{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_38430/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_38430","id":"gutenberg_38430","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/38430","summary":null,"tags":["Category: Philosophy & Ethics","Russell, Bertrand, 1872-1970"],"text":"RTR*ND R*SS*LL ***\n\n\n\nProduced by Adrian Mastronardi and the Online Distributed\nProofreading Team at http://www.pgdp.net (This file was\nproduced from images generously made available by The\nInternet Archive/Canadian Libraries)\n\n\n\n\n\n\n\n\n\n  THE PHILOSOPHY OF\n  MR. B*RTR*ND R*SS*LL\n\n  WITH AN APPENDIX OF LEADING\n  PASSAGES FROM CERTAIN OTHER WORKS\n\n  EDITED BY\n\n  PHILIP E. B. JOURDAIN\n\n  [Illustration]\n\n  LONDON: GEORGE ALLEN & UNWIN LTD.\n  RUSKIN HOUSE 40 MUSEUM STREET, W.C. 1\n  CHICAGO: THE OPEN COURT PUBLISHING CO.\n\n\n\n  _First published in 1918_\n\n  (_All rights reserved_)\n\n\n\n\nEDITOR'S NOTE\n\n\nWhen Mr. B*rtr*nd R*ss*ll, following the advice of Mr. W*ll**m J*m*s,\nagain \"got into touch with reality\" and in July 1911 was torn to pieces\nby Anti-Suffragists, many of whom were political opponents of Mr.\nR*ss*ll and held strong views on the Necessity of Protection of Trade\nand person, a manuscript which was almost ready for the press was\nfortunately saved from the flames on the occasion when a body of eager\nchampions of the Sacredness of Personal Property burnt the late Mr.\nR*ss*ll's house. This manuscript, together with some further fragments\nfound in the late Mr. R*ss*ll's own interleaved copy of his _Prayer-Book\nof Free Man's Worship_, which was fortunately rescued with a few of the\ngreat author's other belongings, was first given to the world in the\n_Monist_ for October 1911 and January 1916, and has here been arranged\nand completed by some other hitherto undecipherable manuscripts. The\ntitle of the above-mentioned _Prayer-Book_, it may perhaps be mentioned,\nwas apparently suggested to Mr. R*ss*ll by that of the Essay on\n\"The Free Man's Worship\" in the _Philosophical Essays_ (London, 1910,\npp. 59-70[1]) of Mr. R*ss*ll's distinguished contemporary, Mr. Bertrand\nRussell, from whom much of Mr. R*ss*ll's philosophy was derived. And,\nindeed, the influence of Mr. Russell extended even beyond philosophical\nviews to arrangement and literary style. The method of arrangement of\nthe present work seems to have been borrowed from Mr. Russell's\n_Philosophy of Leibniz_ of 1900; in the selection of subjects dealt\nwith, Mr. R*ss*ll seems to have been guided by Mr. Russell's _Principles\nof Mathematics_ of 1903; while Mr. R*ss*ll's literary style fortunately\nreminds us more of Mr. Russell's later clear and charming subtleties\nthan his earlier brilliant and no less subtle obscurities. But, on the\nother hand, some important points of Mr. Russell's doctrine, which first\nappeared in books published after Mr. R*ss*ll's death, were anticipated\nin Mr. R*ss*ll's notes, and these anticipations, so interesting for\nfuture historians of philosophy, have been provided by the editor with\nreferences to the later works of Mr. Russell. All editorial notes are\nenclosed in square brackets, to indicate that they were not written by\nthe late Mr. R*ss*ll.\n\nAt the present time we have come to take a calm view of the question so\nmuch debated seven years ago as to the legitimacy of logical arguments\nin political discussions. No longer, fortunately, can that intense\nfeeling be roused which then found expression in the famous cry,\n\"Justice--right or wrong,\" and which played such a large part in the\npolitics of that time. Thus it will not be out of place in this\nunimpassioned record of some of the truths and errors in the world to\nrefer briefly to Mr. R*ss*ll's short and stormy career. Before he was\ntorn to pieces, he had been forbidden to lecture on philosophy or\nmathematics by some well-intentioned advocates of freedom in speech who\nthought that the cause of freedom might be endangered by allowing Mr.\nR*ss*ll to speak freely on points of logic, on the grounds, apparently,\nthat logic is both harmful and unnecessary and might be applied to\npolitics unless strong measures were taken for its suppression. On much\nthe same grounds, his liberty was taken from him by those who remarked\nthat, if necessary, they would die in defence of the sacred principle of\nliberty; and it was in prison that the greater part of the present work\nwas written. Shortly after his liberation, which, like all actions of\npublic bodies, was brought about by the combined honour and interests of\nthose in authority, occurred his lamentable death to which we have\nreferred above.\n\nMr. R*ss*ll maintained that the chief use of \"implication\" in politics\nis to draw conclusions, which are thought to be true, and which are\nconsequently false, from identical propositions, and we can see these\nviews expressed in Chapters III and XIX of the present work. These\nchapters were apparently written before the Government, in the spring\nof 1910, arrived at the famous secret decision that only \"certain\nimplications\" are permitted in discussion. Naturally the secret decision\ngave rise to much speculation among logicians as to which kinds of\nimplication were barred, and Mr. R*ss*ll and Mr. Bertrand Russell had\nmany arguments on the subject, which naturally could not be published at\nthe time. However, after Mr. R*ss*ll's death, successive prosecutions\nwhich were made by the Government at last made it quite clear that the\nopinion held by Mr. R*ss*ll was the correct one. There had been numerous\nprosecutions of people who, from true but not identical premisses, had\ndeduced true conclusions, so that the possible legitimate forms of\n\"implication\" were reduced. Further, the other doubtful cases were\ncleared up in course of time by the prosecution of (1) members of the\nAristotelian Society for deducing true conclusions from false premisses;\n(2) members of the _Mind_ Association for deducing false conclusions\nfrom false premisses; and also by the attempted prosecution of an\neminent lady for deducing true conclusions from identities. Fortunately\nthis lady was able to defend herself successfully by pleading that one\neminent philosopher believed them to be true--which, of course, means\nthat the conclusions are false. Thus appeared the true nature of\nlegitimate political arguments.\n\nFOOTNOTES:\n\n[1] [This Essay is also reprinted in Mr. Russell's _Mysticism and\nLogic_, London and New York, 1918, pp. 46-57.--ED.]\n\n\n\n\n    \"Even a joke should have some meaning....\"\n\n                              (The Red Queen, _T. L. G._, p. 105).\n\n\n\n\nCONTENTS\n\n\n                                                                PAGE\n  EDITOR'S NOTE                                                    3\n  ABBREVIATIONS                                                    9\n  CHAPTER\n      I. THE INDEFINABLES OF LOGIC                                11\n     II. OBJECTIVE VALIDITY OF THE \"LAWS OF THOUGHT\"              15\n    III. IDENTITY                                                 16\n     IV. IDENTITY OF CLASSES                                      18\n      V. ETHICAL APPLICATIONS OF THE LAW OF IDENTITY              19\n     VI. THE LAW OF CONTRADICTION IN MODERN LOGIC                 21\n    VII. SYMBOLISM AND MEANING                                    22\n   VIII. NOMINALISM                                               24\n     IX. AMBIGUITY AND SYMBOLIC LOGIC                             26\n      X. LOGICAL ADDITION AND THE UTILITY OF SYMBOLISM            27\n     XI. CRITICISM                                                29\n    XII. HISTORICAL CRITICISM                                     30\n   XIII. IS THE MIND IN THE HEAD?                                 31\n    XIV. THE PRAGMATIST THEORY OF TRUTH                           32\n     XV. ASSERTION                                                34\n    XVI. THE COMMUTATIVE LAW                                      35\n   XVII. UNIVERSAL AND PARTICULAR PROPOSITIONS                    36\n  XVIII. DENIAL OF GENERALITY AND GENERALITY OF DENIAL            37\n    XIX. IMPLICATION                                              39\n     XX. DIGNITY                                                  43\n    XXI. THE SYNTHETIC NATURE OF DEDUCTION                        45\n   XXII. THE MORTALITY OF SOCRATES                                48\n    XXIII. DENOTING                                               53\n     XXIV. THE                                                    54\n      XXV. NON-ENTITY                                             56\n     XXVI. IS                                                     58\n    XXVII. AND AND OR                                             59\n   XXVIII. THE CONVERSION OF RELATIONS                            60\n     XXIX. PREVIOUS PHILOSOPHICAL THEORIES OF MATHEMATICS         61\n      XXX. FINITE AND INFINITE                                    63\n     XXXI. THE MATHEMATICAL ATTAINMENTS OF TRISTRAM SHANDY        64\n    XXXII. THE HARDSHIPS OF A MAN WITH AN UNLIMITED INCOME        66\n   XXXIII. THE RELATIONS OF MAGNITUDE OF CARDINAL NUMBERS         69\n    XXXIV. THE UNKNOWABLE                                         70\n     XXXV. MR. SPENCER, THE ATHANASIAN CREED, AND THE ARTICLES    73\n    XXXVI. THE HUMOUR OF MATHEMATICIANS                           74\n   XXXVII. THE PARADOXES OF LOGIC                                 75\n  XXXVIII. MODERN LOGIC AND SOME PHILOSOPHICAL ARGUMENTS          79\n    XXXIX. THE HIERARCHY OF JOKES                                 81\n       XL. THE EVIDENCE OF GEOMETRICAL PROPOSITIONS               83\n      XLI. ABSOLUTE AND RELATIVE POSITION                         84\n     XLII. LAUGHTER                                               86\n    XLIII. \"GEDANKENEXPERIMENTE\" AND EVOLUTIONARY ETHICS          88\n           APPENDIXES                                             89\n\n\n\n\nABBREVIATIONS\n\n\n  _A. A. W._  Lewis Carroll: _Alice's Adventures in Wonderland_, London,\n              1908. [This book was first published much earlier, but\n              this was the edition used by Mr. R*ss*ll. The same applies\n              to _H. S._ and _T. L. G._]\n\n  _A. C. P._  John Henry Blunt (ed. by): _The Annotated Book of Common\n              Prayer_, London, new edition, 1888.\n\n  _A. d. L._  Ernst Schroeder: _Vorlesungen ueber die Algebra der Logik,\n              Leipzig_, vol. i., 1890; vol. ii. (two parts), 1891 and\n              1905; vol. iii.: _Algebra und Logik der Relative_, 1895.\n\n  _E. N._     Richard Dedekind: _Essays on the Theory of Numbers_,\n              Chicago and London, 1901.\n\n  _E. L. L._  William Stanley Jevons: _Elementary Lessons in Logic,\n              Deductive and Inductive. With copious Questions and\n              Examples, and a Vocabulary of Logical Terms_, London,\n              24th ed., 1907 [first published in 1870].\n\n  _E. u. I._  Ernst Mach: _Erkenntnis und Irrtum: Skizzen zur\n              Psychologie der Forschung_, Leipzig, 1906.\n\n  _F. L._     Augustus De Morgan: _Formal Logic: or The Calculus of\n              Inference, Necessary and Probable_, London, 1847.\n\n  _Fm. L._    John Neville Keynes: _Studies and Exercises in Formal\n              Logic_, 4th ed., London, 1906.\n\n  _Gg._       Gottlob Frege: _Grundgesetze der Arithmetik\n              begriffschriftlich abgeleitet_, Jena, vol. i., 1893;\n              vol. ii., 1903.\n\n  _Gl._       Gottlob Frege: _Die Grundlagen der Arithmetik: eine\n              logisch-mathematische Untersuchung ueber den Begriff der\n              Zahl_, Breslau, 1884.\n\n  _G. u. E._  G. Heymans: _Die Gesetze und Elemente des\n              wisenschaftlichen Denkens_, Leiden, vol. i., 1890;\n              vol. ii., 1894.\n\n  _H. J._     _The Hibbert Journal: a Quarterly Review of Religion,\n              Theology and Philosophy_, London and New York.\n\n  _H. S._     Lewis Carroll: _The Hunting of the Snark: an Agony in\n              Eight Fits_, London, 1911.\n\n  _M._        _The Monist: a Quarterly Magazine Devoted to Science and\n              Philosophy_, Chicago and London.\n\n  _Md._       _Mind: a Quarterly Review of Psychology and Philosophy_,\n              London and New York.\n\n  _Pa. Ma._   Alfred North Whitehead and Bertrand Russell: _Principia\n              Mathematica_, vol. i., Cambridge, 1910. [Other volumes\n              were published in 1912 and 1913.]\n\n  _P. E._     Bertrand Russell: _Philosophical Essays_, London and New\n              York, 1910.\n\n  _Ph. L._    Bertrand Russell: _A Critical Exposition of the Philosophy\n              of Leibniz, with an Appendix of Leading Passages_,\n              Cambridge, 1900.\n\n  _P. M._     Bertrand Russell: _The Principles of Mathematics_,\n              vol. i., Cambridge, 1903.\n\n  _R. M. M._  _Revue de Metaphysique et de Morale_, Paris.\n\n  _S. B._     Lewis Carroll: _Sylvie and Bruno_, London, 1889.\n\n  _S. L._     John Venn: _Symbolic Logic_, London, 1881; 2nd ed., 1894.\n\n  _S. o. S._  William Stanley Jevons: _The Substitution of Similars, the\n              True Principle of Reasoning derived from a Modification of\n              Aristotle's Dictum_, London, 1869.\n\n  _T. L. G._  Lewis Carroll: _Through the Looking-Glass, and what Alice\n              found there_, London, 1911.\n\n  _Z. S._     Gottlob Frege: _Ueber die Zahlen des Herrn H. Schubert_,\n              Jena, 1899.\n\n\n\n\nCHAPTER I\n\nTHE INDEFINABLES OF LOGIC\n\n\nThe view that the fundamental principles of logic consist solely of the\nlaw of identity was held by Leibniz,[2] Drobisch, Uberweg,[3] and\nTweedledee. Tweedledee, it may be remembered,[4] remarked that certain\nidentities \"are\" logic. Now, there is some doubt as to whether he, like\nJevons,[5] understood \"are\" to mean what mathematicians mean by \"=,\" or,\nlike Schroeder[6] and most logicians, to have the same meaning as the\nrelation of subsumption. The first alternative alone would justify our\ncontention; and we may, I think, conclude from an opposition to\nauthority that may have been indicated by Tweedledee's frequent use of\nthe word \"contrariwise\" that he did not follow the majority of\nlogicians, but held, like Jevons,[7] the mistaken[8] view that the\nquantification of the predicate is relevant to symbolic logic.\n\nIt may be mentioned, by the way, that it is probable that\nHumpty-Dumpty's \"is\" is the \"is\" of identity. In fact, it is not\nunlikely that Humpty-Dumpty was a Hegelian; for, although his ability\nfor clear explanation may seem to militate against this, yet his\ninability to understand mathematics,[9] together with his synthesis of a\ncravat and a belt, which usually serve different purposes,[10] and his\nproclivity towards riddles seem to make out a good case for those who\nhold that he was in fact a Hegelian. Indeed, riddles are very closely\nallied to puns, and it was upon a pun, consisting of the confusion of\nthe \"is\" of predication with the \"is\" of identity--so that, for example,\n\"Socrates\" was identified with \"mortal\" and more generally the\nparticular with the universal--that Hegel's system of philosophy was\nfounded.[11] But the question of Humpty-Dumpty's philosophical opinions\nmust be left for final verification to the historians of philosophy:\nhere I am only concerned with an _a priori_ logical construction of what\nhis views might have been if they formed a consistent whole.[12]\n\nIf the principle of identity were indeed the sole principle of logic,\nthe principles of logic could hardly be said to be, as in fact they are,\na body of propositions whose consistency it is impossible to prove.[13]\nThis characteristic is important and one of the marks of the greatest\npossible security. For example, while a great achievement of late years\nhas been to prove the consistency of the principles of arithmetic, a\nscience which is unreservedly accepted except by some empiricists,[14]\nit can be proved formally that one foundation of arithmetic is\nshattered.[15] It is true that, quite lately, it has been shown that\nthis conclusion may be avoided, and, by a re-moulding of logic, we can\ndraw instead the paradoxical conclusion that the opinions held by\ncommon-sense for so many years are, in part, justified. But it is quite\ncertain that, with the principles of logic, no such proof of\nconsistency, and no such paradoxical result of further investigations is\nto be feared.\n\nStill, this re-moulding has had the result of bringing logic into a\nfuller agreement with common-sense than might be expected. There were\nonly two alternatives: if we chose principles in accordance with\ncommon-sense, we arrived at conclusions which shocked common-sense; by\nstarting with paradoxical principles, we arrived at ordinary\nconclusions. Like the White Knight, we have dyed our whiskers an unusual\ncolour and then hidden them.[16]\n\nThe quaint name of \"Laws of Thought,\" which is often applied to the\nprinciples of Logic, has given rise to confusion in two ways: in the\nfirst place, the \"Laws,\" unlike other laws, cannot be broken, even in\nthought; and, in the second place, people think that the \"Laws\" have\nsomething to do with holding for the operations of their minds, just as\nlaws of nature hold for events in the world around us.[17] But that the\nlaws are not psychological laws follows from the facts that a thing may\nbe true even if nobody believes it, and something else may be false if\neverybody believes it. Such, it may be remarked, is usually the case.\n\nPerhaps the most frequent instance of the assumption that the laws of\nlogic are mental is the treatment of an identity as if its validity were\nan affair of our permission. Some people suggest to others that they\nshould \"let bygones be bygones.\" Another important piece of evidence\nthat the truth of propositions has nothing to do with mind is given by\nthe phrase \"it is morally certain that such-and-such a proposition is\ntrue.\" Now, in the first place, morality, curiously enough, seems to be\nclosely associated with mental acts: we have professorships and\nlectureships of, and examinations in, \"mental and moral philosophy.\" In\nthe second place, it is plain that a \"morally certain\" proposition is a\nhighly doubtful one. Thus it is as vain to expect any information about\nour minds from a study of the \"Laws of Thought\" as it would be to expect\na description of a certain social event from Miss E. E. C. Jones's book\n_An Introduction to General Logic_.\n\nFortunately, the principles or laws of Logic are not a matter of\nphilosophical discussion. Idealists like Tweedledum and Tweedledee, and\neven practical idealists like the White Knight, explicitly accept laws\nlike the law of identity and the excluded middle.[18] In fact,\nthroughout all logic and mathematics, the existence of the human or any\nother mind is totally irrelevant; mental processes are studied by means\nof logic, but the subject-matter of logic does not presuppose mental\nprocesses, and would be equally true if there were no mental processes.\nIt is true that, in that case, we should not know logic; but our\nknowledge must not be confounded with the truths which we know.[19] An\napple is not confused with the eating of it except by savages,\nidealists, and people who are too hungry to think.\n\nFOOTNOTES:\n\n[2] Russell, _Ph. L._, pp. 17, 19, 207-8.\n\n[3] Schroeder, _A. d. L._, i. p. 4.\n\n[4] See Appendix A. This Appendix also illustrates the importance\nattached to the Principle of Identity by the Professor and Bruno.