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Title: The philosophy of Mr. B*rtr*nd R*ss*ll

Editor: Philip E. B. Jourdain

Release date: December 28, 2011 [eBook #38430]

Language: English

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*** START OF THE PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF MR. B*RTR*ND R*SS*LL ***




Produced by Adrian Mastronardi and the Online Distributed
Proofreading Team at http://www.pgdp.net (This file was
produced from images generously made available by The
Internet Archive/Canadian Libraries)









  THE PHILOSOPHY OF
  MR. B*RTR*ND R*SS*LL

  WITH AN APPENDIX OF LEADING
  PASSAGES FROM CERTAIN OTHER WORKS

  EDITED BY

  PHILIP E. B. JOURDAIN

  [Illustration]

  LONDON: GEORGE ALLEN & UNWIN LTD.
  RUSKIN HOUSE 40 MUSEUM STREET, W.C. 1
  CHICAGO: THE OPEN COURT PUBLISHING CO.



  _First published in 1918_

  (_All rights reserved_)




EDITOR'S NOTE


When Mr. B*rtr*nd R*ss*ll, following the advice of Mr. W*ll**m J*m*s,
again "got into touch with reality" and in July 1911 was torn to pieces
by Anti-Suffragists, many of whom were political opponents of Mr.
R*ss*ll and held strong views on the Necessity of Protection of Trade
and person, a manuscript which was almost ready for the press was
fortunately saved from the flames on the occasion when a body of eager
champions of the Sacredness of Personal Property burnt the late Mr.
R*ss*ll's house. This manuscript, together with some further fragments
found in the late Mr. R*ss*ll's own interleaved copy of his _Prayer-Book
of Free Man's Worship_, which was fortunately rescued with a few of the
great author's other belongings, was first given to the world in the
_Monist_ for October 1911 and January 1916, and has here been arranged
and completed by some other hitherto undecipherable manuscripts. The
title of the above-mentioned _Prayer-Book_, it may perhaps be mentioned,
was apparently suggested to Mr. R*ss*ll by that of the Essay on
"The Free Man's Worship" in the _Philosophical Essays_ (London, 1910,
pp. 59-70[1]) of Mr. R*ss*ll's distinguished contemporary, Mr. Bertrand
Russell, from whom much of Mr. R*ss*ll's philosophy was derived. And,
indeed, the influence of Mr. Russell extended even beyond philosophical
views to arrangement and literary style. The method of arrangement of
the present work seems to have been borrowed from Mr. Russell's
_Philosophy of Leibniz_ of 1900; in the selection of subjects dealt
with, Mr. R*ss*ll seems to have been guided by Mr. Russell's _Principles
of Mathematics_ of 1903; while Mr. R*ss*ll's literary style fortunately
reminds us more of Mr. Russell's later clear and charming subtleties
than his earlier brilliant and no less subtle obscurities. But, on the
other hand, some important points of Mr. Russell's doctrine, which first
appeared in books published after Mr. R*ss*ll's death, were anticipated
in Mr. R*ss*ll's notes, and these anticipations, so interesting for
future historians of philosophy, have been provided by the editor with
references to the later works of Mr. Russell. All editorial notes are
enclosed in square brackets, to indicate that they were not written by
the late Mr. R*ss*ll.

At the present time we have come to take a calm view of the question so
much debated seven years ago as to the legitimacy of logical arguments
in political discussions. No longer, fortunately, can that intense
feeling be roused which then found expression in the famous cry,
"Justice--right or wrong," and which played such a large part in the
politics of that time. Thus it will not be out of place in this
unimpassioned record of some of the truths and errors in the world to
refer briefly to Mr. R*ss*ll's short and stormy career. Before he was
torn to pieces, he had been forbidden to lecture on philosophy or
mathematics by some well-intentioned advocates of freedom in speech who
thought that the cause of freedom might be endangered by allowing Mr.
R*ss*ll to speak freely on points of logic, on the grounds, apparently,
that logic is both harmful and unnecessary and might be applied to
politics unless strong measures were taken for its suppression. On much
the same grounds, his liberty was taken from him by those who remarked
that, if necessary, they would die in defence of the sacred principle of
liberty; and it was in prison that the greater part of the present work
was written. Shortly after his liberation, which, like all actions of
public bodies, was brought about by the combined honour and interests of
those in authority, occurred his lamentable death to which we have
referred above.

Mr. R*ss*ll maintained that the chief use of "implication" in politics
is to draw conclusions, which are thought to be true, and which are
consequently false, from identical propositions, and we can see these
views expressed in Chapters III and XIX of the present work. These
chapters were apparently written before the Government, in the spring
of 1910, arrived at the famous secret decision that only "certain
implications" are permitted in discussion. Naturally the secret decision
gave rise to much speculation among logicians as to which kinds of
implication were barred, and Mr. R*ss*ll and Mr. Bertrand Russell had
many arguments on the subject, which naturally could not be published at
the time. However, after Mr. R*ss*ll's death, successive prosecutions
which were made by the Government at last made it quite clear that the
opinion held by Mr. R*ss*ll was the correct one. There had been numerous
prosecutions of people who, from true but not identical premisses, had
deduced true conclusions, so that the possible legitimate forms of
"implication" were reduced. Further, the other doubtful cases were
cleared up in course of time by the prosecution of (1) members of the
Aristotelian Society for deducing true conclusions from false premisses;
(2) members of the _Mind_ Association for deducing false conclusions
from false premisses; and also by the attempted prosecution of an
eminent lady for deducing true conclusions from identities. Fortunately
this lady was able to defend herself successfully by pleading that one
eminent philosopher believed them to be true--which, of course, means
that the conclusions are false. Thus appeared the true nature of
legitimate political arguments.

FOOTNOTES:

[1] [This Essay is also reprinted in Mr. Russell's _Mysticism and
Logic_, London and New York, 1918, pp. 46-57.--ED.]




    "Even a joke should have some meaning...."

                              (The Red Queen, _T. L. G._, p. 105).




