THE PHILOSOPHY OF 
 MR, B*RTR*ND R*SS*L 
 
 EDITED BY 
 
 PHILIP E. B, JOURDAIN
 
 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 
 
 LONDON : GEORGE ALLEN & UNWIN LTD. 
 RUSKIN HOUSE 40 MUSEUM STREET, W.C. i 
 
 CHICAGO : THE OPEN COURT PUBLISHING CO.
 
 First published in 
 
 (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 manu- 
 script, together with some further fragments found in the 
 late Mr. R*ss*H'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 *) of Mr. R*ss*ll's distin- 
 guished 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 philo- 
 sophical 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 for- 
 tunately reminds us more of Mr. Russell's later clear and 
 
 1 [This Essay is also reprinted in Mr. Russell's Mysticism and Logic, 
 London and New York, 1918, pp. 46-57. ED.]
 
 4 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 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 pro- 
 positions, and we can see these views expressed in 
 Chapters III and XIX of the present work. These 
 chapters were apparently written before the Government,
 
 EDITOR'S NOTE 5 
 
 in the spring of 1910, arrived at the famous secret decision 
 that only " certain implications " are permitted in dis- 
 cussion. Naturally the secret decision gave rise to much 
 speculation among logicians as to which kinds of im- 
 plication 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 (i) 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.
 
 " Even a joke should have some meaning . . ." 
 
 (The Red Queen, T. L. G., p. 105).
 
 CONTENTS 
 
 MM 
 
 EDITOR'S NOTE 3 
 
 ABBREVIATIONS ....... 9 
 
 CHAPTER 
 
 I. THE INDEFINABIES OF LOGIC . . . . il 
 
 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 . ig 
 
 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 
 
 XL CRITICISM ........ 29 
 
 XII. HISTORICAL CRITICISM 30 
 
 XIII. IS THE MlND 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
 
 8 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 CHAPTER PACE 
 
 XXIII. DENOTING . . 53 
 
 XXIV. THE 54 
 
 XXV. NON-ENTITY 56 
 
 XXVI. Is 58 
 
 XXVII. AND AND OR 59 
 
 XXV1IT. 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 ARGU- 
 MENTS ........ 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, I-ondon, 
 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 Schroder: Vorlesungen Uber 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 g'.tr Psycho- 
 
 logie 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 begriffschtift- 
 
 lich abgeleitet, Jena, vol. i., 1893 ; vol. ii., 1903. 
 
 Gl. Gottlob Frege : Die Grundlagen der Arithmetik ; eine 
 
 logisch-mathematische Untersuchung ttber den Begriff der 
 Zahl, Breslau, 1884. 
 
 G. u. E. G. Heymans : Die Gesetze und Elemente des wisenschafllichen 
 
 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. 
 9
 
 io THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 H. S. Lewis Carroll : Tht Hunting of the Snark t an Agony in 
 Eight Fits, London, 1911. 
 
 M. The Monist t a Quarterly Magazine Devoted to Science and 
 
 Philosophy, Chicago and London. 
 
 Md. Mind t a Quarterly Review of Psychology and Philosophy, 
 
 Ixmdon 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 Mttaphysique 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-Giass, 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, 1 Drobisch, 
 Uberweg,* and Tweedledee. Tweedledee, it may be re- 
 membered,3 remarked that certain identities " are " logic. 
 Now, there is some doubt as to whether he, like Jevons,4 
 understood " are " to mean what mathematicians mean by 
 " =," or, h'ke Schroder 5 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, h'ke Jevons, 6 the mistaken 7 view 
 that the quantification of the predicate is relevant to sym- 
 bolic 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 mathe- 
 matics, 8 together with his synthesis of a cravat and a belt, 
 
 1 Russell, Ph. L., pp. 17, 19, 207-8. 
 
 * SchrOder, A . d. L,, i. p. 4. 
 
 * See Appendix A. This Appendix also illustrates the importance 
 attached to the Principle of Identity by the Professor and Bruno. 
 
 S. o. S. t pp. 9-15. 
 
 6 A. d. L., i. p. 132. 
 
 8 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. 
 
