree, though she did not actually +state it, that all the things which were done in bad form would shock +her. Consequently she asserted that the class of things which shocked +her was the class of actions in bad form. Consequently the statement of +this lady that some or all of the actions done in bad form shocked her +is an identical proposition of the form "nothing shocks me, except, of +course, the things which do, in fact, shock me"; and this statement the +lady certainly did not intend to make. + +This excellent lady, had she but known it, was logically justified in +making any statement whatever about her unconventionality. For the class +of her unconventional actions was the null class. Thus she might +logically have made inconsistent statements about this class of actions. +As a matter of fact she did make inconsistent statements, but +unfortunately she justified them by stating that, "It is the privilege +of woman to be inconsistent." She was one of those persons who say +things like that. + + + + +CHAPTER V + +ETHICAL APPLICATIONS OF THE LAW OF IDENTITY + + +It may be remembered that Mr. Podsnap remarked, with sadness tempered by +satisfaction, that he regretted to say that "Foreign nations do as they +do do." Besides aiding the comforting expression of moral disapproval, +the law of identity has yet another useful purpose in practical ethics: +It serves the welcome purpose of providing an excuse for infractions of +the moral law. There was once a man who treated his wife badly, was +unfaithful to her, was dishonest in business, and was not particular in +his use of language; and yet his life on earth was described in the +lines: + + This man maintained a wife's a wife, + Men are as they are made, + Business is business, life is life; + And called a spade a spade. + +One of the objects of Dr. G. E. Moore's _Principia Ethica_[22] was to +argue that the word "good" means simply good, and not pleasant or +anything else. Appropriately enough, this book bore on its title-page +the quotation from the preface to the _Sermons_, published in 1726, of +Bishop Joseph Butler, the author of the _Analogy_: "Everything is what +it is and not another thing." + +But another famous Butler--Samuel Butler, the author of _Hudibras_--went +farther than this, and maintained that identities were the highest +attainment of metaphysics itself. At the beginning of the first Canto of +_Hudibras_, in the description of Hudibras himself, Butler wrote: + + He knew what's what, and that's as high + As metaphysic wit can fly. + +I once conducted what I imagined to be an aesthetic investigation for the +purpose of discovery, by the continual use of the word "Why?"[23] the +grounds upon which certain people choose to put milk into a tea-cup +before the tea. I was surprised to discover that it was an ethical, and +not an aesthetic problem; for I soon elicited the fact that it was done +because it was "right." A continuance of my patient questioning elicited +further evidence of the fundamental character of the principle of +identity in ethics; for it was right, I learned, because "right is +right." + +It appears that some people unconsciously think that the principle of +identity is the foundation, in certain religions, of the reasons which +can be alleged for moral conduct, and are surprised when this fact is +pointed out to them. The late Sir Leslie Stephen, when travelling by +railway, fell into conversation with an officer of the Salvation Army, +who tried hard to convert him. Failing in this laudable endeavour, the +Salvationist at last remarked: "But if you aren't saved, you can't go to +heaven!" "That, my friend," replied Stephen, "is an identical +proposition." + +FOOTNOTES: + +[22] Cambridge, 1903. + +[23] Cf. _P. E._, p. 2. + + + + +CHAPTER VI + +THE LAW OF CONTRADICTION IN MODERN LOGIC + + +Considering the important place assigned by philosophers and logicians +to the law of contradiction, the remark will naturally be resented by +many of the older schools of philosophy, and especially by Kantians, +that "in spite of its fame we have found few occasions for its use."[24] +Also in modern times, Benedetto Croce, an opponent of both traditional +logic and mathematical logic, began the preface of the book of 1908 on +Logic[25] by saying that that volume "is and is not" a certain memoir of +his which had been published in 1905. + +FOOTNOTES: + +[24] _Pa. Ma._, p. 116. + +[25] [English translation of the third Italian edition by Douglas +Ainslie, under the title: _Logic as the Science of the Pure Concept_, +London 1917.--ED.] + + + + +CHAPTER VII + +SYMBOLISM AND MEANING + + +When people write down any statement such as "The curfew tolls the knell +of parting day,"[26] which we will call "C" for shortness, what they +mean is not "C" but the _meaning_ of "C"; and not "the meaning of 'C'" +but the _meaning_ of "the meaning of 'C'." And so on, _ad infinitum_. +Thus, in writing or in speech, we always fail to state the meaning of +any proposition whatever. Sometimes, indeed, we succeed in _conveying_ +it; but there is danger in too great a disregard of statement and +preoccupation with conveyance of meaning. Thus many mathematicians have +been so anxious to convey to us a perfectly distinct and unmetaphysical +concept of number that they have stripped away from it everything that +they considered unessential (like its logical nature) and have finally +delivered it to us as a mere _sign_. By the labours of Helmholtz, +Kronecker, Heine, Stolz, Thomae, Pringsheim, and Schubert, many people +were persuaded that, when they said "'2' is a number" they were speaking +the truth, and hold that "Paris" is a town containing the letter "P." +When Frege pointed out[27] this difficulty he was almost universally +denounced in Germany as "_spitzfindig_." In fact, Germans seem to have +been influenced perhaps by that great contemner of "_Spitzfindigkeit_," +Kant, to reject the White Knight's[28] distinctions between words and +their denotations and to regard subtlety with disfavour to such a degree +that their only mathematical logician except Frege, namely Schroeder--the +least subtle of mortals, by the way--seems to have been filled with such +fear of being thought subtle, that he made his books so prolix that +nobody has read them. + +Another term which, as we shall see when discussing the paradoxes of +logic, mathematicians are accustomed to apply to thought which is more +exact than any to which they are accustomed is "scholastic."[29] By +this, I suppose, they mean that the pursuits of certain acute people of +the Middle Ages are unimportant in contrast with the great achievements +of modern thought, as exemplified by a method of making plausible +guesses known as induction,[30] the bicycle, and the gramophone--all of +them instruments of doubtful merit. + +FOOTNOTES: + +[26] Cf. _Md_, N. S., vol. xiv., October 1905, p. 486. + +[27] In _Z. S._, for example. + +[28] See Appendix G. + +[29] Cf. Chapter XXXVII below. + +[30] Cf. _P. M._, p. 11, note. + + + + +CHAPTER VIII + +NOMINALISM + + +De Morgan[31] said that, "if all mankind had spoken one language, we +cannot doubt that there would have been a powerful, perhaps universal, +school of philosophers who would have believed in the inherent connexion +between names and things; who would have taken the sound _man_ to be the +mode of agitating the air which is essentially communicative of the +ideas of reason, cookery, bipedality, etc.... 'The French,' said the +sailor, 'call a cabbage a _shoe_; the fools! Why can't they call it a +cabbage, when they must know it is one?'" + +One of the chief differences between logicians and men of letters is +that the latter mean many different things by one word, whereas the +former do not--at least nowadays. Most mathematicians belong to the +class of men of letters. + +I once had a manservant who told me on a certain occasion that he "never +thought a word about it." I was doubtful whether to class him with such +eminent mathematicians as are mentioned in the last chapter, or as a +supporter of Max Mueller's theory of the identity of thought and +language. However, since the man was very untruthful, and he told me +that he meant what he said and said what he meant,[32] the conclusion is +probably correct that he really believed that the meanings of his words +were not the words themselves. Thus I think it most probable that my +manservant had been a mathematician but had escaped by the aid of logic. + +As regards his remark that he meant what he said and said what he +meant, he plainly wished to pride himself on certain virtues which he +did not possess, and was not indifferent to applause, which, however, +was never evoked. The virtues, if so they be, and the applause were +withheld for other reasons than that the above statements are either +nonsensical or false. Suppose that "I say what I mean" expresses a +truth. What I say (or write) is always a symbol--words (or marks); and +what I mean by the symbol is the meaning of the symbol and not the +symbol itself. So the remark cannot express a truth, any more than the +name "Wellington" won the battle of Waterloo. + +FOOTNOTES: + +[31] _F. L._, pp. 246-7. + +[32] The Hatter (see Appendix H) pointed out that there is a difference +between these two assertions. Thus, he clearly showed that he was a +nominalist, and philosophically opposed to the March Hare who had +recommended Alice to say what she meant. + + + + +CHAPTER IX + +AMBIGUITY AND SYMBOLIC LOGIC + + +The universal use of some system of Symbolic Logic would not only enable +everybody easily to deal with exceedingly complicated arguments, but +would prevent ambiguous arguments. In denying the indispensability of +Symbolic Logic in the former state of things, Keynes[33] is probably +alone, against the need strongly felt by Alice when speaking to the +Duchess,[34] and most modern logicians. It may be noticed that the +Duchess is more consistent than Keynes, for Keynes really uses the signs +for logical multiplication and addition of Boole and Venn under the +different shapes of the words "and" and "or." + +As regards ambiguity, a translation of _Hymns Ancient and Modern_ into, +say, Peanesque, would prevent the puzzle of childhood as to whether the +"his" in + + And Satan trembles when he sees + The weakest saint upon his knees + +refers to the saint's knees or Satan's. + +FOOTNOTES: + +[33] In his _Fm. L._ + +[34] See Appendix I. + + + + +CHAPTER X + +LOGICAL ADDITION AND THE UTILITY OF SYMBOLISM + + +Frequently ordinary language contains subtle psychological implications +which cannot be translated into symbolic logic except at great length. +Thus if a man (say Mr. Jones) wishes to speak collectively of himself +and his wife, the order of mentioning the terms in the class considered +and the names applied to these terms are, logically speaking, +irrelevant. And yet more or less definite information is given about Mr. +Jones according as he talks to his friends of: + + (1) Mrs. Jones and I, + (2) I (or me) and my wife (or missus), + (3) My wife and I, + or (4) I (or me) and Mrs. Jones. + +In case (1) one is probably correct in placing Mr. Jones among the +clergy or the small professional men who make up the bulk of the +middle-class; in case (2) one would conclude that Mr. Jones belonged to +the lower middle-class; the form (3) would be used by Mr. Jones if he +were a member of the upper, upper middle, or lower class; while form (4) +is only used by retired shopkeepers of the lower middle-class, of which +a male member usually combines belief in the supremacy of man with +belief in the dignity of his wife as well as himself. A further +complication is introduced if a wife is referred to as "the wife."[35] +Cases (2) and (3) then each give rise to one more case. Cases (1) and +(4) do not, since nobody has hitherto referred to his wife as "the Mrs. +Jones"--at least without a qualifying adjective before the "Mrs." + +On the other hand, certain descriptive phrases and certain propositions +can be expressed more shortly and more accurately by means of symbolic +logic. Let us consider the proposition "No man marries his deceased +wife's sister." If we assume, as a first approximation, that all +marriages are fertile and that all children are legitimate, then, with +only four primitive ideas: the relation of parent to child (P) and the +three classes of males, females, and dead people, we can define "wife" +(a female who has the relation formed by taking the relative product of +P and [vP][36] to a male), "sister," "deceased wife," and "deceased +wife's sister" in terms of these ideas and of the fundamental notions of +logic. Then the proposition "No man marries his deceased wife's sister" +can be expressed unambiguously by about twenty-nine simple signs on +paper, whereas, in words, the unasserted statement consists of no less +than thirty-four letters. Although, legally speaking, we should have to +adopt somewhat different definitions and possibly increase the +complications of our proposition, it must be remembered that, on the +other hand, we always reduce the number of symbols in any proposition by +increasing the number of definitions in the preliminaries to it. + +But the utility of symbolic logic should not be estimated by the brevity +with which propositions may sometimes be expressed by its means. Logical +simplicity, in fact, can very often only be obtained by apparently +complicated statements. For example, the logical interpretation of "The +father of Charles II was executed" is, "It is not always false of _x_ +that _x_ begat Charles II, and that _x_ was executed and that 'if _y_ +begat Charles II, _y_ is identical with _x_' is always true of _y_."[37] +From the point of view of logic, we may say that the apparently simple +is most often very complicated, and, even if it is not so, symbolism +will make it seem so,[38] and thus draw attention to what might +otherwise easily be overlooked. + +FOOTNOTES: + +[35] Cf. Chapter XXIV below. + +[36] C. S. Peirce's notation for the relation "converse of P." + +[37] Russell, _Md._, N. S., vol. xiv., October 1905, p. 482. + +[38] Russell, _International Monthly_, vol. iv., 1901, pp. 85-6; cf. +_M._, vol. xxii., 1912, p. 153. [This essay is reprinted in _Mysticism +and Logic_, London and New York, 1918, pp. 74-96.--ED.] + + + + +CHAPTER XI + +CRITICISM + + +Those people who think that it is more godlike to seem to turn water +into wine than to seem to turn wine into water surprise me. I cannot +imagine an intolerable critic. It seems to me that, if A resents B's +criticism in trying to put his (A's) discovery in the right or wrong +place, A acts as if he thought he had some private property in truth. +The White Queen seems to have shared the popular misconception as to the +nature of criticism.[39] + +FOOTNOTES: + +[39] See Appendix J. + + + + +CHAPTER XII + +HISTORICAL CRITICISM + + +From a problem in Diophantus's _Arithmetic_ about the price of some wine +it would seem that the wine was of poor quality, and Paul Tannery has +suggested that the prices mentioned for such a wine are higher than were +usual until after the end of the second century. He therefore rejected +the view which was formerly held that Diophantus lived in that +century.