CONTENTS


                                                                PAGE
  EDITOR'S NOTE                                                    3
  ABBREVIATIONS                                                    9
  CHAPTER
      I. THE INDEFINABLES OF LOGIC                                11
     II. OBJECTIVE VALIDITY OF THE "LAWS OF THOUGHT"              15
    III. IDENTITY                                                 16
     IV. IDENTITY OF CLASSES                                      18
      V. ETHICAL APPLICATIONS OF THE LAW OF IDENTITY              19
     VI. THE LAW OF CONTRADICTION IN MODERN LOGIC                 21
    VII. SYMBOLISM AND MEANING                                    22
   VIII. NOMINALISM                                               24
     IX. AMBIGUITY AND SYMBOLIC LOGIC                             26
      X. LOGICAL ADDITION AND THE UTILITY OF SYMBOLISM            27
     XI. CRITICISM                                                29
    XII. HISTORICAL CRITICISM                                     30
   XIII. IS THE MIND IN THE HEAD?                                 31
    XIV. THE PRAGMATIST THEORY OF TRUTH                           32
     XV. ASSERTION                                                34
    XVI. THE COMMUTATIVE LAW                                      35
   XVII. UNIVERSAL AND PARTICULAR PROPOSITIONS                    36
  XVIII. DENIAL OF GENERALITY AND GENERALITY OF DENIAL            37
    XIX. IMPLICATION                                              39
     XX. DIGNITY                                                  43
    XXI. THE SYNTHETIC NATURE OF DEDUCTION                        45
   XXII. THE MORTALITY OF SOCRATES                                48
    XXIII. DENOTING                                               53
     XXIV. THE                                                    54
      XXV. NON-ENTITY                                             56
     XXVI. IS                                                     58
    XXVII. AND AND OR                                             59
   XXVIII. THE CONVERSION OF RELATIONS                            60
     XXIX. PREVIOUS PHILOSOPHICAL THEORIES OF MATHEMATICS         61
      XXX. FINITE AND INFINITE                                    63
     XXXI. THE MATHEMATICAL ATTAINMENTS OF TRISTRAM SHANDY        64
    XXXII. THE HARDSHIPS OF A MAN WITH AN UNLIMITED INCOME        66
   XXXIII. THE RELATIONS OF MAGNITUDE OF CARDINAL NUMBERS         69
    XXXIV. THE UNKNOWABLE                                         70
     XXXV. MR. SPENCER, THE ATHANASIAN CREED, AND THE ARTICLES    73
    XXXVI. THE HUMOUR OF MATHEMATICIANS                           74
   XXXVII. THE PARADOXES OF LOGIC                                 75
  XXXVIII. MODERN LOGIC AND SOME PHILOSOPHICAL ARGUMENTS          79
    XXXIX. THE HIERARCHY OF JOKES                                 81
       XL. THE EVIDENCE OF GEOMETRICAL PROPOSITIONS               83
      XLI. ABSOLUTE AND RELATIVE POSITION                         84
     XLII. LAUGHTER                                               86
    XLIII. "GEDANKENEXPERIMENTE" AND EVOLUTIONARY ETHICS          88
           APPENDIXES                                             89




ABBREVIATIONS


  _A. A. W._  Lewis Carroll: _Alice's Adventures in Wonderland_, London,
              1908. [This book was first published much earlier, but
              this was the edition used by Mr. R*ss*ll. The same applies
              to _H. S._ and _T. L. G._]

  _A. C. P._  John Henry Blunt (ed. by): _The Annotated Book of Common
              Prayer_, London, new edition, 1888.

  _A. d. L._  Ernst Schröder: _Vorlesungen über die Algebra der Logik,
              Leipzig_, vol. i., 1890; vol. ii. (two parts), 1891 and
              1905; vol. iii.: _Algebra und Logik der Relative_, 1895.

  _E. N._     Richard Dedekind: _Essays on the Theory of Numbers_,
              Chicago and London, 1901.

  _E. L. L._  William Stanley Jevons: _Elementary Lessons in Logic,
              Deductive and Inductive. With copious Questions and
              Examples, and a Vocabulary of Logical Terms_, London,
              24th ed., 1907 [first published in 1870].

  _E. u. I._  Ernst Mach: _Erkenntnis und Irrtum: Skizzen zur
              Psychologie der Forschung_, Leipzig, 1906.

  _F. L._     Augustus De Morgan: _Formal Logic: or The Calculus of
              Inference, Necessary and Probable_, London, 1847.

  _Fm. L._    John Neville Keynes: _Studies and Exercises in Formal
              Logic_, 4th ed., London, 1906.

  _Gg._       Gottlob Frege: _Grundgesetze der Arithmetik
              begriffschriftlich abgeleitet_, Jena, vol. i., 1893;
              vol. ii., 1903.

  _Gl._       Gottlob Frege: _Die Grundlagen der Arithmetik: eine
              logisch-mathematische Untersuchung über den Begriff der
              Zahl_, Breslau, 1884.

  _G. u. E._  G. Heymans: _Die Gesetze und Elemente des
              wisenschaftlichen Denkens_, Leiden, vol. i., 1890;
              vol. ii., 1894.

  _H. J._     _The Hibbert Journal: a Quarterly Review of Religion,
              Theology and Philosophy_, London and New York.

  _H. S._     Lewis Carroll: _The Hunting of the Snark: an Agony in
              Eight Fits_, London, 1911.

  _M._        _The Monist: a Quarterly Magazine Devoted to Science and
              Philosophy_, Chicago and London.

  _Md._       _Mind: a Quarterly Review of Psychology and Philosophy_,
              London and New York.

  _Pa. Ma._   Alfred North Whitehead and Bertrand Russell: _Principia
              Mathematica_, vol. i., Cambridge, 1910. [Other volumes
              were published in 1912 and 1913.]

  _P. E._     Bertrand Russell: _Philosophical Essays_, London and New
              York, 1910.

  _Ph. L._    Bertrand Russell: _A Critical Exposition of the Philosophy
              of Leibniz, with an Appendix of Leading Passages_,
              Cambridge, 1900.

  _P. M._     Bertrand Russell: _The Principles of Mathematics_,
              vol. i., Cambridge, 1903.

  _R. M. M._  _Revue de Métaphysique et de Morale_, Paris.

  _S. B._     Lewis Carroll: _Sylvie and Bruno_, London, 1889.

  _S. L._     John Venn: _Symbolic Logic_, London, 1881; 2nd ed., 1894.

  _S. o. S._  William Stanley Jevons: _The Substitution of Similars, the
              True Principle of Reasoning derived from a Modification of
              Aristotle's Dictum_, London, 1869.

  _T. L. G._  Lewis Carroll: _Through the Looking-Glass, and what Alice
              found there_, London, 1911.

  _Z. S._     Gottlob Frege: _Ueber die Zahlen des Herrn H. Schubert_,
              Jena, 1899.




CHAPTER I

THE INDEFINABLES OF LOGIC


The view that the fundamental principles of logic consist solely of the
law of identity was held by Leibniz,[2] Drobisch, Uberweg,[3] and
Tweedledee. Tweedledee, it may be remembered,[4] remarked that certain
identities "are" logic. Now, there is some doubt as to whether he, like
Jevons,[5] understood "are" to mean what mathematicians mean by "=," or,
like Schröder[6] and most logicians, to have the same meaning as the
relation of subsumption. The first alternative alone would justify our
contention; and we may, I think, conclude from an opposition to
authority that may have been indicated by Tweedledee's frequent use of
the word "contrariwise" that he did not follow the majority of
logicians, but held, like Jevons,[7] the mistaken[8] view that the
quantification of the predicate is relevant to symbolic logic.

It may be mentioned, by the way, that it is probable that
Humpty-Dumpty's "is" is the "is" of identity. In fact, it is not
unlikely that Humpty-Dumpty was a Hegelian; for, although his ability
for clear explanation may seem to militate against this, yet his
inability to understand mathematics,[9] together with his synthesis of a
cravat and a belt, which usually serve different purposes,[10] and his
proclivity towards riddles seem to make out a good case for those who
hold that he was in fact a Hegelian. Indeed, riddles are very closely
allied to puns, and it was upon a pun, consisting of the confusion of
the "is" of predication with the "is" of identity--so that, for example,
"Socrates" was identified with "mortal" and more generally the
particular with the universal--that Hegel's system of philosophy was
founded.[11] But the question of Humpty-Dumpty's philosophical opinions
must be left for final verification to the historians of philosophy:
here I am only concerned with an _a priori_ logical construction of what
his views might have been if they formed a consistent whole.[12]

If the principle of identity were indeed the sole principle of logic,
the principles of logic could hardly be said to be, as in fact they are,
a body of propositions whose consistency it is impossible to prove.[13]
This characteristic is important and one of the marks of the greatest
possible security. For example, while a great achievement of late years
has been to prove the consistency of the principles of arithmetic, a
science which is unreservedly accepted except by some empiricists,[14]
it can be proved formally that one foundation of arithmetic is
shattered.[15] It is true that, quite lately, it has been shown that
this conclusion may be avoided, and, by a re-moulding of logic, we can
draw instead the paradoxical conclusion that the opinions held by
common-sense for so many years are, in part, justified. But it is quite
certain that, with the principles of logic, no such proof of
consistency, and no such paradoxical result of further investigations is
to be feared.