 7 S. L., 1881, pp. 173-5, 324-5 ; 1894, pp. 194-6. 
 
 ' 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.
 
 12 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 which usually serve different putposes,9 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. 10 
 But the question of Humpty-Dumpty's philosophical opinions 
 must be left for final verification to the historians of philo- 
 sophy : here I am only concerned with an a priori logical 
 construction of what his views might have been if they 
 formed a consistent whole. 11 
 
 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. 12 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,^ it can be proved formally that one foundation 
 of arithmetic is shattered. J 4 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 consis- 
 tency, and no such paradoxical result of further investi- 
 gations is to be feared. 
 
 9 See Appendix B. 
 
 10 [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.] 
 
 11 Cf. Ph. L., pp. v.-vi. 3. 
 
 12 Cf. Fieri, R. M. M., March 1906, p. 199. 
 
 13 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). 
 
 " See Frege, Gg., ii. p. 253.
 
 THE INDEFINABLES OF LOGIC 13 
 
 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. x 5 
 
 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. 16 
 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 " propo- 
 sition 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 Tweedle- 
 
 15 See Appendix E. 
 
 " See Frege, Gg. t i. p. 15.
 
 14 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 dum and Tweedledee, and even practical idealists like the 
 White Knight, explicitly accept laws like the law of identity 
 and the excluded middle. 1 ? 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. 18 An 
 apple is not confused with the eating of it except by savages, 
 idealists, and people who are too hungry to think. 
 
 17 See the above references and also Appendix F. 
 
 18 Cf. B. Russell, H. /., 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, hi a doubtful fashion, that she 
 thought that her mistress was in unless she was out. I con- 
 cluded 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 tha 
 identities are fundamental to all logic. We will now con 
 sider some other views of the value of identities. 
 
 Identities are frequently used in common life by peoph 
 who seem to imagine that they can draw important con 
 elusions respecting conduct or matters of fact from them 
 I have heard of a man who gained the double reputatioi 
 of being a philosopher and a fatalist by the repeated enun 
 ciation of the identity " Whatever will be, will be " ; anc 
 the Italian equivalent of this makes up an appreciable part 
 of one of Mr. Robert Hichens' novels. Further, the identit} 
 " Life is Life " has not only been often accepted as an ex^ 
 planation for a particular way of living but has even beer 
 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 dishonest} 
 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 intel- 
 ligent child as invalid x what is called the " West " does 
 in fact, merge into the " East." Even if we are to take 
 
 1 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. Traditiona 
 geography books, no doubt, gave rise to the opinions held by Lady 
 Blount and the Zetetic Society. 
 
 16
 
 IDENTITY 17 
 
 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 l that any true propo- 
 sition 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 O. 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 con- 
 clusion 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 
 Ibelief in the truth of his premisses by putting limits to the 
 -validity of the principle of identity. In the course of the 
 IDebate 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 
 iplausible opinion that a joke is a joke under all circumstances. 
 
 * 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 ac- 
 quired 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 satis- 
 faction 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 tilings which shocked her was 
 the class of actions in bad form. Consequently the state- 
 ment 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 uncon- 
 ventionality. 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. 
 

 
 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 Eihica l 
 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 descrip- 
 tion of Hudibras himself, Butler wrote : 
 
 He knew what's what, and that's as high 
 As metaphysic wit can fly. 
 
 1 Cambridge, 1903. 
 19
 
 20 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 I once conducted what I imagined to be an aesthetic 
 investigation for the purpose of discovery, by the continual 
 use of the word " Why ? " * 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 funda- 
 mental 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." 
 
 2 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 philo- 
 sophy, and especially by Kantians, that " in spite of its fame 
 we have found few occasions for its use." * 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 z by saying that that volume " is and is 
 not " a certain memoir of his which had been published in 
 1905. 
 