\n\n[5] _S. o. S._, pp. 9-15.\n\n[6] _A. d. L._, i. p. 132.\n\n[7] Cf., besides the reference in the last note but one, _E. L. L._, pp.\n183, 191. \"Contrariwise,\" it may be remarked, is not a term used in\ntraditional logic.\n\n[8] _S. L._, 1881, pp. 173-5, 324-5; 1894, pp. 194-6.\n\n[9] Cf. Appendix C, and William Robertson Smith, \"Hegel and the\nMetaphysics of the Fluxional Calculus,\" _Trans. Roy. Soc., Edinb._, vol.\nxxv., 1869, pp. 491-511.\n\n[10] See Appendix B.\n\n[11] [This is a remarkable anticipation of the note on pp. 39-40 of Mr.\nRussell's book, published about three years after the death of Mr.\nR*ss*ll, and entitled _Our Knowledge of the External World as a Field\nfor Scientific Method in Philosophy_, Chicago and London, 1914.--ED.]\n\n[12] Cf. _Ph. L._, pp. v.-vi. 3.\n\n[13] Cf. Pieri, _R. M. M._, March 1906, p. 199.\n\n[14] As a type of these, Humpty-Dumpty, with his inability to admit\nanything not empirically given and his lack of comprehension of pure\nmathematics, may be taken (see Appendix C). In his (correct) thesis that\ndefinitions are nominal, too, Humpty-Dumpty reminds one of J. S. Mill\n(see Appendix D).\n\n[15] See Frege, _Gg._, ii. p. 253.\n\n[16] See Appendix E.\n\n[17] See Frege, _Gg._, i. p. 15.\n\n[18] See the above references and also Appendix F.\n\n[19] Cf. B. Russell, _H. J._, July 1904, p. 812.\n\n\n\n\nCHAPTER II\n\nOBJECTIVE VALIDITY OF THE \"LAWS OF THOUGHT\"\n\n\nI once inquired of a maid-servant whether her mistress was at home. She\nreplied, in a doubtful fashion, that she _thought_ that her mistress was\nin unless she was out. I concluded that the maid was uncertain as to the\nobjective validity of the law of excluded middle, and remarked that to\nher mistress. But since I used the phrase \"laws of thought,\" the\nmistress perhaps supposed that a \"law of thought\" has something to do\nwith thinking and seemed to imagine that I wished to impute to the maid\nsome moral defect of an unimportant nature. Thus she remonstrated with\nme in an amused way, since she probably imagined that I meant to find\nfault with the maid's capacity for thinking.\n\n\n\n\nCHAPTER III\n\nIDENTITY\n\n\nIn the first chapter we have noticed the opinion that identities are\nfundamental to all logic. We will now consider some other views of the\nvalue of identities.\n\nIdentities are frequently used in common life by people who seem to\nimagine that they can draw important conclusions respecting conduct or\nmatters of fact from them. I have heard of a man who gained the double\nreputation of being a philosopher and a fatalist by the repeated\nenunciation of the identity \"Whatever will be, will be\"; and the Italian\nequivalent of this makes up an appreciable part of one of Mr. Robert\nHichens' novels. Further, the identity \"Life is Life\" has not only been\noften accepted as an explanation for a particular way of living but has\neven been considered by an authoress who calls herself \"Zack\" to be an\nappropriate title for a novel; while \"Business is Business\" is\nfrequently thought to provide an excuse for dishonesty in trading, for\nwhich purpose it is plainly inadequate.\n\nAnother example is given by a poem of Mr. Kipling, where he seems to\nassert that \"East is East\" and \"West is West\" imply that \"never the\ntwain shall meet.\" The conclusion, now, is false; for, since the world\nis round--as geography books still maintain by arguments which strike\nevery intelligent child as invalid[20]--what is called the \"West\" does,\nin fact, merge into the \"East.\" Even if we are to take the statement\nmetaphorically, it is still untrue, as the Japanese nation has shown.\n\nThe law-courts are often rightly blamed for being strenuous opponents of\nthe spread of modern logic: the frequent misuse of _and_, _or_, _the_,\nand _provided that_ in them is notorious. But the fault seems partly to\nlie in the uncomplicated nature of the logical problems which are dealt\nwith in them. Thus it is no uncommon thing for somebody to appear there\nwho is unable to establish his own identity, or for A to assert that B\nwas \"not himself\" when he made a will leaving his money to C.\n\nThe chief use of identities is in implication. Since, in logic, we so\nunderstand _implication_[21] that any true proposition implies and is\nimplied by any other true proposition; if one is convinced of the truth\nof the proposition Q, it is advisable to choose one or more identities\nP, whose truth is undoubted, and say that P implies Q. Thus, Mr. Austen\nChamberlain, according to _The Times_ of March 27, 1909, professed to\ndeduce the conclusion that it is not right that women should have votes\nfrom the premisses that \"man is man\" and \"woman is woman.\" This method\nrequires that one should have made up one's mind about the conclusion\nbefore discovering the premisses--by what, no doubt, Jevons would call\nan \"inverse or inductive method.\" Thus the method is of use only in\nspeeches and in giving good advice.\n\nMr. Austen Chamberlain afterwards rather destroyed one's belief in the\ntruth of his premisses by putting limits to the validity of the\nprinciple of identity. In the course of the Debate on the Budget of\n1909, he maintained, against Mr. Lloyd George, that a joke was a joke\nexcept when it was an untruth: Mr. Lloyd George, apparently, being of\nthe plausible opinion that a joke is a joke under all circumstances.\n\nFOOTNOTES:\n\n[20] The argument about the hull of a ship disappearing first is not\nconvincing, since it would equally well prove that the surface of the\nearth was, for example, corrugated on a large scale. If the common-sense\nof the reader were supposed to dismiss the possibility of water clinging\nto such corrugations, it might equally be supposed to dismiss the\npossibility of water clinging to a spherical earth. Traditional\ngeography books, no doubt, gave rise to the opinions held by Lady Blount\nand the Zetetic Society.\n\n[21] The subject of Implication will be further considered in Chapter\nXIX.\n\n\n\n\nCHAPTER IV\n\nIDENTITY OF CLASSES\n\n\nI once heard of a meritorious lady who was extremely conventional; on\nthe slender grounds of carefully acquired habits of preferring the word\n\"woman\" to the word \"lady\" and of going to the post-office without a\nhat, imagined that she was unconventional and altogether a remarkable\nperson; and who once remarked with great satisfaction that she was a\n\"very queer person,\" and that nothing shocked her \"except, of course,\nbad form.\"\n\nThus, she asserted that all the things which shocked her were actions in\nbad form; and she would undoubtedly agree, though she did not actually\nstate it, that all the things which were done in bad form would shock\nher. Consequently she asserted that the class of things which shocked\nher was the class of actions in bad form. Consequently the statement of\nthis lady that some or all of the actions done in bad form shocked her\nis an identical proposition of the form \"nothing shocks me, except, of\ncourse, the things which do, in fact, shock me\"; and this statement the\nlady certainly did not intend to make.\n\nThis excellent lady, had she but known it, was logically justified in\nmaking any statement whatever about her unconventionality. For the class\nof her unconventional actions was the null class. Thus she might\nlogically have made inconsistent statements about this class of actions.\nAs a matter of fact she did make inconsistent statements, but\nunfortunately she justified them by stating that, \"It is the privilege\nof woman to be inconsistent.\" She was one of those persons who say\nthings like that.\n\n\n\n\nCHAPTER V\n\nETHICAL APPLICATIONS OF THE LAW OF IDENTITY\n\n\nIt may be remembered that Mr. Podsnap remarked, with sadness tempered by\nsatisfaction, that he regretted to say that \"Foreign nations do as they\ndo do.\" Besides aiding the comforting expression of moral disapproval,\nthe law of identity has yet another useful purpose in practical ethics:\nIt serves the welcome purpose of providing an excuse for infractions of\nthe moral law. There was once a man who treated his wife badly, was\nunfaithful to her, was dishonest in business, and was not particular in\nhis use of language; and yet his life on earth was described in the\nlines:\n\n  This man maintained a wife's a wife,\n  Men are as they are made,\n  Business is business, life is life;\n  And called a spade a spade.\n\nOne of the objects of Dr. G. E. Moore's _Principia Ethica_[22] was to\nargue that the word \"good\" means simply good, and not pleasant or\nanything else. Appropriately enough, this book bore on its title-page\nthe quotation from the preface to the _Sermons_, published in 1726, of\nBishop Joseph Butler, the author of the _Analogy_: \"Everything is what\nit is and not another thing.\"\n\nBut another famous Butler--Samuel Butler, the author of _Hudibras_--went\nfarther than this, and maintained that identities were the highest\nattainment of metaphysics itself. At the beginning of the first Canto of\n_Hudibras_, in the description of Hudibras himself, Butler wrote:\n\n  He knew what's what, and that's as high\n  As metaphysic wit can fly.\n\nI once conducted what I imagined to be an aesthetic investigation for the\npurpose of discovery, by the continual use of the word \"Why?\"[23] the\ngrounds upon which certain people choose to put milk into a tea-cup\nbefore the tea. I was surprised to discover that it was an ethical, and\nnot an aesthetic problem; for I soon elicited the fact that it was done\nbecause it was \"right.\" A continuance of my patient questioning elicited\nfurther evidence of the fundamental character of the principle of\nidentity in ethics; for it was right, I learned, because \"right is\nright.\"\n\nIt appears that some people unconsciously think that the principle of\nidentity is the foundation, in certain religions, of the reasons which\ncan be alleged for moral conduct, and are surprised when this fact is\npointed out to them. The late Sir Leslie Stephen, when travelling by\nrailway, fell into conversation with an officer of the Salvation Army,\nwho tried hard to convert him. Failing in this laudable endeavour, the\nSalvationist at last remarked: \"But if you aren't saved, you can't go to\nheaven!\" \"That, my friend,\" replied Stephen, \"is an identical\nproposition.\"\n\nFOOTNOTES:\n\n[22] Cambridge, 1903.\n\n[23] Cf. _P. E._, p. 2.\n\n\n\n\nCHAPTER VI\n\nTHE LAW OF CONTRADICTION IN MODERN LOGIC\n\n\nConsidering the important place assigned by philosophers and logicians\nto the law of contradiction, the remark will naturally be resented by\nmany of the older schools of philosophy, and especially by Kantians,\nthat \"in spite of its fame we have found few occasions for its use.\"[24]\nAlso in modern times, Benedetto Croce, an opponent of both traditional\nlogic and mathematical logic, began the preface of the book of 1908 on\nLogic[25] by saying that that volume \"is and is not\" a certain memoir of\nhis which had been published in 1905.\n\nFOOTNOTES:\n\n[24] _Pa. Ma._, p. 116.\n\n[25] [English translation of the third Italian edition by Douglas\nAinslie, under the title: _Logic as the Science of the Pure Concept_,\nLondon 1917.--ED.]\n\n\n\n\nCHAPTER VII\n\nSYMBOLISM AND MEANING\n\n\nWhen people write down any statement such as \"The curfew tolls the knell\nof parting day,\"[26] which we will call \"C\" for shortness, what they\nmean is not \"C\" but the _meaning_ of \"C\"; and not \"the meaning of 'C'\"\nbut the _meaning_ of \"the meaning of 'C'.\" And so on, _ad infinitum_.\nThus, in writing or in speech, we always fail to state the meaning of\nany proposition whatever. Sometimes, indeed, we succeed in _conveying_\nit; but there is danger in too great a disregard of statement and\npreoccupation with conveyance of meaning. Thus many mathematicians have\nbeen so anxious to convey to us a perfectly distinct and unmetaphysical\nconcept of number that they have stripped away from it everything that\nthey considered unessential (like its logical nature) and have finally\ndelivered it to us as a mere _sign_. By the labours of Helmholtz,\nKronecker, Heine, Stolz, Thomae, Pringsheim, and Schubert, many people\nwere persuaded that, when they said \"'2' is a number\" they were speaking\nthe truth, and hold that \"Paris\" is a town containing the letter \"P.\"\nWhen Frege pointed out[27] this difficulty he was almost universally\ndenounced in Germany as \"_spitzfindig_.\" In fact, Germans seem to have\nbeen influenced perhaps by that great contemner of \"_Spitzfindigkeit_,\"\nKant, to reject the White Knight's[28] distinctions between words and\ntheir denotations and to regard subtlety with disfavour to such a degree\nthat their only mathematical logician except Frege, namely Schroeder--the\nleast subtle of mortals, by the way--seems to have been filled with such\nfear of being thought subtle, that he made his books so prolix that\nnobody has read them.\n\nAnother term which, as we shall see when discussing the paradoxes of\nlogic, mathematicians are accustomed to apply to thought which is more\nexact than any to which they are accustomed is \"scholastic.\"[29] By\nthis, I suppose, they mean that the pursuits of certain acute people of\nthe Middle Ages are unimportant in contrast with the great achievements\nof modern thought, as exemplified by a method of making plausible\nguesses known as induction,[30] the bicycle, and the gramophone--all of\nthem instruments of doubtful merit.\n\nFOOTNOTES:\n\n[26] Cf. _Md_, N. S., vol. xiv., October 1905, p. 486.\n\n[27] In _Z. S._, for example.\n\n[28] See Appendix G.\n\n[29] Cf. Chapter XXXVII below.\n\n[30] Cf. _P. M._, p. 11, note.\n\n\n\n\nCHAPTER VIII\n\nNOMINALISM\n\n\nDe Morgan[31] said that, \"if all mankind had spoken one language, we\ncannot doubt that there would have been a powerful, perhaps universal,\nschool of philosophers who would have believed in the inherent connexion\nbetween names and things; who would have taken the sound _man_ to be the\nmode of agitating the air which is essentially communicative of the\nideas of reason, cookery, bipedality, etc.... 'The French,' said the\nsailor, 'call a cabbage a _shoe_; the fools! Why can't they call it a\ncabbage, when they must know it is one?'\"\n\nOne of the chief differences between logicians and men of letters is\nthat the latter mean many different things by one word, whereas the\nformer do not--at least nowadays. Most mathematicians belong to the\nclass of men of letters.\n\nI once had a manservant who told me on a certain occasion that he \"never\nthought a word about it.\" I was doubtful whether to class him with such\neminent mathematicians as are mentioned in the last chapter, or as a\nsupporter of Max Mueller's theory of the identity of thought and\nlanguage. However, since the man was very untruthful, and he told me\nthat he meant what he said and said what he meant,[32] the conclusion is\nprobably correct that he really believed that the meanings of his words\nwere not the words themselves. Thus I think it most probable that my\nmanservant had been a mathematician but had escaped by the aid of logic.\n\nAs regards his remark that he meant what he said and said what he\nmeant, he plainly wished to pride himself on certain virtues which he\ndid not possess, and was not indifferent to applause, which, however,\nwas never evoked. The virtues, if so they be, and the applause were\nwithheld for other reasons than that the above statements are either\nnonsensical or false. Suppose that \"I say what I mean\" expresses a\ntruth. What I say (or write) is always a symbol--words (or marks); and\nwhat I mean by the symbol is the meaning of the symbol and not the\nsymbol itself. So the remark cannot express a truth, any more than the\nname \"Wellington\" won the battle of Waterloo.\n\nFOOTNOTES:\n\n[31] _F. L._, pp. 246-7.\n\n[32] The Hatter (see Appendix H) pointed out that there is a difference\nbetween these two assertions. Thus, he clearly showed that he was a\nnominalist, and philosophically opposed to the March Hare who had\nrecommended Alice to say what she meant.\n\n\n\n\nCHAPTER IX\n\nAMBIGUITY AND SYMBOLIC LOGIC\n\n\nThe universal use of some system of Symbolic Logic would not only enable\neverybody easily to deal with exceedingly complicated arguments, but\nwould prevent ambiguous arguments. In denying the indispensability of\nSymbolic Logic in the former state of things, Keynes[33] is probably\nalone, against the need strongly felt by Alice when speaking to the\nDuchess,[34] and most modern logicians. It may be noticed that the\nDuchess is more consistent than Keynes, for Keynes really uses the signs\nfor logical multiplication and addition of Boole and Venn under the\ndifferent shapes of the words \"and\" and \"or.\"\n\nAs regards ambiguity, a translation of _Hymns Ancient and Modern_ into,\nsay, Peanesque, would prevent the puzzle of childhood as to whether the\n\"his\" in\n\n  And Satan trembles when he sees\n  The weakest saint upon his knees\n\nrefers to the saint's knees or Satan's.\n\nFOOTNOTES:\n\n[33] In his _Fm. L._\n\n[34] See Appendix I.\n\n\n\n\nCHAPTER X\n\nLOGICAL ADDITION AND THE UTILITY OF SYMBOLISM\n\n\nFrequently ordinary language contains subtle psychological implications\nwhich cannot be translated into symbolic logic except at great length.\nThus if a man (say Mr. Jones) wishes to speak collectively of himself\nand his wife, the order of mentioning the terms in the class considered\nand the names applied to these terms are, logically speaking,\nirrelevant. And yet more or less definite information is given about Mr.\nJones according as he talks to his friends of:\n\n     (1) Mrs. Jones and I,\n     (2) I (or me) and my wife (or missus),\n     (3) My wife and I,\n  or (4) I (or me) and Mrs. Jones.\n\nIn case (1) one is probably correct in placing Mr. Jones among the\nclergy or the small professional men who make up the bulk of the\nmiddle-class; in case (2) one would conclude that Mr. Jones belonged to\nthe lower middle-class; the form (3) would be used by Mr. Jones if he\nwere a member of the upper, upper middle, or lower class; while form (4)\nis only used by retired shopkeepers of the lower middle-class, of which\na male member usually combines belief in the supremacy of man with\nbelief in the dignity of his wife as well as himself. A further\ncomplication is introduced if a wife is referred to as \"the wife.\"[35]\nCases (2) and (3) then each give rise to one more case. Cases (1) and\n(4) do not, since nobody has hitherto referred to his wife as \"the Mrs.\nJones\"--at least without a qualifying adjective before the \"Mrs.\"\n\nOn the other hand, certain descriptive phrases and certain propositions\ncan be expressed more shortly and more accurately by means of symbolic\nlogic. Let us consider the proposition \"No man marries his deceased\nwife's sister.\" If we assume, as a first approximation, that all\nmarriages are fertile and that all children are legitimate, then, with\nonly four primitive ideas: the relation of parent to child (P) and the\nthree classes of males, females, and dead people, we can define \"wife\"\n(a female who has the relation formed by taking the relative product of\nP and [vP][36] to a male), \"sister,\" \"deceased wife,\" and \"deceased\nwife's sister\" in terms of these ideas and of the fundamental notions of\nlogic. Then the proposition \"No man marries his deceased wife's sister\"\ncan be expressed unambiguously by about twenty-nine simple signs on\npaper, whereas, in words, the unasserted statement consists of no less\nthan thirty-four letters. Although, legally speaking, we should have to\nadopt somewhat different definitions and possibly increase the\ncomplications of our proposition, it must be remembered that, on the\nother hand, we always reduce the number of symbols in any proposition by\nincreasing the number of definitions in the preliminaries to it.