Still, this re-moulding has had the result of bringing logic into a
fuller agreement with common-sense than might be expected. There were
only two alternatives: if we chose principles in accordance with
common-sense, we arrived at conclusions which shocked common-sense; by
starting with paradoxical principles, we arrived at ordinary
conclusions. Like the White Knight, we have dyed our whiskers an unusual
colour and then hidden them.[16]

The quaint name of "Laws of Thought," which is often applied to the
principles of Logic, has given rise to confusion in two ways: in the
first place, the "Laws," unlike other laws, cannot be broken, even in
thought; and, in the second place, people think that the "Laws" have
something to do with holding for the operations of their minds, just as
laws of nature hold for events in the world around us.[17] But that the
laws are not psychological laws follows from the facts that a thing may
be true even if nobody believes it, and something else may be false if
everybody believes it. Such, it may be remarked, is usually the case.

Perhaps the most frequent instance of the assumption that the laws of
logic are mental is the treatment of an identity as if its validity were
an affair of our permission. Some people suggest to others that they
should "let bygones be bygones." Another important piece of evidence
that the truth of propositions has nothing to do with mind is given by
the phrase "it is morally certain that such-and-such a proposition is
true." Now, in the first place, morality, curiously enough, seems to be
closely associated with mental acts: we have professorships and
lectureships of, and examinations in, "mental and moral philosophy." In
the second place, it is plain that a "morally certain" proposition is a
highly doubtful one. Thus it is as vain to expect any information about
our minds from a study of the "Laws of Thought" as it would be to expect
a description of a certain social event from Miss E. E. C. Jones's book
_An Introduction to General Logic_.

Fortunately, the principles or laws of Logic are not a matter of
philosophical discussion. Idealists like Tweedledum and Tweedledee, and
even practical idealists like the White Knight, explicitly accept laws
like the law of identity and the excluded middle.[18] In fact,
throughout all logic and mathematics, the existence of the human or any
other mind is totally irrelevant; mental processes are studied by means
of logic, but the subject-matter of logic does not presuppose mental
processes, and would be equally true if there were no mental processes.
It is true that, in that case, we should not know logic; but our
knowledge must not be confounded with the truths which we know.[19] An
apple is not confused with the eating of it except by savages,
idealists, and people who are too hungry to think.

FOOTNOTES:

[2] Russell, _Ph. L._, pp. 17, 19, 207-8.

[3] Schröder, _A. d. L._, i. p. 4.

[4] See Appendix A. This Appendix also illustrates the importance
attached to the Principle of Identity by the Professor and Bruno.

[5] _S. o. S._, pp. 9-15.

[6] _A. d. L._, i. p. 132.

[7] Cf., besides the reference in the last note but one, _E. L. L._, pp.
183, 191. "Contrariwise," it may be remarked, is not a term used in
traditional logic.

[8] _S. L._, 1881, pp. 173-5, 324-5; 1894, pp. 194-6.

[9] Cf. Appendix C, and William Robertson Smith, "Hegel and the
Metaphysics of the Fluxional Calculus," _Trans. Roy. Soc., Edinb._, vol.
xxv., 1869, pp. 491-511.

[10] See Appendix B.

[11] [This is a remarkable anticipation of the note on pp. 39-40 of Mr.
Russell's book, published about three years after the death of Mr.
R*ss*ll, and entitled _Our Knowledge of the External World as a Field
for Scientific Method in Philosophy_, Chicago and London, 1914.--ED.]

[12] Cf. _Ph. L._, pp. v.-vi. 3.

[13] Cf. Pieri, _R. M. M._, March 1906, p. 199.

[14] As a type of these, Humpty-Dumpty, with his inability to admit
anything not empirically given and his lack of comprehension of pure
mathematics, may be taken (see Appendix C). In his (correct) thesis that
definitions are nominal, too, Humpty-Dumpty reminds one of J. S. Mill
(see Appendix D).

[15] See Frege, _Gg._, ii. p. 253.

[16] See Appendix E.

[17] See Frege, _Gg._, i. p. 15.

[18] See the above references and also Appendix F.

[19] Cf. B. Russell, _H. J._, July 1904, p. 812.




CHAPTER II

OBJECTIVE VALIDITY OF THE "LAWS OF THOUGHT"


I once inquired of a maid-servant whether her mistress was at home. She
replied, in a doubtful fashion, that she _thought_ that her mistress was
in unless she was out. I concluded that the maid was uncertain as to the
objective validity of the law of excluded middle, and remarked that to
her mistress. But since I used the phrase "laws of thought," the
mistress perhaps supposed that a "law of thought" has something to do
with thinking and seemed to imagine that I wished to impute to the maid
some moral defect of an unimportant nature. Thus she remonstrated with
me in an amused way, since she probably imagined that I meant to find
fault with the maid's capacity for thinking.




CHAPTER III

IDENTITY


In the first chapter we have noticed the opinion that identities are
fundamental to all logic. We will now consider some other views of the
value of identities.

Identities are frequently used in common life by people who seem to
imagine that they can draw important conclusions respecting conduct or
matters of fact from them. I have heard of a man who gained the double
reputation of being a philosopher and a fatalist by the repeated
enunciation of the identity "Whatever will be, will be"; and the Italian
equivalent of this makes up an appreciable part of one of Mr. Robert
Hichens' novels. Further, the identity "Life is Life" has not only been
often accepted as an explanation for a particular way of living but has
even been considered by an authoress who calls herself "Zack" to be an
appropriate title for a novel; while "Business is Business" is
frequently thought to provide an excuse for dishonesty in trading, for
which purpose it is plainly inadequate.

Another example is given by a poem of Mr. Kipling, where he seems to
assert that "East is East" and "West is West" imply that "never the
twain shall meet." The conclusion, now, is false; for, since the world
is round--as geography books still maintain by arguments which strike
every intelligent child as invalid[20]--what is called the "West" does,
in fact, merge into the "East." Even if we are to take the statement
metaphorically, it is still untrue, as the Japanese nation has shown.

The law-courts are often rightly blamed for being strenuous opponents of
the spread of modern logic: the frequent misuse of _and_, _or_, _the_,
and _provided that_ in them is notorious. But the fault seems partly to
lie in the uncomplicated nature of the logical problems which are dealt
with in them. Thus it is no uncommon thing for somebody to appear there
who is unable to establish his own identity, or for A to assert that B
was "not himself" when he made a will leaving his money to C.