 1 Pa. Ma., p. 116. 
 
 J [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," * 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 speak- 
 ing the truth, and hold that " Paris " is a town containing 
 the letter "P." When Frege pointed out * this difficulty 
 he was almost universally denounced in Germany as " spitz- 
 findig." In fact, Germans seem to have been influenced 
 perhaps by that great contemner of " Spitz findigkeit," Kant, 
 to reject the White Knight's 3 distinctions between words 
 and their denotations and to regard subtlety with disfavour 
 to such a degree that their only mathematical logician except 
 Frege, namely Schroder the least subtle of mortals, by 
 the way seems to have been filled with such fear of being 
 
 1 Cf. Md, N. S., vol. xiv., October 1905, p. 486, 
 
 2 In Z. S., for example. 
 ' See Appendix G,
 
 SYMBOLISM AND MEANING 23 
 
 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." 4 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, 5 
 the bicycle, and the gramophone all of them instruments 
 of doubtful merit. 
 
 Cf. Chapter XXXVII below. 
 5 Cf. P. M., p. n, note.
 
 CHAPTER VIII 
 
 NOMINALISM 
 
 DE MORGAN l 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 com- 
 municative 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 Miiller'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, 2 
 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 
 
 1 F. L., pp. 246-7. 
 
 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. 
 
 24
 
 NOMINALISM 25 
 
 said what he meant, he plainly wished to pride himself on 
 certain virtues which he did not possess, and was not in- 
 different 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.
 
 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 com- 
 plicated arguments, but would prevent ambiguous arguments. 
 In denying the indispensability of Symbolic Logic in the 
 former state of things, Keynes I is probably alone, against 
 the need strongly felt by Alice when speaking to the Duchess, 2 
 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. 
 
 1 In his Fm. L. 
 
 2 See Appendix I.
 
 CHAPTER X 
 
 LOGICAL ADDITION AND THE UTILITY OF 
 SYMBOLISM 
 
 FREyuENTLY ordinary language contains subtle psycho- 
 logical 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 (i) 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 con- 
 clude 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 supre- 
 macy 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." l Cases (2) and (3) then each 
 give rise to one more case. Cases (i) 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." 
 
 Cf. Chapter XXIV below. 
 27
 
 28 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 On the other hand, certain descriptive phrases and certain 
 propositions can be expressed more shortly arid more accu- 
 rately 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 P * 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 some- 
 what different definitions and possibly increase the compli- 
 cations of our proposition, it must be remembered that, 
 on the other hand, we always reduce the number of symbols 
 hi 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." 3 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,4 
 and thus draw attention to what might otherwise easily 
 be overlooked. 
 
 1 C. S. Peirce's notation for the relation " converse of P." 
 8 Russell, Md., N. 3-, vol. xiv., October 1905, p. 482. 
 4 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 miscon- 
 ception as to the nature of criticism. 1 
 
 1 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 pi ices 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. 1 
 
 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 : 2 " 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." 
 
 1 W. W. Rouse Ball, A Short Account of the History of Mathematics, 
 4th edition, London, 1908, p. 109. 
 
 2 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. 
 
 / 
 
 1 
 
 , * +. ik 
 
 31 

 
 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 ; s 
 
 1 Both cases cannot occur ; the question is similar to that arising 
 in the discussion of the mortality of Socrates (see Chapter XXII). 
 
 3*
 
 THE PRAGMATIST THEORY OF TRUTH 33 
 
 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,* that what he said 
 three times is true, and to give results just as insecure. 
 
 1 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," l 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 hypothetical 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. 
 
 1 Md, N. S., vol. iv., 1895, PP- 278-80. Cf. Russell, P. M., p. 35. 
 
 34
 
 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. 
 
 15
 
 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 tres- 
 passers 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, 1 considered 
 universal propositions to be existential, like particular ones. 
 
 If the Gnat z 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 3 objected to 
 other symbolic logicians that their premisses imply that all 
 Centaurs are flower-pots, they could reply that their pre- 
 misses also imply the more usual view that Centaurs are 
 not flower-pots. 
 
 1 Cf., e.g., F. L., p. 4. 
 
 J See Appendix L. 
 