[40] + +The same method applied to a problem given by the ancient Hindu +algebraist Brahmagupta, who lived in the seventh century after Christ, +might result in placing Brahmagupta in prehistoric times. This is the +problem:[41] "Two apes lived at the top of a cliff of height _h_, whose +base was distant _mh_ from a neighbouring village. One descended the +cliff and walked to the village, the other flew up a height _x_ and then +flew in a straight line to the village. The distance traversed by each +was the same. Find _x_." + +FOOTNOTES: + +[40] W. W. Rouse Ball, _A Short Account of the History of Mathematics_, +4th edition, London, 1908, p. 109. + +[41] _Ibid._, pp. 148-9. + + + + +CHAPTER XIII + +IS THE MIND IN THE HEAD? + + +The contrary opinion has been maintained by idealists and a certain +election agent with whom I once had to deal and who remarked that +something slipped his mind and then went out of his head altogether. At +some period, then, a remembrance was in his head and out of his mind; +his mind was not, then, wholly within his head. Also, one is sometimes +assured that with certain people "out of sight is out of mind." What is +in their minds is therefore in sight, and cannot therefore be inside +their heads. + + + + +CHAPTER XIV + +THE PRAGMATIST THEORY OF TRUTH + + +The pragmatist theory that "truth" is a belief which works well +sometimes conflicts with common-sense and not with logic. It is commonly +supposed that it is always better to be sometimes right than to be never +right. But this is by no means true. For example, consider the case of a +watch which has stopped; it is exactly right twice every day. A watch, +on the other hand, which is always five minutes slow is never exactly +right. And yet there can be no question but that a belief in the +accuracy of the watch which was never right would, on the whole, produce +better results than such a belief in the one which had altogether +stopped. The pragmatist would, then, conclude that the watch which was +always inaccurate gave truer results than the one which was sometimes +accurate. In this conclusion the pragmatist would seem to be correct, +and this is an instance of how the false premisses of pragmatism may +give rise to true conclusions. + +From the text written above the church clock in a certain English +village, "Be ye ready, for ye know not the time," it would be concluded +that the clock never stopped for a period as long as twelve hours. For +the text is rather a vague symbolical expression of a propositional +function which is asserted to be true at all instants. The proposition +that a presumably not illiterate and credulous observer of the clock at +any definite instant does not know the time implies, then, that the +clock is always wrong. Now, if the clock stopped for twelve hours, it +would be absolutely right at least once. It must be right twice if it +were right at the first instant it stopped or the last instant at which +it went;[42] but the second possibility is excluded by hypothesis, and +the occurrence of the first possibility--or of the analogous possibility +of the stopped clock being right three times in twenty-four hours--does +not affect the present question. Hence the clock can never stop for +twelve hours. + +The pragmatist's criterion of truth appears to be far more difficult to +apply than the Bellman's,[43] that what he said three times is true, and +to give results just as insecure. + +FOOTNOTES: + +[42] Both cases cannot occur; the question is similar to that arising in +the discussion of the mortality of Socrates (see Chapter XXII). + +[43] See Appendix K. + + + + +CHAPTER XV + +ASSERTION + + +The subject of the present chapter must not be confused with the +assertion of ordinary life. Commonly, an unasserted proposition is +synonymous with a probably false statement, while an asserted +proposition is synonymous with one that is certainly false. But in logic +we apply assertion also to true propositions, and, as Lewis Carroll +showed in his version of "What the Tortoise said to Achilles,"[44] +usually pass over unconsciously an infinite series of implications in so +doing. If _p_ and _q_ are propositions, _p_ is true, and _p_ implies +_q_, then, at first sight, one would think that one might assert _q_. +But, from (A) _p_ is true, and (B) _p_ implies _q_, we must, in order to +deduce (Z) _q_ is true, accept the hypothetical: (C) If A and B are +true, Z must be true. And then, in order to deduce Z from A, B, and C, +we must accept another hypothetical: (D) If A, B, and C are true, Z must +be true; and so on _ad infinitum_. Thus, in deducing Z, we pass over an +infinite series of hypotheticals which increase in complexity. Thus we +need a new principle to be able to assert _q_. + +Frege was the first logician sharply to distinguish between an asserted +proposition, like "A is greater than B," and one which is merely +considered, like "A's being greater than B," although an analogous +distinction had been made in our common discourse on certain +psychological grounds, for long previously. In fact, soon after the +invention of speech, the necessity of distinguishing between a +considered proposition and an asserted one became evident, on account of +the state of things referred to at the beginning of this chapter. + +FOOTNOTES: + +[44] _Md._ N. S., vol. iv., 1895, pp. 278-80. Cf. Russell, _P. M._, p. +35. + + + + +CHAPTER XVI + +THE COMMUTATIVE LAW + + +Often the meaning of a sentence tacitly implies that the commutative law +does not hold. We are all familiar with the passage in which Macaulay +pointed out that, by using the commutative law because of exigencies of +metre, Robert Montgomery unintentionally made Creation tremble at the +Atheist's nod instead of the Almighty's. This use of the commutative law +by writers of verse renders it doubtful whether, in the hymn-line: + + The humble poor believe, + +we are to understand a statement about the humble poor, or a doubtful +maxim as to the attitude of our minds to statements made by the humble +poor. + +The non-commutativity of English titles offers difficulties to some +novelists and Americans who refer to Mary Lady So-and-So as Lady Mary +So-and-So, and _vice versa_. + + + + +CHAPTER XVII + +UNIVERSAL AND PARTICULAR PROPOSITIONS + + +People who are cynical as to the morality of the English are often +unpleasantly surprised to learn that "All trespassers will be +prosecuted" does not necessarily imply that "some trespassers will be +prosecuted." The view that universal propositions are non-existential is +now generally held: Bradley and Venn seem to have been the first to hold +this, while older logicians, such as De Morgan,[45] considered universal +propositions to be existential, like particular ones. + +If the Gnat[46] had been content to affirm his proposition about the +means of subsistence of Bread-and-Butter flies, in consequence of their +lack of which such flies always die, without pointing out such an insect +and thereby proving that the class of them is not null, Alice's doubt as +to the existence of the class in question, even if it were proved to be +well founded, would not have affected the validity of the proposition. + +This brings us to a great convenience in treating universal propositions +as non-existential: we can maintain that all _x_'s are _y_'s at the same +time as that no _x_'s are _y_'s, if only _x_ is the null-class. Thus, +when Mr. MacColl[47] objected to other symbolic logicians that their +premisses imply that all Centaurs are flower-pots, they could reply that +their premisses also imply the more usual view that Centaurs are not +flower-pots. + +FOOTNOTES: + +[45] Cf., e.g., _F. L._, p. 4. + +[46] See Appendix L. + +[47] Cf., e.g., _Md._, N. S., vol. xiv., July, 1905, pp. 399-400. + + + + +CHAPTER XVIII + +DENIAL OF GENERALITY AND GENERALITY OF DENIAL + + +The conclusion of a certain song[48] about a young man who poisoned his +sweetheart with sheep's-head broth, and was frightened to death by a +voice exclaiming: + + "Where's that young maid + What you did poison with my head?" + +at his bedside, gives rise to difficulties which are readily solved by a +symbolism that brings into relief the principle that the denial of a +universal and non-existential proposition is a particular and +existential one. The conclusion of the song is: + + Now all young men, both high and low, + Take warning by this dismal go! + For if he'd never done nobody no wrong, + He might have been here to have heard this song. + +It is an obvious error, say Whitehead and Russell,[49] though one easy +to commit, to assume that the cases: (1) all the propositions of a +certain class are true; and (2) no proposition of the class is true; are +each other's contradictories. However, in the modification[50] of +Frege's symbolism which was used by Russell + + (1) is (_x_). _x_, + and (2) is (_x_). not _x_; + +while the contradictory of (1) is: + + not (_x_). _x_. + +The last line but one of the above verse may, then, be written: + + (_t_). not (_x_). not not [Greek: ph] (_x_, _t_), + +where "[Greek: ph] (_x_, _t_)" denotes the unasserted propositional +function "the doing wrong to the person _x_ at the instant _t_." By +means of the principle of double negation we can at once simplify the +above expression into: + + (_t_). not (_x_). [Greek: ph] (_x_, _t_); + +which can be thus read: "If at every instant of his life there was at +least one person _x_ to whom he did no wrong (at that instant)." It is +difficult to imagine any one so sunk in iniquity that he would not +satisfy this hypothesis. We are forced, then, unless our imagination for +evil is to be distrusted, to conclude that any one might have been there +to have heard that song. Now this conclusion is plainly false, possibly +on physical grounds, and certainly on aesthetic grounds. It may be added, +by the way, that it is quite possible that De Morgan was mistaken in his +interpretation of the above proposition owing to the fact that he was +unacquainted with Frege's work. In fact, if he had not noticed the fact +that _any_ two of the "not's" cannot be cancelled against one another he +would have concluded that the interpretation was: "If he had never done +any wrong to anybody." + +According as the symbol for "not" comes before the (_x_) or between the +(_x_) and the [Greek: ph], we have an expression of what Frege called +respectively the denial of generality, and the generality of denial. The +denial of the generality of a denial is the form of all existential +propositions, while the assertion of or denial of generality is the +general form of all non-existential or universal propositions. + +FOOTNOTES: + +[48] To which De Morgan drew attention in a letter; see (Mrs.) S. E. De +Morgan, _Memoir of Augustus De Morgan_, London, 1882, p. 324. + +[49] _Pa. Ma._, p. 16. + +[50] However, here, for the printer's convenience, we depart from Mr. +Russell's usage so far as to write "not" for a curly minus sign. + + + + +CHAPTER XIX + +IMPLICATION + + +A good illustration of the fact that what is called "implication" in +logic is such that a false proposition implies any other proposition, +true or false, is given by Lewis Carroll's puzzle of the three +barbers.[51] + +Allen, Brown, and Carr keep a barber's shop together; so that one of +them must be in during working hours. Allen has lately had an illness of +such a nature that, if Allen is out, Brown must be accompanying him. +Further, if Carr is out, then, if Allen is out, Brown must be in for +obvious business reasons. The problem is, may Carr ever go out? + +Putting _p_ for "Carr is out," _q_ for "Allen is out" and _r_ for "Brown +is out," we have: + + (1) _q_ implies _r_, + (2) _p_ implies that _q_ implies not-_r_. + +Lewis Carroll supposed that "_q_ implies _r_" and "_q_ implies not-_r_" +are inconsistent, and hence that _p_ must be false. But these +propositions are not inconsistent, and are, in fact, both true if _q_ is +false. The contradictory of "_q_ implies _r_" is "_q_ does not imply +_r_" which is not a consequence of "_q_ implies not-_r_." It seems to be +true theoretically that, if Mr. X is a Christian, he is not an Atheist, +but we cannot conclude from this alone that his being a Christian does +not imply that he is an Atheist, unless we assume that the class of +Christians is not null. Thus, if _p_ is true, _q_ is false; or, if Carr +is out, Allen is in. The odd part of this conclusion is that it is the +one which common-sense would have drawn in that particular case. + +A distinguished philosopher (M) once thought that the logical use of the +word "implication"--any false proposition being said to "imply" any +proposition true or false--is absurd, on the grounds that it is +ridiculous to suppose that the proposition "2 and 2 make 5" implies the +proposition "M is the Pope." This is a most unfortunate instance, +because it so happens that the false proposition that 2 and 2 make 5 can +rigorously be proved to imply that M, or anybody else other than the +Pope, is the Pope. For if 2 and 2 make 5, since they also make 4, we +would conclude that 5 is equal to 4. Consequently, subtracting 3 from +both sides, we conclude that 2 would be equal to 1. But if this were +true, since M and the Pope are two, they would be one, and obviously +then M would be the Pope. + +The principle that the false implies the true has very important +applications in political arguments. In fact, it is hard to find a +single principle of politics of which false propositions are not the +main support. + +If _p_ and _q_ are two propositions, and _p_ implies _q_; then, if, and +only if, _q_ and _p_ are both false or both true, we also have: _q_ +implies _p_. The most important applications of this invertibility were +made by the late Samuel Butler[52] and Mr. G. B. Shaw. A political +application may be made as follows: In a country where only those with +middling-sized incomes are taxed, conservative and _bourgeois_ +politicians would still maintain that the proposition "the rich are +taxed" implies the proposition "the poor are taxed," and this +implication, which is true because both premiss and conclusion are +false, would be quite unnecessarily supported by many false practical +arguments. It is equally true that "the poor are taxed" implies that +"the rich are taxed." And this can be proved, in certain cases, on other +grounds. For the taxation of the poor would imply, ultimately, that the +poor could not afford to pay a little more for the necessities of life +than, in strict justice, they ought; and this would mean the cessation +of one of the chief means of production of individual wealth. + +We also see why a valuable means for the discovery of truth is given by +the inversion of platitudinous implications. It may happen that another +platitude is the result of inversion; but it is the fate of any true +remark, especially if it is easy to remember by reason of a paradoxical +form, to become a platitude in course of time. There are rare cases of a +platitude remaining unrepeated for so long that, by a converse process, +it has become paradoxical. Such, for example, is Plato's remark that a +lie is less important than an error in thought. + +Of late years, a method of disguising platitudes as paradoxes has been +too extensively used by Mr. G. K. Chesterton. The method is as follows. +Take any proposition _p_ which holds of an entity _a_; choose _p_ so +that it seems plausible that _p_ also holds of at least two other +entities _b_ and _c_; call _a_, _b_, _c_, and any others for which _p_ +holds or seems to hold, the class A, and _p_ the "A-ness" or "A-ity" of +A; let _d_ be an entity for which _p_ does not hold; and put _d_ among +the A's when you think that nobody is looking. Then state your paradox: +"Some A's do not have A-ness." By further manipulation you can get the +proposition "No A's have A-ness." But it is possible to make a very +successful _coup_ if A is the null-class, which has the advantage that +manipulation is unnecessary. Thus, Mr. Chesterton, in his _Orthodoxy_ +put A for the class of doubters who doubt the possibility of logic, and +proved that such agnostics refuted themselves--a conclusion which seems +to have pleased many clergymen. + +In this way, Mr. Chesterton has been enabled readily to write many books +and to maintain, on almost every page, such theses as that simplicity is +not simple, heterodoxy is not heterodox, poets are not poetical, and so +on; thereby building up the gigantic platitude that Mr. Chesterton is +Chestertonian. + +In the chapter on Identity we have illustrated the use of a case of the +principle that any proposition implies any true proposition. This +important principle may be called _the principle of the irrelevant +premiss_;[53] and is of great service in oratory, because it does not +matter what the premiss is, true or false. There is a _principle of the +irrelevant conclusion_, but, except in law-courts, interruptions of +meetings, and family life, this is seldom used, partly because of the +limitation involved in the logical impossibility for the conclusion to +be false if the premiss be true, but chiefly because the conclusion is +more important than the premiss, being usually a matter of prejudice. + +Certain modern logicians, such as Frege, have found it necessary so to +extend the meaning of implication of _q_ by _p_ that it holds when _p_ +is not a proposition at all. Hitherto, politicians, finding that either +identical or false propositions are sufficient for their needs, have +made no use of this principle; but it is obvious that their stock of +arguments would be vastly increased thereby. + +Logical implication is often an enemy of dignity and eloquence. De +Morgan[54] relates "a tradition of a Cambridge professor who was once +asked in a mathematical discussion, 'I suppose you will admit that the +whole is greater than its part?' and who answered, 'Not I, until I see +what use you are going to make of it.'" And the care displayed by +cautious mathematicians like Poincare, Schoenflies, Borel, Hobson, and +Baire in abstaining from pushing their arguments to their logical +conclusions is probably founded on the unconscious--but no less +well-grounded--fear of appearing ridiculous if they dealt with such +extreme cases as "the series of all ordinal numbers."