The chief use of identities is in implication. Since, in logic, we so
understand _implication_[21] that any true proposition implies and is
implied by any other true proposition; if one is convinced of the truth
of the proposition Q, it is advisable to choose one or more identities
P, whose truth is undoubted, and say that P implies Q. Thus, Mr. Austen
Chamberlain, according to _The Times_ of March 27, 1909, professed to
deduce the conclusion that it is not right that women should have votes
from the premisses that "man is man" and "woman is woman." This method
requires that one should have made up one's mind about the conclusion
before discovering the premisses--by what, no doubt, Jevons would call
an "inverse or inductive method." Thus the method is of use only in
speeches and in giving good advice.

Mr. Austen Chamberlain afterwards rather destroyed one's belief in the
truth of his premisses by putting limits to the validity of the
principle of identity. In the course of the Debate on the Budget of
1909, he maintained, against Mr. Lloyd George, that a joke was a joke
except when it was an untruth: Mr. Lloyd George, apparently, being of
the plausible opinion that a joke is a joke under all circumstances.

FOOTNOTES:

[20] The argument about the hull of a ship disappearing first is not
convincing, since it would equally well prove that the surface of the
earth was, for example, corrugated on a large scale. If the common-sense
of the reader were supposed to dismiss the possibility of water clinging
to such corrugations, it might equally be supposed to dismiss the
possibility of water clinging to a spherical earth. Traditional
geography books, no doubt, gave rise to the opinions held by Lady Blount
and the Zetetic Society.

[21] The subject of Implication will be further considered in Chapter
XIX.




CHAPTER IV

IDENTITY OF CLASSES


I once heard of a meritorious lady who was extremely conventional; on
the slender grounds of carefully acquired habits of preferring the word
"woman" to the word "lady" and of going to the post-office without a
hat, imagined that she was unconventional and altogether a remarkable
person; and who once remarked with great satisfaction that she was a
"very queer person," and that nothing shocked her "except, of course,
bad form."

Thus, she asserted that all the things which shocked her were actions in
bad form; and she would undoubtedly agree, though she did not actually
state it, that all the things which were done in bad form would shock
her. Consequently she asserted that the class of things which shocked
her was the class of actions in bad form. Consequently the statement of
this lady that some or all of the actions done in bad form shocked her
is an identical proposition of the form "nothing shocks me, except, of
course, the things which do, in fact, shock me"; and this statement the
lady certainly did not intend to make.

This excellent lady, had she but known it, was logically justified in
making any statement whatever about her unconventionality. For the class
of her unconventional actions was the null class. Thus she might
logically have made inconsistent statements about this class of actions.
As a matter of fact she did make inconsistent statements, but
unfortunately she justified them by stating that, "It is the privilege
of woman to be inconsistent." She was one of those persons who say
things like that.




CHAPTER V

ETHICAL APPLICATIONS OF THE LAW OF IDENTITY


It may be remembered that Mr. Podsnap remarked, with sadness tempered by
satisfaction, that he regretted to say that "Foreign nations do as they
do do." Besides aiding the comforting expression of moral disapproval,
the law of identity has yet another useful purpose in practical ethics:
It serves the welcome purpose of providing an excuse for infractions of
the moral law. There was once a man who treated his wife badly, was
unfaithful to her, was dishonest in business, and was not particular in
his use of language; and yet his life on earth was described in the
lines:

  This man maintained a wife's a wife,
  Men are as they are made,
  Business is business, life is life;
  And called a spade a spade.

One of the objects of Dr. G. E. Moore's _Principia Ethica_[22] was to
argue that the word "good" means simply good, and not pleasant or
anything else. Appropriately enough, this book bore on its title-page
the quotation from the preface to the _Sermons_, published in 1726, of
Bishop Joseph Butler, the author of the _Analogy_: "Everything is what
it is and not another thing."

But another famous Butler--Samuel Butler, the author of _Hudibras_--went
farther than this, and maintained that identities were the highest
attainment of metaphysics itself. At the beginning of the first Canto of
_Hudibras_, in the description of Hudibras himself, Butler wrote:

  He knew what's what, and that's as high
  As metaphysic wit can fly.

I once conducted what I imagined to be an æsthetic investigation for the
purpose of discovery, by the continual use of the word "Why?"[23] the
grounds upon which certain people choose to put milk into a tea-cup
before the tea. I was surprised to discover that it was an ethical, and
not an æsthetic problem; for I soon elicited the fact that it was done
because it was "right." A continuance of my patient questioning elicited
further evidence of the fundamental character of the principle of
identity in ethics; for it was right, I learned, because "right is
right."

It appears that some people unconsciously think that the principle of
identity is the foundation, in certain religions, of the reasons which
can be alleged for moral conduct, and are surprised when this fact is
pointed out to them. The late Sir Leslie Stephen, when travelling by
railway, fell into conversation with an officer of the Salvation Army,
who tried hard to convert him. Failing in this laudable endeavour, the
Salvationist at last remarked: "But if you aren't saved, you can't go to
heaven!" "That, my friend," replied Stephen, "is an identical
proposition."

FOOTNOTES:

[22] Cambridge, 1903.

[23] Cf. _P. E._, p. 2.




CHAPTER VI

THE LAW OF CONTRADICTION IN MODERN LOGIC


Considering the important place assigned by philosophers and logicians
to the law of contradiction, the remark will naturally be resented by
many of the older schools of philosophy, and especially by Kantians,
that "in spite of its fame we have found few occasions for its use."[24]
Also in modern times, Benedetto Croce, an opponent of both traditional
logic and mathematical logic, began the preface of the book of 1908 on
Logic[25] by saying that that volume "is and is not" a certain memoir of
his which had been published in 1905.

FOOTNOTES:

[24] _Pa. Ma._, p. 116.

[25] [English translation of the third Italian edition by Douglas
Ainslie, under the title: _Logic as the Science of the Pure Concept_,
London 1917.--ED.]




CHAPTER VII

SYMBOLISM AND MEANING


When people write down any statement such as "The curfew tolls the knell
of parting day,"[26] which we will call "C" for shortness, what they
mean is not "C" but the _meaning_ of "C"; and not "the meaning of 'C'"
but the _meaning_ of "the meaning of 'C'." And so on, _ad infinitum_.
Thus, in writing or in speech, we always fail to state the meaning of
any proposition whatever. Sometimes, indeed, we succeed in _conveying_
it; but there is danger in too great a disregard of statement and
preoccupation with conveyance of meaning. Thus many mathematicians have
been so anxious to convey to us a perfectly distinct and unmetaphysical
concept of number that they have stripped away from it everything that
they considered unessential (like its logical nature) and have finally
delivered it to us as a mere _sign_. By the labours of Helmholtz,
Kronecker, Heine, Stolz, Thomae, Pringsheim, and Schubert, many people
were persuaded that, when they said "'2' is a number" they were speaking
the truth, and hold that "Paris" is a town containing the letter "P."
When Frege pointed out[27] this difficulty he was almost universally
denounced in Germany as "_spitzfindig_." In fact, Germans seem to have
been influenced perhaps by that great contemner of "_Spitzfindigkeit_,"
Kant, to reject the White Knight's[28] distinctions between words and
their denotations and to regard subtlety with disfavour to such a degree
that their only mathematical logician except Frege, namely Schröder--the
least subtle of mortals, by the way--seems to have been filled with such
fear of being thought subtle, that he made his books so prolix that
nobody has read them.