 8 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 1 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 propo- 
 sition 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 I 
 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, * though 
 one easy to commit, to assume that the cases : (i) all the 
 propositions of a certain class are true ; and (2) no propo- 
 sition of the class is true; are each other's contradictories. 
 However, in the modification 3 of Frege's symbolism which 
 was used by Russell 
 
 (i) is (*) x, 
 and (2) is (x) . not x ; 
 
 1 To which De Morgan drew attention in a letter ; see (Mrs.) S. E. 
 De Morgan, Memoir of Augustus De Morgan, London, 1882, p. 324. 
 
 * Pa. Ma., p. 16. 
 
 * 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. 
 
 37
 
 38 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 while the contradictory of (i) is : 
 
 not (x) . x. 
 
 The last line but one of the above verse may, then, be 
 written : 
 
 (t) . not (*) . not not (x, (], 
 
 where " $ (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 : 
 
 (0 . not (x) . $ (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 0, 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.
 
 CHAPTER XIX 
 
 IMPLICATION 
 
 A GOOD illustration of the fact that what is called " impli- 
 cation " 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. 1 
 
 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. 
 Bui 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. 
 
 1 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). 
 
 39
 
 40 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 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 any- 
 body 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 I. 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 * 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 neces- 
 sities 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. 
 
 1 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.
 
 IMPLICATION 41 
 
 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 in- 
 version ; 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 tune. 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 para- 
 doxes 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 manipu- 
 lation 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 ; 3 and is of great service 
 
 8 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.
 
 42 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 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 limita- 
 tion 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 prin- 
 ciple ; 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 4 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 un- 
 conscious but no less well-grounded fear of appearing 
 ridiculous if they dealt with such extreme cases as " the 
 series of all ordinal numbers." 5 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. 
 
 4 F. L., p. 264. 
 
 6 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 mathe- 
 matics 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 
 
 43
 
 44 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 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, con- 
 sidered as an argument to prove the conclusion, there is 
 a petitio principii. When we say 
 
 All men are mortal, 
 Socrates is a mail, 
 
 therefore 
 
 Socrates is mortal, 
 
 it is unanswerably urged by the adversaries of the syllogistic 
 theory, that the proposition, Socrates is mortal, is presup- 
 posed 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 
 
 45
 
 46 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 doubt which could affect any case comprised with it is dis- 
 pelled 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. 1 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 reason- 
 able 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 was 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 
 
 1 [The following passage is almost word for word the same as a 
 passage on pp. 123-5 f 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. ED.]
 
 THE SYNTHETIC NATURE OF DEDUCTION 47 
 
 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 prob- 
 ability 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 doubt- 
 fully 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. 2 
 
 [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 assump- 
 tion 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 l that there must have been either 
 a first moment of his non-existence or a last one of his exist- 
 ence, but not both ; just as there seems to be either a greatest 
 weight that a man can lift or a least weight that he cannot 
 lift, but not both. a 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-analyticai 
 
 1 By " Dedekind's Axiom," E. N., p. u. 
 vol. xx., 1910, pp. 134-5. 
 
 4 8 
 
 1 By 
 
 M.,
 
 THE MORTALITY OF SOCRATES 49 
 
 method to one of the problems of our knowledge of the 
 eternal world. 3 
 
 We will start from the generally accepted proposition that 
 all men are mortal. Clearly, if we could know each indi- 
 vidual 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 intro- 
 duce 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 pro- 
 fessions." 
 
 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 
 
 1 [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.] 
 
 4
 
 50 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 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 : (i) 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 propo- 
 sition 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 4 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 
 
 " 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' Enclyclopadia of Religion and Ethics, 
 vol. iv., Part 2, Edinburgh, 1912, p. 96. ED.] ; Robert Latta, Leibniz : 
 The Monadology and other Philosophical Writings, Oxford, 1 898, pp. 2 1 ff , 
 29 (note) ; Couturat, La Logique de Leibniz d'apres des documents intdits, 
 Paris, 1901, pp. 130, 132 ; and Russell, Ph. L., pp. 108-16, 243-9.
 
