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+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: 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)
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+
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+
+
+
+
+  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
+
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