Another term which, as we shall see when discussing the paradoxes of
logic, mathematicians are accustomed to apply to thought which is more
exact than any to which they are accustomed is "scholastic."[29] By
this, I suppose, they mean that the pursuits of certain acute people of
the Middle Ages are unimportant in contrast with the great achievements
of modern thought, as exemplified by a method of making plausible
guesses known as induction,[30] the bicycle, and the gramophone--all of
them instruments of doubtful merit.

FOOTNOTES:

[26] Cf. _Md_, N. S., vol. xiv., October 1905, p. 486.

[27] In _Z. S._, for example.

[28] See Appendix G.

[29] Cf. Chapter XXXVII below.

[30] Cf. _P. M._, p. 11, note.




CHAPTER VIII

NOMINALISM


De Morgan[31] said that, "if all mankind had spoken one language, we
cannot doubt that there would have been a powerful, perhaps universal,
school of philosophers who would have believed in the inherent connexion
between names and things; who would have taken the sound _man_ to be the
mode of agitating the air which is essentially communicative of the
ideas of reason, cookery, bipedality, etc.... 'The French,' said the
sailor, 'call a cabbage a _shoe_; the fools! Why can't they call it a
cabbage, when they must know it is one?'"

One of the chief differences between logicians and men of letters is
that the latter mean many different things by one word, whereas the
former do not--at least nowadays. Most mathematicians belong to the
class of men of letters.

I once had a manservant who told me on a certain occasion that he "never
thought a word about it." I was doubtful whether to class him with such
eminent mathematicians as are mentioned in the last chapter, or as a
supporter of Max Müller's theory of the identity of thought and
language. However, since the man was very untruthful, and he told me
that he meant what he said and said what he meant,[32] the conclusion is
probably correct that he really believed that the meanings of his words
were not the words themselves. Thus I think it most probable that my
manservant had been a mathematician but had escaped by the aid of logic.

As regards his remark that he meant what he said and said what he
meant, he plainly wished to pride himself on certain virtues which he
did not possess, and was not indifferent to applause, which, however,
was never evoked. The virtues, if so they be, and the applause were
withheld for other reasons than that the above statements are either
nonsensical or false. Suppose that "I say what I mean" expresses a
truth. What I say (or write) is always a symbol--words (or marks); and
what I mean by the symbol is the meaning of the symbol and not the
symbol itself. So the remark cannot express a truth, any more than the
name "Wellington" won the battle of Waterloo.

FOOTNOTES:

[31] _F. L._, pp. 246-7.

[32] The Hatter (see Appendix H) pointed out that there is a difference
between these two assertions. Thus, he clearly showed that he was a
nominalist, and philosophically opposed to the March Hare who had
recommended Alice to say what she meant.




CHAPTER IX

AMBIGUITY AND SYMBOLIC LOGIC


The universal use of some system of Symbolic Logic would not only enable
everybody easily to deal with exceedingly complicated arguments, but
would prevent ambiguous arguments. In denying the indispensability of
Symbolic Logic in the former state of things, Keynes[33] is probably
alone, against the need strongly felt by Alice when speaking to the
Duchess,[34] and most modern logicians. It may be noticed that the
Duchess is more consistent than Keynes, for Keynes really uses the signs
for logical multiplication and addition of Boole and Venn under the
different shapes of the words "and" and "or."

As regards ambiguity, a translation of _Hymns Ancient and Modern_ into,
say, Peanesque, would prevent the puzzle of childhood as to whether the
"his" in

  And Satan trembles when he sees
  The weakest saint upon his knees

refers to the saint's knees or Satan's.

FOOTNOTES:

[33] In his _Fm. L._

[34] See Appendix I.




CHAPTER X

LOGICAL ADDITION AND THE UTILITY OF SYMBOLISM


Frequently ordinary language contains subtle psychological implications
which cannot be translated into symbolic logic except at great length.
Thus if a man (say Mr. Jones) wishes to speak collectively of himself
and his wife, the order of mentioning the terms in the class considered
and the names applied to these terms are, logically speaking,
irrelevant. And yet more or less definite information is given about Mr.
Jones according as he talks to his friends of:

     (1) Mrs. Jones and I,
     (2) I (or me) and my wife (or missus),
     (3) My wife and I,
  or (4) I (or me) and Mrs. Jones.

In case (1) one is probably correct in placing Mr. Jones among the
clergy or the small professional men who make up the bulk of the
middle-class; in case (2) one would conclude that Mr. Jones belonged to
the lower middle-class; the form (3) would be used by Mr. Jones if he
were a member of the upper, upper middle, or lower class; while form (4)
is only used by retired shopkeepers of the lower middle-class, of which
a male member usually combines belief in the supremacy of man with
belief in the dignity of his wife as well as himself. A further
complication is introduced if a wife is referred to as "the wife."[35]
Cases (2) and (3) then each give rise to one more case. Cases (1) and
(4) do not, since nobody has hitherto referred to his wife as "the Mrs.
Jones"--at least without a qualifying adjective before the "Mrs."

On the other hand, certain descriptive phrases and certain propositions
can be expressed more shortly and more accurately by means of symbolic
logic. Let us consider the proposition "No man marries his deceased
wife's sister." If we assume, as a first approximation, that all
marriages are fertile and that all children are legitimate, then, with
only four primitive ideas: the relation of parent to child (P) and the
three classes of males, females, and dead people, we can define "wife"
(a female who has the relation formed by taking the relative product of
P and [vP][36] to a male), "sister," "deceased wife," and "deceased
wife's sister" in terms of these ideas and of the fundamental notions of
logic. Then the proposition "No man marries his deceased wife's sister"
can be expressed unambiguously by about twenty-nine simple signs on
paper, whereas, in words, the unasserted statement consists of no less
than thirty-four letters. Although, legally speaking, we should have to
adopt somewhat different definitions and possibly increase the
complications of our proposition, it must be remembered that, on the
other hand, we always reduce the number of symbols in any proposition by
increasing the number of definitions in the preliminaries to it.

But the utility of symbolic logic should not be estimated by the brevity
with which propositions may sometimes be expressed by its means. Logical
simplicity, in fact, can very often only be obtained by apparently
complicated statements. For example, the logical interpretation of "The
father of Charles II was executed" is, "It is not always false of _x_
that _x_ begat Charles II, and that _x_ was executed and that 'if _y_
begat Charles II, _y_ is identical with _x_' is always true of _y_."[37]
From the point of view of logic, we may say that the apparently simple
is most often very complicated, and, even if it is not so, symbolism
will make it seem so,[38] and thus draw attention to what might
otherwise easily be overlooked.

FOOTNOTES:

[35] Cf. Chapter XXIV below.

[36] C. S. Peirce's notation for the relation "converse of P."

[37] Russell, _Md._, N. S., vol. xiv., October 1905, p. 482.

[38] Russell, _International Monthly_, vol. iv., 1901, pp. 85-6; cf.
_M._, vol. xxii., 1912, p. 153. [This essay is reprinted in _Mysticism
and Logic_, London and New York, 1918, pp. 74-96.--ED.]




CHAPTER XI

CRITICISM


Those people who think that it is more godlike to seem to turn water
into wine than to seem to turn wine into water surprise me. I cannot
imagine an intolerable critic. It seems to me that, if A resents B's
criticism in trying to put his (A's) discovery in the right or wrong
place, A acts as if he thought he had some private property in truth.
The White Queen seems to have shared the popular misconception as to the
nature of criticism.[39]

FOOTNOTES:

[39] See Appendix J.