[55] They are, +probably, as unconscious of implication as Gibbon, when he remarked that +he always had a copy of Horace in his pocket, and often in his hand, was +of the necessary implication of these propositions that his hand was +sometimes in his pocket. + +FOOTNOTES: + +[51] _Md._, N. S., vol. iii., 1894, pp. 436-8. Cf. the discussions by W. +E. Johnson (_ibid._, p. 583) and Russell (_P. M._, p. 18, note, and +_Md._, N. S., vol. xiv., 1905, pp. 400-1). + +[52] The inhabitants of "Erewhon" punished invalids more severely than +criminals. In modern times, one frequently hears the statement that +crime is a disease; and if so, it is surely false that criminals ought +to be punished. + +[53] _Irrelevant_ in a popular sense; one would not say, speaking +loosely, that the fact that Brutus killed Caesar implies that the sea is +salt; and yet this conclusion is implied both by the above premiss, and +the premiss that Caesar killed Brutus. Cf. on such questions Venn, +_S. L._, 2nd ed., pp. 240-4. + +[54] _F. L._, p. 264. + +[55] Cf. Chapters XXIX and XXXVII. + + + + +CHAPTER XX + +DIGNITY + + +We have seen, at the end of the preceding chapter, that logical +implication is often an enemy of dignity. The subject of dignity is not +usually considered in treatises on logic, but, as we have remarked, many +mathematicians implicitly or explicitly seem to fear either that the +dignity of mathematics will be impaired if she follows out conclusions +logically, or that only an act of faith can save us from the belief +that, if we followed out conclusions logically, we should find out +something alarming about the past, present, or future of mathematics. + +Thus it seems necessary to inquire rather more closely into the nature +of dignity, with a view to the discovery of whether it is, as is +commonly supposed, a merit in life and logic. + +The chief use of dignity is to veil ignorance. Thus, it is well known +that the most dignified people, as a rule, are schoolmasters, and +schoolmasters are usually so occupied with teaching that they have no +time to learn anything. And because dignity is used to hide ignorance, +it is plain that impudence is not always the opposite of dignity, but +that dignity is sometimes impudence. Dignity is said to inspire respect; +and this may be in part why respect for others is an error of judgment +and self-respect is ridiculous. + +Self-respect is, of course, self-esteem. William James has remarked that +self-esteem depends, not simply upon our success, but upon the ratio of +our success to our pretensions, and can therefore be increased by +diminishing our pretensions. Thus if a man is successful, but only then, +can he be both ambitious and dignified. James also implies that +happiness increases with self-esteem. Likeness of thought with one's +friends, then, does not make one happy, for otherwise a man who esteemed +himself little would be indeed happy. Also if a man is unhappy he could +not, from our premisses, by the principles of the syllogism and of +contraposition, be dignified--a conclusion which should be fatal to many +novelists' heroes. + +A reflection on pessimism to which this discussion gives rise is the +following: It would appear that a man's self-esteem would be increased +by a conviction of the unworthiness of his neighbours. A man, therefore, +who thinks that the world and all its inhabitants, except himself, are +very bad, should be extremely happy. In fact, the effects would hardly +be distinguishable from those of optimism. And optimism, as everybody +knows, is a state of mind induced by stupidity. + + + + +CHAPTER XXI + +THE SYNTHETIC NATURE OF DEDUCTION + + +Doubt has often been expressed as to whether a syllogism can add to +our knowledge in any way. John Stuart Mill and Henri Poincare, in +particular, held the opinion that the conclusion of a syllogism is an +"analytic" judgment in the sense of Kant, and therefore could be +obtained by the mere dissection of the premisses. Any one, then, who +maintains that mathematics is founded solely on logical principles would +appear to maintain that mathematics, in the last instance, reduces to a +huge tautology. + +Mill, in Chapter III of Book II of his _System of Logic_, said that "it +must be granted that in every syllogism, considered as an argument to +prove the conclusion, there is a _petitio principii_. When we say + + All men are mortal, + Socrates is a man, + +therefore + + Socrates is mortal, + +it is unanswerably urged by the adversaries of the syllogistic theory, +that the proposition, Socrates is mortal, is presupposed in the more +general assumption, All men are mortal; that we cannot be assured of the +mortality of all men unless we are already certain of the mortality of +every individual man; that if it be still doubtful whether Socrates, or +any other individual we choose to name, be mortal or not, the same +degree of uncertainty must hang over the assertion, All men are mortal; +that the general principle, instead of being given as evidence of the +particular case, cannot itself be taken for true without exception until +every shadow of doubt which could affect any case comprised with it is +dispelled by evidence _aliunde_; and then what remains for the syllogism +to prove? That, in short, no reasoning from general to particular can, +as such, prove anything, since from a general principle we cannot infer +any particulars but those which the principle itself assumes as known. +This doctrine appears to me irrefragable...." + +But it is not difficult to see that in certain cases at least deduction +gives us _new_ knowledge.[56] If we already know that two and two always +make four, and that Asquith and Lloyd George are two and so are the +German Emperor and the Crown Prince, we can deduce that Asquith and +Lloyd George and the German Emperor and the Crown Prince are four. This +is new knowledge, not contained in our premisses, because the general +proposition, "two and two are four," never told us there were such +people as Asquith and Lloyd George and the German Emperor and the Crown +Prince, and the particular premisses did not tell us that there were +four of them, whereas the particular proposition deduced does tell us +both these things. But the newness of the knowledge is much less certain +if we take the stock instance of deduction that is always given in books +on logic, namely "All men are mortal; Socrates is a man, therefore +Socrates is mortal." In this case what we really know beyond reasonable +doubt is that certain men, A, B, C, were mortal, since, in fact, they +have died. If Socrates is one of these men, it is foolish to go the +roundabout way through "all men are mortal" to arrive at the conclusion +that _probably_ Socrates is mortal. If Socrates is not one of the men on +whom our induction is based, we shall still do better to argue straight +from our A, B, C, to Socrates, than to go round by the general +proposition, "all men are mortal." For the probability that Socrates is +mortal is greater, on our data, than the probability that all men are +mortal. This is obvious, because if all men are mortal, so is Socrates; +but if Socrates is mortal, it does not follow that all men are mortal. +Hence we shall reach the conclusion that Socrates is mortal, with a +greater approach to certainty if we make our argument purely inductive +than if we go by way of "all men are mortal" and then use deduction. + +Many years ago there appeared, principally owing to the initiative of +Dr. F. C. S. Schiller of Oxford, a comic number of _Mind_. The idea was +extraordinarily good, not so the execution. A German friend of Dr. +Schiller was puzzled by the appearance of the advertisements, which were +doubtfully humorous. However, by a syllogistic process, he acquired +information which was new and useful to him, and thus incidentally +refuted Mill. Presumably he started from the title of the magazine +(_Mind!_), for a mark of exclamation seems nearly always in German to be +a sign of an intended joke (including of course the mark after the +politeness expressed in the first sentence of a private letter or a +public address). There would be, then, the following syllogism: + + This is a book of would-be jokes (i.e. everything in this book is a + would-be joke); + This advertisement is in this book; + Therefore, this advertisement is a would-be joke. + +Thus the syllogism may be almost as powerful an agent in the detection +of humour as M. Bergson's criterion, to be described in a future +chapter.[57] + +FOOTNOTES: + +[56] [The following passage is almost word for word the same as a +passage on pp. 123-5 of Mr. Russell's _Problems of Philosophy_, first +published in 1912, a year after Mr. R*ss*ll's death. It is easy hastily +to conclude that Mr. Russell was indebted to Mr. R*ss*ll to a greater +degree than is usually supposed. But an examination of the internal +evidence leads us to another conclusion. The two texts, it will be +found, differ only in the names of the German Emperor, the Crown Prince +and the other personages being replaced, in the book of 1912, by those +of Messrs. Brown, Jones, Smith, and Robinson. Now, Mr. Russell, in a new +edition of his _Problems_ issued near the beginning of the European war +and before the Russian revolution, substituted "the Emperor of Russia" +for "the Emperor of China" of the first edition. Hence it seems quite +likely that Mr. Russell, who has always shown a tendency to substitute +existents for nonentities, wrote Mr. R*ss*ll's notes.--<sc>Ed.</sc>] + +[57] [See Chapter XLII.--ED.] + + + + +CHAPTER XXII + +THE MORTALITY OF SOCRATES + + +The mortality of Socrates is so often asserted in books on logic that it +may be as well briefly to consider what it means. The phrase "Socrates +is mortal" may be thus defined: "There is at least one instant _t_ such +that _t_ has not to Socrates the one-many relation R which is the +converse of the relation 'exists at,' and all instants following _t_ +have not the relation R to Socrates, and there is at least one instant +_t'_ such that neither _t'_ nor any instant preceding _t'_ has the +relation R to Socrates." + +This definition has many merits. In the first place, no assumption is +made that Socrates ever lived at all. In the second place, no assumption +is made that the instants of time form a continuous series. In the third +place, no assumption is made as to whether Socrates had a first or last +moment of his existence. If time be indeed a continuous series, then we +can easily deduce[58] that there must have been _either_ a first moment +of his non-existence _or_ a last one of his existence, but not both; +just as there seems to be either a greatest weight that a man can lift +or a least weight that he cannot lift, but not both.[59] This may be set +forth as follows: for the present we will not concern ourselves with +evidence for or against human immortality; I will merely try to present +some logical questions which persistently arise whenever we think of +eternal life. One of the greatest merits of modern logic is that it has +allowed us to give precision to such problems, while definitely +abandoning any pretensions of solving them; and I will now apply the +logico-analytical method to one of the problems of our knowledge of the +eternal world.[60] + +We will start from the generally accepted proposition that all men are +mortal. Clearly, if we could know each individual man, and know that he +was mortal, that would not enable us to know that all men are mortal, +unless we knew, in addition, that those were all the men there are. But +we need not here assume any such knowledge of general propositions; and, +though most of us will admit that the proposition in question has great +intrinsic plausibility, it is not strictly necessary for our present +purpose to assume anything more than the still more probable proposition +"Socrates is mortal." This last proposition, quite apart from the fact +that we have a large amount of historical evidence for its truth, has +been repeated so often in books on logic that it has taken on the +respectable air of a platitude while preserving the character of an +exceedingly probable truth. The truth also results from the fact that it +is used as the conclusion of a syllogism. For it is a well-known fact +that syllogisms can only be regarded as forming part of a sound +education if the conclusions are obviously true. The use of a syllogism +of the form "All cats are ducks and all ducks are mice, therefore all +cats are mice," would introduce grave doubts into the University of +Oxford as to whether logic could any longer be considered as a valuable +mental training for what are amusingly called the "learned professions." + +If, then, we divide all the instants of time, whether past, present, or +future, into two series--those instants at which Socrates was alive, and +those instants at which he was not alive--and leave out of +consideration, for the sake of greater simplicity, all those instants +before he lived, we see at once, by the simple application of Dedekind's +Axiom, that, if Socrates entered into eternal life after his death, +there must have been either a last moment of his earthly life _or_ a +first moment of his eternal life, but not both. + +Logic alone can give us no information as to which of these cases +actually occurred, and we are thrown back on to a discussion of +empirical evidence. It is no unusual thing to read of people who thought +"that every moment would be their last." In this case it is quite +obvious that they consequently thought that eternity would have no +beginning. + +Now here we must consider two things: (1) It is plainly unsafe to +conclude from what people think will happen to what will happen; (2) +even if we could so conclude, it would be unsafe to deduce that there +was a last moment in the life of Socrates: we could only make the guess +plausible, as we should be using the inductive method. + +There are two other pieces of evidence that there is a last moment of +any earthly existence, which we may now briefly consider. That this was +so was held by Carlo Michaelstaedter; but since he apparently only +believed this because he wanted, by attributing a supposed ethical value +to that moment, to give support to his theory of suicide, we ought not +to give great weight to this evidence. Secondly, Thomas Hobbes objected +to the principle "that a quantity may grow less and less eternally, so +as at last to be equal to another quantity; or, which is all one, that +there is a last in eternity" as "void of sense." Now, the principle +meant is true, so that, although the other proposition mentioned by +Hobbes does not follow logically from the first, there is some evidence +that this other is true. In fact, that Hobbes thought that such-and-such +a proposition followed from another proposition which he wrongly +believed to be false, is far better evidence for the truth of +such-and-such a proposition than any we have for the truth of