 THE MORTALITY OF SOCRATES 51 
 
 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. Tnis 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 Sep- 
 tember 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.5 
 
 [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 Black- 
 burn, in his decision (9 App. Cos., 371, 373) that a man born on
 
 52 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 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 possi- 
 bility of infinite number to conclude that Socrates, like the 
 worm, can never die. Thus is it quite plain that the diffi- 
 culties 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 argu- 
 ment 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 there- 
 fore 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. 
 
 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 con- 
 nected 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, 1 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 * 
 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 Tweededum's house leads " and " the house to 
 which the road that leads to Tweedledee 's house leads " have 
 different Sinn, but the same Bedeutung. 
 
 1 Cf. P. M., pp. 53-4. 
 9 See Appendix Mi 
 
 \ 
 
 )
 
 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. 1 People who refer to " the 
 Oxford Movement " imply that Oxford only moved once ; 
 and those quaint people who say that " A is quite the gentle- 
 man " 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. 2 
 
 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" * s a verv misleading equation, 
 
 1 Cf. MA., N. S., vol. xiv., 1905, pp. 481, 484. 
 ' Cf, ibid., p. 491, note.
 
 THE 55 
 
 arid what we really mean by that faultily abbreviated state- 
 ment 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 con- 
 versely), 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.
 
 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 ' 
 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 com- 
 plain, 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, 1 
 in round squares and fabulous monsters) against the dis- 
 honest 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," 3 
 and took up the position of the White King in opposition 
 to Alice's later assertions.4 
 
 Venn's experience illustrates another characteristic of 
 mathematical logic. It is necessary, in order to make our 
 
 1 5. L., 1881, p. 339, note. 
 
 2 See Appendix N. 
 
 8 Md., N. S., vol. xiv., October 1905, pp. 479-93. 
 See Appendix N. 
 
 56
 
 NON-ENTITY 57 
 
 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.5 
 1 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 : * "... 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." 
 
 Schroder J 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 *s or are where they cannot (such as : Englishmen 
 are Britons), but failed to see 3 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 
 Encyclopedia 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. 
 
 1 F. L., p. 268. 
 
 1 A. d. L., i. pp. 127 sqq. 
 
 * Ibid., vol. ii. pp. 461, 597. 
 
 58
 
 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 x 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 alter- 
 natives is almost implied. Thus, a reviewer in Mind* 
 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 psycho- 
 logists : 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. 
 
 1 Pure Logic . . ., London, 1864, pp. 76-9. Cf. Venn, S. L., 2nd ed., 
 pp. 40-8. 
 
 1 N. S., vol. iv. p. 261. 
 
 59
 
 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 ft, is called the converse of R. 
 As De Morgan * 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. . . ." 
 
 1 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 : ' 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 prin- 
 ciple 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 weak- 
 nesses 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 
 
 1 These principles, after many attempts to state them by Peacock 
 the Red and the White Queen (see Appendix P), Hankel, Schroder, 
 and Schubert had been made, were first precisely formulated by Frege- 
 in Z. S.; cf. also Chapter VII. 
 
 61
 
 62 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 Poincare", have produced such absurd discussions on the 
 fundamental principles of mathematics, 3 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 : i 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.3 
 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 4 and 
 modern mathematical logicians. 
 
 * See Couturat, It. M. M., vol. xiv., March, 1906, pp. 208-50, and 
 Russell, ibid., September, 1906, pp. 627-34. 
 3 See Appendix P. 
 See ibid.
 
 CHAPTER XXX 
 FINITE AND INFINITE 
 
 I WAS once shown a statement made by an eminent mathe- 
 matician 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 = <x> as a value of the variable."
 
 CHAPTER XXXI 
 
 THE MATHEMATICAL ATTAINMENTS OF 
 TRISTRAM SHANDY 
 
 TRISTRAM SHANDY r 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 3 ), apparently made I5A to result, and this 
 can hardly be maintained to be the half of loA. It is 
 possible, however, that Tristram Shandy succeeded in proving 
 the apparently paradoxical equation. 
 
 i5A = 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 3 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 
 
 1 Cf. a letter of De Morgan in Mrs. De Morgan's Memoir of Augustus 
 De Morgan, p. 324. 
 
 * Kant's tract was published in 1763, while Tristram Shandy was 
 published in 1760. 
 