\n\nBut the utility of symbolic logic should not be estimated by the brevity\nwith which propositions may sometimes be expressed by its means. Logical\nsimplicity, in fact, can very often only be obtained by apparently\ncomplicated statements. For example, the logical interpretation of \"The\nfather of Charles II was executed\" is, \"It is not always false of _x_\nthat _x_ begat Charles II, and that _x_ was executed and that 'if _y_\nbegat Charles II, _y_ is identical with _x_' is always true of _y_.\"[37]\nFrom the point of view of logic, we may say that the apparently simple\nis most often very complicated, and, even if it is not so, symbolism\nwill make it seem so,[38] and thus draw attention to what might\notherwise easily be overlooked.\n\nFOOTNOTES:\n\n[35] Cf. Chapter XXIV below.\n\n[36] C. S. Peirce's notation for the relation \"converse of P.\"\n\n[37] Russell, _Md._, N. S., vol. xiv., October 1905, p. 482.\n\n[38] Russell, _International Monthly_, vol. iv., 1901, pp. 85-6; cf.\n_M._, vol. xxii., 1912, p. 153. [This essay is reprinted in _Mysticism\nand Logic_, London and New York, 1918, pp. 74-96.--ED.]\n\n\n\n\nCHAPTER XI\n\nCRITICISM\n\n\nThose people who think that it is more godlike to seem to turn water\ninto wine than to seem to turn wine into water surprise me. I cannot\nimagine an intolerable critic. It seems to me that, if A resents B's\ncriticism in trying to put his (A's) discovery in the right or wrong\nplace, A acts as if he thought he had some private property in truth.\nThe White Queen seems to have shared the popular misconception as to the\nnature of criticism.[39]\n\nFOOTNOTES:\n\n[39] See Appendix J.\n\n\n\n\nCHAPTER XII\n\nHISTORICAL CRITICISM\n\n\nFrom a problem in Diophantus's _Arithmetic_ about the price of some wine\nit would seem that the wine was of poor quality, and Paul Tannery has\nsuggested that the prices mentioned for such a wine are higher than were\nusual until after the end of the second century. He therefore rejected\nthe view which was formerly held that Diophantus lived in that\ncentury.[40]\n\nThe same method applied to a problem given by the ancient Hindu\nalgebraist Brahmagupta, who lived in the seventh century after Christ,\nmight result in placing Brahmagupta in prehistoric times. This is the\nproblem:[41] \"Two apes lived at the top of a cliff of height _h_, whose\nbase was distant _mh_ from a neighbouring village. One descended the\ncliff and walked to the village, the other flew up a height _x_ and then\nflew in a straight line to the village. The distance traversed by each\nwas the same. Find _x_.\"\n\nFOOTNOTES:\n\n[40] W. W. Rouse Ball, _A Short Account of the History of Mathematics_,\n4th edition, London, 1908, p. 109.\n\n[41] _Ibid._, pp. 148-9.\n\n\n\n\nCHAPTER XIII\n\nIS THE MIND IN THE HEAD?\n\n\nThe contrary opinion has been maintained by idealists and a certain\nelection agent with whom I once had to deal and who remarked that\nsomething slipped his mind and then went out of his head altogether. At\nsome period, then, a remembrance was in his head and out of his mind;\nhis mind was not, then, wholly within his head. Also, one is sometimes\nassured that with certain people \"out of sight is out of mind.\" What is\nin their minds is therefore in sight, and cannot therefore be inside\ntheir heads.\n\n\n\n\nCHAPTER XIV\n\nTHE PRAGMATIST THEORY OF TRUTH\n\n\nThe pragmatist theory that \"truth\" is a belief which works well\nsometimes conflicts with common-sense and not with logic. It is commonly\nsupposed that it is always better to be sometimes right than to be never\nright. But this is by no means true. For example, consider the case of a\nwatch which has stopped; it is exactly right twice every day. A watch,\non the other hand, which is always five minutes slow is never exactly\nright. And yet there can be no question but that a belief in the\naccuracy of the watch which was never right would, on the whole, produce\nbetter results than such a belief in the one which had altogether\nstopped. The pragmatist would, then, conclude that the watch which was\nalways inaccurate gave truer results than the one which was sometimes\naccurate. In this conclusion the pragmatist would seem to be correct,\nand this is an instance of how the false premisses of pragmatism may\ngive rise to true conclusions.\n\nFrom the text written above the church clock in a certain English\nvillage, \"Be ye ready, for ye know not the time,\" it would be concluded\nthat the clock never stopped for a period as long as twelve hours. For\nthe text is rather a vague symbolical expression of a propositional\nfunction which is asserted to be true at all instants. The proposition\nthat a presumably not illiterate and credulous observer of the clock at\nany definite instant does not know the time implies, then, that the\nclock is always wrong. Now, if the clock stopped for twelve hours, it\nwould be absolutely right at least once. It must be right twice if it\nwere right at the first instant it stopped or the last instant at which\nit went;[42] but the second possibility is excluded by hypothesis, and\nthe occurrence of the first possibility--or of the analogous possibility\nof the stopped clock being right three times in twenty-four hours--does\nnot affect the present question. Hence the clock can never stop for\ntwelve hours.\n\nThe pragmatist's criterion of truth appears to be far more difficult to\napply than the Bellman's,[43] that what he said three times is true, and\nto give results just as insecure.\n\nFOOTNOTES:\n\n[42] Both cases cannot occur; the question is similar to that arising in\nthe discussion of the mortality of Socrates (see Chapter XXII).\n\n[43] See Appendix K.\n\n\n\n\nCHAPTER XV\n\nASSERTION\n\n\nThe subject of the present chapter must not be confused with the\nassertion of ordinary life. Commonly, an unasserted proposition is\nsynonymous with a probably false statement, while an asserted\nproposition is synonymous with one that is certainly false. But in logic\nwe apply assertion also to true propositions, and, as Lewis Carroll\nshowed in his version of \"What the Tortoise said to Achilles,\"[44]\nusually pass over unconsciously an infinite series of implications in so\ndoing. If _p_ and _q_ are propositions, _p_ is true, and _p_ implies\n_q_, then, at first sight, one would think that one might assert _q_.\nBut, from (A) _p_ is true, and (B) _p_ implies _q_, we must, in order to\ndeduce (Z) _q_ is true, accept the hypothetical: (C) If A and B are\ntrue, Z must be true. And then, in order to deduce Z from A, B, and C,\nwe must accept another hypothetical: (D) If A, B, and C are true, Z must\nbe true; and so on _ad infinitum_. Thus, in deducing Z, we pass over an\ninfinite series of hypotheticals which increase in complexity. Thus we\nneed a new principle to be able to assert _q_.\n\nFrege was the first logician sharply to distinguish between an asserted\nproposition, like \"A is greater than B,\" and one which is merely\nconsidered, like \"A's being greater than B,\" although an analogous\ndistinction had been made in our common discourse on certain\npsychological grounds, for long previously. In fact, soon after the\ninvention of speech, the necessity of distinguishing between a\nconsidered proposition and an asserted one became evident, on account of\nthe state of things referred to at the beginning of this chapter.\n\nFOOTNOTES:\n\n[44] _Md._ N. S., vol. iv., 1895, pp. 278-80. Cf. Russell, _P. M._, p.\n35.\n\n\n\n\nCHAPTER XVI\n\nTHE COMMUTATIVE LAW\n\n\nOften the meaning of a sentence tacitly implies that the commutative law\ndoes not hold. We are all familiar with the passage in which Macaulay\npointed out that, by using the commutative law because of exigencies of\nmetre, Robert Montgomery unintentionally made Creation tremble at the\nAtheist's nod instead of the Almighty's. This use of the commutative law\nby writers of verse renders it doubtful whether, in the hymn-line:\n\n  The humble poor believe,\n\nwe are to understand a statement about the humble poor, or a doubtful\nmaxim as to the attitude of our minds to statements made by the humble\npoor.\n\nThe non-commutativity of English titles offers difficulties to some\nnovelists and Americans who refer to Mary Lady So-and-So as Lady Mary\nSo-and-So, and _vice versa_.\n\n\n\n\nCHAPTER XVII\n\nUNIVERSAL AND PARTICULAR PROPOSITIONS\n\n\nPeople who are cynical as to the morality of the English are often\nunpleasantly surprised to learn that \"All trespassers will be\nprosecuted\" does not necessarily imply that \"some trespassers will be\nprosecuted.\" The view that universal propositions are non-existential is\nnow generally held: Bradley and Venn seem to have been the first to hold\nthis, while older logicians, such as De Morgan,[45] considered universal\npropositions to be existential, like particular ones.\n\nIf the Gnat[46] had been content to affirm his proposition about the\nmeans of subsistence of Bread-and-Butter flies, in consequence of their\nlack of which such flies always die, without pointing out such an insect\nand thereby proving that the class of them is not null, Alice's doubt as\nto the existence of the class in question, even if it were proved to be\nwell founded, would not have affected the validity of the proposition.\n\nThis brings us to a great convenience in treating universal propositions\nas non-existential: we can maintain that all _x_'s are _y_'s at the same\ntime as that no _x_'s are _y_'s, if only _x_ is the null-class. Thus,\nwhen Mr. MacColl[47] objected to other symbolic logicians that their\npremisses imply that all Centaurs are flower-pots, they could reply that\ntheir premisses also imply the more usual view that Centaurs are not\nflower-pots.\n\nFOOTNOTES:\n\n[45] Cf., e.g., _F. L._, p. 4.\n\n[46] See Appendix L.\n\n[47] Cf., e.g., _Md._, N. S., vol. xiv., July, 1905, pp. 399-400.\n\n\n\n\nCHAPTER XVIII\n\nDENIAL OF GENERALITY AND GENERALITY OF DENIAL\n\n\nThe conclusion of a certain song[48] about a young man who poisoned his\nsweetheart with sheep's-head broth, and was frightened to death by a\nvoice exclaiming:\n\n  \"Where's that young maid\n  What you did poison with my head?\"\n\nat his bedside, gives rise to difficulties which are readily solved by a\nsymbolism that brings into relief the principle that the denial of a\nuniversal and non-existential proposition is a particular and\nexistential one. The conclusion of the song is:\n\n  Now all young men, both high and low,\n  Take warning by this dismal go!\n  For if he'd never done nobody no wrong,\n  He might have been here to have heard this song.\n\nIt is an obvious error, say Whitehead and Russell,[49] though one easy\nto commit, to assume that the cases: (1) all the propositions of a\ncertain class are true; and (2) no proposition of the class is true; are\neach other's contradictories. However, in the modification[50] of\nFrege's symbolism which was used by Russell\n\n      (1) is (_x_). _x_,\n  and (2) is (_x_). not _x_;\n\nwhile the contradictory of (1) is:\n\n  not (_x_). _x_.\n\nThe last line but one of the above verse may, then, be written:\n\n  (_t_). not (_x_). not not [Greek: ph] (_x_, _t_),\n\nwhere \"[Greek: ph] (_x_, _t_)\" denotes the unasserted propositional\nfunction \"the doing wrong to the person _x_ at the instant _t_.\" By\nmeans of the principle of double negation we can at once simplify the\nabove expression into:\n\n  (_t_). not (_x_). [Greek: ph] (_x_, _t_);\n\nwhich can be thus read: \"If at every instant of his life there was at\nleast one person _x_ to whom he did no wrong (at that instant).\" It is\ndifficult to imagine any one so sunk in iniquity that he would not\nsatisfy this hypothesis. We are forced, then, unless our imagination for\nevil is to be distrusted, to conclude that any one might have been there\nto have heard that song. Now this conclusion is plainly false, possibly\non physical grounds, and certainly on aesthetic grounds. It may be added,\nby the way, that it is quite possible that De Morgan was mistaken in his\ninterpretation of the above proposition owing to the fact that he was\nunacquainted with Frege's work. In fact, if he had not noticed the fact\nthat _any_ two of the \"not's\" cannot be cancelled against one another he\nwould have concluded that the interpretation was: \"If he had never done\nany wrong to anybody.\"\n\nAccording as the symbol for \"not\" comes before the (_x_) or between the\n(_x_) and the [Greek: ph], we have an expression of what Frege called\nrespectively the denial of generality, and the generality of denial. The\ndenial of the generality of a denial is the form of all existential\npropositions, while the assertion of or denial of generality is the\ngeneral form of all non-existential or universal propositions.\n\nFOOTNOTES:\n\n[48] To which De Morgan drew attention in a letter; see (Mrs.) S. E. De\nMorgan, _Memoir of Augustus De Morgan_, London, 1882, p. 324.\n\n[49] _Pa. Ma._, p. 16.\n\n[50] However, here, for the printer's convenience, we depart from Mr.\nRussell's usage so far as to write \"not\" for a curly minus sign.\n\n\n\n\nCHAPTER XIX\n\nIMPLICATION\n\n\nA good illustration of the fact that what is called \"implication\" in\nlogic is such that a false proposition implies any other proposition,\ntrue or false, is given by Lewis Carroll's puzzle of the three\nbarbers.[51]\n\nAllen, Brown, and Carr keep a barber's shop together; so that one of\nthem must be in during working hours. Allen has lately had an illness of\nsuch a nature that, if Allen is out, Brown must be accompanying him.\nFurther, if Carr is out, then, if Allen is out, Brown must be in for\nobvious business reasons. The problem is, may Carr ever go out?\n\nPutting _p_ for \"Carr is out,\" _q_ for \"Allen is out\" and _r_ for \"Brown\nis out,\" we have:\n\n  (1) _q_ implies _r_,\n  (2) _p_ implies that _q_ implies not-_r_.\n\nLewis Carroll supposed that \"_q_ implies _r_\" and \"_q_ implies not-_r_\"\nare inconsistent, and hence that _p_ must be false. But these\npropositions are not inconsistent, and are, in fact, both true if _q_ is\nfalse. The contradictory of \"_q_ implies _r_\" is \"_q_ does not imply\n_r_\" which is not a consequence of \"_q_ implies not-_r_.\" It seems to be\ntrue theoretically that, if Mr. X is a Christian, he is not an Atheist,\nbut we cannot conclude from this alone that his being a Christian does\nnot imply that he is an Atheist, unless we assume that the class of\nChristians is not null. Thus, if _p_ is true, _q_ is false; or, if Carr\nis out, Allen is in. The odd part of this conclusion is that it is the\none which common-sense would have drawn in that particular case.\n\nA distinguished philosopher (M) once thought that the logical use of the\nword \"implication\"--any false proposition being said to \"imply\" any\nproposition true or false--is absurd, on the grounds that it is\nridiculous to suppose that the proposition \"2 and 2 make 5\" implies the\nproposition \"M is the Pope.\" This is a most unfortunate instance,\nbecause it so happens that the false proposition that 2 and 2 make 5 can\nrigorously be proved to imply that M, or anybody else other than the\nPope, is the Pope. For if 2 and 2 make 5, since they also make 4, we\nwould conclude that 5 is equal to 4. Consequently, subtracting 3 from\nboth sides, we conclude that 2 would be equal to 1. But if this were\ntrue, since M and the Pope are two, they would be one, and obviously\nthen M would be the Pope.\n\nThe principle that the false implies the true has very important\napplications in political arguments. In fact, it is hard to find a\nsingle principle of politics of which false propositions are not the\nmain support.\n\nIf _p_ and _q_ are two propositions, and _p_ implies _q_; then, if, and\nonly if, _q_ and _p_ are both false or both true, we also have: _q_\nimplies _p_. The most important applications of this invertibility were\nmade by the late Samuel Butler[52] and Mr. G. B. Shaw. A political\napplication may be made as follows: In a country where only those with\nmiddling-sized incomes are taxed, conservative and _bourgeois_\npoliticians would still maintain that the proposition \"the rich are\ntaxed\" implies the proposition \"the poor are taxed,\" and this\nimplication, which is true because both premiss and conclusion are\nfalse, would be quite unnecessarily supported by many false practical\narguments. It is equally true that \"the poor are taxed\" implies that\n\"the rich are taxed.\" And this can be proved, in certain cases, on other\ngrounds. For the taxation of the poor would imply, ultimately, that the\npoor could not afford to pay a little more for the necessities of life\nthan, in strict justice, they ought; and this would mean the cessation\nof one of the chief means of production of individual wealth.\n\nWe also see why a valuable means for the discovery of truth is given by\nthe inversion of platitudinous implications. It may happen that another\nplatitude is the result of inversion; but it is the fate of any true\nremark, especially if it is easy to remember by reason of a paradoxical\nform, to become a platitude in course of time. There are rare cases of a\nplatitude remaining unrepeated for so long that, by a converse process,\nit has become paradoxical. Such, for example, is Plato's remark that a\nlie is less important than an error in thought.\n\nOf late years, a method of disguising platitudes as paradoxes has been\ntoo extensively used by Mr. G. K. Chesterton. The method is as follows.\nTake any proposition _p_ which holds of an entity _a_; choose _p_ so\nthat it seems plausible that _p_ also holds of at least two other\nentities _b_ and _c_; call _a_, _b_, _c_, and any others for which _p_\nholds or seems to hold, the class A, and _p_ the \"A-ness\" or \"A-ity\" of\nA; let _d_ be an entity for which _p_ does not hold; and put _d_ among\nthe A's when you think that nobody is looking. Then state your paradox:\n\"Some A's do not have A-ness.\" By further manipulation you can get the\nproposition \"No A's have A-ness.\" But it is possible to make a very\nsuccessful _coup_ if A is the null-class, which has the advantage that\nmanipulation is unnecessary. Thus, Mr. Chesterton, in his _Orthodoxy_\nput A for the class of doubters who doubt the possibility of logic, and\nproved that such agnostics refuted themselves--a conclusion which seems\nto have pleased many clergymen.\n\nIn this way, Mr. Chesterton has been enabled readily to write many books\nand to maintain, on almost every page, such theses as that simplicity is\nnot simple, heterodoxy is not heterodox, poets are not poetical, and so\non; thereby building up the gigantic platitude that Mr. Chesterton is\nChestertonian.\n\nIn the chapter on Identity we have illustrated the use of a case of the\nprinciple that any proposition implies any true proposition. This\nimportant principle may be called _the principle of the irrelevant\npremiss_;[53] and is of great service in oratory, because it does not\nmatter what the premiss is, true or false. There is a _principle of the\nirrelevant conclusion_, but, except in law-courts, interruptions of\nmeetings, and family life, this is seldom used, partly because of the\nlimitation involved in the logical impossibility for the conclusion to\nbe false if the premiss be true, but chiefly because the conclusion is\nmore important than the premiss, being usually a matter of prejudice.