most of +our most cherished beliefs. + +Thirdly, Leibniz, in a dialogue[61] written on his journey of 1676 to +visit Spinoza, raised the question whether the moment at which a man +dies may be regarded as both the last moment at which he is alive and +the first at which he is dead, as it must be by Aristotle's theory of +continuity. Agreement with this view violates the law of contradiction; +denial of it implies that two moments can be immediately adjacent. By +the denial, then, we are led to regard space and time as made up of +indivisible points and moments, and thus, since we can draw one and only +one parallel from any point in the diagonal of a square to a given side, +the diagonal will contain the same (infinite) number of points as that +side, and will therefore be equal to it. In this Leibniz repeated an +argument used by the ancient Arabs, Roger Bacon, and William of Occam. +This Leibniz considered to be a proof that a line cannot be an aggregate +of points. Indeed, their number would be "the number of all numbers" of +the greatest possible integer, which _is_ not. + +It does not seem, further, that any light is thrown on the logical +question of human mortality or immortality by legal decisions. It would +appear that one can, legally speaking, be alive for any period less than +twenty-four hours after one is dead and be dead for any period less than +twenty-four hours before one's death. At least, according to _Salkeld_, +i. 44, it was "adjudged that if one be born the first of February at +eleven at night, and the last of January in the twenty-first year of his +age, at one of the clock in the morning, he makes his will of lands, and +dies, it is a good will, for he was then of age." In Sir Robert Howard's +case (_ibid._, ii. 625) it was held by Chief Justice Holt that "if A be +born on the third day of September; and on the second day of September +twenty-one years afterwards he make his will, this is a good will; for +the law will make no fraction of a day, and by consequence he was of +age." But it is hardly necessary to remark that in this way the problem +with which we are concerned is merely shifted and not solved. For the +question as to whether there is or is not a last moment of a man's life +is not answered by the decision that he dies legally twenty-four hours +before or after he dies in the usual sense of the word, and the problem +arises as to whether there is or is not a last moment of his legal +age.[62] + +So assuming that there was a last moment of Socrates's earthly life, and +consequently no first moment of his eternal life, we see, further, that, +unless the possibility of infinite numbers is granted, it would be quite +possible for us logically to doubt the possibility of an eternal life +for Socrates on the same grounds as those which led Zeno to assert that +motion was impossible and that Achilles could never overtake the +Tortoise. If, on the other hand, it be admitted that eternity, at least +in the case of Socrates, had a beginning, these same arguments of Zeno +would lead any one who denies the possibility of infinite number to +conclude that Socrates, like the worm, can never die. Thus is it quite +plain that the difficulties about immortality which meet us at the very +outset of our inquiry can partly be solved only by the help of the +theory of infinite numbers and partly, it would seem, not at all. + +There is another difficulty about immortality which is quite distinct +from this and is analogous to another argument of Zeno. If, indeed, all +the instants of time be divided, as before, into the two series of +instants at which Socrates was alive and instants at which he was not +alive, it follows at once that no instant of time is not accounted for. +At none of these instants, however, does Socrates die; obviously he +cannot die either when he is alive or when he is dead. Thus it would +appear that Socrates never died, and that we ought to re-define the term +"mortal" to mean "a human being who is alive at some moments and dead at +some." Consequently we must avoid the very tempting conclusion that, +because Socrates never died, he was therefore immortal. + +It is very important carefully to distinguish between the two arguments +I have just set forth. The second argument proves quite rigidly that +Socrates and, indeed, anybody else, never dies, whether there is or is +not a last moment of his life on earth. The first argument proves that, +if there is a first moment of Socrates's eternal life, his life on earth +never ends. But we have seen that we cannot conclude that this unending +life proves that he never is or will be in a state of eternity. + +FOOTNOTES: + +[58] By "Dedekind's Axiom," _E. N._, p. 11. + +[59] _M._, vol. xx., 1910, pp. 134-5. + +[60] [Here, again, Mr. R*ss*ll's work seems to anticipate some of Mr. +Russell's later work, e.g. in _Our Knowledge of the External World as a +Field for Scientific Method in Philosophy_, Chicago and London, 1914, +pp. 3-4, 55-6, _et passim._--ED.] + +[61] "Pacidius Philalethi" in Louis Couturat, _Opuscules et Fragments +inedits de Leibniz_, Paris, 1903, pp. 594-627, especially pp. 599, 601, +608, 611. Cf. [A. E. Taylor, Hastings' _Encyclopaedia of Religion and +Ethics_, vol. iv., Part 2, Edinburgh, 1912, p. 96.--ED.]; Robert Latta, +_Leibniz: The Monadology and other Philosophical Writings_, Oxford, +1898, pp. 21 ff, 29 (note); Couturat, _La Logique de Leibniz d'apres des +documents inedits_, Paris, 1901, pp. 130, 132; and Russell, _Ph. L._, +pp. 108-16, 243-9. + +[62] [It may be remarked that, according to _The Times_ of December 20, +1917, Mr. Justice Sargant, in the Chancery Division, also held that "the +law did not recognize fractions of a day," and that Lord Blackburn, in +his decision (9 _App. Cas._, 371, 373) that a man born on the thirteenth +of May 1853 attained the age of twenty-one on the thirteenth of May 1874 +"was not speaking strictly."--ED.] + + + + +CHAPTER XXIII + +DENOTING + + +A concept _denotes_ when, if it occurs in a proposition, the proposition +is not about the concept, but _about_ a term connected in a certain +peculiar way with the concept. Some people often assert that man is +mortal, and yet we never see announced in _The Times_ that Man died on a +certain day at his villa residence "Camelot" at Upper Tooting,[63] nor +do we hear that Procrastination was again the butt of Mr. Plowden's +jokes at Marylebone Police Court last week. + +That two phrases may have different _meanings_ and the same _denotation_ +was discovered by Alice and Frege. Alice[64] observed that the road +which led to Tweedledum's house was that which led to the house of +Tweedledee; and Frege pointed out that the phrases "the house to which +the road that leads to Tweedledum's house leads" and "the house to which +the road that leads to Tweedledee's house leads" have different _Sinn_, +but the same _Bedeutung_. + +FOOTNOTES: + +[63] Cf. _P. M._, pp. 53-4. + +[64] See Appendix M. + + + + +CHAPTER XXIV + +THE + + +The word "the" implies existence and uniqueness; it is a mistake to talk +of "the son of So-and-So" if So-and-So has a fine family of ten +sons.[65] People who refer to "the Oxford Movement" imply that Oxford +only moved once; and those quaint people who say that "A is quite the +gentleman" imply both the doubtful proposition that there is only one +gentleman in the world, and the indubitably false proposition that he is +that man. Probably A is one of those persons who add to the confusion in +the use of the definite article by speaking of his wife as "the wife." + +In a certain Children's Hymn Book one reads: + + The river vast and small. + +Few would deny that there is not more than one such river, but +unfortunately it is doubtful if there is such a river at all. The case +is exactly the same with the ontological proof of the existence of the +most perfect being.[66] + +According to the _Daily Mail_ of October 9, 1906, Judge Russell decided +against a claim brought by an agent against his company for appointing +another agent, the claim being on the ground that he was appointed as +"the" agent. + +Most people admit that the number 2 can be added to the number 2 to give +the number 4, but this is a mistake. They concede, when they use _the_, +that there is only one number 2, and yet they imagine that, when they +consider it apart as the first term of our above sum, they can find +another to add to it, and thereby form the third term. The truth is that +"2 + 2 = 4" is a very misleading equation, and what we really mean by +that faultily abbreviated statement is more precisely: If _x_ and _y_ +denote any things which form a class B, and _x'_ and _y'_ any other +things that form a class (A) which, like that of _x_ and _y_, is a +member of the class (which we call "2") of those classes which have a +one-one correspondence with B (so that any member of A corresponds to +one, and only one, member of B, and conversely), the class of all the +terms of A and B together is a member of that class of classes which, +analogously, we call "4." In this, for the sake of shortness, we have +introduced abbreviations which should not be used in a rigorous logical +statement. + +FOOTNOTES: + +[65] Cf. _Md._, N. S., vol. xiv., 1905, pp. 481, 484. + +[66] Cf. _ibid._, p. 491, note. + + + + +CHAPTER XXV + +NON-ENTITY + + +When people say that such-and-such a thing "is non-existent" they +usually mean that there is not any "thing" of the kind spoken of. Venn +meant this when he described[67] his encounter with what he imagined to +be a very ingenious tradesman: "I once had some strawberry plants +furnished me which the vendor admitted would not bear many berries. But +he assured me that this did not matter, since they made up in their size +what they lost in their number. (He gave me, in fact, the hyperbolic +formula, _xy = c_, to connect the number and magnitude.) When summer +came, _no_ fruit whatever appeared. I saw that it would be no use to +complain, because the man would urge that the size of the non-existent +berry was infinite, which I could not see my way to disprove. I had +forgotten to bar zero values of either variable." + +It is to be regretted that this useful note was omitted in the second +edition of Venn's book; one can imagine that it might have protected Mr. +MacColl and Herr Meinong (who believed, unlike Alice in what may be +called her first theory,[68] in round squares and fabulous monsters) +against the dishonest practices of traders who were too ready with +promises. For the death-blow to this kind of trade was not given until +1905, when Mr. Russell published his article "On Denoting,"[69] and took +up the position of the White King in opposition to Alice's later +assertions.[70] + +Venn's experience illustrates another characteristic of mathematical +logic. It is necessary, in order to make our arguments conclusive, to +devote great care to the elimination of difficulties which rarely occur. +The White Knight--who was like Boole in being a pioneer of mathematical +logic in this way, and yet seems to have held, like Boole, those +philosophical opinions which would base logic on psychology--recognized +the necessity of taking precautions against any unusual appearance of +mice on a horse's back.[71] + +FOOTNOTES: + +[67] _S. L._, 1881, p. 339, note. + +[68] See Appendix N. + +[69] _Md._, N. S., vol. xiv., October 1905, pp. 479-93. + +[70] See Appendix N. + +[71] See Appendix O. + + + + +CHAPTER XXVI + +IS + + +_Is_ has four perfectly distinct meanings in English, besides misuses of +the word. Among the misuses, perhaps the most important are those +referred to by De Morgan:[72] "... We say 'murder _is_ death to the +perpetrator' where the copula is _brings_; 'two and two _are_ four,' the +copula being 'have the value of,' etc." + +Schroeder[73] quite satisfactorily pointed out the well-known distinction +between an _is_ where subject and predicate can be interchanged (such +as: "the class whose members are Shem, Ham and Japhet is the class of +the sons of Noah") and an _is_ or _are_ where they cannot (such as: +Englishmen are Britons), but failed to see[74] the more important +distinction (made by Peano) of is in the sense of "is a member of." If +Englishmen are Britons, and Britons are civilized people, it follows +that Englishmen are civilized people; but, though the _Harmsworth +Encyclopaedia_ is a member of the class Book (of one or more volumes), +and this class is the member of a class A of which it is the only +member, yet the _Harmsworth Encyclopaedia_ is not a member of A, for it +is not true that it is the whole class of books; and such a statement +would not even be made except possibly in the form of an advertisement. + +The fourth meaning of _is_ is _exists_; it is in certain rare moods a +matter for regret that there are difficulties in the way of using one +word to denote four different things. For, if there were not, we might +prove the existence of any thing we please by making it the subject of a +proposition, and thereby earn the gratitude of theologians. + +FOOTNOTES: + +[72] _F. L._, p. 268. + +[73] _A. d. L._, i. pp. 127 sqq. + +[74] _Ibid._, vol. ii. pp. 461, 597. + + + + +CHAPTER XXVII + +_AND_ AND _OR_ + + +When, with Boole, alternatives (A, B) are considered as mutually +exclusive, logical addition may be described as the process of taking A +_and_ B or A _or_ B. It is a great and rare convenience to have two +terms for denoting the same thing: commonly, people denote several +things by the same term, and only the Germans have the privilege of +referring to, say, _continuity_ as _Stetigkeit_ or _Kontinuierlichkeit_. +But Jevons[75] quoted Milton, Shakespeare, and Darwin to prove that +alternatives are not exclusive, and so attained first to recognized +views by arguments which were plainly irrelevant. + +Of course, _and_ is often used as the sign of logical addition: thus one +may speak of one's brothers _and_ sisters, without being understood to +mean the null-class (as should be the case), or pray for one's +"relations and friends," without being sure that one's prayer would be +answered,--as it certainly would if one meant to pray for the +null-class, this being the class indicated. And a word like _while_ is +often used for a logical addition, when exclusiveness of the +alternatives is almost implied. Thus, a reviewer in _Mind_,[76] noticing +the translation of Mach's _Popular Scientific Lectures_ into American, +said of the lectures that: "Most of them will be familiar ... to +epistemologists and experimental psychologists: while the remainder, +which deal with physical questions, are well worth reading." The reader +has the impression, probably given unintentionally, that Professor +Mach's epistemological and psychological lectures are not, in the +reviewer's opinion, worth reading. + +FOOTNOTES: + +[75] _Pure Logic_ ..., London, 1864, pp. 76-9. Cf. Venn, _S. L._, 2nd +ed., pp. 40-8. + +[76] N. S., vol. iv. p. 261. + + + + +CHAPTER XXVIII + +THE CONVERSION OF RELATIONS + + +The "Conversion of Relations" does not mean what it might be supposed to +mean; it has nothing to do with what Kant called "the wholesome art of +persuasion." What concerns us here is the convertibility of a logical +relation. If A has a certain relation R to B, the relation of B to A, +which may be denoted by [vR], is called the _converse_ of R. As De +Morgan[77] remarked, this conversion may sometimes present difficulties. +The following is De Morgan's example: + +"Teacher: 'Now, boys, Shem, Ham and Japheth were Noah's sons; who was +the father of Shem, Ham and Japheth?' No answer. + +"Teacher: 'Boys, you know Mr. Smith, the carpenter, opposite; has he any +sons?' + +"Boys: 'Oh! yes, sir! there's Bill and Ben.' + +"Teacher: 'And who is the father of Bill and Ben Smith?' + +"Boys: 'Why, Mr. Smith, to be sure.' + +"Teacher: 'Well, then, once more, Shem, Ham and Japheth were _Noah's_ +sons; who was the father of Shem, Ham and Japheth?' + +"A long pause; at last a boy, indignant at what he thought the attempted +trick, cried out: 'It _couldn't_ have been Mr. Smith.' These boys had +never converted the relation of father and son...." + +FOOTNOTES: + +[77] _Trans. Camb. Phil. Soc._, vol. x., 1864, part ii., note on page +334. + + + + +CHAPTER XXIX + +PREVIOUS PHILOSOPHICAL THEORIES OF MATHEMATICS + + +Mathematicians usually try to found mathematics on two principles:[78] +one is the principle of confusion between the sign and the thing +signified (they call this principle the foundation-stone of the formal +theory), and the other is the Principle of the Identity of Discernibles +(which they call the principle of the permanence of equivalent forms). + +But the truth is that if we set sail on a voyage of discovery with Logic +alone at the helm, we must either throw such principles as "the identity +of those conceptions which have in common the properties that interest +us" and "the principle of permanence" overboard, or, if we do not like +to act in such a way to old companions with whom we are so familiar that +we can hardly feel contempt for them, at least recognize them clearly as +having no logical validity and merely as psychological principles, and +reduce them to the humble rank of stewards, to minister to our human +weaknesses on the voyage. And then, if we adopt the wise policy of +keeping our axioms down to the minimum number, we must refrain from +creating or thinking that we are creating new numbers to fill up gaps +among the older ones, and thence recognize that our rational numbers are +not particular cases of "real" numbers, and so on. + +We thus get a world of conceptions which looks, and is, very different +from that which ordinary mathematicians think they see; and perhaps this +is the reason why some mathematicians of great eminence, such as Hilbert +and Poincare, have produced such absurd discussions on the fundamental +principles of mathematics,[79] showing once more the truth of the not +quite original remark of Aunt Jane, who + + ... observed, the second time + She tumbled off a 'bus: + "The step is short from the sublime + To the ridiculous." + +In their readiness to consider many different things as one thing--to +consider, for example, the ratio 2:1 as the same thing as the cardinal +number 2--such mathematicians as Peacock, Hankel, and Schubert were +forestalled by the Pigeon, who thought that Alice and the Serpent were +the same creature, because both had long necks and ate eggs.