 8 P. M., pp. 358-9 [Cf. M., vol. xxii., January 1912, p. 187. ED.] 
 
 6 4
 
 THE ATTAINMENTS OF TRISTRAM SHANDY 65 
 
 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. 4 " 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 denned 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, 5 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 
 
 ioA = 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. 6 
 
 Cf. P. M., pp. 350, 358-9 ; M., vol. xxii., 1912, p. 157. 
 
 [Cf. for example, Cosmo Guastella, Dell' infinite, Palermo, 1912. 
 ED.] 
 
 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 fortu- 
 nately 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 philo- 
 sophers 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 
 
 66
 
 AN UNLIMITED INCOME 67 
 
 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 
 paiis 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 Konig 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
 
 68 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 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 sup- 
 pose, 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 Schroder- 
 Bernstein theorem was used, long before Bernstein or 
 Schroder, by Edward Thurlow, afterward the law-lord Lord 
 Thurlow, when an undergraduate of Caius College, Cam- 
 bridge. 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 Schroder-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, 1 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 Kebaldnvayi, i.e. covers the whole 
 sphere of thought and existence. . . . ' Knowable ' and 
 ' Nameable ' are among the examples of Kebaldnvayi 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 speak- 
 ing 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 : 3 
 
 "It is doubtless true that saying what a thing is not, is, 
 in some measure, saying what it is ; ... Hence it cannot 
 
 1 Md., N. S., vol. iv., 1895, p. 168. 
 
 * First Principles, 6th ed., 1900, pp. 107-10. The first edition was 
 published in 1862. 
 
 70
 
 THE UNKNOWABLE 71 
 
 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 3 description of the argu- 
 ments 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, 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. 4 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 ") 
 
 * Note on p. 6 of his paper : " On Infinity ; and on the Sign of 
 Equality," Trans. Camb. Phil. Soc. t vol. xi., part i., pp. 1-45 (read 
 May 1 6, 1864). 
 
 * The assertion of the finitude of a man's mind appears to be non- 
 sense ; 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.).
 
 72 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 are analogous to the difficulty arising from a man's state- 
 ment : " I am lying." 5 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 : 6 
 
 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 Person, 
 who remarked that all Germans are ignorant of Greek metres, 
 
 All, save only Hermann ; 
 And Hermann's a German. 
 
 6 Russell, R. M. M., vol. xiv., September 1906, pp. 632-3, 640-4. 
 6 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 incomprehen- 
 sible," and so on, they are not falling into the same error 
 as Mr. Spencer, for the Latin equivalent for " incompre- 
 hensible " is merely " immensus," and Bishop Hilsey trans- 
 lated it more correctly as " immeasurable." ' It is a 
 regrettable fact that Dr. Blunt, 1 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 state- 
 ments 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. 
 
 1 A. C. P., p. 217. 
 8 Ibid., p. 2 1 3. 
 
 73
 
 CHAPTER XXXVI 
 
 THE HUMOUR OF MATHEMATICIANS 
 
 BRAHMAGUPTA'S problem x appears to be the earliest instance 
 of a kind of joke which has been much used by mathe- 
 maticians. 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. Stackel 2 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 mathe- 
 maticians has removed this danger. Still to many mathe- 
 maticians 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. 
 
 1 See Chapter XII. 
 
 * Encykl. der math. Wiss., vol. iv., part i., p. 474. 
 
 74
 
 CHAPTER XXXVII 
 THE PARADOXES OF LOGIC 
 
 WE have already l 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 iiber Mengenlehre* Here 3 a 
 battle-cry in italics 
 
 " Against all resignation, but also against all scholasticism I " 
 
 found utterance. Later on, Schoenflies 4 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, 5 blinded 
 
 1 Chapters VII and XXXVI. 
 
 1 Die Entwickelung der Lehre von den Punktmannigfaltigkeiten. 
 Bericht, erstattet der deutschen Mathematiker-Vereinigung, Leipzig, 
 1908. 
 