\n\nCertain modern logicians, such as Frege, have found it necessary so to\nextend the meaning of implication of _q_ by _p_ that it holds when _p_\nis not a proposition at all. Hitherto, politicians, finding that either\nidentical or false propositions are sufficient for their needs, have\nmade no use of this principle; but it is obvious that their stock of\narguments would be vastly increased thereby.\n\nLogical implication is often an enemy of dignity and eloquence. De\nMorgan[54] relates \"a tradition of a Cambridge professor who was once\nasked in a mathematical discussion, 'I suppose you will admit that the\nwhole is greater than its part?' and who answered, 'Not I, until I see\nwhat use you are going to make of it.'\" And the care displayed by\ncautious mathematicians like Poincare, Schoenflies, Borel, Hobson, and\nBaire in abstaining from pushing their arguments to their logical\nconclusions is probably founded on the unconscious--but no less\nwell-grounded--fear of appearing ridiculous if they dealt with such\nextreme cases as \"the series of all ordinal numbers.\"[55] They are,\nprobably, as unconscious of implication as Gibbon, when he remarked that\nhe always had a copy of Horace in his pocket, and often in his hand, was\nof the necessary implication of these propositions that his hand was\nsometimes in his pocket.\n\nFOOTNOTES:\n\n[51] _Md._, N. S., vol. iii., 1894, pp. 436-8. Cf. the discussions by W.\nE. Johnson (_ibid._, p. 583) and Russell (_P. M._, p. 18, note, and\n_Md._, N. S., vol. xiv., 1905, pp. 400-1).\n\n[52] The inhabitants of \"Erewhon\" punished invalids more severely than\ncriminals. In modern times, one frequently hears the statement that\ncrime is a disease; and if so, it is surely false that criminals ought\nto be punished.\n\n[53] _Irrelevant_ in a popular sense; one would not say, speaking\nloosely, that the fact that Brutus killed Caesar implies that the sea is\nsalt; and yet this conclusion is implied both by the above premiss, and\nthe premiss that Caesar killed Brutus. Cf. on such questions Venn,\n_S. L._, 2nd ed., pp. 240-4.\n\n[54] _F. L._, p. 264.\n\n[55] Cf. Chapters XXIX and XXXVII.\n\n\n\n\nCHAPTER XX\n\nDIGNITY\n\n\nWe have seen, at the end of the preceding chapter, that logical\nimplication is often an enemy of dignity. The subject of dignity is not\nusually considered in treatises on logic, but, as we have remarked, many\nmathematicians implicitly or explicitly seem to fear either that the\ndignity of mathematics will be impaired if she follows out conclusions\nlogically, or that only an act of faith can save us from the belief\nthat, if we followed out conclusions logically, we should find out\nsomething alarming about the past, present, or future of mathematics.\n\nThus it seems necessary to inquire rather more closely into the nature\nof dignity, with a view to the discovery of whether it is, as is\ncommonly supposed, a merit in life and logic.\n\nThe chief use of dignity is to veil ignorance. Thus, it is well known\nthat the most dignified people, as a rule, are schoolmasters, and\nschoolmasters are usually so occupied with teaching that they have no\ntime to learn anything. And because dignity is used to hide ignorance,\nit is plain that impudence is not always the opposite of dignity, but\nthat dignity is sometimes impudence. Dignity is said to inspire respect;\nand this may be in part why respect for others is an error of judgment\nand self-respect is ridiculous.\n\nSelf-respect is, of course, self-esteem. William James has remarked that\nself-esteem depends, not simply upon our success, but upon the ratio of\nour success to our pretensions, and can therefore be increased by\ndiminishing our pretensions. Thus if a man is successful, but only then,\ncan he be both ambitious and dignified. James also implies that\nhappiness increases with self-esteem. Likeness of thought with one's\nfriends, then, does not make one happy, for otherwise a man who esteemed\nhimself little would be indeed happy. Also if a man is unhappy he could\nnot, from our premisses, by the principles of the syllogism and of\ncontraposition, be dignified--a conclusion which should be fatal to many\nnovelists' heroes.\n\nA reflection on pessimism to which this discussion gives rise is the\nfollowing: It would appear that a man's self-esteem would be increased\nby a conviction of the unworthiness of his neighbours. A man, therefore,\nwho thinks that the world and all its inhabitants, except himself, are\nvery bad, should be extremely happy. In fact, the effects would hardly\nbe distinguishable from those of optimism. And optimism, as everybody\nknows, is a state of mind induced by stupidity.\n\n\n\n\nCHAPTER XXI\n\nTHE SYNTHETIC NATURE OF DEDUCTION\n\n\nDoubt has often been expressed as to whether a syllogism can add to\nour knowledge in any way. John Stuart Mill and Henri Poincare, in\nparticular, held the opinion that the conclusion of a syllogism is an\n\"analytic\" judgment in the sense of Kant, and therefore could be\nobtained by the mere dissection of the premisses. Any one, then, who\nmaintains that mathematics is founded solely on logical principles would\nappear to maintain that mathematics, in the last instance, reduces to a\nhuge tautology.\n\nMill, in Chapter III of Book II of his _System of Logic_, said that \"it\nmust be granted that in every syllogism, considered as an argument to\nprove the conclusion, there is a _petitio principii_. When we say\n\n  All men are mortal,\n  Socrates is a man,\n\ntherefore\n\n  Socrates is mortal,\n\nit is unanswerably urged by the adversaries of the syllogistic theory,\nthat the proposition, Socrates is mortal, is presupposed in the more\ngeneral assumption, All men are mortal; that we cannot be assured of the\nmortality of all men unless we are already certain of the mortality of\nevery individual man; that if it be still doubtful whether Socrates, or\nany other individual we choose to name, be mortal or not, the same\ndegree of uncertainty must hang over the assertion, All men are mortal;\nthat the general principle, instead of being given as evidence of the\nparticular case, cannot itself be taken for true without exception until\nevery shadow of doubt which could affect any case comprised with it is\ndispelled by evidence _aliunde_; and then what remains for the syllogism\nto prove? That, in short, no reasoning from general to particular can,\nas such, prove anything, since from a general principle we cannot infer\nany particulars but those which the principle itself assumes as known.\nThis doctrine appears to me irrefragable....\"\n\nBut it is not difficult to see that in certain cases at least deduction\ngives us _new_ knowledge.[56] If we already know that two and two always\nmake four, and that Asquith and Lloyd George are two and so are the\nGerman Emperor and the Crown Prince, we can deduce that Asquith and\nLloyd George and the German Emperor and the Crown Prince are four. This\nis new knowledge, not contained in our premisses, because the general\nproposition, \"two and two are four,\" never told us there were such\npeople as Asquith and Lloyd George and the German Emperor and the Crown\nPrince, and the particular premisses did not tell us that there were\nfour of them, whereas the particular proposition deduced does tell us\nboth these things. But the newness of the knowledge is much less certain\nif we take the stock instance of deduction that is always given in books\non logic, namely \"All men are mortal; Socrates is a man, therefore\nSocrates is mortal.\" In this case what we really know beyond reasonable\ndoubt is that certain men, A, B, C, were mortal, since, in fact, they\nhave died. If Socrates is one of these men, it is foolish to go the\nroundabout way through \"all men are mortal\" to arrive at the conclusion\nthat _probably_ Socrates is mortal. If Socrates is not one of the men on\nwhom our induction is based, we shall still do better to argue straight\nfrom our A, B, C, to Socrates, than to go round by the general\nproposition, \"all men are mortal.\" For the probability that Socrates is\nmortal is greater, on our data, than the probability that all men are\nmortal. This is obvious, because if all men are mortal, so is Socrates;\nbut if Socrates is mortal, it does not follow that all men are mortal.\nHence we shall reach the conclusion that Socrates is mortal, with a\ngreater approach to certainty if we make our argument purely inductive\nthan if we go by way of \"all men are mortal\" and then use deduction.\n\nMany years ago there appeared, principally owing to the initiative of\nDr. F. C. S. Schiller of Oxford, a comic number of _Mind_. The idea was\nextraordinarily good, not so the execution. A German friend of Dr.\nSchiller was puzzled by the appearance of the advertisements, which were\ndoubtfully humorous. However, by a syllogistic process, he acquired\ninformation which was new and useful to him, and thus incidentally\nrefuted Mill. Presumably he started from the title of the magazine\n(_Mind!_), for a mark of exclamation seems nearly always in German to be\na sign of an intended joke (including of course the mark after the\npoliteness expressed in the first sentence of a private letter or a\npublic address). There would be, then, the following syllogism:\n\n  This is a book of would-be jokes (i.e. everything in this book is a\n          would-be joke);\n  This advertisement is in this book;\n  Therefore, this advertisement is a would-be joke.\n\nThus the syllogism may be almost as powerful an agent in the detection\nof humour as M. Bergson's criterion, to be described in a future\nchapter.[57]\n\nFOOTNOTES:\n\n[56] [The following passage is almost word for word the same as a\npassage on pp. 123-5 of Mr. Russell's _Problems of Philosophy_, first\npublished in 1912, a year after Mr. R*ss*ll's death. It is easy hastily\nto conclude that Mr. Russell was indebted to Mr. R*ss*ll to a greater\ndegree than is usually supposed. But an examination of the internal\nevidence leads us to another conclusion. The two texts, it will be\nfound, differ only in the names of the German Emperor, the Crown Prince\nand the other personages being replaced, in the book of 1912, by those\nof Messrs. Brown, Jones, Smith, and Robinson. Now, Mr. Russell, in a new\nedition of his _Problems_ issued near the beginning of the European war\nand before the Russian revolution, substituted \"the Emperor of Russia\"\nfor \"the Emperor of China\" of the first edition. Hence it seems quite\nlikely that Mr. Russell, who has always shown a tendency to substitute\nexistents for nonentities, wrote Mr. R*ss*ll's notes.--<sc>Ed.</sc>]\n\n[57] [See Chapter XLII.--ED.]\n\n\n\n\nCHAPTER XXII\n\nTHE MORTALITY OF SOCRATES\n\n\nThe mortality of Socrates is so often asserted in books on logic that it\nmay be as well briefly to consider what it means. The phrase \"Socrates\nis mortal\" may be thus defined: \"There is at least one instant _t_ such\nthat _t_ has not to Socrates the one-many relation R which is the\nconverse of the relation 'exists at,' and all instants following _t_\nhave not the relation R to Socrates, and there is at least one instant\n_t'_ such that neither _t'_ nor any instant preceding _t'_ has the\nrelation R to Socrates.\"\n\nThis definition has many merits. In the first place, no assumption is\nmade that Socrates ever lived at all. In the second place, no assumption\nis made that the instants of time form a continuous series. In the third\nplace, no assumption is made as to whether Socrates had a first or last\nmoment of his existence. If time be indeed a continuous series, then we\ncan easily deduce[58] that there must have been _either_ a first moment\nof his non-existence _or_ a last one of his existence, but not both;\njust as there seems to be either a greatest weight that a man can lift\nor a least weight that he cannot lift, but not both.[59] This may be set\nforth as follows: for the present we will not concern ourselves with\nevidence for or against human immortality; I will merely try to present\nsome logical questions which persistently arise whenever we think of\neternal life. One of the greatest merits of modern logic is that it has\nallowed us to give precision to such problems, while definitely\nabandoning any pretensions of solving them; and I will now apply the\nlogico-analytical method to one of the problems of our knowledge of the\neternal world.[60]\n\nWe will start from the generally accepted proposition that all men are\nmortal. Clearly, if we could know each individual man, and know that he\nwas mortal, that would not enable us to know that all men are mortal,\nunless we knew, in addition, that those were all the men there are. But\nwe need not here assume any such knowledge of general propositions; and,\nthough most of us will admit that the proposition in question has great\nintrinsic plausibility, it is not strictly necessary for our present\npurpose to assume anything more than the still more probable proposition\n\"Socrates is mortal.\" This last proposition, quite apart from the fact\nthat we have a large amount of historical evidence for its truth, has\nbeen repeated so often in books on logic that it has taken on the\nrespectable air of a platitude while preserving the character of an\nexceedingly probable truth. The truth also results from the fact that it\nis used as the conclusion of a syllogism. For it is a well-known fact\nthat syllogisms can only be regarded as forming part of a sound\neducation if the conclusions are obviously true. The use of a syllogism\nof the form \"All cats are ducks and all ducks are mice, therefore all\ncats are mice,\" would introduce grave doubts into the University of\nOxford as to whether logic could any longer be considered as a valuable\nmental training for what are amusingly called the \"learned professions.\"\n\nIf, then, we divide all the instants of time, whether past, present, or\nfuture, into two series--those instants at which Socrates was alive, and\nthose instants at which he was not alive--and leave out of\nconsideration, for the sake of greater simplicity, all those instants\nbefore he lived, we see at once, by the simple application of Dedekind's\nAxiom, that, if Socrates entered into eternal life after his death,\nthere must have been either a last moment of his earthly life _or_ a\nfirst moment of his eternal life, but not both.\n\nLogic alone can give us no information as to which of these cases\nactually occurred, and we are thrown back on to a discussion of\nempirical evidence. It is no unusual thing to read of people who thought\n\"that every moment would be their last.\" In this case it is quite\nobvious that they consequently thought that eternity would have no\nbeginning.\n\nNow here we must consider two things: (1) It is plainly unsafe to\nconclude from what people think will happen to what will happen; (2)\neven if we could so conclude, it would be unsafe to deduce that there\nwas a last moment in the life of Socrates: we could only make the guess\nplausible, as we should be using the inductive method.\n\nThere are two other pieces of evidence that there is a last moment of\nany earthly existence, which we may now briefly consider. That this was\nso was held by Carlo Michaelstaedter; but since he apparently only\nbelieved this because he wanted, by attributing a supposed ethical value\nto that moment, to give support to his theory of suicide, we ought not\nto give great weight to this evidence. Secondly, Thomas Hobbes objected\nto the principle \"that a quantity may grow less and less eternally, so\nas at last to be equal to another quantity; or, which is all one, that\nthere is a last in eternity\" as \"void of sense.\" Now, the principle\nmeant is true, so that, although the other proposition mentioned by\nHobbes does not follow logically from the first, there is some evidence\nthat this other is true. In fact, that Hobbes thought that such-and-such\na proposition followed from another proposition which he wrongly\nbelieved to be false, is far better evidence for the truth of\nsuch-and-such a proposition than any we have for the truth of most of\nour most cherished beliefs.\n\nThirdly, Leibniz, in a dialogue[61] written on his journey of 1676 to\nvisit Spinoza, raised the question whether the moment at which a man\ndies may be regarded as both the last moment at which he is alive and\nthe first at which he is dead, as it must be by Aristotle's theory of\ncontinuity. Agreement with this view violates the law of contradiction;\ndenial of it implies that two moments can be immediately adjacent. By\nthe denial, then, we are led to regard space and time as made up of\nindivisible points and moments, and thus, since we can draw one and only\none parallel from any point in the diagonal of a square to a given side,\nthe diagonal will contain the same (infinite) number of points as that\nside, and will therefore be equal to it. In this Leibniz repeated an\nargument used by the ancient Arabs, Roger Bacon, and William of Occam.\nThis Leibniz considered to be a proof that a line cannot be an aggregate\nof points. Indeed, their number would be \"the number of all numbers\" of\nthe greatest possible integer, which _is_ not.\n\nIt does not seem, further, that any light is thrown on the logical\nquestion of human mortality or immortality by legal decisions. It would\nappear that one can, legally speaking, be alive for any period less than\ntwenty-four hours after one is dead and be dead for any period less than\ntwenty-four hours before one's death. At least, according to _Salkeld_,\ni. 44, it was \"adjudged that if one be born the first of February at\neleven at night, and the last of January in the twenty-first year of his\nage, at one of the clock in the morning, he makes his will of lands, and\ndies, it is a good will, for he was then of age.\" In Sir Robert Howard's\ncase (_ibid._, ii. 625) it was held by Chief Justice Holt that \"if A be\nborn on the third day of September; and on the second day of September\ntwenty-one years afterwards he make his will, this is a good will; for\nthe law will make no fraction of a day, and by consequence he was of\nage.\" But it is hardly necessary to remark that in this way the problem\nwith which we are concerned is merely shifted and not solved. For the\nquestion as to whether there is or is not a last moment of a man's life\nis not answered by the decision that he dies legally twenty-four hours\nbefore or after he dies in the usual sense of the word, and the problem\narises as to whether there is or is not a last moment of his legal\nage.[62]\n\nSo assuming that there was a last moment of Socrates's earthly life, and\nconsequently no first moment of his eternal life, we see, further, that,\nunless the possibility of infinite numbers is granted, it would be quite\npossible for us logically to doubt the possibility of an eternal life\nfor Socrates on the same grounds as those which led Zeno to assert that\nmotion was impossible and that Achilles could never overtake the\nTortoise. If, on the other hand, it be admitted that eternity, at least\nin the case of Socrates, had a beginning, these same arguments of Zeno\nwould lead any one who denies the possibility of infinite number to\nconclude that Socrates, like the worm, can never die. Thus is it quite\nplain that the difficulties about immortality which meet us at the very\noutset of our inquiry can partly be solved only by the help of the\ntheory of infinite numbers and partly, it would seem, not at all.