[80] It is, +however, doubtful whether the Pigeon would have followed the example of +the mathematicians just mentioned so far as to embrace the creed of +nominalism and so to feel no difficulty in subtracting from zero--a +difficulty which was pointed out with great acuteness by the Hatter[81] +and modern mathematical logicians. + +FOOTNOTES: + +[78] These principles, after many attempts to state them by Peacock, the +Red and the White Queen (see Appendix P), Hankel, Schroeder, and Schubert +had been made, were first precisely formulated by Frege in _Z. S._; cf. +also Chapter VII. + +[79] See Couturat, _R. M. M._, vol. xiv., March, 1906, pp. 208-50, and +Russell, _ibid._, September, 1906, pp. 627-34. + +[80] See Appendix P. + +[81] See _ibid._ + + + + +CHAPTER XXX + +FINITE AND INFINITE + + +I was once shown a statement made by an eminent mathematician of +Cambridge from which one would conclude that this mathematician thought +that finite distances became infinite when they were great enough. In +one of those splendidly printed books, bound in blue, published by the +University Press, and sold at about a guinea as a guide to some advanced +branch of pure mathematics, one may read, even in the second edition +published in 1900, the words: "Representation [of a complex variable] on +a plane is obviously more effective for points at a finite distance from +the origin than for points at a very great distance." + +Plainly some of the points at a very great distance are at a _finite_ +distance, for the same author mentions that Neumann's sphere for +representing the positions of points on a plane "has the advantage ... +of exhibiting the uniqueness of _z_ = [infinity symbol] as a value of +the variable." + + + + +CHAPTER XXXI + +THE MATHEMATICAL ATTAINMENTS OF TRISTRAM SHANDY + + +Tristram Shandy[82] said that his father was sometimes a gainer by +misfortune; for if the pleasure of haranguing about it was as ten, and +the misfortune itself only as five, he gained "half in half," and was +well off again as if the misfortune had never happened. + +Suppose that the unit (arbitrary) of pleasure is denoted by A, Tristram +Shandy, by neglecting, in this ethical discussion, to introduce negative +quantities (Kant's pamphlet advocating this introduction into philosophy +was made subsequently[83]), apparently made 15A to result, and this can +hardly be maintained to be the half of 10A. It is possible, however, +that Tristram Shandy succeeded in proving the apparently paradoxical +equation + + 15A = 5A + +by remarking that the axiom "the whole is greater than the part" does +not always hold. This remark follows at once from what Mr. Russell[84] +has called "The Paradox of Tristram Shandy." This paradox is described +by Mr. Russell as follows: + +"Tristram Shandy, as we know, took two years writing the history of the +first two days of his life, and lamented that, at this rate, material +would accumulate faster than he could deal with it, so that he could +never come to an end. Now I maintain that, if he had lived for ever, +and not wearied of his task, then, even if his life had continued as +eventfully as it began, no part of his biography would have remained +unwritten." + +This paradox is strictly correlative to the well-known paradox of Zeno +about Achilles and the Tortoise.[85] "The Achilles proves that two +variables in a continuous series, which approach equality from the same +side, cannot ever have a common limit: the Tristram Shandy proves that +two variables which start from a common term, and proceed in the same +direction, but diverge more and more, may yet determine the same +limiting class (which, however, is not necessarily a segment, because +segments were defined as having terms beyond them). The Achilles assumes +that whole and part cannot be similar, and deduces a paradox; the other, +starting from a platitude, deduces that whole and part may be similar. +For common-sense, it must be confessed that it is a most unfortunate +state of things." And Mr. Russell considers that, in the face of proofs, +it ought to commit suicide in despair. + +Now, I suggest the extremely unlikely possibility that Tristram Shandy, +by reflection on his own life and literary labours, was led to the +correct course of accepting the paradox which resulted from this +reflection and rejecting the Achilles. Thus, he concluded that an +infinite whole may be similar (or, in Cantor's terminology, +"equivalent") to a proper part of itself, and hence, by a confusion of +similarity with identity (or equivalence with equality) which he shares +with some subsequent philosophers,[86] that a whole may be equal to a +proper part of itself. If A is an infinite class, it is not difficult to +see that we can have + + 10A = 5A. + +In this way many have avoided an opinion which rests on no better +foundation than that formerly entertained by the inductive philosophers +of Central Africa, that all men are black.[87] + +FOOTNOTES: + +[82] Cf. a letter of De Morgan in Mrs. De Morgan's _Memoir of Augustus +De Morgan_, p. 324. + +[83] Kant's tract was published in 1763, while _Tristram Shandy_ was +published in 1760. + +[84] _P. M._, pp. 358-9 [Cf. _M._, vol. xxii., January 1912, p. +187.--ED.] + +[85] Cf. _P. M._, pp. 350, 358-9; _M._, vol. xxii., 1912, p. 157. + +[86] [Cf. for example, Cosmo Guastella, _Dell' infinito_, Palermo, +1912.--ED.] + +[87] Cf. Russell, _P. M._, p. 360. + + + + +CHAPTER XXXII + +THE HARDSHIPS OF A MAN WITH AN UNLIMITED INCOME + + +I once heard a man refer to his income as limited, in order to +illustrate the hardship of a class of men, of which he of course was +one, in having to pay a somewhat high income-tax. It is obvious that +this man spoke enviously, and consequently admitted the existence of +more fortunately placed individuals who had unlimited incomes. A little +reflection would have shown the man that he was not taking up a +paradoxical attitude. A "paradoxical attitude" is of course the +assertion of one or more propositions of which the truth cannot be +perceived by a philosopher--and particularly an idealist--and can be +perceived by a logician and occasionally, but not always, by a man of +common-sense. Such propositions are: "The cat is hungry," "Columbus +discovered America," and "A thing which is always at rest may move from +the position A to the different position B." + +Now, if a man had an unlimited income, it is an immediate inference +that, however low income-tax might be, he would have to pay annually to +the Exchequer of his nation a sum equal in value to his whole income. +Further, if his income was derived from a capital invested at a finite +rate of interest (as is usual), the annual payments of income-tax would +each be equal in value to the man's whole capital. If, then, the man +with an unlimited income chose to be discontented, he would be sure of a +sympathetic audience among philosophers and business acquaintances; but +discontent could not last long, for the thought of the difficulties he +was putting in the way of the Chancellor of the Exchequer, who would +find the drawing up of his budget most puzzling, would be amusing. +Again, the discovery that, after paying an infinite income-tax, the +income would be quite undiminished, would obviously afford satisfaction, +though perhaps the satisfaction might be mixed with a slight uneasiness +as to any action the Commissioners of Income-Tax might take in view of +this fact. + +A problem of a wholly different nature is connected with the possible +purchase by the man with an unlimited income of an enumerable infinity +of pairs of boots. If he wished to prove that he had an even number of +boots, it would be easy if right boots were distinguishable from left +ones, but if the man were a faddist of such a kind that he insisted that +his left boots should not be made in any way differently from his right +ones, it would not be possible for him to prove the theorem mentioned +unless he assumed what is known as "the multiplicative axiom." In fact +this axiom shows that it is legitimate to pick out an infinite +succession of members of an infinite class in an arbitrary way. In the +case of the pairs of boots, each pair contains two members, and if there +is no means of distinguishing between them, when we wish to pick out one +of them for each of the infinity of pairs, we cannot say which ones we +mean to pick out unless we assume, by means of the above axiom, that a +particularized member can always be found even with things of each of +which it can be said that, like Private James in the _Bab Ballads_, + + No characteristic trait had he + Of any distinctive kind. + +However, a solution of the puzzle was given by Dr. Denes Koenig of +Budapest. You first prove that there are points in space such that, if P +is one of them, not more than a finite number of pairs of boots are such +that each centre of mass of the two members of a pair is equidistant +from P. Taking a point P of this sort, select from each pair the boot +whose centre of mass is nearest P. (There may be a finite number of +pairs left over, but they can be dealt with arbitrarily.) + +Another form of the problem is as follows. Every time the man bought a +pair of boots he also bought a pair of socks to go with it; he had an +enumerable infinity of pairs of each, and the problem is to prove that +he had as many boots as he had socks. In this case the boots, we will +suppose, can be divided into right and left, but the socks cannot. Thus +there are an enumerable infinity of boots, but the number of the socks +cannot be determined without admitting the axiom mentioned above. A +further difficulty might arise if the owner of the boots and socks lost +one leg in some accident, and told his butler to give away half his +socks. Naturally the butler would find great logical difficulties in so +doing, and it would seem to be an interesting ethical problem whether he +should be dismissed from his situation for failing to prove the +multiplicative axiom. Again, if the butler stole a pair of boots, the +millionaire would have as many pairs as before, but might have fewer +boots. There is as yet no evidence that the number of his boots is equal +to or greater than the number of pairs. + + + + +CHAPTER XXXIII + +THE RELATIONS OF MAGNITUDE OF CARDINAL NUMBERS + + +The theorems of cardinal arithmetic are frequently used in ordinary +conversation. What is known as the Schroeder-Bernstein theorem was used, +long before Bernstein or Schroeder, by Edward Thurlow, afterward the +law-lord Lord Thurlow, when an undergraduate of Caius College, +Cambridge. Thurlow was rebuked for idleness by the Master, who said to +him: "Whenever I look out of the window, Mr. Thurlow, I see you crossing +the Court." The provost thus asserted a one-one correspondence between +the class A of his acts of looking out of the window and a part of the +class B of Thurlow's acts of crossing the Court. Thurlow asserted in +reply a one-one correspondence between B and a part of A: "Whenever +I cross the Court I see you looking out of the window." The +Schroeder-Bernstein theorem, then, allows us to conclude that there is a +one-one correspondence between the classes A and B. That A and B were +finite classes is not the fault of the Master or Thurlow; nor is it +relevant logically. + + + + +CHAPTER XXXIV + +THE UNKNOWABLE + + +According to Mr. S. N. Gupta,[88] the first thing that every student of +Hindu logic has to learn when he is said to begin the study of inference +is that "all H is S" is not always equivalent to "No H is not S." "The +latter proposition is an absurdity when S is _Kebalanvayi_, i.e. covers +the whole sphere of thought and existence.... 'Knowable' and 'Nameable' +are among the examples of _Kebalanvayi_ terms. If you say there is a +thing not-knowable, how do you know it? If you say there is a thing +not-nameable, you must point that out, i.e. somehow name it. Thus you +contradict yourself." + +Mr. Herbert Spencer's doctrine of the "Unknowable" gives rise to some +amusing thoughts. To state that all knowledge of such and such a thing +is above a certain person's intelligence is not self-contradictory, but +merely rude: to state that all knowledge of a certain thing is above all +possible human intelligence is nonsense, in spite of its modest, +platitudinous appearance. For the statement seems to show that we do +know something of it, viz. that it is unknowable. + +To the last (1900) edition of _First Principles_ was added a "Postscript +to Part I," in which the justice of this simple and well-known criticism +as to the contradiction involved in speaking of an "Unknowable," which +had been often made during the forty odd years in which the various +editions had been on the market, was grudgingly acknowledged as +follows:[89] + +"It is doubtless true that saying what a thing is not, is, in some +measure, saying what it is;... Hence it cannot be denied that to affirm +of the Ultimate Reality that it is unknowable is, in a remote way, to +assert some knowledge of it, and therefore involves a contradiction." + +The "Postscript" reminds one of the postscript to a certain Irishman's +letter. This Irishman, missing his razors after his return from a visit +to a friend, wrote to his friend, giving precise directions where to +look for the missing razors; but, before posting the letter, added a +postscript to the effect that he had found the razors. + +One is tempted to inquire, analogously, what might be, in view of the +Postscript, the point of much of Spencer's Part I. It is, to use De +Morgan's[90] description of the arguments of some who maintain that we +can know nothing about infinity, of the same force as that of the man +who answered the question how long he had been deaf and dumb. + +But the best part of the joke against Mr. Spencer is that he, as we +shall see in Chapter XXXVIII, was refuted by a fallacious argument, and +thus mistakenly asserted the validity of the refutation of remarks which +happen to be unsound. + +The analogy of the contradiction of Burali-Forti with the contradiction +involved in the notion of an "unknowable" may be set forth as follows. +If A should say to B: "I know things which you never by any possibility +can know," he may be speaking the truth. In the same way, [Greek: o] may +be said, without contradiction, to transcend all the _finite_ integers. +But if some one else, C, should say: "There are some things which no +human being can ever know anything about," he is talking nonsense.