 Ibid., p. 7. The battle-cry is : " Gegen jede Resignation, aber 
 auch gegen jede Scholastik I " 
 
 4 " 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 : " Fur den Cantorismus aber gegen deu Russell- 
 ismus ! " 
 
 6 Ibid., p. 251. " Es ist also," he exclaims with the eloquence of 
 emotion and the emotion of eloquence, " nicht die Geringschatzung 
 der Philosophic, die mich dabei treibt, sondern die Liebe zur Mathe- 
 matik ; . . ." 
 
 75
 
 76 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 by their rather uncritical love of mathematics. Thus it 
 comes about that Schoenflies 6 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.7 
 
 Schoenflies 8 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 atten- 
 tion 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 main- 
 tain 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 9 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. 10 
 
 6 " Ueber die Stellung der Definition in der Axiomatik, " Jahresber, 
 der deutsch. Math.-Ver., vol. xx., 1911, p. 251. 
 
 7 [Cf. for this, M., vol. xxii., January 1912, pp. 149-58. ED.] 
 
 8 Bericht, 1908, p. 76, note ; cf. p. 72. 
 
 * E.g. in F. Hausdorff's review of Russell's Principles of 1903 in 
 the Vierteljahrsschr. Jiir wiss. Philos. und Soziologie. 
 10 [Cf. M., vol. xxv., 1915, pp. 333-8. ED.]
 
 THE PARADOXES OF LOGIC 77 
 
 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 " 
 " one might as well, in talking to a man with a long nose, 
 say : ' When I speak of noses. I except such as are inordin- 
 ately 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 " of the theory of aggre- 
 gates 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, '3 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 continu- 
 ation, 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, lightly 
 showed great astonishment that butter did not help to cure 
 
 11 Russell, A. ]. M., vol. xxx., 1908, p. 226. 
 11 Loc. cit., p. 222. 
 " See Appendix Q.
 
 78 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 both hunger and watches that would not go.M 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. *s 
 
 14 See "Appendix R. 
 16 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, 
 Konig, 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 prin- 
 ciple 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 : x " 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 prin- 
 ciple 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, 
 1 Pa. Ma., p. 40. 
 
 79
 
 8o THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 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 falla- 
 ciously 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 prepositional function <p(x) 
 there is a range of values of x for which fy(x) has a sig- 
 nification 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 prepositional function possessed the corre- 
 sponding property or not. Thus, Jevons, in early days, 2 
 was of opinion that virtue is neither black nor not-black 
 because it is not coloured, but rather later 3 he admitted that 
 virtue is not triangular.4 
 
 a s. o. S. pp. 36-7-. 
 
 3 E. L. L., pp. 120-1. 
 
 4 [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 uor false^ . . ." EEK]
 
 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 suc- 
 ceeded, 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, 
 lie would reflect and afterwards pronounce on the ques- 
 tion 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. 1 The person (A) who told this joke told 
 the story of how the joke was received to another Scotchman 
 thereby making a joke about a joke of the first order, and 
 thus making a joke of the second order. A remarked on 
 this joke that no joke could penetrate the head of the 
 Scotchman to whom the joke of the first order was told, 
 even if it were fired into his head with a gun. The Scotch- 
 
 1 [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.] 
 
 6 8l
 
 82 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 man, after severe thought, replied : " But ye couldn't do 
 that, ye know ! " A repeated the whole story, which 
 constituted a joke of the third order, to a third Scotchman. 
 This last Scotchman again, after prolonged thought, replied ; 
 "He had ye there!" This whole story is a joke of the 
 fourth order. 
 
 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 trans- 
 finite order presumably only excite the inaudible laughter 
 of the gods.
 
 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. 
 
 83
 
 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, 1 " 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 
 1 Md., N. S., vol. x., July, 1901, pp. 313-14.
 
 ABSOLUTE AND RELATIVE POSITION 85 
 
 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 ; * but also that the foundations 
 of logic are inductive in their nature. 3 " We cannot," says 
 Mill,4 " 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." 
 