\n\nThere is another difficulty about immortality which is quite distinct\nfrom this and is analogous to another argument of Zeno. If, indeed, all\nthe instants of time be divided, as before, into the two series of\ninstants at which Socrates was alive and instants at which he was not\nalive, it follows at once that no instant of time is not accounted for.\nAt none of these instants, however, does Socrates die; obviously he\ncannot die either when he is alive or when he is dead. Thus it would\nappear that Socrates never died, and that we ought to re-define the term\n\"mortal\" to mean \"a human being who is alive at some moments and dead at\nsome.\" Consequently we must avoid the very tempting conclusion that,\nbecause Socrates never died, he was therefore immortal.\n\nIt is very important carefully to distinguish between the two arguments\nI have just set forth. The second argument proves quite rigidly that\nSocrates and, indeed, anybody else, never dies, whether there is or is\nnot a last moment of his life on earth. The first argument proves that,\nif there is a first moment of Socrates's eternal life, his life on earth\nnever ends. But we have seen that we cannot conclude that this unending\nlife proves that he never is or will be in a state of eternity.\n\nFOOTNOTES:\n\n[58] By \"Dedekind's Axiom,\" _E. N._, p. 11.\n\n[59] _M._, vol. xx., 1910, pp. 134-5.\n\n[60] [Here, again, Mr. R*ss*ll's work seems to anticipate some of Mr.\nRussell's later work, e.g. in _Our Knowledge of the External World as a\nField for Scientific Method in Philosophy_, Chicago and London, 1914,\npp. 3-4, 55-6, _et passim._--ED.]\n\n[61] \"Pacidius Philalethi\" in Louis Couturat, _Opuscules et Fragments\ninedits de Leibniz_, Paris, 1903, pp. 594-627, especially pp. 599, 601,\n608, 611. Cf. [A. E. Taylor, Hastings' _Encyclopaedia of Religion and\nEthics_, vol. iv., Part 2, Edinburgh, 1912, p. 96.--ED.]; Robert Latta,\n_Leibniz: The Monadology and other Philosophical Writings_, Oxford,\n1898, pp. 21 ff, 29 (note); Couturat, _La Logique de Leibniz d'apres des\ndocuments inedits_, Paris, 1901, pp. 130, 132; and Russell, _Ph. L._,\npp. 108-16, 243-9.\n\n[62] [It may be remarked that, according to _The Times_ of December 20,\n1917, Mr. Justice Sargant, in the Chancery Division, also held that \"the\nlaw did not recognize fractions of a day,\" and that Lord Blackburn, in\nhis decision (9 _App. Cas._, 371, 373) that a man born on the thirteenth\nof May 1853 attained the age of twenty-one on the thirteenth of May 1874\n\"was not speaking strictly.\"--ED.]\n\n\n\n\nCHAPTER XXIII\n\nDENOTING\n\n\nA concept _denotes_ when, if it occurs in a proposition, the proposition\nis not about the concept, but _about_ a term connected in a certain\npeculiar way with the concept. Some people often assert that man is\nmortal, and yet we never see announced in _The Times_ that Man died on a\ncertain day at his villa residence \"Camelot\" at Upper Tooting,[63] nor\ndo we hear that Procrastination was again the butt of Mr. Plowden's\njokes at Marylebone Police Court last week.\n\nThat two phrases may have different _meanings_ and the same _denotation_\nwas discovered by Alice and Frege. Alice[64] observed that the road\nwhich led to Tweedledum's house was that which led to the house of\nTweedledee; and Frege pointed out that the phrases \"the house to which\nthe road that leads to Tweedledum's house leads\" and \"the house to which\nthe road that leads to Tweedledee's house leads\" have different _Sinn_,\nbut the same _Bedeutung_.\n\nFOOTNOTES:\n\n[63] Cf. _P. M._, pp. 53-4.\n\n[64] See Appendix M.\n\n\n\n\nCHAPTER XXIV\n\nTHE\n\n\nThe word \"the\" implies existence and uniqueness; it is a mistake to talk\nof \"the son of So-and-So\" if So-and-So has a fine family of ten\nsons.[65] People who refer to \"the Oxford Movement\" imply that Oxford\nonly moved once; and those quaint people who say that \"A is quite the\ngentleman\" imply both the doubtful proposition that there is only one\ngentleman in the world, and the indubitably false proposition that he is\nthat man. Probably A is one of those persons who add to the confusion in\nthe use of the definite article by speaking of his wife as \"the wife.\"\n\nIn a certain Children's Hymn Book one reads:\n\n  The river vast and small.\n\nFew would deny that there is not more than one such river, but\nunfortunately it is doubtful if there is such a river at all. The case\nis exactly the same with the ontological proof of the existence of the\nmost perfect being.[66]\n\nAccording to the _Daily Mail_ of October 9, 1906, Judge Russell decided\nagainst a claim brought by an agent against his company for appointing\nanother agent, the claim being on the ground that he was appointed as\n\"the\" agent.\n\nMost people admit that the number 2 can be added to the number 2 to give\nthe number 4, but this is a mistake. They concede, when they use _the_,\nthat there is only one number 2, and yet they imagine that, when they\nconsider it apart as the first term of our above sum, they can find\nanother to add to it, and thereby form the third term. The truth is that\n\"2 + 2 = 4\" is a very misleading equation, and what we really mean by\nthat faultily abbreviated statement is more precisely: If _x_ and _y_\ndenote any things which form a class B, and _x'_ and _y'_ any other\nthings that form a class (A) which, like that of _x_ and _y_, is a\nmember of the class (which we call \"2\") of those classes which have a\none-one correspondence with B (so that any member of A corresponds to\none, and only one, member of B, and conversely), the class of all the\nterms of A and B together is a member of that class of classes which,\nanalogously, we call \"4.\" In this, for the sake of shortness, we have\nintroduced abbreviations which should not be used in a rigorous logical\nstatement.\n\nFOOTNOTES:\n\n[65] Cf. _Md._, N. S., vol. xiv., 1905, pp. 481, 484.\n\n[66] Cf. _ibid._, p. 491, note.\n\n\n\n\nCHAPTER XXV\n\nNON-ENTITY\n\n\nWhen people say that such-and-such a thing \"is non-existent\" they\nusually mean that there is not any \"thing\" of the kind spoken of. Venn\nmeant this when he described[67] his encounter with what he imagined to\nbe a very ingenious tradesman: \"I once had some strawberry plants\nfurnished me which the vendor admitted would not bear many berries. But\nhe assured me that this did not matter, since they made up in their size\nwhat they lost in their number. (He gave me, in fact, the hyperbolic\nformula, _xy = c_, to connect the number and magnitude.) When summer\ncame, _no_ fruit whatever appeared. I saw that it would be no use to\ncomplain, because the man would urge that the size of the non-existent\nberry was infinite, which I could not see my way to disprove. I had\nforgotten to bar zero values of either variable.\"\n\nIt is to be regretted that this useful note was omitted in the second\nedition of Venn's book; one can imagine that it might have protected Mr.\nMacColl and Herr Meinong (who believed, unlike Alice in what may be\ncalled her first theory,[68] in round squares and fabulous monsters)\nagainst the dishonest practices of traders who were too ready with\npromises. For the death-blow to this kind of trade was not given until\n1905, when Mr. Russell published his article \"On Denoting,\"[69] and took\nup the position of the White King in opposition to Alice's later\nassertions.[70]\n\nVenn's experience illustrates another characteristic of mathematical\nlogic. It is necessary, in order to make our arguments conclusive, to\ndevote great care to the elimination of difficulties which rarely occur.\nThe White Knight--who was like Boole in being a pioneer of mathematical\nlogic in this way, and yet seems to have held, like Boole, those\nphilosophical opinions which would base logic on psychology--recognized\nthe necessity of taking precautions against any unusual appearance of\nmice on a horse's back.[71]\n\nFOOTNOTES:\n\n[67] _S. L._, 1881, p. 339, note.\n\n[68] See Appendix N.\n\n[69] _Md._, N. S., vol. xiv., October 1905, pp. 479-93.\n\n[70] See Appendix N.\n\n[71] See Appendix O.\n\n\n\n\nCHAPTER XXVI\n\nIS\n\n\n_Is_ has four perfectly distinct meanings in English, besides misuses of\nthe word. Among the misuses, perhaps the most important are those\nreferred to by De Morgan:[72] \"... We say 'murder _is_ death to the\nperpetrator' where the copula is _brings_; 'two and two _are_ four,' the\ncopula being 'have the value of,' etc.\"\n\nSchroeder[73] quite satisfactorily pointed out the well-known distinction\nbetween an _is_ where subject and predicate can be interchanged (such\nas: \"the class whose members are Shem, Ham and Japhet is the class of\nthe sons of Noah\") and an _is_ or _are_ where they cannot (such as:\nEnglishmen are Britons), but failed to see[74] the more important\ndistinction (made by Peano) of is in the sense of \"is a member of.\" If\nEnglishmen are Britons, and Britons are civilized people, it follows\nthat Englishmen are civilized people; but, though the _Harmsworth\nEncyclopaedia_ is a member of the class Book (of one or more volumes),\nand this class is the member of a class A of which it is the only\nmember, yet the _Harmsworth Encyclopaedia_ is not a member of A, for it\nis not true that it is the whole class of books; and such a statement\nwould not even be made except possibly in the form of an advertisement.\n\nThe fourth meaning of _is_ is _exists_; it is in certain rare moods a\nmatter for regret that there are difficulties in the way of using one\nword to denote four different things. For, if there were not, we might\nprove the existence of any thing we please by making it the subject of a\nproposition, and thereby earn the gratitude of theologians.\n\nFOOTNOTES:\n\n[72] _F. L._, p. 268.\n\n[73] _A. d. L._, i. pp. 127 sqq.\n\n[74] _Ibid._, vol. ii. pp. 461, 597.\n\n\n\n\nCHAPTER XXVII\n\n_AND_ AND _OR_\n\n\nWhen, with Boole, alternatives (A, B) are considered as mutually\nexclusive, logical addition may be described as the process of taking A\n_and_ B or A _or_ B. It is a great and rare convenience to have two\nterms for denoting the same thing: commonly, people denote several\nthings by the same term, and only the Germans have the privilege of\nreferring to, say, _continuity_ as _Stetigkeit_ or _Kontinuierlichkeit_.\nBut Jevons[75] quoted Milton, Shakespeare, and Darwin to prove that\nalternatives are not exclusive, and so attained first to recognized\nviews by arguments which were plainly irrelevant.\n\nOf course, _and_ is often used as the sign of logical addition: thus one\nmay speak of one's brothers _and_ sisters, without being understood to\nmean the null-class (as should be the case), or pray for one's\n\"relations and friends,\" without being sure that one's prayer would be\nanswered,--as it certainly would if one meant to pray for the\nnull-class, this being the class indicated. And a word like _while_ is\noften used for a logical addition, when exclusiveness of the\nalternatives is almost implied. Thus, a reviewer in _Mind_,[76] noticing\nthe translation of Mach's _Popular Scientific Lectures_ into American,\nsaid of the lectures that: \"Most of them will be familiar ... to\nepistemologists and experimental psychologists: while the remainder,\nwhich deal with physical questions, are well worth reading.\" The reader\nhas the impression, probably given unintentionally, that Professor\nMach's epistemological and psychological lectures are not, in the\nreviewer's opinion, worth reading.\n\nFOOTNOTES:\n\n[75] _Pure Logic_ ..., London, 1864, pp. 76-9. Cf. Venn, _S. L._, 2nd\ned., pp. 40-8.\n\n[76] N. S., vol. iv. p. 261.\n\n\n\n\nCHAPTER XXVIII\n\nTHE CONVERSION OF RELATIONS\n\n\nThe \"Conversion of Relations\" does not mean what it might be supposed to\nmean; it has nothing to do with what Kant called \"the wholesome art of\npersuasion.\" What concerns us here is the convertibility of a logical\nrelation. If A has a certain relation R to B, the relation of B to A,\nwhich may be denoted by [vR], is called the _converse_ of R. As De\nMorgan[77] remarked, this conversion may sometimes present difficulties.\nThe following is De Morgan's example:\n\n\"Teacher: 'Now, boys, Shem, Ham and Japheth were Noah's sons; who was\nthe father of Shem, Ham and Japheth?' No answer.\n\n\"Teacher: 'Boys, you know Mr. Smith, the carpenter, opposite; has he any\nsons?'\n\n\"Boys: 'Oh! yes, sir! there's Bill and Ben.'\n\n\"Teacher: 'And who is the father of Bill and Ben Smith?'\n\n\"Boys: 'Why, Mr. Smith, to be sure.'\n\n\"Teacher: 'Well, then, once more, Shem, Ham and Japheth were _Noah's_\nsons; who was the father of Shem, Ham and Japheth?'\n\n\"A long pause; at last a boy, indignant at what he thought the attempted\ntrick, cried out: 'It _couldn't_ have been Mr. Smith.' These boys had\nnever converted the relation of father and son....\"\n\nFOOTNOTES:\n\n[77] _Trans. Camb. Phil. Soc._, vol. x., 1864, part ii., note on page\n334.\n\n\n\n\nCHAPTER XXIX\n\nPREVIOUS PHILOSOPHICAL THEORIES OF MATHEMATICS\n\n\nMathematicians usually try to found mathematics on two principles:[78]\none is the principle of confusion between the sign and the thing\nsignified (they call this principle the foundation-stone of the formal\ntheory), and the other is the Principle of the Identity of Discernibles\n(which they call the principle of the permanence of equivalent forms).\n\nBut the truth is that if we set sail on a voyage of discovery with Logic\nalone at the helm, we must either throw such principles as \"the identity\nof those conceptions which have in common the properties that interest\nus\" and \"the principle of permanence\" overboard, or, if we do not like\nto act in such a way to old companions with whom we are so familiar that\nwe can hardly feel contempt for them, at least recognize them clearly as\nhaving no logical validity and merely as psychological principles, and\nreduce them to the humble rank of stewards, to minister to our human\nweaknesses on the voyage. And then, if we adopt the wise policy of\nkeeping our axioms down to the minimum number, we must refrain from\ncreating or thinking that we are creating new numbers to fill up gaps\namong the older ones, and thence recognize that our rational numbers are\nnot particular cases of \"real\" numbers, and so on.\n\nWe thus get a world of conceptions which looks, and is, very different\nfrom that which ordinary mathematicians think they see; and perhaps this\nis the reason why some mathematicians of great eminence, such as Hilbert\nand Poincare, have produced such absurd discussions on the fundamental\nprinciples of mathematics,[79] showing once more the truth of the not\nquite original remark of Aunt Jane, who\n\n  ... observed, the second time\n  She tumbled off a 'bus:\n  \"The step is short from the sublime\n  To the ridiculous.\"\n\nIn their readiness to consider many different things as one thing--to\nconsider, for example, the ratio 2:1 as the same thing as the cardinal\nnumber 2--such mathematicians as Peacock, Hankel, and Schubert were\nforestalled by the Pigeon, who thought that Alice and the Serpent were\nthe same creature, because both had long necks and ate eggs.[80] It is,\nhowever, doubtful whether the Pigeon would have followed the example of\nthe mathematicians just mentioned so far as to embrace the creed of\nnominalism and so to feel no difficulty in subtracting from zero--a\ndifficulty which was pointed out with great acuteness by the Hatter[81]\nand modern mathematical logicians.\n\nFOOTNOTES:\n\n[78] These principles, after many attempts to state them by Peacock, the\nRed and the White Queen (see Appendix P), Hankel, Schroeder, and Schubert\nhad been made, were first precisely formulated by Frege in _Z. S._; cf.\nalso Chapter VII.\n\n[79] See Couturat, _R. M. M._, vol. xiv., March, 1906, pp. 208-50, and\nRussell, _ibid._, September, 1906, pp. 627-34.\n\n[80] See Appendix P.\n\n[81] See _ibid._\n\n\n\n\nCHAPTER XXX\n\nFINITE AND INFINITE\n\n\nI was once shown a statement made by an eminent mathematician of\nCambridge from which one would conclude that this mathematician thought\nthat finite distances became infinite when they were great enough. In\none of those splendidly printed books, bound in blue, published by the\nUniversity Press, and sold at about a guinea as a guide to some advanced\nbranch of pure mathematics, one may read, even in the second edition\npublished in 1900, the words: \"Representation [of a complex variable] on\na plane is obviously more effective for points at a finite distance from\nthe origin than for points at a very great distance.\"\n\nPlainly some of the points at a very great distance are at a _finite_\ndistance, for the same author mentions that Neumann's sphere for\nrepresenting the positions of points on a plane \"has the advantage ...\nof exhibiting the uniqueness of _z_ = [infinity symbol] as a value of\nthe variable.\"\n\n\n\n\nCHAPTER XXXI\n\nTHE MATHEMATICAL ATTAINMENTS OF TRISTRAM SHANDY\n\n\nTristram Shandy[82] said that his father was sometimes a gainer by\nmisfortune; for if the pleasure of haranguing about it was as ten, and\nthe misfortune itself only as five, he gained \"half in half,\" and was\nwell off again as if the misfortune had never happened.\n\nSuppose that the unit (arbitrary) of pleasure is denoted by A, Tristram\nShandy, by neglecting, in this ethical discussion, to introduce negative\nquantities (Kant's pamphlet advocating this introduction into philosophy\nwas made subsequently[83]), apparently made 15A to result, and this can\nhardly be maintained to be the half of 10A. It is possible, however,\nthat Tristram Shandy succeeded in proving the apparently paradoxical\nequation\n\n  15A = 5A\n\nby remarking that the axiom \"the whole is greater than the part\" does\nnot always hold. This remark follows at once from what Mr. Russell[84]\nhas called \"The Paradox of Tristram Shandy.\" This paradox is described\nby Mr. Russell as follows:\n\n\"Tristram Shandy, as we know, took two years writing the history of the\nfirst two days of his life, and lamented that, at this rate, material\nwould accumulate faster than he could deal with it, so that he could\nnever come to an end. Now I maintain that, if he had lived for ever,\nand not wearied of his task, then, even if his life had continued as\neventfully as it began, no part of his biography would have remained\nunwritten.\"\n\nThis paradox is strictly correlative to the well-known paradox of Zeno\nabout Achilles and the Tortoise.[85] \"The Achilles proves that two\nvariables in a continuous series, which approach equality from the same\nside, cannot ever have a common limit: the Tristram Shandy proves that\ntwo variables which start from a common term, and proceed in the same\ndirection, but diverge more and more, may yet determine the same\nlimiting class (which, however, is not necessarily a segment, because\nsegments were defined as having terms beyond them). The Achilles assumes\nthat whole and part cannot be similar, and deduces a paradox; the other,\nstarting from a platitude, deduces that whole and part may be similar.\nFor common-sense, it must be confessed that it is a most unfortunate\nstate of things.\" And Mr. Russell considers that, in the face of proofs,\nit ought to commit suicide in despair.