[91] +And in the same way if we succeeded in imagining a number which +transcends _all_ numbers, we have succeeded in imagining the absurdity +of a number which transcends itself. + +All the paradoxes of logic (or "the theory of aggregates") are +analogous to the difficulty arising from a man's statement: "I am +lying."[92] In fact, if this is true, it is false, and _vice versa_. If +such a statement is spread out a little, it becomes an amusing hoax or +an epigram. Thus, one may present to a friend a card bearing on both +sides the words: "The statement on the other side of this card is +false"; while the first of the epigrams derived from this principle +seems to have been written by a Greek satirist:[93] + + Lerians are bad; not _some_ bad and some _not_; + But all; there's not a Lerian in the lot, + Save Procles, that you could a good man call;-- + And Procles--is a Lerian after all. + +This is the original of a well-known epigram by Porson, who remarked +that all Germans are ignorant of Greek metres, + + All, save only Hermann;-- + And Hermann's a German. + +FOOTNOTES: + +[88] _Md._, N. S., vol. iv., 1895, p. 168. + +[89] _First Principles_, 6th ed., 1900, pp. 107-10. The first edition +was published in 1862. + +[90] Note on p. 6 of his paper: "On Infinity; and on the Sign of +Equality," _Trans. Camb. Phil. Soc._, vol. xi., part i., pp. 1-45 (read +May 16, 1864). + +[91] The assertion of the finitude of a man's mind appears to be +nonsense; both because, if we say that the mind of man is limited we +tacitly postulate an "unknowable," and because, even if the human mind +were finite, there is no more reason against its conceiving the infinite +than there is for a mind to be blue in order to conceive a pair of blue +eyes (cf. De Morgan, _loc. cit._). + +[92] Russell, _R. M. M._, vol. xiv., September 1906, pp. 632-3, 640-4. + +[93] _The Greek Anthology_, by Lord Neaves (Ancient Classics for English +Readers), Edinburgh and London, 1897, p. 194. + + + + +CHAPTER XXXV + +MR. SPENCER, THE ATHANASIAN CREED AND THE ARTICLES + + +When, in what I believe is misleadingly known as "The Athanasian Creed," +people say "The Father incomprehensible," and so on, they are not +falling into the same error as Mr. Spencer, for the Latin equivalent for +"incomprehensible" is merely "_immensus_," and Bishop Hilsey translated +it more correctly as "immeasurable."[94] It is a regrettable fact that +Dr. Blunt,[95] in his mistaken modesty, has added a note to this passage +that: "Yet it is true that a meaning not intended in the Creed has +developed itself through this change of language, for the Nature of God +is as far beyond the grasp of the mind as it is beyond the possibility +of being contained within local bounds." + +Mr. Spencer seems no happier when we compare his statements with those +in the Anglican Articles of Religion. There God is never referred to as +infinite. It is true that His power and goodness are so referred to; but +this deficiency was presumably brought about intentionally, so that +faith might gain in meaning as time went on. + +FOOTNOTES: + +[94] _A. C. P._, p. 217. + +[95] _Ibid._, p. 218. + + + + +CHAPTER XXXVI + +THE HUMOUR OF MATHEMATICIANS + + +Brahmagupta's problem[96] appears to be the earliest instance of a kind +of joke which has been much used by mathematicians. For the sake of +giving a certain picturesqueness to the data of problems, and so to +excite that sort of interest which is partly expressed by a smile, +mathematicians have got into the habit of talking, for example, of +monkeys in the form of geometrical points climbing up massless ropes. +Professor P. Staeckel[97] truly remarked that physiological +mechanics--the mechanics of bones, muscles, and so on--is wholly +different from this. There was once a lecturer on mathematics at +Cambridge who used yearly to propound to his pupils a problem in rigid +dynamics which related to the motion of a garden roller supposed to be +without mass or friction, when a heavy and perfectly rough insect walked +round the interior of it in the direction of normal rolling. + +Hitherto this has been the only mathematical outlet for the humour of +mathematicians; and those who really had the interests of mathematics at +heart saw with alarm the growing tendency towards scholasticism in +mathematical jokes. Fortunately the discovery of logic by some +mathematicians has removed this danger. Still to many mathematicians +logic is still unknown, and to them--to Professor A. Schoenflies for +example--modern mathematics, owing to its alliance with logic, appears +to be sinking into scholasticism. It is true that the word +"scholasticism" is not used by Professor Schoenflies in any +intentionally precise signification, but merely as a vague epithet of +disapproval, as the word "socialism" is used by the ordinary philistine, +and this would certainly serve as a sufficient excuse. But no excuse is +needed: these opinions are themselves a source of mathematical jokes. + +FOOTNOTES: + +[96] See Chapter XII. + +[97] _Encykl. der math. Wiss._, vol. iv., part i., p. 474. + + + + +CHAPTER XXXVII + +THE PARADOXES OF LOGIC + + +We have already[98] referred to the contempt shown by some +mathematicians for exact thought, which they condemn under the name +of "scholasticism." An example of this is given by Schoenflies in +the second part of his publication usually known as the _Bericht +ueber Mengenlehre_.[99] Here[100] a battle-cry in italics-- + + "_Against all resignation, but also against all scholasticism!_"-- + +found utterance. Later on, Schoenflies[101] became bolder and adopted a +more personal battle-cry, also in italics, and with a whole line to +itself: + + "_For Cantorism but against Russellism!_" + +"Cantorism" means the theory of transfinite aggregates and numbers +erected for the most part by Georg Cantor. Shortly speaking, the great +sin of "Russellism" is to have gone too far in the chain of logical +deduction for many mathematicians, who were perhaps, like +Schoenflies,[102] blinded by their rather uncritical love of +mathematics. Thus it comes about that Schoenflies[103] denounces +Russellism as "scholastic and unhealthy." This queer blend of qualities +would surely arouse the curiosity of the most _blase_ as to what strange +thing Russellism must be.[104] + +Schoenflies[105] said that some mathematicians attributed to the logical +paradoxes which have given Russell so much trouble to clear up, +"especially to those that are artificially constructed, a significance +that they do not have." Yet no grounds were given for this assertion, +from which it might be concluded that the rigid examination of any +concept was unimportant. The paradoxes are simply the necessary results +of certain logical views which are currently held, which views do not, +except when they are examined rather closely, appear to contain any +difficulty. The contradiction is not felt, as it happens, by people who +confine their attention to the first few number-classes of Cantor, and +this seems to have given rise to the opinion, which it is a little +surprising to find that some still hold, that cases not usually met +with, though falling under the same concept as those usually met with, +are of little importance. One might just as well maintain that +continuous but not differentiable functions are unimportant because they +are artificially constructed--a term which I suppose means that they do +not present themselves when unasked for. Rather should we say that it is +by the discovery and investigation of such cases that the concept in +question can alone be judged, and the validity of certain theorems--if +they are valid--conclusively proved. That this has been done, chiefly by +the work of Russell, is simply a fact; that this work has been and is +misunderstood by many[106] is regrettable for this reason, among others, +that it proves that, at the present time, as in the days in which +_Gulliver's Travels_ were written, some mathematicians are bad +reasoners.[107] + +Nearly all mathematicians agreed that the way to solve these paradoxes +was simply not to mention them; but there was some divergence of opinion +as to how they were to be unmentioned. It was clearly unsatisfactory +merely not to mention them. Thus Poincare was apparently of opinion that +the best way of avoiding such awkward subjects was to mention that they +were not to be mentioned. But[108] "one might as well, in talking to a +man with a long nose, say: 'When I speak of noses, I except such as are +inordinately long,' which would not be a very successful effort to avoid +a painful topic." + +Schoenflies, in his paper of 1911 mentioned above, adopted the +convenient plan of referring these logical difficulties at the root of +mathematics to a department of knowledge which he called "philosophy." +He said[109] of the theory of aggregates that though "born of the +acuteness of the mathematical spirit, it has gradually fallen into +philosophical ways, and has lost to some extent the compelling force +which dwells in the mathematical process of conclusion." + +The majority of mathematicians have followed Schoenflies rather than +Poincare, and have thus adopted tactics rather like those of the March +Hare and the Gryphon,[110] who promptly changed the subject when Alice +raised awkward questions. Indeed, the process of the first of these +creatures of a child's dream is rather preferable to that of +Schoenflies. The March Hare refused to discuss the subject because he +was bored when difficulties arose. Schoenflies would not say that he was +bored--he professed interest in philosophical matters, but simply called +the logical continuation of a subject by another name when he did not +wish to discuss the continuation, and thus implied that he had discussed +the whole subject. Further, Schoenflies would not apparently admit that +the one method of logic could be applied to the solution of both +mathematical and philosophical problems, in so far as these problems are +soluble at all; but the March Hare, shortly before the remark we have +just quoted, rightly showed great astonishment that butter did not help +to cure both hunger and watches that would not go.[111] The judgment of +Schoenflies by which certain apparently mathematical questions were +condemned as "philosophical," rested on grounds as flimsy as those in +the Dreyfus Case, or the Trial in _Wonderland_.[112] + +FOOTNOTES: + +[98] Chapters VII and XXXVI. + +[99] _Die Entwickelung der Lehre von den Punktmannigfaltigkeiten._ +Bericht, erstattet der deutschen Mathematiker-Vereinigung, Leipzig, +1908. + +[100] _Ibid._, p. 7. The battle-cry is: "_Gegen jede Resignation, aber +auch gegen jede Scholastik!_" + +[101] "Ueber die Stellung der Definition in der Axiomatik," _Jahresber, +der deutsch. Math.-Ver._, vol. xx., 1911, pp. 222-5. The battle-cry is +on p. 256 and is: "Fuer den Cantorismus aber gegen den Russellismus!" + +[102] _Ibid._, p. 251. "Es ist also," he exclaims with the eloquence of +emotion and the emotion of eloquence, "nicht die Geringschaetzung der +Philosophie, die mich dabei treibt, sondern die Liebe zur +Mathematik;..." + +[103] "Ueber die Stellung der Definition in der Axiomatik," _Jahresber, +der deutsch. Math.-Ver._, vol. xx., 1911, p. 251. + +[104] [Cf. for this, _M._, vol. xxii., January 1912, pp. 149-58.--ED.] + +[105] _Bericht_, 1908, p. 76, note; cf. p. 72. + +[106] E.g. in F. Hausdorff's review of Russell's _Principles_ of 1903 in +the _Vierteljahrsschr. fuer wiss. Philos. und Soziologie_. + +[107] [Cf. _M._, vol. xxv., 1915, pp. 333-8.--ED.] + +[108] Russell, _A. J. M._, vol. xxx., 1908, p. 226. + +[109] _Loc. cit._, p. 222. + +[110] See Appendix Q. + +[111] See Appendix R. + +[112] See Appendix S. + + + + +CHAPTER XXXVIII + +MODERN LOGIC AND SOME PHILOSOPHICAL ARGUMENTS + + +The most noteworthy reformation of recent years in logic is the +discovery and development by Mr. Bertrand Russell of the fact that the +paradoxes--of Burali-Forti, Russell, Koenig, Richard, and others--which +have appeared of late years in the mathematical theory of aggregates and +have just been referred to, are of an entirely _logical_ nature, and +that their avoidance requires us to take account of a principle which +has been hitherto unrecognized, and which renders invalid several +well-known arguments in refutation of scepticism, agnosticism, and the +statement of a man that he asserts nothing. + +Dr. Whitehead and Mr. Russell say:[113] "The principle which enables us +to avoid illegitimate totalities may be stated as follows: 'Whatever +involves _all_ of a collection must not be one of the collection,' or +conversely: 'If, provided a certain collection had a total, it would +have members only definable in terms of that total, then the said +collection has no total.' We shall call this the 'vicious-circle +principle,' because it enables us to avoid the vicious circles involved +in the assumption of illegitimate totalities. Arguments which are +condemned by the vicious-circle principle will be called 'vicious-circle +fallacies.' Such arguments, in certain circumstances, may lead to +contradictions, but it often happens that the conclusions to which they +lead are in fact true, though the arguments are fallacious. Take, for +example, the law of excluded middle in the form 'all propositions are +true or false.' If from this law we argue that, because the law of +excluded middle is a proposition, therefore the law of excluded middle +is true or false, we incur a vicious-circle fallacy. 'All propositions' +must be in some way limited before it becomes a legitimate totality, and +any limitation which makes it legitimate must make any statement about +the totality fall outside the totality. Similarly the imaginary sceptic +who asserts that he knows nothing and is refuted by being asked if he +knows that he knows nothing, has asserted nonsense, and has been +fallaciously refuted by an argument which involves a vicious-circle +fallacy. In order that the sceptic's assertion may become significant it +is necessary to place some limitation upon the things of which he is +asserting his ignorance; the proposition that he is ignorant of every +member of this collection must not itself be one of the collection. +Hence any significant scepticism is not open to the above form of +refutation." + +In fact, the world of things falls into various sets of things of the +same "type." For every propositional function [Greek: ph](_x_) there is +a range of values of _x_ for which [Greek: ph](_x_) has a signification +as a true or a false proposition. Until this theory was brought forward, +there were occasionally discussions as to whether an object which did +not belong to the range of a certain propositional function possessed +the corresponding property or not. Thus, Jevons, in early days,[114] was +of opinion that virtue is neither black nor not-black because it is not +coloured, but rather later[115] he admitted that virtue is not +triangular.[116] + +FOOTNOTES: + +[113] _Pa. Ma._, p. 40. + +[114] _S. o. S._ pp. 36-7. + +[115] _E. L. L._, pp. 120-1. + +[116] [It may perhaps be added that, some years after Mr. R*ss*ll's +death, Dr. Whitehead stated, in an address delivered in 1916 and +reprinted in his book on _The Organisation of Thought_ (London, 1917, p. +120), that "the specific heat of virtue is 0.003 is, I should imagine, +not a proposition at all, so that it is neither true nor +false...."