 1 J. B. Stallo, The Concepts and Theories of Modern Physics, 4th ed., 
 London, 1900, pp. 217-27. 
 
 3 Ibid., pp. 140-4. 
 
 Examination of the Philosophy of Sir William Hamilton, vol. i. 
 p. 88, Amer. ed.
 
 CHAPTER XLII 
 
 LAUGHTER 
 
 [Ix seemed advisable to give here T 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 3 of 
 M. Bergson's book on Laughter,! 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 unin- 
 telligible 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 
 
 1 From a remark on p. 47 above, it is evident that Mr. R*ss*ll 
 intended to write some such chapter as this. 
 
 * The Professor's Guide to Laughter, The Cambridge Pevieiv, vol. 
 xxxii., 1912, pp. 193-4. 
 
 3 Laughter, an Essay on the Meaning of the Comic, English transla- 
 tion by C. Brereton and F. Rothwell, London, 1911. 
 
 86
 
 LAUGHTER 87 
 
 movement is even more completely mechanical. When the 
 clown makes a bad joke for the first time, we keep our coun- 
 tenance, 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*U would hardly have committed. ED.]
 
 CHAPTER XLIII 
 
 "GEDANKENEXPERIMENTE" AND EVOLUTIONARY 
 
 ETHICS 
 
 THE " Gedankenexperimente," upon which so much weight 
 has been laid by Mach I and Heymans, 2 had already been 
 investigated by the White Queen.3 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 his- 
 torical 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. 
 
 1 See, e.g., E. u. I., pp. 183-200. 
 
 * G. u. E., vol. i.{ 
 
 * 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 1 ' " 
 
 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. I.. 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, 1 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.' "
 
 90 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 E. CONFORMITY OF A PARADOXICAL LOGIC WITH 
 
 COMMON-SENSE. 
 T. L. G., p. 100 : 
 
 " But I was thinking of a plan 
 To dye OPC'S whiskers green. 
 And always use so large a Ian 
 That they couid 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 Gale ". . . .' "
 
 APPENDIXES 91 
 
 H. NOMINALISM. 
 
 A. A. W. t 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 " I ' 
 
 " ' It is the same thing with you,' said the Hatter ; and here the 
 conversation dropped, ..." 
 
 I. UTILITY OF SYMBOLIC LOGIC. 
 
 A. A. W. t 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 I Keep count, I entreat ; 
 You will find I have told it you twice. 
 'Tis the song of the Jubjub 1 The proof is complete, 
 If only I've stated it thrice."
 
 92 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 L. UNIVERSAL AND PARTICULAR PROPOSITIONS. 
 
 T. L. G., p. 40 : The Gnat had told Alice that the Bread-and-butter- 
 fly 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 / 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 / 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 I 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."
 
 APPENDIXES 93 
 
 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.' 
 
 1 ' Nine from eight I can't, you know,' Alice replied very readily 
 ' but ' 
 
 ' ' She can't do Substraction.' said the White Qneen." 
 
 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.' 
 
 1 ' 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 arc put out here ? ' she asked. 
 
 4 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.'
 
 94 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 " ' 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).
 
 APPENDIXES 95 
 
 " ' 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 ' 
 
 1 ' It doesn't prove anything of the sort 1 ' said Alice. ' Why, you 
 don't even know what they're about I ' 
 
 ' 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 I 
 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 inter- 
 rupting him), ' I'll give him sixpence. / don't believe there's an atom 
 of meaning in it.'
 
 96 THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL 
 
 " 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 ' // she should push the matter on ' that must be the 
 Queen ' What would become of you ? ' What indeed ! ' / 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 1 ' said the Queen. ' Sentence first verdict afterwards.' 
 
 " ' Stuff and nonsense ! ' said Alice loudly. ' The idea of having 
 the sentence first 1 ' 
 
 " ' 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
 
 > 
 
 THE LIBRARY 
 UNIVERSITY OF CALIFORNIA 
 
 Santa Barbara 
 
 THIS BOOK IS DUE ON THE LAST DATE 
 STAMPED BELOW.