\n\nNow, I suggest the extremely unlikely possibility that Tristram Shandy,\nby reflection on his own life and literary labours, was led to the\ncorrect course of accepting the paradox which resulted from this\nreflection and rejecting the Achilles. Thus, he concluded that an\ninfinite whole may be similar (or, in Cantor's terminology,\n\"equivalent\") to a proper part of itself, and hence, by a confusion of\nsimilarity with identity (or equivalence with equality) which he shares\nwith some subsequent philosophers,[86] that a whole may be equal to a\nproper part of itself. If A is an infinite class, it is not difficult to\nsee that we can have\n\n  10A = 5A.\n\nIn this way many have avoided an opinion which rests on no better\nfoundation than that formerly entertained by the inductive philosophers\nof Central Africa, that all men are black.[87]\n\nFOOTNOTES:\n\n[82] Cf. a letter of De Morgan in Mrs. De Morgan's _Memoir of Augustus\nDe Morgan_, p. 324.\n\n[83] Kant's tract was published in 1763, while _Tristram Shandy_ was\npublished in 1760.\n\n[84] _P. M._, pp. 358-9 [Cf. _M._, vol. xxii., January 1912, p.\n187.--ED.]\n\n[85] Cf. _P. M._, pp. 350, 358-9; _M._, vol. xxii., 1912, p. 157.\n\n[86] [Cf. for example, Cosmo Guastella, _Dell' infinito_, Palermo,\n1912.--ED.]\n\n[87] Cf. Russell, _P. M._, p. 360.\n\n\n\n\nCHAPTER XXXII\n\nTHE HARDSHIPS OF A MAN WITH AN UNLIMITED INCOME\n\n\nI once heard a man refer to his income as limited, in order to\nillustrate the hardship of a class of men, of which he of course was\none, in having to pay a somewhat high income-tax. It is obvious that\nthis man spoke enviously, and consequently admitted the existence of\nmore fortunately placed individuals who had unlimited incomes. A little\nreflection would have shown the man that he was not taking up a\nparadoxical attitude. A \"paradoxical attitude\" is of course the\nassertion of one or more propositions of which the truth cannot be\nperceived by a philosopher--and particularly an idealist--and can be\nperceived by a logician and occasionally, but not always, by a man of\ncommon-sense. Such propositions are: \"The cat is hungry,\" \"Columbus\ndiscovered America,\" and \"A thing which is always at rest may move from\nthe position A to the different position B.\"\n\nNow, if a man had an unlimited income, it is an immediate inference\nthat, however low income-tax might be, he would have to pay annually to\nthe Exchequer of his nation a sum equal in value to his whole income.\nFurther, if his income was derived from a capital invested at a finite\nrate of interest (as is usual), the annual payments of income-tax would\neach be equal in value to the man's whole capital. If, then, the man\nwith an unlimited income chose to be discontented, he would be sure of a\nsympathetic audience among philosophers and business acquaintances; but\ndiscontent could not last long, for the thought of the difficulties he\nwas putting in the way of the Chancellor of the Exchequer, who would\nfind the drawing up of his budget most puzzling, would be amusing.\nAgain, the discovery that, after paying an infinite income-tax, the\nincome would be quite undiminished, would obviously afford satisfaction,\nthough perhaps the satisfaction might be mixed with a slight uneasiness\nas to any action the Commissioners of Income-Tax might take in view of\nthis fact.\n\nA problem of a wholly different nature is connected with the possible\npurchase by the man with an unlimited income of an enumerable infinity\nof pairs of boots. If he wished to prove that he had an even number of\nboots, it would be easy if right boots were distinguishable from left\nones, but if the man were a faddist of such a kind that he insisted that\nhis left boots should not be made in any way differently from his right\nones, it would not be possible for him to prove the theorem mentioned\nunless he assumed what is known as \"the multiplicative axiom.\" In fact\nthis axiom shows that it is legitimate to pick out an infinite\nsuccession of members of an infinite class in an arbitrary way. In the\ncase of the pairs of boots, each pair contains two members, and if there\nis no means of distinguishing between them, when we wish to pick out one\nof them for each of the infinity of pairs, we cannot say which ones we\nmean to pick out unless we assume, by means of the above axiom, that a\nparticularized member can always be found even with things of each of\nwhich it can be said that, like Private James in the _Bab Ballads_,\n\n  No characteristic trait had he\n  Of any distinctive kind.\n\nHowever, a solution of the puzzle was given by Dr. Denes Koenig of\nBudapest. You first prove that there are points in space such that, if P\nis one of them, not more than a finite number of pairs of boots are such\nthat each centre of mass of the two members of a pair is equidistant\nfrom P. Taking a point P of this sort, select from each pair the boot\nwhose centre of mass is nearest P. (There may be a finite number of\npairs left over, but they can be dealt with arbitrarily.)\n\nAnother form of the problem is as follows. Every time the man bought a\npair of boots he also bought a pair of socks to go with it; he had an\nenumerable infinity of pairs of each, and the problem is to prove that\nhe had as many boots as he had socks. In this case the boots, we will\nsuppose, can be divided into right and left, but the socks cannot. Thus\nthere are an enumerable infinity of boots, but the number of the socks\ncannot be determined without admitting the axiom mentioned above. A\nfurther difficulty might arise if the owner of the boots and socks lost\none leg in some accident, and told his butler to give away half his\nsocks. Naturally the butler would find great logical difficulties in so\ndoing, and it would seem to be an interesting ethical problem whether he\nshould be dismissed from his situation for failing to prove the\nmultiplicative axiom. Again, if the butler stole a pair of boots, the\nmillionaire would have as many pairs as before, but might have fewer\nboots. There is as yet no evidence that the number of his boots is equal\nto or greater than the number of pairs.\n\n\n\n\nCHAPTER XXXIII\n\nTHE RELATIONS OF MAGNITUDE OF CARDINAL NUMBERS\n\n\nThe theorems of cardinal arithmetic are frequently used in ordinary\nconversation. What is known as the Schroeder-Bernstein theorem was used,\nlong before Bernstein or Schroeder, by Edward Thurlow, afterward the\nlaw-lord Lord Thurlow, when an undergraduate of Caius College,\nCambridge. Thurlow was rebuked for idleness by the Master, who said to\nhim: \"Whenever I look out of the window, Mr. Thurlow, I see you crossing\nthe Court.\" The provost thus asserted a one-one correspondence between\nthe class A of his acts of looking out of the window and a part of the\nclass B of Thurlow's acts of crossing the Court. Thurlow asserted in\nreply a one-one correspondence between B and a part of A: \"Whenever\nI cross the Court I see you looking out of the window.\" The\nSchroeder-Bernstein theorem, then, allows us to conclude that there is a\none-one correspondence between the classes A and B. That A and B were\nfinite classes is not the fault of the Master or Thurlow; nor is it\nrelevant logically.\n\n\n\n\nCHAPTER XXXIV\n\nTHE UNKNOWABLE\n\n\nAccording to Mr. S. N. Gupta,[88] the first thing that every student of\nHindu logic has to learn when he is said to begin the study of inference\nis that \"all H is S\" is not always equivalent to \"No H is not S.\" \"The\nlatter proposition is an absurdity when S is _Kebalanvayi_, i.e. covers\nthe whole sphere of thought and existence.... 'Knowable' and 'Nameable'\nare among the examples of _Kebalanvayi_ terms. If you say there is a\nthing not-knowable, how do you know it? If you say there is a thing\nnot-nameable, you must point that out, i.e. somehow name it. Thus you\ncontradict yourself.\"\n\nMr. Herbert Spencer's doctrine of the \"Unknowable\" gives rise to some\namusing thoughts. To state that all knowledge of such and such a thing\nis above a certain person's intelligence is not self-contradictory, but\nmerely rude: to state that all knowledge of a certain thing is above all\npossible human intelligence is nonsense, in spite of its modest,\nplatitudinous appearance. For the statement seems to show that we do\nknow something of it, viz. that it is unknowable.\n\nTo the last (1900) edition of _First Principles_ was added a \"Postscript\nto Part I,\" in which the justice of this simple and well-known criticism\nas to the contradiction involved in speaking of an \"Unknowable,\" which\nhad been often made during the forty odd years in which the various\neditions had been on the market, was grudgingly acknowledged as\nfollows:[89]\n\n\"It is doubtless true that saying what a thing is not, is, in some\nmeasure, saying what it is;... Hence it cannot be denied that to affirm\nof the Ultimate Reality that it is unknowable is, in a remote way, to\nassert some knowledge of it, and therefore involves a contradiction.\"\n\nThe \"Postscript\" reminds one of the postscript to a certain Irishman's\nletter. This Irishman, missing his razors after his return from a visit\nto a friend, wrote to his friend, giving precise directions where to\nlook for the missing razors; but, before posting the letter, added a\npostscript to the effect that he had found the razors.\n\nOne is tempted to inquire, analogously, what might be, in view of the\nPostscript, the point of much of Spencer's Part I. It is, to use De\nMorgan's[90] description of the arguments of some who maintain that we\ncan know nothing about infinity, of the same force as that of the man\nwho answered the question how long he had been deaf and dumb.\n\nBut the best part of the joke against Mr. Spencer is that he, as we\nshall see in Chapter XXXVIII, was refuted by a fallacious argument, and\nthus mistakenly asserted the validity of the refutation of remarks which\nhappen to be unsound.\n\nThe analogy of the contradiction of Burali-Forti with the contradiction\ninvolved in the notion of an \"unknowable\" may be set forth as follows.\nIf A should say to B: \"I know things which you never by any possibility\ncan know,\" he may be speaking the truth. In the same way, [Greek: o] may\nbe said, without contradiction, to transcend all the _finite_ integers.\nBut if some one else, C, should say: \"There are some things which no\nhuman being can ever know anything about,\" he is talking nonsense.[91]\nAnd in the same way if we succeeded in imagining a number which\ntranscends _all_ numbers, we have succeeded in imagining the absurdity\nof a number which transcends itself.\n\nAll the paradoxes of logic (or \"the theory of aggregates\") are\nanalogous to the difficulty arising from a man's statement: \"I am\nlying.\"[92] In fact, if this is true, it is false, and _vice versa_. If\nsuch a statement is spread out a little, it becomes an amusing hoax or\nan epigram. Thus, one may present to a friend a card bearing on both\nsides the words: \"The statement on the other side of this card is\nfalse\"; while the first of the epigrams derived from this principle\nseems to have been written by a Greek satirist:[93]\n\n  Lerians are bad; not _some_ bad and some _not_;\n  But all; there's not a Lerian in the lot,\n  Save Procles, that you could a good man call;--\n  And Procles--is a Lerian after all.\n\nThis is the original of a well-known epigram by Porson, who remarked\nthat all Germans are ignorant of Greek metres,\n\n  All, save only Hermann;--\n  And Hermann's a German.\n\nFOOTNOTES:\n\n[88] _Md._, N. S., vol. iv., 1895, p. 168.\n\n[89] _First Principles_, 6th ed., 1900, pp. 107-10. The first edition\nwas published in 1862.\n\n[90] Note on p. 6 of his paper: \"On Infinity; and on the Sign of\nEquality,\" _Trans. Camb. Phil. Soc._, vol. xi., part i., pp. 1-45 (read\nMay 16, 1864).\n\n[91] The assertion of the finitude of a man's mind appears to be\nnonsense; both because, if we say that the mind of man is limited we\ntacitly postulate an \"unknowable,\" and because, even if the human mind\nwere finite, there is no more reason against its conceiving the infinite\nthan there is for a mind to be blue in order to conceive a pair of blue\neyes (cf. De Morgan, _loc. cit._).\n\n[92] Russell, _R. M. M._, vol. xiv., September 1906, pp. 632-3, 640-4.\n\n[93] _The Greek Anthology_, by Lord Neaves (Ancient Classics for English\nReaders), Edinburgh and London, 1897, p. 194.\n\n\n\n\nCHAPTER XXXV\n\nMR. SPENCER, THE ATHANASIAN CREED AND THE ARTICLES\n\n\nWhen, in what I believe is misleadingly known as \"The Athanasian Creed,\"\npeople say \"The Father incomprehensible,\" and so on, they are not\nfalling into the same error as Mr. Spencer, for the Latin equivalent for\n\"incomprehensible\" is merely \"_immensus_,\" and Bishop Hilsey translated\nit more correctly as \"immeasurable.\"[94] It is a regrettable fact that\nDr. Blunt,[95] in his mistaken modesty, has added a note to this passage\nthat: \"Yet it is true that a meaning not intended in the Creed has\ndeveloped itself through this change of language, for the Nature of God\nis as far beyond the grasp of the mind as it is beyond the possibility\nof being contained within local bounds.\"\n\nMr. Spencer seems no happier when we compare his statements with those\nin the Anglican Articles of Religion. There God is never referred to as\ninfinite. It is true that His power and goodness are so referred to; but\nthis deficiency was presumably brought about intentionally, so that\nfaith might gain in meaning as time went on.\n\nFOOTNOTES:\n\n[94] _A. C. P._, p. 217.\n\n[95] _Ibid._, p. 218.\n\n\n\n\nCHAPTER XXXVI\n\nTHE HUMOUR OF MATHEMATICIANS\n\n\nBrahmagupta's problem[96] appears to be the earliest instance of a kind\nof joke which has been much used by mathematicians. For the sake of\ngiving a certain picturesqueness to the data of problems, and so to\nexcite that sort of interest which is partly expressed by a smile,\nmathematicians have got into the habit of talking, for example, of\nmonkeys in the form of geometrical points climbing up massless ropes.\nProfessor P. Staeckel[97] truly remarked that physiological\nmechanics--the mechanics of bones, muscles, and so on--is wholly\ndifferent from this. There was once a lecturer on mathematics at\nCambridge who used yearly to propound to his pupils a problem in rigid\ndynamics which related to the motion of a garden roller supposed to be\nwithout mass or friction, when a heavy and perfectly rough insect walked\nround the interior of it in the direction of normal rolling.\n\nHitherto this has been the only mathematical outlet for the humour of\nmathematicians; and those who really had the interests of mathematics at\nheart saw with alarm the growing tendency towards scholasticism in\nmathematical jokes. Fortunately the discovery of logic by some\nmathematicians has removed this danger. Still to many mathematicians\nlogic is still unknown, and to them--to Professor A. Schoenflies for\nexample--modern mathematics, owing to its alliance with logic, appears\nto be sinking into scholasticism. It is true that the word\n\"scholasticism\" is not used by Professor Schoenflies in any\nintentionally precise signification, but merely as a vague epithet of\ndisapproval, as the word \"socialism\" is used by the ordinary philistine,\nand this would certainly serve as a sufficient excuse. But no excuse is\nneeded: these opinions are themselves a source of mathematical jokes.\n\nFOOTNOTES:\n\n[96] See Chapter XII.\n\n[97] _Encykl. der math. Wiss._, vol. iv., part i., p. 474.\n\n\n\n\nCHAPTER XXXVII\n\nTHE PARADOXES OF LOGIC\n\n\nWe have already[98] referred to the contempt shown by some\nmathematicians for exact thought, which they condemn under the name\nof \"scholasticism.\" An example of this is given by Schoenflies in\nthe second part of his publication usually known as the _Bericht\nueber Mengenlehre_.[99] Here[100] a battle-cry in italics--\n\n  \"_Against all resignation, but also against all scholasticism!_\"--\n\nfound utterance. Later on, Schoenflies[101] became bolder and adopted a\nmore personal battle-cry, also in italics, and with a whole line to\nitself:\n\n  \"_For Cantorism but against Russellism!_\"\n\n\"Cantorism\" means the theory of transfinite aggregates and numbers\nerected for the most part by Georg Cantor. Shortly speaking, the great\nsin of \"Russellism\" is to have gone too far in the chain of logical\ndeduction for many mathematicians, who were perhaps, like\nSchoenflies,[102] blinded by their rather uncritical love of\nmathematics. Thus it comes about that Schoenflies[103] denounces\nRussellism as \"scholastic and unhealthy.\" This queer blend of qualities\nwould surely arouse the curiosity of the most _blase_ as to what strange\nthing Russellism must be.[104]\n\nSchoenflies[105] said that some mathematicians attributed to the logical\nparadoxes which have given Russell so much trouble to clear up,\n\"especially to those that are artificially constructed, a significance\nthat they do not have.\" Yet no grounds were given for this assertion,\nfrom which it might be concluded that the rigid examination of any\nconcept was unimportant. The paradoxes are simply the necessary results\nof certain logical views which are currently held, which views do not,\nexcept when they are examined rather closely, appear to contain any\ndifficulty. The contradiction is not felt, as it happens, by people who\nconfine their attention to the first few number-classes of Cantor, and\nthis seems to have given rise to the opinion, which it is a little\nsurprising to find that some still hold, that cases not usually met\nwith, though falling under the same concept as those usually met with,\nare of little importance. One might just as well maintain that\ncontinuous but not differentiable functions are unimportant because they\nare artificially constructed--a term which I suppose means that they do\nnot present themselves when unasked for. Rather should we say that it is\nby the discovery and investigation of such cases that the concept in\nquestion can alone be judged, and the validity of certain theorems--if\nthey are valid--conclusively proved. That this has been done, chiefly by\nthe work of Russell, is simply a fact; that this work has been and is\nmisunderstood by many[106] is regrettable for this reason, among others,\nthat it proves that, at the present time, as in the days in which\n_Gulliver's Travels_ were written, some mathematicians are bad\nreasoners.[107]\n\nNearly all mathematicians agreed that the way to solve these paradoxes\nwas simply not to mention them; but there was some divergence of opinion\nas to how they were to be unmentioned. It was clearly unsatisfactory\nmerely not to mention them. Thus Poincare was apparently of opinion that\nthe best way of avoiding such awkward subjects was to mention that they\nwere not to be mentioned. But[108] \"one might as well, in talking to a\nman with a long nose, say: 'When I speak of noses, I except such as are\ninordinately long,' which would not be a very successful effort to avoid\na painful topic.\"\n\nSchoenflies, in his paper of 1911 mentioned above, adopted the\nconvenient plan of referring these logical difficulties at the root of\nmathematics to a department of knowledge which he called \"philosophy.\"\nHe said[109] of the theory of aggregates that though \"born of the\nacuteness of the mathematical spirit, it has gradually fallen into\nphilosophical ways, and has lost to some extent the compelling force\nwhich dwells in the mathematical process of conclusion.