CHAPTER XII

HISTORICAL CRITICISM


From a problem in Diophantus's _Arithmetic_ about the price of some wine
it would seem that the wine was of poor quality, and Paul Tannery has
suggested that the prices mentioned for such a wine are higher than were
usual until after the end of the second century. He therefore rejected
the view which was formerly held that Diophantus lived in that
century.[40]

The same method applied to a problem given by the ancient Hindu
algebraist Brahmagupta, who lived in the seventh century after Christ,
might result in placing Brahmagupta in prehistoric times. This is the
problem:[41] "Two apes lived at the top of a cliff of height _h_, whose
base was distant _mh_ from a neighbouring village. One descended the
cliff and walked to the village, the other flew up a height _x_ and then
flew in a straight line to the village. The distance traversed by each
was the same. Find _x_."

FOOTNOTES:

[40] W. W. Rouse Ball, _A Short Account of the History of Mathematics_,
4th edition, London, 1908, p. 109.

[41] _Ibid._, pp. 148-9.




CHAPTER XIII

IS THE MIND IN THE HEAD?


The contrary opinion has been maintained by idealists and a certain
election agent with whom I once had to deal and who remarked that
something slipped his mind and then went out of his head altogether. At
some period, then, a remembrance was in his head and out of his mind;
his mind was not, then, wholly within his head. Also, one is sometimes
assured that with certain people "out of sight is out of mind." What is
in their minds is therefore in sight, and cannot therefore be inside
their heads.




CHAPTER XIV

THE PRAGMATIST THEORY OF TRUTH


The pragmatist theory that "truth" is a belief which works well
sometimes conflicts with common-sense and not with logic. It is commonly
supposed that it is always better to be sometimes right than to be never
right. But this is by no means true. For example, consider the case of a
watch which has stopped; it is exactly right twice every day. A watch,
on the other hand, which is always five minutes slow is never exactly
right. And yet there can be no question but that a belief in the
accuracy of the watch which was never right would, on the whole, produce
better results than such a belief in the one which had altogether
stopped. The pragmatist would, then, conclude that the watch which was
always inaccurate gave truer results than the one which was sometimes
accurate. In this conclusion the pragmatist would seem to be correct,
and this is an instance of how the false premisses of pragmatism may
give rise to true conclusions.

From the text written above the church clock in a certain English
village, "Be ye ready, for ye know not the time," it would be concluded
that the clock never stopped for a period as long as twelve hours. For
the text is rather a vague symbolical expression of a propositional
function which is asserted to be true at all instants. The proposition
that a presumably not illiterate and credulous observer of the clock at
any definite instant does not know the time implies, then, that the
clock is always wrong. Now, if the clock stopped for twelve hours, it
would be absolutely right at least once. It must be right twice if it
were right at the first instant it stopped or the last instant at which
it went;[42] but the second possibility is excluded by hypothesis, and
the occurrence of the first possibility--or of the analogous possibility
of the stopped clock being right three times in twenty-four hours--does
not affect the present question. Hence the clock can never stop for
twelve hours.

The pragmatist's criterion of truth appears to be far more difficult to
apply than the Bellman's,[43] that what he said three times is true, and
to give results just as insecure.

FOOTNOTES:

[42] Both cases cannot occur; the question is similar to that arising in
the discussion of the mortality of Socrates (see Chapter XXII).

[43] See Appendix K.




CHAPTER XV

ASSERTION


The subject of the present chapter must not be confused with the
assertion of ordinary life. Commonly, an unasserted proposition is
synonymous with a probably false statement, while an asserted
proposition is synonymous with one that is certainly false. But in logic
we apply assertion also to true propositions, and, as Lewis Carroll
showed in his version of "What the Tortoise said to Achilles,"[44]
usually pass over unconsciously an infinite series of implications in so
doing. If _p_ and _q_ are propositions, _p_ is true, and _p_ implies
_q_, then, at first sight, one would think that one might assert _q_.
But, from (A) _p_ is true, and (B) _p_ implies _q_, we must, in order to
deduce (Z) _q_ is true, accept the hypothetical: (C) If A and B are
true, Z must be true. And then, in order to deduce Z from A, B, and C,
we must accept another hypothetical: (D) If A, B, and C are true, Z must
be true; and so on _ad infinitum_. Thus, in deducing Z, we pass over an
infinite series of hypotheticals which increase in complexity. Thus we
need a new principle to be able to assert _q_.

Frege was the first logician sharply to distinguish between an asserted
proposition, like "A is greater than B," and one which is merely
considered, like "A's being greater than B," although an analogous
distinction had been made in our common discourse on certain
psychological grounds, for long previously. In fact, soon after the
invention of speech, the necessity of distinguishing between a
considered proposition and an asserted one became evident, on account of
the state of things referred to at the beginning of this chapter.

FOOTNOTES:

[44] _Md._ N. S., vol. iv., 1895, pp. 278-80. Cf. Russell, _P. M._, p.
35.




CHAPTER XVI

THE COMMUTATIVE LAW


Often the meaning of a sentence tacitly implies that the commutative law
does not hold. We are all familiar with the passage in which Macaulay
pointed out that, by using the commutative law because of exigencies of
metre, Robert Montgomery unintentionally made Creation tremble at the
Atheist's nod instead of the Almighty's. This use of the commutative law
by writers of verse renders it doubtful whether, in the hymn-line:

  The humble poor believe,

we are to understand a statement about the humble poor, or a doubtful
maxim as to the attitude of our minds to statements made by the humble
poor.

The non-commutativity of English titles offers difficulties to some
novelists and Americans who refer to Mary Lady So-and-So as Lady Mary
So-and-So, and _vice versa_.




CHAPTER XVII

UNIVERSAL AND PARTICULAR PROPOSITIONS


People who are cynical as to the morality of the English are often
unpleasantly surprised to learn that "All trespassers will be
prosecuted" does not necessarily imply that "some trespassers will be
prosecuted." The view that universal propositions are non-existential is
now generally held: Bradley and Venn seem to have been the first to hold
this, while older logicians, such as De Morgan,[45] considered universal
propositions to be existential, like particular ones.

If the Gnat[46] had been content to affirm his proposition about the
means of subsistence of Bread-and-Butter flies, in consequence of their
lack of which such flies always die, without pointing out such an insect
and thereby proving that the class of them is not null, Alice's doubt as
to the existence of the class in question, even if it were proved to be
well founded, would not have affected the validity of the proposition.

This brings us to a great convenience in treating universal propositions
as non-existential: we can maintain that all _x_'s are _y_'s at the same
time as that no _x_'s are _y_'s, if only _x_ is the null-class. Thus,
when Mr. MacColl[47] objected to other symbolic logicians that their
premisses imply that all Centaurs are flower-pots, they could reply that
their premisses also imply the more usual view that Centaurs are not
flower-pots.

FOOTNOTES:

[45] Cf., e.g., _F. L._, p. 4.

[46] See Appendix L.

[47] Cf., e.g., _Md._, N. S., vol. xiv., July, 1905, pp. 399-400.




CHAPTER XVIII

DENIAL OF GENERALITY AND GENERALITY OF DENIAL


The conclusion of a certain song[48] about a young man who poisoned his
sweetheart with sheep's-head broth, and was frightened to death by a
voice exclaiming:

  "Where's that young maid
  What you did poison with my head?"

at his bedside, gives rise to difficulties which are readily solved by a
symbolism that brings into relief the principle that the denial of a
universal and non-existential proposition is a particular and
existential one. The conclusion of the song is:

  Now all young men, both high and low,
  Take warning by this dismal go!
  For if he'd never done nobody no wrong,
  He might have been here to have heard this song.