--ED.] + + + + +CHAPTER XXXIX + +THE HIERARCHY OF JOKES + + +Jokes may be divided into various types. Thus a joke or class of jokes +can only be the subject of a joke of higher order. Otherwise we would +get the same vicious-circle fallacy which gives rise to so many +paradoxes in logic and mathematics. A certain Oxford scholar succeeded, +to his own satisfaction, in reducing all jokes to primitive types, +consisting of thirty-seven proto-Aryan jokes. When any proposition was +propounded to him, he would reflect and afterwards pronounce on the +question as to whether the proposition was a joke or not. If he decided, +by his theory, that it was a joke, he would solemnly say: "There _is_ +that joke." If this narration is accepted as a joke, since it cannot be +reduced to one of the proto-Aryan jokes under pain of leading us to +commit a vicious-circle fallacy, we must conclude that there is at least +one joke which is not proto-Aryan; and, in fact, is of a higher type. +There is no great difficulty in forming a hierarchy of jokes of various +types. Thus a joke of the fourth type (or order) is as follows: A joke +of the first order was told to a Scotchman, who, as we would expect, was +unable to see it.[117] The person (A) who told this joke told the story +of how the joke was received to another Scotchman thereby making a joke +about a joke of the first order, and thus making a joke of the second +order. A remarked on this joke that no joke could penetrate the head of +the Scotchman to whom the joke of the first order was told, even if it +were fired into his head with a gun. The Scotchman, after severe +thought, replied: "But ye couldn't do that, ye know!" A repeated the +whole story, which constituted a joke of the third order, to a third +Scotchman. This last Scotchman again, after prolonged thought, replied: +"He had ye there!" This whole story is a joke of the fourth order. + +Most known jokes are of the first order, for the simple reason that the +majority of people find that the slightest mental effort effectually +destroys any perception of humour. It seems to me that a joke becomes +more pleasurable in proportion as logical faculties are brought into +play by it; and hence that logical power is allied, or possibly +identical, with the power of grasping more subtle jokes. The jokes which +amuse the frequenters of music-halls, Conservatives, and M. Bergson--and +which usually deal with accidents, physical defects, mothers-in-law, +foreigners, or over-ripe cheese--are usually jokes of the first order. +Jokes of the second, and even of the third, order appeal to ordinary +well-educated people; jokes of higher order require either special +ability or a sound logical training on the part of the hearer if the +joke is to be appreciated; while jokes of transfinite order presumably +only excite the inaudible laughter of the gods. + +FOOTNOTES: + +[117] [It may be that, like certain remarks about cheese and +mothers-in-law (see below), the statement that Scotchmen cannot see +jokes is a joke of the first order.--ED.] + + + + +CHAPTER XL + +THE EVIDENCE OF GEOMETRICAL PROPOSITIONS + + +It has often been maintained that the twentieth proposition of the first +book of Euclid--that two sides of a triangle are together greater than +the third side--is evident even to asses. This does not, however, seem +to me generally true. I once asked a coastguardsman the distance from A +to B; he replied: "Eight miles." On further inquiry I elicited the fact +that the distance from A to C was two miles and the distance from C to B +was twenty-two miles. Now the paths from A to B and from C to B were by +sea; while the path from A to C was by land. Hence if the path by land +was rugged and the distance along the road was two miles, it would +appear that the coastguardsman believed that not only could one side of +a triangle be greater than the other two, but that one straight side of +a triangle might be greater than one straight side and any curvilinear +side of the same triangle. The only escape from part of this astonishing +creed would be by assuming that the distance of two miles from A to C +was measured "as the crow flies," while the road A to C was so hilly +that a pedestrian would traverse more than fourteen miles when +proceeding from A to C. Then indeed the coastguardsman could maintain +the true proposition that there is at least one triangle ABC, with the +side AC curvilinear, such that the sum of the lengths of AB and AC is +greater than the length of BC, and only deny the twentieth proposition +of the first book of Euclid. + +Reasoning with the coastguardsman only had the effect of his adducing +the authority of one Captain Jones in support of the accuracy of his +data. Possibly Captain Jones held strange views as to the influence of +temperature or other physical circumstances, or even the nature of space +itself, on the lengths of lines in the neighbourhood of the triangle +ABC. + + + + +CHAPTER XLI + +ABSOLUTE AND RELATIVE POSITION + + +Some people maintain that position in space or time must be relative +because, if we try to determine the position of a body A, if bodies B, +C, D with respect to which the position of A could be determined were +not present, we should be trying to determine something about A without +having our senses affected by other things. These people seem to me to +be like the cautious guest who refused to say anything about his host's +port-wine until he had tasted red ink. + +"Wherein, then," says Mr. Russell,[118] "lies the plausibility of the +notion that all points are exactly alike? This notion is, I believe, a +psychological illusion, due to the fact that we cannot remember a point +so as to know it when we meet again. Among simultaneously presented +points it is easy to distinguish; but though we are perpetually moving, +and thus being brought among new points, we are quite unable to detect +this fact by our senses, and we recognize places only by the objects +they contain. But this seems to be a mere blindness on our part--there +is no difficulty, so far as I can see, in supposing an immediate +difference between points, as between colours, but a difference which +our senses are not constructed to be aware of. Let us take an analogy: +Suppose a man with a very bad memory for faces; he would be able to +know, at any moment, whether he saw one face or many, but he would not +be aware whether he had seen any of the faces before. Thus he might be +led to define people by the rooms in which he saw them, and to suppose +it self-contradictory that new people should come to his lectures, or +that old people should cease to do so. In the latter point at least it +will be admitted by lecturers that he would be mistaken. And as with +faces, so with points--inability to recognize them must be attributed, +not to the absence of individuality, but merely to our incapacity." + +Another form of this tendency is shown by Kronecker, Borel, Poincare, +and many other mathematicians, who refuse mere logical determination of +a conception and require that it be actually described in a finite +number of terms. These eminent mathematicians were anticipated by the +empirical philosopher who would not pronounce that the "law of thought" +that A is either in the place B or not is true until he had looked to +make sure. This philosopher was of the same school as J. S. Mill and +Buckle, who seem to have maintained implicitly not only that, in view of +the fact that the breadth of a geometrical line depends upon the +material out of which it is constructed, or upon which it is drawn, that +there ought to be a paste-board geometry, a stone geometry, and so +on;[119] but also that the foundations of logic are inductive in their +nature.[120] "We cannot," says Mill,[121] "conceive a round square, not +merely because no such object has ever presented itself in our +experience, for that would not be enough. Neither, for anything we know, +are the two ideas in themselves incompatible. To conceive a body all +black and yet white would only be to conceive two different sensations +as produced in us simultaneously by the same object--a conception +familiar to our experience--and we should probably be as well able to +conceive a round square as a hard square, or a heavy square, if it were +not that in our uniform experience, at the instant when a thing begins +to be round, it ceases to be square, so that the beginning of the one +impression is inseparably associated with the departure or cessation of +the other. Thus our inability to form a conception always arises from +our being compelled to form another contradictory to it." + +FOOTNOTES: + +[118] _Md._, N. S., vol. x., July, 1901, pp. 313-14. + +[119] J. B. Stallo, _The Concepts and Theories of Modern Physics_, 4th +ed., London, 1900, pp. 217-27. + +[120] _Ibid._, pp. 140-4. + +[121] _Examination of the Philosophy of Sir William Hamilton_, vol. i. +p. 88, Amer. ed. + + + + +CHAPTER XLII + +LAUGHTER + + +[It seemed advisable to give here[122] some views on laughter, most of +which were also held by Mr. R*ss*ll, though no written expression of his +views has yet been found. In a review[123] of M. Bergson's book on +_Laughter_,[124] Mr. Russell has remarked: + +"It has long been recognized by publishers that everybody desires to be +a perfect lady or gentleman (as the case may be); to this fact we owe +the constant stream of etiquette-books. But if there is one thing which +people desire even more, it is to have a faultless sense of humour. Yet +so far as I know, there is no book called 'Jokes without Tears, by Mr. +McQuedy.' This extraordinary lacuna has now been filled. Those to whom +laughter has hitherto been an unintelligible vagary, in which one must +join, though one could never tell when it would break out, need only +study M. Bergson's book to acquire the finest flower of Parisian wit. By +observing a very simple formula they will know infallibly what is funny +and what is not; if they sometimes surprise their unlearned friends, +they have only to mention their authority in order to silence doubt. +'The attitudes, gestures and movements of the human body,' says M. +Bergson, 'are laughable in exact proportion as that body reminds us of a +mere machine.' When an elderly gentleman slips on a piece of orange-peel +and falls, we laugh, because his body follows the laws of dynamics +instead of a human purpose. When a man falls from a scaffolding and +breaks his neck on the pavement, we presumably laugh even more, since +the movement is even more completely mechanical. When the clown makes a +bad joke for the first time, we keep our countenance, but at the fifth +repetition we smile, and at the tenth we roar with laughter, because we +begin to feel him a mere automaton. We laugh at Moliere's misers, +misanthropists and hypocrites, because they are mere types mechanically +dominated by a master impulse. Presumably we laugh at Balzac's +characters for the same reason; and presumably we never smile at +Falstaff, because he is individual throughout." + +The review concludes with the reflection that "it would seem to be +impossible to find any such formula as M. Bergson seeks. Every formula +treats what is living as if it were mechanical, and is therefore by his +own rule a fitting object of laughter." Now, this undoubtedly true +conclusion has been obtained, as is readily seen, by a vicious-circle +fallacy which Mr. R*ss*ll would hardly have committed.--ED.] + +FOOTNOTES: + +[122] From a remark on p. 47 above, it is evident that Mr. R*ss*ll +intended to write some such chapter as this. + +[123] _The Professor's Guide to Laughter, The Cambridge Review_, vol. +xxxii., 1912, pp. 193-4. + +[124] _Laughter, an Essay on the Meaning of the Comic_, English +translation by C. Brereton and F. Rothwell, London, 1911. + + + + +CHAPTER XLIII + +"GEDANKENEXPERIMENTE" AND EVOLUTIONARY ETHICS + + +The "Gedankenexperimente," upon which so much weight has been laid by +Mach[125] and Heymans,[126] had already been investigated by the White +Queen,[127] who, however, seems to have perceived that the results of +such experiments are not always logically valid. The psychological +founding of logic appears to be not without analogy with the surprising +method of advocates of evolutionary ethics, who expect to discover what +_is_ good by inquiring what cannibals have _thought_ good. I sometimes +feel inclined to apply the historical method to the multiplication +table. I should make a statistical inquiry among school-children, before +their pristine wisdom had been biassed by teachers. I should put down +their answers as to what 6 times 9 amounts to, I should work out the +average of their answers to six places of decimals, and should then +decide that, at the present stage of human development, this average is +the value of 6 times 9. + +FOOTNOTES: + +[125] See, e.g., _E. u. I._, pp. 183-200. + +[126] _G. u. E._, vol. i. + +[127] See Appendix T. + + + + +APPENDIXES + + +A. LOGIC AND THE PRINCIPLE OF IDENTITY. + +_T. L. G._, p. 45: "'Contrariwise," continued Tweedledee, "if it was so, +it might be; and if it were so, it would be: but as it isn't, it ain't. +That's logic." + + * * * * * + +_S. B._, p. 159: The Professor said: "The day is the same length as +anything that is the same length as _it_." + + * * * * * + +_S. B._, p. 161: Bruno observed that, when the Other Professor lost +himself, he should shout: "He'd be sure to hear hisself, 'cause he +couldn't be far off." + + +B. SYNTHESIS OF CONTRADICTORIES. + +_T. L. G._, p. 71: "'What a beautiful belt you've got on!' Alice +suddenly remarked.... 'At least,' she corrected herself on second +thoughts, 'a beautiful cravat, I should have said--no, a belt, I mean--I +beg your pardon!' she added in dismay, for Humpty-Dumpty looked +thoroughly offended, and she began to wish she hadn't chosen that +subject. 'If only I knew,' she thought to herself, 'which was neck and +which was waist!'" + + +C. EMPIRICAL PHILOSOPHERS AND MATHEMATICS. + +_T. L. G._, p. 79: "'... Now if you had the two eyes on the same side of +the nose, for instance--or the mouth at the top--that would be _some_ +help.' + +"'It wouldn't look nice,' Alice objected. But Humpty-Dumpty only shut +his eyes and said: 'Wait till you've tried.'" + + * * * * * + +_T. L. G._, p. 72: "'And if you take one from three hundred and +sixty-five, what remains?' + +"'Three hundred and sixty-four, of course.' + +"Humpty-Dumpty looked doubtful. 'I'd rather see that done on paper,' +he said." + + +D. NOMINAL DEFINITION. + +_T. L. G._, p. 73: "'When _I_ used a word,' Humpty-Dumpty said in rather +a scornful tone, 'it means just what I choose it to mean--neither more +nor less.' + +"'The question is,' said Alice, 'whether you _can_ make words mean +different things.' + +"'The question is,' said Humpty-Dumpty, 'which is to be master--that's +all.'" + + +E. CONFORMITY OF A PARADOXICAL LOGIC WITH COMMON-SENSE. + +_T. L. G._, p. 100: + + "But I was thinking of a plan + To dye one's whiskers green, + And always use so large a fan + That they could not be seen." + (Verse from White Knight's song.) + + +F. IDEALISTS AND THE LAWS OF LOGIC. + +_T. L. G._, p. 52-3: Tweedledee exclaimed: "'... if he [the Red King] +left off dreaming about you [Alice], where do you suppose you'd be?' + +"'Where I am now, of course,' said Alice. + +"'Not you!' Tweedledee retorted contemptuously. 'You'd be nowhere. Why, +you're only a sort of thing in his dream!' + +"'If that there King was to wake,' added Tweedledum, 'you'd go +out--bang!--just like a candle!' + +"'I shouldn't!' Alice exclaimed indignantly. 'Besides, if _I'm_ only a +sort of thing in his dream, what are _you_, I should like to know?' + +"'Ditto,' said Tweedledum...; 'you know very well you're not real.' + +"'I _am_ real!' said Alice, and began to cry." + + * * * * * + +_T. L. G._, p. 97: "'How _can_ you go on talking so quietly, head +downwards?' Alice asked, as she dragged him out by the feet, and laid +him in a heap on the bank. + +"The Knight looked surprised at the question. 'What does it matter where +my body happens to be?' he said. 