\"\n\nThe majority of mathematicians have followed Schoenflies rather than\nPoincare, and have thus adopted tactics rather like those of the March\nHare and the Gryphon,[110] who promptly changed the subject when Alice\nraised awkward questions. Indeed, the process of the first of these\ncreatures of a child's dream is rather preferable to that of\nSchoenflies. The March Hare refused to discuss the subject because he\nwas bored when difficulties arose. Schoenflies would not say that he was\nbored--he professed interest in philosophical matters, but simply called\nthe logical continuation of a subject by another name when he did not\nwish to discuss the continuation, and thus implied that he had discussed\nthe whole subject. Further, Schoenflies would not apparently admit that\nthe one method of logic could be applied to the solution of both\nmathematical and philosophical problems, in so far as these problems are\nsoluble at all; but the March Hare, shortly before the remark we have\njust quoted, rightly showed great astonishment that butter did not help\nto cure both hunger and watches that would not go.[111] The judgment of\nSchoenflies by which certain apparently mathematical questions were\ncondemned as \"philosophical,\" rested on grounds as flimsy as those in\nthe Dreyfus Case, or the Trial in _Wonderland_.[112]\n\nFOOTNOTES:\n\n[98] Chapters VII and XXXVI.\n\n[99] _Die Entwickelung der Lehre von den Punktmannigfaltigkeiten._\nBericht, erstattet der deutschen Mathematiker-Vereinigung, Leipzig,\n1908.\n\n[100] _Ibid._, p. 7. The battle-cry is: \"_Gegen jede Resignation, aber\nauch gegen jede Scholastik!_\"\n\n[101] \"Ueber die Stellung der Definition in der Axiomatik,\" _Jahresber,\nder deutsch. Math.-Ver._, vol. xx., 1911, pp. 222-5. The battle-cry is\non p. 256 and is: \"Fuer den Cantorismus aber gegen den Russellismus!\"\n\n[102] _Ibid._, p. 251. \"Es ist also,\" he exclaims with the eloquence of\nemotion and the emotion of eloquence, \"nicht die Geringschaetzung der\nPhilosophie, die mich dabei treibt, sondern die Liebe zur\nMathematik;...\"\n\n[103] \"Ueber die Stellung der Definition in der Axiomatik,\" _Jahresber,\nder deutsch. Math.-Ver._, vol. xx., 1911, p. 251.\n\n[104] [Cf. for this, _M._, vol. xxii., January 1912, pp. 149-58.--ED.]\n\n[105] _Bericht_, 1908, p. 76, note; cf. p. 72.\n\n[106] E.g. in F. Hausdorff's review of Russell's _Principles_ of 1903 in\nthe _Vierteljahrsschr. fuer wiss. Philos. und Soziologie_.\n\n[107] [Cf. _M._, vol. xxv., 1915, pp. 333-8.--ED.]\n\n[108] Russell, _A. J. M._, vol. xxx., 1908, p. 226.\n\n[109] _Loc. cit._, p. 222.\n\n[110] See Appendix Q.\n\n[111] See Appendix R.\n\n[112] See Appendix S.\n\n\n\n\nCHAPTER XXXVIII\n\nMODERN LOGIC AND SOME PHILOSOPHICAL ARGUMENTS\n\n\nThe most noteworthy reformation of recent years in logic is the\ndiscovery and development by Mr. Bertrand Russell of the fact that the\nparadoxes--of Burali-Forti, Russell, Koenig, Richard, and others--which\nhave appeared of late years in the mathematical theory of aggregates and\nhave just been referred to, are of an entirely _logical_ nature, and\nthat their avoidance requires us to take account of a principle which\nhas been hitherto unrecognized, and which renders invalid several\nwell-known arguments in refutation of scepticism, agnosticism, and the\nstatement of a man that he asserts nothing.\n\nDr. Whitehead and Mr. Russell say:[113] \"The principle which enables us\nto avoid illegitimate totalities may be stated as follows: 'Whatever\ninvolves _all_ of a collection must not be one of the collection,' or\nconversely: 'If, provided a certain collection had a total, it would\nhave members only definable in terms of that total, then the said\ncollection has no total.' We shall call this the 'vicious-circle\nprinciple,' because it enables us to avoid the vicious circles involved\nin the assumption of illegitimate totalities. Arguments which are\ncondemned by the vicious-circle principle will be called 'vicious-circle\nfallacies.' Such arguments, in certain circumstances, may lead to\ncontradictions, but it often happens that the conclusions to which they\nlead are in fact true, though the arguments are fallacious. Take, for\nexample, the law of excluded middle in the form 'all propositions are\ntrue or false.' If from this law we argue that, because the law of\nexcluded middle is a proposition, therefore the law of excluded middle\nis true or false, we incur a vicious-circle fallacy. 'All propositions'\nmust be in some way limited before it becomes a legitimate totality, and\nany limitation which makes it legitimate must make any statement about\nthe totality fall outside the totality. Similarly the imaginary sceptic\nwho asserts that he knows nothing and is refuted by being asked if he\nknows that he knows nothing, has asserted nonsense, and has been\nfallaciously refuted by an argument which involves a vicious-circle\nfallacy. In order that the sceptic's assertion may become significant it\nis necessary to place some limitation upon the things of which he is\nasserting his ignorance; the proposition that he is ignorant of every\nmember of this collection must not itself be one of the collection.\nHence any significant scepticism is not open to the above form of\nrefutation.\"\n\nIn fact, the world of things falls into various sets of things of the\nsame \"type.\" For every propositional function [Greek: ph](_x_) there is\na range of values of _x_ for which [Greek: ph](_x_) has a signification\nas a true or a false proposition. Until this theory was brought forward,\nthere were occasionally discussions as to whether an object which did\nnot belong to the range of a certain propositional function possessed\nthe corresponding property or not. Thus, Jevons, in early days,[114] was\nof opinion that virtue is neither black nor not-black because it is not\ncoloured, but rather later[115] he admitted that virtue is not\ntriangular.[116]\n\nFOOTNOTES:\n\n[113] _Pa. Ma._, p. 40.\n\n[114] _S. o. S._ pp. 36-7.\n\n[115] _E. L. L._, pp. 120-1.\n\n[116] [It may perhaps be added that, some years after Mr. R*ss*ll's\ndeath, Dr. Whitehead stated, in an address delivered in 1916 and\nreprinted in his book on _The Organisation of Thought_ (London, 1917, p.\n120), that \"the specific heat of virtue is 0.003 is, I should imagine,\nnot a proposition at all, so that it is neither true nor\nfalse....\"--ED.]\n\n\n\n\nCHAPTER XXXIX\n\nTHE HIERARCHY OF JOKES\n\n\nJokes may be divided into various types. Thus a joke or class of jokes\ncan only be the subject of a joke of higher order. Otherwise we would\nget the same vicious-circle fallacy which gives rise to so many\nparadoxes in logic and mathematics. A certain Oxford scholar succeeded,\nto his own satisfaction, in reducing all jokes to primitive types,\nconsisting of thirty-seven proto-Aryan jokes. When any proposition was\npropounded to him, he would reflect and afterwards pronounce on the\nquestion as to whether the proposition was a joke or not. If he decided,\nby his theory, that it was a joke, he would solemnly say: \"There _is_\nthat joke.\" If this narration is accepted as a joke, since it cannot be\nreduced to one of the proto-Aryan jokes under pain of leading us to\ncommit a vicious-circle fallacy, we must conclude that there is at least\none joke which is not proto-Aryan; and, in fact, is of a higher type.\nThere is no great difficulty in forming a hierarchy of jokes of various\ntypes. Thus a joke of the fourth type (or order) is as follows: A joke\nof the first order was told to a Scotchman, who, as we would expect, was\nunable to see it.[117] The person (A) who told this joke told the story\nof how the joke was received to another Scotchman thereby making a joke\nabout a joke of the first order, and thus making a joke of the second\norder. A remarked on this joke that no joke could penetrate the head of\nthe Scotchman to whom the joke of the first order was told, even if it\nwere fired into his head with a gun. The Scotchman, after severe\nthought, replied: \"But ye couldn't do that, ye know!\" A repeated the\nwhole story, which constituted a joke of the third order, to a third\nScotchman. This last Scotchman again, after prolonged thought, replied:\n\"He had ye there!\" This whole story is a joke of the fourth order.\n\nMost known jokes are of the first order, for the simple reason that the\nmajority of people find that the slightest mental effort effectually\ndestroys any perception of humour. It seems to me that a joke becomes\nmore pleasurable in proportion as logical faculties are brought into\nplay by it; and hence that logical power is allied, or possibly\nidentical, with the power of grasping more subtle jokes. The jokes which\namuse the frequenters of music-halls, Conservatives, and M. Bergson--and\nwhich usually deal with accidents, physical defects, mothers-in-law,\nforeigners, or over-ripe cheese--are usually jokes of the first order.\nJokes of the second, and even of the third, order appeal to ordinary\nwell-educated people; jokes of higher order require either special\nability or a sound logical training on the part of the hearer if the\njoke is to be appreciated; while jokes of transfinite order presumably\nonly excite the inaudible laughter of the gods.\n\nFOOTNOTES:\n\n[117] [It may be that, like certain remarks about cheese and\nmothers-in-law (see below), the statement that Scotchmen cannot see\njokes is a joke of the first order.--ED.]\n\n\n\n\nCHAPTER XL\n\nTHE EVIDENCE OF GEOMETRICAL PROPOSITIONS\n\n\nIt has often been maintained that the twentieth proposition of the first\nbook of Euclid--that two sides of a triangle are together greater than\nthe third side--is evident even to asses. This does not, however, seem\nto me generally true. I once asked a coastguardsman the distance from A\nto B; he replied: \"Eight miles.\" On further inquiry I elicited the fact\nthat the distance from A to C was two miles and the distance from C to B\nwas twenty-two miles. Now the paths from A to B and from C to B were by\nsea; while the path from A to C was by land. Hence if the path by land\nwas rugged and the distance along the road was two miles, it would\nappear that the coastguardsman believed that not only could one side of\na triangle be greater than the other two, but that one straight side of\na triangle might be greater than one straight side and any curvilinear\nside of the same triangle. The only escape from part of this astonishing\ncreed would be by assuming that the distance of two miles from A to C\nwas measured \"as the crow flies,\" while the road A to C was so hilly\nthat a pedestrian would traverse more than fourteen miles when\nproceeding from A to C. Then indeed the coastguardsman could maintain\nthe true proposition that there is at least one triangle ABC, with the\nside AC curvilinear, such that the sum of the lengths of AB and AC is\ngreater than the length of BC, and only deny the twentieth proposition\nof the first book of Euclid.\n\nReasoning with the coastguardsman only had the effect of his adducing\nthe authority of one Captain Jones in support of the accuracy of his\ndata. Possibly Captain Jones held strange views as to the influence of\ntemperature or other physical circumstances, or even the nature of space\nitself, on the lengths of lines in the neighbourhood of the triangle\nABC.\n\n\n\n\nCHAPTER XLI\n\nABSOLUTE AND RELATIVE POSITION\n\n\nSome people maintain that position in space or time must be relative\nbecause, if we try to determine the position of a body A, if bodies B,\nC, D with respect to which the position of A could be determined were\nnot present, we should be trying to determine something about A without\nhaving our senses affected by other things. These people seem to me to\nbe like the cautious guest who refused to say anything about his host's\nport-wine until he had tasted red ink.\n\n\"Wherein, then,\" says Mr. Russell,[118] \"lies the plausibility of the\nnotion that all points are exactly alike? This notion is, I believe, a\npsychological illusion, due to the fact that we cannot remember a point\nso as to know it when we meet again. Among simultaneously presented\npoints it is easy to distinguish; but though we are perpetually moving,\nand thus being brought among new points, we are quite unable to detect\nthis fact by our senses, and we recognize places only by the objects\nthey contain. But this seems to be a mere blindness on our part--there\nis no difficulty, so far as I can see, in supposing an immediate\ndifference between points, as between colours, but a difference which\nour senses are not constructed to be aware of. Let us take an analogy:\nSuppose a man with a very bad memory for faces; he would be able to\nknow, at any moment, whether he saw one face or many, but he would not\nbe aware whether he had seen any of the faces before. Thus he might be\nled to define people by the rooms in which he saw them, and to suppose\nit self-contradictory that new people should come to his lectures, or\nthat old people should cease to do so. In the latter point at least it\nwill be admitted by lecturers that he would be mistaken. And as with\nfaces, so with points--inability to recognize them must be attributed,\nnot to the absence of individuality, but merely to our incapacity.\"\n\nAnother form of this tendency is shown by Kronecker, Borel, Poincare,\nand many other mathematicians, who refuse mere logical determination of\na conception and require that it be actually described in a finite\nnumber of terms. These eminent mathematicians were anticipated by the\nempirical philosopher who would not pronounce that the \"law of thought\"\nthat A is either in the place B or not is true until he had looked to\nmake sure. This philosopher was of the same school as J. S. Mill and\nBuckle, who seem to have maintained implicitly not only that, in view of\nthe fact that the breadth of a geometrical line depends upon the\nmaterial out of which it is constructed, or upon which it is drawn, that\nthere ought to be a paste-board geometry, a stone geometry, and so\non;[119] but also that the foundations of logic are inductive in their\nnature.[120] \"We cannot,\" says Mill,[121] \"conceive a round square, not\nmerely because no such object has ever presented itself in our\nexperience, for that would not be enough. Neither, for anything we know,\nare the two ideas in themselves incompatible. To conceive a body all\nblack and yet white would only be to conceive two different sensations\nas produced in us simultaneously by the same object--a conception\nfamiliar to our experience--and we should probably be as well able to\nconceive a round square as a hard square, or a heavy square, if it were\nnot that in our uniform experience, at the instant when a thing begins\nto be round, it ceases to be square, so that the beginning of the one\nimpression is inseparably associated with the departure or cessation of\nthe other. Thus our inability to form a conception always arises from\nour being compelled to form another contradictory to it.\"\n\nFOOTNOTES:\n\n[118] _Md._, N. S., vol. x., July, 1901, pp. 313-14.\n\n[119] J. B. Stallo, _The Concepts and Theories of Modern Physics_, 4th\ned., London, 1900, pp. 217-27.\n\n[120] _Ibid._, pp. 140-4.\n\n[121] _Examination of the Philosophy of Sir William Hamilton_, vol. i.\np. 88, Amer. ed.\n\n\n\n\nCHAPTER XLII\n\nLAUGHTER\n\n\n[It seemed advisable to give here[122] some views on laughter, most of\nwhich were also held by Mr. R*ss*ll, though no written expression of his\nviews has yet been found. In a review[123] of M. Bergson's book on\n_Laughter_,[124] Mr. Russell has remarked:\n\n\"It has long been recognized by publishers that everybody desires to be\na perfect lady or gentleman (as the case may be); to this fact we owe\nthe constant stream of etiquette-books. But if there is one thing which\npeople desire even more, it is to have a faultless sense of humour. Yet\nso far as I know, there is no book called 'Jokes without Tears, by Mr.\nMcQuedy.' This extraordinary lacuna has now been filled. Those to whom\nlaughter has hitherto been an unintelligible vagary, in which one must\njoin, though one could never tell when it would break out, need only\nstudy M. Bergson's book to acquire the finest flower of Parisian wit. By\nobserving a very simple formula they will know infallibly what is funny\nand what is not; if they sometimes surprise their unlearned friends,\nthey have only to mention their authority in order to silence doubt.\n'The attitudes, gestures and movements of the human body,' says M.\nBergson, 'are laughable in exact proportion as that body reminds us of a\nmere machine.' When an elderly gentleman slips on a piece of orange-peel\nand falls, we laugh, because his body follows the laws of dynamics\ninstead of a human purpose. When a man falls from a scaffolding and\nbreaks his neck on the pavement, we presumably laugh even more, since\nthe movement is even more completely mechanical. When the clown makes a\nbad joke for the first time, we keep our countenance, but at the fifth\nrepetition we smile, and at the tenth we roar with laughter, because we\nbegin to feel him a mere automaton. We laugh at Moliere's misers,\nmisanthropists and hypocrites, because they are mere types mechanically\ndominated by a master impulse. Presumably we laugh at Balzac's\ncharacters for the same reason; and presumably we never smile at\nFalstaff, because he is individual throughout.\"\n\nThe review concludes with the reflection that \"it would seem to be\nimpossible to find any such formula as M. Bergson seeks. Every formula\ntreats what is living as if it were mechanical, and is therefore by his\nown rule a fitting object of laughter.\" Now, this undoubtedly true\nconclusion has been obtained, as is readily seen, by a vicious-circle\nfallacy which Mr. R*ss*ll would hardly have committed.--ED.]\n\nFOOTNOTES:\n\n[122] From a remark on p. 47 above, it is evident that Mr. R*ss*ll\nintended to write some such chapter as this.\n\n[123] _The Professor's Guide to Laughter, The Cambridge Review_, vol.\nxxxii., 1912, pp. 193-4.\n\n[124] _Laughter, an Essay on the Meaning of the Comic_, English\ntranslation by C. Brereton and F. Rothwell, London, 1911.\n\n\n\n\nCHAPTER XLIII\n\n\"GEDANKENEXPERIMENTE\" AND EVOLUTIONARY ETHICS\n\n\nThe \"Gedankenexperimente,\" upon which so much weight has been laid by\nMach[125] and Heymans,[126] had already been investigated by the White\nQueen,[127] who, however, seems to have perceived that the results of\nsuch experiments are not always logically valid. The psychological\nfounding of logic appears to be not without analogy with the surprising\nmethod of advocates of evolutionary ethics, who expect to discover what\n_is_ good by inquiring what cannibals have _thought_ good. I sometimes\nfeel inclined to apply the historical method to the multiplication\ntable. I should make a statistical inquiry among school-children, before\ntheir pristine wisdom had been biassed by teachers. I should put down\ntheir answers as to what 6 times 9 amounts to, I should work out the\naverage of their answers to six places of decimals, and should then\ndecide that, at the present stage of human development, this average is\nthe value of 6 times 9.\n\nFOOTNOTES:\n\n[125] See, e.g., _E. u. I._, pp. 183-200.\n\n[126] _G. u. E._, vol. i.\n\n[127] See Appendix T.\n\n\n\n\nAPPENDIXES\n\n\nA. LOGIC AND THE PRINCIPLE OF IDENTITY.\n\n_T. L. G._, p. 45: \"'Contrariwise,\" continued Tweedledee, \"if it was so,\nit might be; and if it were so, it would be: but as it isn't, it ain't.\nThat's logic.