It is an obvious error, say Whitehead and Russell,[49] though one easy
to commit, to assume that the cases: (1) all the propositions of a
certain class are true; and (2) no proposition of the class is true; are
each other's contradictories. However, in the modification[50] of
Frege's symbolism which was used by Russell

      (1) is (_x_). _x_,
  and (2) is (_x_). not _x_;

while the contradictory of (1) is:

  not (_x_). _x_.

The last line but one of the above verse may, then, be written:

  (_t_). not (_x_). not not [Greek: ph] (_x_, _t_),

where "[Greek: ph] (_x_, _t_)" denotes the unasserted propositional
function "the doing wrong to the person _x_ at the instant _t_." By
means of the principle of double negation we can at once simplify the
above expression into:

  (_t_). not (_x_). [Greek: ph] (_x_, _t_);

which can be thus read: "If at every instant of his life there was at
least one person _x_ to whom he did no wrong (at that instant)." It is
difficult to imagine any one so sunk in iniquity that he would not
satisfy this hypothesis. We are forced, then, unless our imagination for
evil is to be distrusted, to conclude that any one might have been there
to have heard that song. Now this conclusion is plainly false, possibly
on physical grounds, and certainly on æsthetic grounds. It may be added,
by the way, that it is quite possible that De Morgan was mistaken in his
interpretation of the above proposition owing to the fact that he was
unacquainted with Frege's work. In fact, if he had not noticed the fact
that _any_ two of the "not's" cannot be cancelled against one another he
would have concluded that the interpretation was: "If he had never done
any wrong to anybody."

According as the symbol for "not" comes before the (_x_) or between the
(_x_) and the [Greek: ph], we have an expression of what Frege called
respectively the denial of generality, and the generality of denial. The
denial of the generality of a denial is the form of all existential
propositions, while the assertion of or denial of generality is the
general form of all non-existential or universal propositions.

FOOTNOTES:

[48] To which De Morgan drew attention in a letter; see (Mrs.) S. E. De
Morgan, _Memoir of Augustus De Morgan_, London, 1882, p. 324.

[49] _Pa. Ma._, p. 16.

[50] However, here, for the printer's convenience, we depart from Mr.
Russell's usage so far as to write "not" for a curly minus sign.




CHAPTER XIX

IMPLICATION


A good illustration of the fact that what is called "implication" in
logic is such that a false proposition implies any other proposition,
true or false, is given by Lewis Carroll's puzzle of the three
barbers.[51]

Allen, Brown, and Carr keep a barber's shop together; so that one of
them must be in during working hours. Allen has lately had an illness of
such a nature that, if Allen is out, Brown must be accompanying him.
Further, if Carr is out, then, if Allen is out, Brown must be in for
obvious business reasons. The problem is, may Carr ever go out?

Putting _p_ for "Carr is out," _q_ for "Allen is out" and _r_ for "Brown
is out," we have:

  (1) _q_ implies _r_,
  (2) _p_ implies that _q_ implies not-_r_.

Lewis Carroll supposed that "_q_ implies _r_" and "_q_ implies not-_r_"
are inconsistent, and hence that _p_ must be false. But these
propositions are not inconsistent, and are, in fact, both true if _q_ is
false. The contradictory of "_q_ implies _r_" is "_q_ does not imply
_r_" which is not a consequence of "_q_ implies not-_r_." It seems to be
true theoretically that, if Mr. X is a Christian, he is not an Atheist,
but we cannot conclude from this alone that his being a Christian does
not imply that he is an Atheist, unless we assume that the class of
Christians is not null. Thus, if _p_ is true, _q_ is false; or, if Carr
is out, Allen is in. The odd part of this conclusion is that it is the
one which common-sense would have drawn in that particular case.

A distinguished philosopher (M) once thought that the logical use of the
word "implication"--any false proposition being said to "imply" any
proposition true or false--is absurd, on the grounds that it is
ridiculous to suppose that the proposition "2 and 2 make 5" implies the
proposition "M is the Pope." This is a most unfortunate instance,
because it so happens that the false proposition that 2 and 2 make 5 can
rigorously be proved to imply that M, or anybody else other than the
Pope, is the Pope. For if 2 and 2 make 5, since they also make 4, we
would conclude that 5 is equal to 4. Consequently, subtracting 3 from
both sides, we conclude that 2 would be equal to 1. But if this were
true, since M and the Pope are two, they would be one, and obviously
then M would be the Pope.

The principle that the false implies the true has very important
applications in political arguments. In fact, it is hard to find a
single principle of politics of which false propositions are not the
main support.

If _p_ and _q_ are two propositions, and _p_ implies _q_; then, if, and
only if, _q_ and _p_ are both false or both true, we also have: _q_
implies _p_. The most important applications of this invertibility were
made by the late Samuel Butler[52] and Mr. G. B. Shaw. A political
application may be made as follows: In a country where only those with
middling-sized incomes are taxed, conservative and _bourgeois_
politicians would still maintain that the proposition "the rich are
taxed" implies the proposition "the poor are taxed," and this
implication, which is true because both premiss and conclusion are
false, would be quite unnecessarily supported by many false practical
arguments. It is equally true that "the poor are taxed" implies that
"the rich are taxed." And this can be proved, in certain cases, on other
grounds. For the taxation of the poor would imply, ultimately, that the
poor could not afford to pay a little more for the necessities of life
than, in strict justice, they ought; and this would mean the cessation
of one of the chief means of production of individual wealth.

We also see why a valuable means for the discovery of truth is given by
the inversion of platitudinous implications. It may happen that another
platitude is the result of inversion; but it is the fate of any true
remark, especially if it is easy to remember by reason of a paradoxical
form, to become a platitude in course of time. There are rare cases of a
platitude remaining unrepeated for so long that, by a converse process,
it has become paradoxical. Such, for example, is Plato's remark that a
lie is less important than an error in thought.

Of late years, a method of disguising platitudes as paradoxes has been
too extensively used by Mr. G. K. Chesterton. The method is as follows.
Take any proposition _p_ which holds of an entity _a_; choose _p_ so
that it seems plausible that _p_ also holds of at least two other
entities _b_ and _c_; call _a_, _b_, _c_, and any others for which _p_
holds or seems to hold, the class A, and _p_ the "A-ness" or "A-ity" of
A; let _d_ be an entity for which _p_ does not hold; and put _d_ among
the A's when you think that nobody is looking. Then state your paradox:
"Some A's do not have A-ness." By further manipulation you can get the
proposition "No A's have A-ness." But it is possible to make a very
successful _coup_ if A is the null-class, which has the advantage that
manipulation is unnecessary. Thus, Mr. Chesterton, in his _Orthodoxy_
put A for the class of doubters who doubt the possibility of logic, and
proved that such agnostics refuted themselves--a conclusion which seems
to have pleased many clergymen.

In this way, Mr. Chesterton has been enabled readily to write many books
and to maintain, on almost every page, such theses as that simplicity is
not simple, heterodoxy is not heterodox, poets are not poetical, and so
on; thereby building up the gigantic platitude that Mr. Chesterton is
Chestertonian.