'My mind goes on working all the same. +In fact, the more head downwards I am, the more I keep inventing new +things.'" + + * * * * * + +_T. L. G._, p. 98: "'... Everybody that hears me sing--either it brings +the _tears_ into their eyes, or else----' + +"'Or else what?' said Alice, for the Knight had made a sudden pause. + +"'Or else it doesn't, you know.'" + + +G. DISTINCTION BETWEEN SIGN AND SIGNIFICATION. + +_T. L. G._, pp. 98-9: "'The name of the song is called "_Haddocks' +Eyes_."' + +"'Oh, that's the name of the song, is it?' Alice said, trying to feel +interested. + +"'No, you don't understand,' the Knight said looking a little vexed. +'That's what the name is _called_. The name really _is_ "_The Aged Aged +Man_."' + +"'Then I ought to have said "That's what the _song_ is called"?' Alice +corrected herself. + +"'No, you oughtn't: that's another thing. The _song_ is called "_Ways +and Means_": but that's only what it's _called_, you know!' + +"'Well, what _is_ the song, then?' said Alice, who was by this time +completely bewildered. + +"'I was coming to that,' the Knight said. 'The song really _is +"A-sitting on a Gate_"....'" + + +H. NOMINALISM. + +_A. A. W._, p. 70: "'Then you should say what you mean,' the March Hare +went on. + +"'I do,' Alice hastily replied; 'at least--at least I mean what I +say--that's the same thing, you know.' + +"'Not the same thing a bit!' said the Hatter. 'Why, you might just as +well say that "I see what I eat" is the same thing as "I eat what I +see."' + +"'You might just as well say,' added the March Hare, 'that "I like what +I get" is the same thing as "I get what I like"!' + +"'You might just as well say,' added the Dormouse, which seemed to be +talking in its sleep, 'that "I breathe when I sleep" is the same as "I +sleep when I breathe"!' + +"'It _is_ the same thing with you,' said the Hatter; and here the +conversation dropped,..." + + +I. UTILITY OF SYMBOLIC LOGIC. + +_A. A. W._, p. 92: "'I quite agree with you,' said the Duchess, 'and the +moral of that is--"Be what you would seem to be"--or if you'd like it +put more simply--"Never imagine yourself not to be otherwise than what +it might appear to others that what you were or might have been was not +otherwise than what you had been would have appeared to them to be +otherwise."' + +"'I think I should understand that better,' Alice said very politely, +'if I had it written down: but I can't quite follow it as you say it.' + +"'That's nothing to what I could say if I chose,' the Duchess replied, +in a pleased tone." + + +J. MISTAKE AS TO THE NATURE OF CRITICISM. + +_T. L. G._, p. 105: "'She's in that state of mind,' said the White +Queen, 'that she wants to deny _something_--only she doesn't know what +to deny.' + +"'A nasty, vicious temper,' the White Queen remarked; and then there was +an uncomfortable silence for a minute or two." + + +K. A CRITERION OF TRUTH. + +_H. S._, p. 3: + + "Just the place for a Snark! I have said it twice: + That alone should encourage the crew. + Just the place for a Snark! I have said it thrice: + What I tell you three times is true." + + * * * * * + +_H. S._, p. 50: + + "'Tis the note of the Jubjub! Keep count. I entreat; + You will find I have told it you twice. + 'Tis the song of the Jubjub! The proof is complete, + If only I've stated it thrice." + + +L. UNIVERSAL AND PARTICULAR PROPOSITIONS. + +_T. L. G._, p. 40: The Gnat had told Alice that the Bread-and-butterfly +lives on weak tea with cream in it; so: + +"'Supposing it couldn't find any?' she suggested. + +"'Then it would die, of course.' + +"'But that must happen very often,' Alice remarked thoughtfully. + +"'It always happens,' said the Gnat." + + +M. DENOTING. + +_T. L. G._, p. 43: Tweedledum and Tweedledee were, in many respects, +indistinguishable, and Alice, walking along the road, noticed that +"whenever the road divided there were sure to be two finger-posts +pointing the same way, one marked 'TO TWEEDLEDUM'S HOUSE' and the other +'TO THE HOUSE OF TWEEDLEDEE.' + +"'I do believe,' said Alice at last, 'that they live in the same +house!...'" + + +N. NON-ENTITY. + +_T. L. G._, p. 87: "'I always thought they [human children] were +fabulous monsters!' said the Unicorn.... + +"'Do you know [said Alice], I always thought Unicorns were fabulous +monsters, too! I never saw one alive before!' + +"'Well, now that we _have_ seen each other,' said the Unicorn, 'if +you'll believe in me, I'll believe in you. Is that a bargain?'" + + * * * * * + +_T. L. G._, pp. 80-1: "'I see nobody on the road,' said Alice. + +"'I only wish _I_ had such eyes,' the [White] King remarked in a fretful +tone. 'To be able to see Nobody! And at that distance, too! Why, it's as +much as _I_ can do to see real people by this light!'" + + * * * * * + +_A. A. W._, p. 17: "And she [Alice] tried to fancy what the flame of a +candle looks like after the candle is blown out, for she could not +remember ever having seen such a thing." + + * * * * * + +_A. A. W._, p. 68: "... This time it [the Cheshire Cat] vanished quite +slowly, beginning with the end of the tail, and ending with the grin, +which remained some time after the rest of it had gone. + +"'Well! I've often seen a cat without a grin,' thought Alice; 'but a +grin without a cat! It's the most curious thing I ever saw in all my +life!'" + + * * * * * + +_A. A. W._, p. 77: "... The Dormouse went on,...; 'and they drew all +manner of things--everything that begins with an M.' + +"'Why with an M?' said Alice. + +"'Why not?' said the March Hare. + +"Alice was silent. + +"... [The Dormouse] went on: '--that begins with an M, such as +mouse-traps, and the moon, and memory, and muchness, you know you say +things are "much of a muchness"--did you ever see such a thing as a +drawing of a muchness?' + +"'Really, now you ask me,' said Alice, very much confused, 'I don't +think----' + +"'Then you shouldn't talk,' said the Hatter." + + +O. OBJECTS OF MATHEMATICAL LOGIC. + +_T. L. G._, p. 93: "'I was wondering what the mouse-trap [fastened to +the White Knight's saddle] was for,' said Alice. 'It isn't very likely +there would be any mice on the horse's back.' + +"'Not very likely, perhaps,' said the Knight, 'but, if they _do_ come, I +don't choose to have them running all about.' + +"'You see,' he went on after a pause, 'it's as well to be provided for +_everything_. That's the reason the horse has all these anklets round +his feet.' + +"'But what are they for?' Alice asked in a tone of great curiosity. + +"'To guard against the bites of sharks,' the Knight replied." + + +P. THE PRINCIPLE OF PERMANENCE. + +_T. L. G._, p. 106: "'Can you do Subtraction? [said the Red Queen] Take +nine from eight.' + +"'Nine from eight I can't, you know,' Alice replied very readily 'but--' + +"'She can't do Substraction,' said the White Queen." + + * * * * * + +_A. A. W._, p. 56: [Said the Pigeon to Alice]: "'... No, no! You're a +serpent; and there's no use denying it. I suppose you'll be telling me +next that you never tasted an egg!' + +"'I _have_ tasted eggs certainly,' said Alice, who was a very truthful +child; 'but little girls eat eggs quite as much as serpents do, you +know.' + +"'I don't believe it,' said the Pigeon; 'but if they do, why then +they're a kind of serpent, that's all I can say.' + +"This was such a new idea to Alice, that she was quite silent for a +minute or two, which gave the Pigeon the opportunity of adding, 'You're +looking for eggs, I know _that_ well enough; and what does it matter to +me whether you're a little girl or a serpent?' + +"'It matters a good deal to _me_,' said Alice hastily;..." + + * * * * * + +_A. A. W._, p. 75: "'But why [asked Alice] did they live at the bottom +of a well?' + +"'Take some more tea,' the March Hare said to Alice, very earnestly. + +"'I've had nothing yet,' Alice replied in an offended tone, 'so I can't +take more.' + +"'You mean you can't take _less_,' said the Hatter: 'it's very easy to +take _more_ than nothing.'" + + +Q. MATHEMATICIANS' TREATMENT OF LOGIC. + +_A. A. W._, p. 74: The Hatter had told of his quarrel with Time, and of +Time's refusal now to do anything he asked: "'... It's always six +o'clock now!' + +"A bright idea came into Alice's head. 'Is that the reason so many tea +things are put out here?' she asked. + +"'Yes, that's it,' said the Hatter, with a sigh: 'it's always tea time, +and we've no time to wash the things between whiles.' + +"'Then you keep moving round, I suppose?' said Alice. + +"'Exactly so,' said the Hatter: 'as the things get used up.' + +"'But what happens when you come to the beginning again?' Alice ventured +to ask. + +"'Suppose we change the subject,' the March Hare interrupted, yawning. +'I'm getting tired of this.'" + + * * * * * + +_A. A. W._, p. 99: "'And how many hours a day did you do lessons?' said +Alice, in a hurry to change the subject. + +"'Ten hours the first day,' said the Mock Turtle, 'nine the next, and so +on.' + +"'What a curious plan!' exclaimed Alice. + +"'That's the reason they're called lessons,' the Gryphon remarked, +'because they lessen from day to day.' + +"This was quite a new idea to Alice, and she thought it over a little +before she made her next remark. 'Then the eleventh day must have been a +holiday.' + +"'Of course it was,' said the Mock Turtle. + +"'And how did you manage on the twelfth?' Alice went on eagerly. + +"'That's enough about lessons,' the Gryphon interrupted in a very +decided tone...." + + +R. METHOD IN MATHEMATICS AND LOGIC. + +_A. A. W._, p. 71: "'Two days wrong!' sighed the Hatter. 'I told you +butter wouldn't suit the works!' he added, looking angrily at the March +Hare. + +"'It was the _best_ butter,' the March Hare meekly replied. + +"'Yes, but some crumbs must have got in as well,' the Hatter grumbled; +'you shouldn't have put it in with the bread-knife.' + +"The March Hare took the watch and looked at it gloomily: then he dipped +it into his cup of tea, and looked at it again: but he could think of +nothing better to say than his first remark, 'It was the _best_ butter, +you know.'" + + +S. VERDICT THAT LOGIC IS PHILOSOPHY. + +_A. A. W._, pp. 119-23: "... 'Consider your verdict,' he [the King] said +to the jury, in a low trembling voice. + +"'There's more evidence to come yet, please your Majesty,' said the +White Rabbit, jumping up in a great hurry: 'this paper has just been +picked up.' + +"'What's in it?' said the Queen. + +"'I haven't opened it yet,' said the White Rabbit, 'but it seems to be a +letter written by the prisoner to--to somebody.' + +"'It must have been that,' said the King, 'unless it was written to +nobody, which isn't usual, you know.' + +"'Who is it directed to?' said one of the jurymen. + +"'It isn't directed at all,' said the White Rabbit, 'in fact there's +nothing written on the _outside_.' He unfolded the paper as he spoke, +and added, 'It isn't a letter, after all: it's a set of verses.' + +"'Are they in the prisoner's handwriting?' asked another of the jurymen. + +"'No they're not,' said the White Rabbit, 'and that's the queerest thing +about it.' (The jury all looked puzzled). + +"'He must have imitated somebody else's hand,' said the King. (The jury +brightened up again.) + +"'Please your Majesty,' said the Knave, 'I didn't write it, and they +can't prove that I did: there's no name signed at the end.' + +"'If you didn't sign it, said the King, that only makes the matter +worse. You _must_ have meant some mischief, or else you'd have signed +your name like an honest man.' + +"There was a general clapping of hands at this: it was the first really +clever thing the King had said that day. + +"'That _proves_ his guilt, of course,' said the Queen, 'so, off +with----' + +"'It doesn't prove anything of the sort!' said Alice. 'Why, you don't +even know what they're about!' + +"'Read them,' said the King. + +"The White Rabbit put on his spectacles. 'Where shall I begin, please +your Majesty?' he asked. + +"'Begin at the beginning,' the King said very gravely, 'and go on till +you come to the end: then stop.' + +"There was dead silence in the court, whilst the White Rabbit read out +these verses: + + "'_They told me you had been to her, + And mentioned me to him; + She gave me a good character, + But said I could not swim._ + + _He sent them word I had not gone + (We know it to be true): + If she should push the matter on, + What would become of you?_ + + _I gave her one, they gave him two, + You gave us three or more; + They all returned from him to you, + Though they were mine before._ + + _If I or she should chance to be + Involved in this affair, + He trusts to you to set them free + Exactly as they were._ + + _My notion was that you had been + (Before she had this fit) + An obstacle that came between + Him, and ourselves, and it._ + + _Don't let him know she liked them best, + For this must ever be + A secret kept from all the rest, + Between yourself and me._' + +"'That's the most important piece of evidence we've heard yet,' said the +King, rubbing his hands, 'so now let the jury----' + +"'If any one of them can explain it,' said Alice (she had grown so large +in the last few minutes that she wasn't a bit afraid of interrupting +him), 'I'll give him sixpence. _I_ don't believe there's an atom of +meaning in it.' + +"The jury all wrote down on their slates, 'She doesn't believe there's +an atom of meaning in it,' but none of them attempted to explain the +paper. + +"'If there's no meaning in it,' said the King, 'that saves a world of +trouble, you know, as we needn't try to find any. And yet I don't know,' +he went on, spreading out the verses on his knee and looking at them +with one eye; 'I seem to see some meaning in them after all. "_-- said +I could not swim_"; you can't swim, can you?' he added, turning to the +Knave. + +"The Knave shook his head sadly. 'Do I look like it?' he said. (Which he +certainly did _not_, being made entirely of cardboard.) + +"'All right, so far,' said the King; and he went on muttering over the +verses to himself: ''_We know it to be true_'--that's the jury, of +course--'_If she should push the matter on_'--that must be the +Queen--'_What would become of you?_' What indeed!--'_I gave her one, +they gave him two!_' why, that must be what he did with the tarts, you +know----' + +"'But it goes on, '_They all returned from him to you_,'' said Alice. + +"'Why, there they are!' said the King, triumphantly pointing to the +tarts on the table. 'Nothing can be clearer than that. Then +again--'_Before she had this fit_'--you never had fits, my dear, I +think?' he said to the Queen. + +"'Never!' said the Queen furiously, throwing an inkstand at the Lizard +as she spoke. (The unfortunate little Bill had left off writing on his +slate with one finger, as he found it made no mark; but he now hastily +began again, using the ink that was trickling down his face, as long as +it lasted.) + +"'Then the words don't _fit_ you,' said the King, looking round the +court with a smile. There was a dead silence. + +"'It's a pun!' the King added in an angry tone, and everybody laughed. + +"'Let the jury consider their verdict,' the King said, for about the +twentieth time that day. + +"'No, no!' said the Queen. 'Sentence first--verdict afterwards.' + +"'Stuff and nonsense!' said Alice loudly. 'The idea of having the +sentence first!' + +"'Hold your tongue!' said the Queen, turning purple...." + + +T. "GEDANKENEXPERIMENTE." + +_T. L. G._, p. 61: "Alice laughed. 'There's no use trying,' she said: +'one _can't_ believe impossible things.' + +"'I daresay you haven't had much practice,' said the [White] Queen. +'When I was your age, I always did it for half-an-hour a day. Why, +sometimes I've believed as many as six impossible things before +breakfast.'" + + +_Printed in Great Britain by_ + +UNWIN BROTHERS, LIMITED, THE GRESHAM PRESS, WOKING AND LONDON + + + + + +End of Project Gutenberg's The philosophy of B*rtr*nd R*ss*ll, by Various + +*** END OF THIS PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF B*RTR*ND R*SS*LL *** + +***** This file should be named 38430.txt or 38430.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/8/4/3/38430/ + +Produced by Adrian Mastronardi and the Online Distributed +Proofreading Team at http://www.pgdp.net (This file was +produced from images generously made available by The +Internet Archive/Canadian Libraries) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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