\"\n\n       *       *       *       *       *\n\n_S. B._, p. 159: The Professor said: \"The day is the same length as\nanything that is the same length as _it_.\"\n\n       *       *       *       *       *\n\n_S. B._, p. 161: Bruno observed that, when the Other Professor lost\nhimself, he should shout: \"He'd be sure to hear hisself, 'cause he\ncouldn't be far off.\"\n\n\nB. SYNTHESIS OF CONTRADICTORIES.\n\n_T. L. G._, p. 71: \"'What a beautiful belt you've got on!' Alice\nsuddenly remarked.... 'At least,' she corrected herself on second\nthoughts, 'a beautiful cravat, I should have said--no, a belt, I mean--I\nbeg your pardon!' she added in dismay, for Humpty-Dumpty looked\nthoroughly offended, and she began to wish she hadn't chosen that\nsubject. 'If only I knew,' she thought to herself, 'which was neck and\nwhich was waist!'\"\n\n\nC. EMPIRICAL PHILOSOPHERS AND MATHEMATICS.\n\n_T. L. G._, p. 79: \"'... Now if you had the two eyes on the same side of\nthe nose, for instance--or the mouth at the top--that would be _some_\nhelp.'\n\n\"'It wouldn't look nice,' Alice objected. But Humpty-Dumpty only shut\nhis eyes and said: 'Wait till you've tried.'\"\n\n       *       *       *       *       *\n\n_T. L. G._, p. 72: \"'And if you take one from three hundred and\nsixty-five, what remains?'\n\n\"'Three hundred and sixty-four, of course.'\n\n\"Humpty-Dumpty looked doubtful. 'I'd rather see that done on paper,'\nhe said.\"\n\n\nD. NOMINAL DEFINITION.\n\n_T. L. G._, p. 73: \"'When _I_ used a word,' Humpty-Dumpty said in rather\na scornful tone, 'it means just what I choose it to mean--neither more\nnor less.'\n\n\"'The question is,' said Alice, 'whether you _can_ make words mean\ndifferent things.'\n\n\"'The question is,' said Humpty-Dumpty, 'which is to be master--that's\nall.'\"\n\n\nE. CONFORMITY OF A PARADOXICAL LOGIC WITH COMMON-SENSE.\n\n_T. L. G._, p. 100:\n\n  \"But I was thinking of a plan\n  To dye one's whiskers green,\n  And always use so large a fan\n  That they could not be seen.\"\n                                   (Verse from White Knight's song.)\n\n\nF. IDEALISTS AND THE LAWS OF LOGIC.\n\n_T. L. G._, p. 52-3: Tweedledee exclaimed: \"'... if he [the Red King]\nleft off dreaming about you [Alice], where do you suppose you'd be?'\n\n\"'Where I am now, of course,' said Alice.\n\n\"'Not you!' Tweedledee retorted contemptuously. 'You'd be nowhere. Why,\nyou're only a sort of thing in his dream!'\n\n\"'If that there King was to wake,' added Tweedledum, 'you'd go\nout--bang!--just like a candle!'\n\n\"'I shouldn't!' Alice exclaimed indignantly. 'Besides, if _I'm_ only a\nsort of thing in his dream, what are _you_, I should like to know?'\n\n\"'Ditto,' said Tweedledum...; 'you know very well you're not real.'\n\n\"'I _am_ real!' said Alice, and began to cry.\"\n\n       *       *       *       *       *\n\n_T. L. G._, p. 97: \"'How _can_ you go on talking so quietly, head\ndownwards?' Alice asked, as she dragged him out by the feet, and laid\nhim in a heap on the bank.\n\n\"The Knight looked surprised at the question. 'What does it matter where\nmy body happens to be?' he said. 'My mind goes on working all the same.\nIn fact, the more head downwards I am, the more I keep inventing new\nthings.'\"\n\n       *       *       *       *       *\n\n_T. L. G._, p. 98: \"'... Everybody that hears me sing--either it brings\nthe _tears_ into their eyes, or else----'\n\n\"'Or else what?' said Alice, for the Knight had made a sudden pause.\n\n\"'Or else it doesn't, you know.'\"\n\n\nG. DISTINCTION BETWEEN SIGN AND SIGNIFICATION.\n\n_T. L. G._, pp. 98-9: \"'The name of the song is called \"_Haddocks'\nEyes_.\"'\n\n\"'Oh, that's the name of the song, is it?' Alice said, trying to feel\ninterested.\n\n\"'No, you don't understand,' the Knight said looking a little vexed.\n'That's what the name is _called_. The name really _is_ \"_The Aged Aged\nMan_.\"'\n\n\"'Then I ought to have said \"That's what the _song_ is called\"?' Alice\ncorrected herself.\n\n\"'No, you oughtn't: that's another thing. The _song_ is called \"_Ways\nand Means_\": but that's only what it's _called_, you know!'\n\n\"'Well, what _is_ the song, then?' said Alice, who was by this time\ncompletely bewildered.\n\n\"'I was coming to that,' the Knight said. 'The song really _is\n\"A-sitting on a Gate_\"....'\"\n\n\nH. NOMINALISM.\n\n_A. A. W._, p. 70: \"'Then you should say what you mean,' the March Hare\nwent on.\n\n\"'I do,' Alice hastily replied; 'at least--at least I mean what I\nsay--that's the same thing, you know.'\n\n\"'Not the same thing a bit!' said the Hatter. 'Why, you might just as\nwell say that \"I see what I eat\" is the same thing as \"I eat what I\nsee.\"'\n\n\"'You might just as well say,' added the March Hare, 'that \"I like what\nI get\" is the same thing as \"I get what I like\"!'\n\n\"'You might just as well say,' added the Dormouse, which seemed to be\ntalking in its sleep, 'that \"I breathe when I sleep\" is the same as \"I\nsleep when I breathe\"!'\n\n\"'It _is_ the same thing with you,' said the Hatter; and here the\nconversation dropped,...\"\n\n\nI. UTILITY OF SYMBOLIC LOGIC.\n\n_A. A. W._, p. 92: \"'I quite agree with you,' said the Duchess, 'and the\nmoral of that is--\"Be what you would seem to be\"--or if you'd like it\nput more simply--\"Never imagine yourself not to be otherwise than what\nit might appear to others that what you were or might have been was not\notherwise than what you had been would have appeared to them to be\notherwise.\"'\n\n\"'I think I should understand that better,' Alice said very politely,\n'if I had it written down: but I can't quite follow it as you say it.'\n\n\"'That's nothing to what I could say if I chose,' the Duchess replied,\nin a pleased tone.\"\n\n\nJ. MISTAKE AS TO THE NATURE OF CRITICISM.\n\n_T. L. G._, p. 105: \"'She's in that state of mind,' said the White\nQueen, 'that she wants to deny _something_--only she doesn't know what\nto deny.'\n\n\"'A nasty, vicious temper,' the White Queen remarked; and then there was\nan uncomfortable silence for a minute or two.\"\n\n\nK. A CRITERION OF TRUTH.\n\n_H. S._, p. 3:\n\n  \"Just the place for a Snark! I have said it twice:\n  That alone should encourage the crew.\n  Just the place for a Snark! I have said it thrice:\n  What I tell you three times is true.\"\n\n       *       *       *       *       *\n\n_H. S._, p. 50:\n\n  \"'Tis the note of the Jubjub! Keep count. I entreat;\n  You will find I have told it you twice.\n  'Tis the song of the Jubjub! The proof is complete,\n  If only I've stated it thrice.\"\n\n\nL. UNIVERSAL AND PARTICULAR PROPOSITIONS.\n\n_T. L. G._, p. 40: The Gnat had told Alice that the Bread-and-butterfly\nlives on weak tea with cream in it; so:\n\n\"'Supposing it couldn't find any?' she suggested.\n\n\"'Then it would die, of course.'\n\n\"'But that must happen very often,' Alice remarked thoughtfully.\n\n\"'It always happens,' said the Gnat.\"\n\n\nM. DENOTING.\n\n_T. L. G._, p. 43: Tweedledum and Tweedledee were, in many respects,\nindistinguishable, and Alice, walking along the road, noticed that\n\"whenever the road divided there were sure to be two finger-posts\npointing the same way, one marked 'TO TWEEDLEDUM'S HOUSE' and the other\n'TO THE HOUSE OF TWEEDLEDEE.'\n\n\"'I do believe,' said Alice at last, 'that they live in the same\nhouse!...'\"\n\n\nN. NON-ENTITY.\n\n_T. L. G._, p. 87: \"'I always thought they [human children] were\nfabulous monsters!' said the Unicorn....\n\n\"'Do you know [said Alice], I always thought Unicorns were fabulous\nmonsters, too! I never saw one alive before!'\n\n\"'Well, now that we _have_ seen each other,' said the Unicorn, 'if\nyou'll believe in me, I'll believe in you. Is that a bargain?'\"\n\n       *       *       *       *       *\n\n_T. L. G._, pp. 80-1: \"'I see nobody on the road,' said Alice.\n\n\"'I only wish _I_ had such eyes,' the [White] King remarked in a fretful\ntone. 'To be able to see Nobody! And at that distance, too! Why, it's as\nmuch as _I_ can do to see real people by this light!'\"\n\n       *       *       *       *       *\n\n_A. A. W._, p. 17: \"And she [Alice] tried to fancy what the flame of a\ncandle looks like after the candle is blown out, for she could not\nremember ever having seen such a thing.\"\n\n       *       *       *       *       *\n\n_A. A. W._, p. 68: \"... This time it [the Cheshire Cat] vanished quite\nslowly, beginning with the end of the tail, and ending with the grin,\nwhich remained some time after the rest of it had gone.\n\n\"'Well! I've often seen a cat without a grin,' thought Alice; 'but a\ngrin without a cat! It's the most curious thing I ever saw in all my\nlife!'\"\n\n       *       *       *       *       *\n\n_A. A. W._, p. 77: \"... The Dormouse went on,...; 'and they drew all\nmanner of things--everything that begins with an M.'\n\n\"'Why with an M?' said Alice.\n\n\"'Why not?' said the March Hare.\n\n\"Alice was silent.\n\n\"... [The Dormouse] went on: '--that begins with an M, such as\nmouse-traps, and the moon, and memory, and muchness, you know you say\nthings are \"much of a muchness\"--did you ever see such a thing as a\ndrawing of a muchness?'\n\n\"'Really, now you ask me,' said Alice, very much confused, 'I don't\nthink----'\n\n\"'Then you shouldn't talk,' said the Hatter.\"\n\n\nO. OBJECTS OF MATHEMATICAL LOGIC.\n\n_T. L. G._, p. 93: \"'I was wondering what the mouse-trap [fastened to\nthe White Knight's saddle] was for,' said Alice. 'It isn't very likely\nthere would be any mice on the horse's back.'\n\n\"'Not very likely, perhaps,' said the Knight, 'but, if they _do_ come, I\ndon't choose to have them running all about.'\n\n\"'You see,' he went on after a pause, 'it's as well to be provided for\n_everything_. That's the reason the horse has all these anklets round\nhis feet.'\n\n\"'But what are they for?' Alice asked in a tone of great curiosity.\n\n\"'To guard against the bites of sharks,' the Knight replied.\"\n\n\nP. THE PRINCIPLE OF PERMANENCE.\n\n_T. L. G._, p. 106: \"'Can you do Subtraction? [said the Red Queen] Take\nnine from eight.'\n\n\"'Nine from eight I can't, you know,' Alice replied very readily 'but--'\n\n\"'She can't do Substraction,' said the White Queen.\"\n\n       *       *       *       *       *\n\n_A. A. W._, p. 56: [Said the Pigeon to Alice]: \"'... No, no! You're a\nserpent; and there's no use denying it. I suppose you'll be telling me\nnext that you never tasted an egg!'\n\n\"'I _have_ tasted eggs certainly,' said Alice, who was a very truthful\nchild; 'but little girls eat eggs quite as much as serpents do, you\nknow.'\n\n\"'I don't believe it,' said the Pigeon; 'but if they do, why then\nthey're a kind of serpent, that's all I can say.'\n\n\"This was such a new idea to Alice, that she was quite silent for a\nminute or two, which gave the Pigeon the opportunity of adding, 'You're\nlooking for eggs, I know _that_ well enough; and what does it matter to\nme whether you're a little girl or a serpent?'\n\n\"'It matters a good deal to _me_,' said Alice hastily;...\"\n\n       *       *       *       *       *\n\n_A. A. W._, p. 75: \"'But why [asked Alice] did they live at the bottom\nof a well?'\n\n\"'Take some more tea,' the March Hare said to Alice, very earnestly.\n\n\"'I've had nothing yet,' Alice replied in an offended tone, 'so I can't\ntake more.'\n\n\"'You mean you can't take _less_,' said the Hatter: 'it's very easy to\ntake _more_ than nothing.'\"\n\n\nQ. MATHEMATICIANS' TREATMENT OF LOGIC.\n\n_A. A. W._, p. 74: The Hatter had told of his quarrel with Time, and of\nTime's refusal now to do anything he asked: \"'... It's always six\no'clock now!'\n\n\"A bright idea came into Alice's head. 'Is that the reason so many tea\nthings are put out here?' she asked.\n\n\"'Yes, that's it,' said the Hatter, with a sigh: 'it's always tea time,\nand we've no time to wash the things between whiles.'\n\n\"'Then you keep moving round, I suppose?' said Alice.\n\n\"'Exactly so,' said the Hatter: 'as the things get used up.'\n\n\"'But what happens when you come to the beginning again?' Alice ventured\nto ask.\n\n\"'Suppose we change the subject,' the March Hare interrupted, yawning.\n'I'm getting tired of this.'\"\n\n       *       *       *       *       *\n\n_A. A. W._, p. 99: \"'And how many hours a day did you do lessons?' said\nAlice, in a hurry to change the subject.\n\n\"'Ten hours the first day,' said the Mock Turtle, 'nine the next, and so\non.'\n\n\"'What a curious plan!' exclaimed Alice.\n\n\"'That's the reason they're called lessons,' the Gryphon remarked,\n'because they lessen from day to day.'\n\n\"This was quite a new idea to Alice, and she thought it over a little\nbefore she made her next remark. 'Then the eleventh day must have been a\nholiday.'\n\n\"'Of course it was,' said the Mock Turtle.\n\n\"'And how did you manage on the twelfth?' Alice went on eagerly.\n\n\"'That's enough about lessons,' the Gryphon interrupted in a very\ndecided tone....\"\n\n\nR. METHOD IN MATHEMATICS AND LOGIC.\n\n_A. A. W._, p. 71: \"'Two days wrong!' sighed the Hatter. 'I told you\nbutter wouldn't suit the works!' he added, looking angrily at the March\nHare.\n\n\"'It was the _best_ butter,' the March Hare meekly replied.\n\n\"'Yes, but some crumbs must have got in as well,' the Hatter grumbled;\n'you shouldn't have put it in with the bread-knife.'\n\n\"The March Hare took the watch and looked at it gloomily: then he dipped\nit into his cup of tea, and looked at it again: but he could think of\nnothing better to say than his first remark, 'It was the _best_ butter,\nyou know.'\"\n\n\nS. VERDICT THAT LOGIC IS PHILOSOPHY.\n\n_A. A. W._, pp. 119-23: \"... 'Consider your verdict,' he [the King] said\nto the jury, in a low trembling voice.\n\n\"'There's more evidence to come yet, please your Majesty,' said the\nWhite Rabbit, jumping up in a great hurry: 'this paper has just been\npicked up.'\n\n\"'What's in it?' said the Queen.\n\n\"'I haven't opened it yet,' said the White Rabbit, 'but it seems to be a\nletter written by the prisoner to--to somebody.'\n\n\"'It must have been that,' said the King, 'unless it was written to\nnobody, which isn't usual, you know.'\n\n\"'Who is it directed to?' said one of the jurymen.\n\n\"'It isn't directed at all,' said the White Rabbit, 'in fact there's\nnothing written on the _outside_.' He unfolded the paper as he spoke,\nand added, 'It isn't a letter, after all: it's a set of verses.'\n\n\"'Are they in the prisoner's handwriting?' asked another of the jurymen.\n\n\"'No they're not,' said the White Rabbit, 'and that's the queerest thing\nabout it.' (The jury all looked puzzled).\n\n\"'He must have imitated somebody else's hand,' said the King. (The jury\nbrightened up again.)\n\n\"'Please your Majesty,' said the Knave, 'I didn't write it, and they\ncan't prove that I did: there's no name signed at the end.'\n\n\"'If you didn't sign it, said the King, that only makes the matter\nworse. You _must_ have meant some mischief, or else you'd have signed\nyour name like an honest man.'\n\n\"There was a general clapping of hands at this: it was the first really\nclever thing the King had said that day.\n\n\"'That _proves_ his guilt, of course,' said the Queen, 'so, off\nwith----'\n\n\"'It doesn't prove anything of the sort!' said Alice. 'Why, you don't\neven know what they're about!'\n\n\"'Read them,' said the King.\n\n\"The White Rabbit put on his spectacles. 'Where shall I begin, please\nyour Majesty?' he asked.\n\n\"'Begin at the beginning,' the King said very gravely, 'and go on till\nyou come to the end: then stop.'\n\n\"There was dead silence in the court, whilst the White Rabbit read out\nthese verses:\n\n  \"'_They told me you had been to her,\n  And mentioned me to him;\n  She gave me a good character,\n  But said I could not swim._\n\n  _He sent them word I had not gone\n  (We know it to be true):\n  If she should push the matter on,\n  What would become of you?_\n\n  _I gave her one, they gave him two,\n  You gave us three or more;\n  They all returned from him to you,\n  Though they were mine before._\n\n  _If I or she should chance to be\n  Involved in this affair,\n  He trusts to you to set them free\n  Exactly as they were._\n\n  _My notion was that you had been\n  (Before she had this fit)\n  An obstacle that came between\n  Him, and ourselves, and it._\n\n  _Don't let him know she liked them best,\n  For this must ever be\n  A secret kept from all the rest,\n  Between yourself and me._'\n\n\"'That's the most important piece of evidence we've heard yet,' said the\nKing, rubbing his hands, 'so now let the jury----'\n\n\"'If any one of them can explain it,' said Alice (she had grown so large\nin the last few minutes that she wasn't a bit afraid of interrupting\nhim), 'I'll give him sixpence. _I_ don't believe there's an atom of\nmeaning in it.'\n\n\"The jury all wrote down on their slates, 'She doesn't believe there's\nan atom of meaning in it,' but none of them attempted to explain the\npaper.\n\n\"'If there's no meaning in it,' said the King, 'that saves a world of\ntrouble, you know, as we needn't try to find any. And yet I don't know,'\nhe went on, spreading out the verses on his knee and looking at them\nwith one eye; 'I seem to see some meaning in them after all. \"_-- said\nI could not swim_\"; you can't swim, can you?' he added, turning to the\nKnave.\n\n\"The Knave shook his head sadly. 'Do I look like it?' he said. (Which he\ncertainly did _not_, being made entirely of cardboard.)\n\n\"'All right, so far,' said the King; and he went on muttering over the\nverses to himself: ''_We know it to be true_'--that's the jury, of\ncourse--'_If she should push the matter on_'--that must be the\nQueen--'_What would become of you?_' What indeed!--'_I gave her one,\nthey gave him two!_' why, that must be what he did with the tarts, you\nknow----'\n\n\"'But it goes on, '_They all returned from him to you_,'' said Alice.\n\n\"'Why, there they are!' said the King, triumphantly pointing to the\ntarts on the table. 'Nothing can be clearer than that. Then\nagain--'_Before she had this fit_'--you never had fits, my dear, I\nthink?' he said to the Queen.\n\n\"'Never!' said the Queen furiously, throwing an inkstand at the Lizard\nas she spoke. (The unfortunate little Bill had left off writing on his\nslate with one finger, as he found it made no mark; but he now hastily\nbegan again, using the ink that was trickling down his face, as long as\nit lasted.)\n\n\"'Then the words don't _fit_ you,' said the King, looking round the\ncourt with a smile. There was a dead silence.\n\n\"'It's a pun!' the King added in an angry tone, and everybody laughed.\n\n\"'Let the jury consider their verdict,' the King said, for about the\ntwentieth time that day.\n\n\"'No, no!' said the Queen. 'Sentence first--verdict afterwards.'\n\n\"'Stuff and nonsense!' said Alice loudly. 'The idea of having the\nsentence first!'\n\n\"'Hold your tongue!' said the Queen, turning purple....\"\n\n\nT. \"GEDANKENEXPERIMENTE.\"\n\n_T. L. G._, p. 61: \"Alice laughed. 'There's no use trying,' she said:\n'one _can't_ believe impossible things.'\n\n\"'I daresay you haven't had much practice,' said the [White] Queen.\n'When I was your age, I always did it for half-an-hour a day. Why,\nsometimes I've believed as many as six impossible things before\nbreakfast.'\"\n\n\n_Printed in Great Britain by_\n\nUNWIN BROTHERS, LIMITED, THE GRESHAM PRESS, WOKING AND LONDON","title":"The philosophy of Mr. B*rtr*nd R*ss*ll","tradition":"philosophy-ethics","translated_text":null,"word_count":25671}