In the chapter on Identity we have illustrated the use of a case of the
principle that any proposition implies any true proposition. This
important principle may be called _the principle of the irrelevant
premiss_;[53] and is of great service in oratory, because it does not
matter what the premiss is, true or false. There is a _principle of the
irrelevant conclusion_, but, except in law-courts, interruptions of
meetings, and family life, this is seldom used, partly because of the
limitation involved in the logical impossibility for the conclusion to
be false if the premiss be true, but chiefly because the conclusion is
more important than the premiss, being usually a matter of prejudice.

Certain modern logicians, such as Frege, have found it necessary so to
extend the meaning of implication of _q_ by _p_ that it holds when _p_
is not a proposition at all. Hitherto, politicians, finding that either
identical or false propositions are sufficient for their needs, have
made no use of this principle; but it is obvious that their stock of
arguments would be vastly increased thereby.

Logical implication is often an enemy of dignity and eloquence. De
Morgan[54] relates "a tradition of a Cambridge professor who was once
asked in a mathematical discussion, 'I suppose you will admit that the
whole is greater than its part?' and who answered, 'Not I, until I see
what use you are going to make of it.'" And the care displayed by
cautious mathematicians like Poincaré, Schoenflies, Borel, Hobson, and
Baire in abstaining from pushing their arguments to their logical
conclusions is probably founded on the unconscious--but no less
well-grounded--fear of appearing ridiculous if they dealt with such
extreme cases as "the series of all ordinal numbers."[55] They are,
probably, as unconscious of implication as Gibbon, when he remarked that
he always had a copy of Horace in his pocket, and often in his hand, was
of the necessary implication of these propositions that his hand was
sometimes in his pocket.

FOOTNOTES:

[51] _Md._, N. S., vol. iii., 1894, pp. 436-8. Cf. the discussions by W.
E. Johnson (_ibid._, p. 583) and Russell (_P. M._, p. 18, note, and
_Md._, N. S., vol. xiv., 1905, pp. 400-1).

[52] The inhabitants of "Erewhon" punished invalids more severely than
criminals. In modern times, one frequently hears the statement that
crime is a disease; and if so, it is surely false that criminals ought
to be punished.

[53] _Irrelevant_ in a popular sense; one would not say, speaking
loosely, that the fact that Brutus killed Cæsar implies that the sea is
salt; and yet this conclusion is implied both by the above premiss, and
the premiss that Cæsar killed Brutus. Cf. on such questions Venn,
_S. L._, 2nd ed., pp. 240-4.

[54] _F. L._, p. 264.

[55] Cf. Chapters XXIX and XXXVII.




CHAPTER XX

DIGNITY


We have seen, at the end of the preceding chapter, that logical
implication is often an enemy of dignity. The subject of dignity is not
usually considered in treatises on logic, but, as we have remarked, many
mathematicians implicitly or explicitly seem to fear either that the
dignity of mathematics will be impaired if she follows out conclusions
logically, or that only an act of faith can save us from the belief
that, if we followed out conclusions logically, we should find out
something alarming about the past, present, or future of mathematics.

Thus it seems necessary to inquire rather more closely into the nature
of dignity, with a view to the discovery of whether it is, as is
commonly supposed, a merit in life and logic.

The chief use of dignity is to veil ignorance. Thus, it is well known
that the most dignified people, as a rule, are schoolmasters, and
schoolmasters are usually so occupied with teaching that they have no
time to learn anything. And because dignity is used to hide ignorance,
it is plain that impudence is not always the opposite of dignity, but
that dignity is sometimes impudence. Dignity is said to inspire respect;
and this may be in part why respect for others is an error of judgment
and self-respect is ridiculous.

Self-respect is, of course, self-esteem. William James has remarked that
self-esteem depends, not simply upon our success, but upon the ratio of
our success to our pretensions, and can therefore be increased by
diminishing our pretensions. Thus if a man is successful, but only then,
can he be both ambitious and dignified. James also implies that
happiness increases with self-esteem. Likeness of thought with one's
friends, then, does not make one happy, for otherwise a man who esteemed
himself little would be indeed happy. Also if a man is unhappy he could
not, from our premisses, by the principles of the syllogism and of
contraposition, be dignified--a conclusion which should be fatal to many
novelists' heroes.

A reflection on pessimism to which this discussion gives rise is the
following: It would appear that a man's self-esteem would be increased
by a conviction of the unworthiness of his neighbours. A man, therefore,
who thinks that the world and all its inhabitants, except himself, are
very bad, should be extremely happy. In fact, the effects would hardly
be distinguishable from those of optimism. And optimism, as everybody
knows, is a state of mind induced by stupidity.




CHAPTER XXI

THE SYNTHETIC NATURE OF DEDUCTION


Doubt has often been expressed as to whether a syllogism can add to
our knowledge in any way. John Stuart Mill and Henri Poincaré, in
particular, held the opinion that the conclusion of a syllogism is an
"analytic" judgment in the sense of Kant, and therefore could be
obtained by the mere dissection of the premisses. Any one, then, who
maintains that mathematics is founded solely on logical principles would
appear to maintain that mathematics, in the last instance, reduces to a
huge tautology.

Mill, in Chapter III of Book II of his _System of Logic_, said that "it
must be granted that in every syllogism, considered as an argument to
prove the conclusion, there is a _petitio principii_. When we say

  All men are mortal,
  Socrates is a man,

therefore

  Socrates is mortal,

it is unanswerably urged by the adversaries of the syllogistic theory,
that the proposition, Socrates is mortal, is presupposed in the more
general assumption, All men are mortal; that we cannot be assured of the
mortality of all men unless we are already certain of the mortality of
every individual man; that if it be still doubtful whether Socrates, or
any other individual we choose to name, be mortal or not, the same
degree of uncertainty must hang over the assertion, All men are mortal;
that the general principle, instead of being given as evidence of the
particular case, cannot itself be taken for true without exception until
every shadow of doubt which could affect any case comprised with it is
dispelled by evidence _aliunde_; and then what remains for the syllogism
to prove? That, in short, no reasoning from general to particular can,
as such, prove anything, since from a general principle we cannot infer
any particulars but those which the principle itself assumes as known.
This doctrine appears to me irrefragable...."

But it is not difficult to see that in certain cases at least deduction
gives us _new_ knowledge.[56] If we already know that two and two always
make four, and that Asquith and Lloyd George are two and so are the
German Emperor and the Crown Prince, we can deduce that Asquith and
Lloyd George and the German Emperor and the Crown Prince are four. This
is new knowledge, not contained in our premisses, because the general
proposition, "two and two are four," never told us there were such
people as Asquith and Lloyd George and the German Emperor and the Crown
Prince, and the particular premisses did not tell us that there were
four of them, whereas the particular proposition deduced does tell us
both these things. But the newness of the knowledge is much less certain
if we take the stock instance of deduction that is always given in books
on logic, namely "All men are mortal; Socrates is a man, therefore
Socrates is mortal." In this case what we really know beyond reasonable
doubt is that certain men, A, B, C, were mortal, since, in fact, they
have died. If Socrates is one of these men, it is foolish to go the
roundabout way through "all men are mortal" to arrive at the conclusion
that _probably_ Socrates is mortal. If Socrates is not one of the men on
whom our induction is based, we shall still do better to argue straight
from our A, B, C, to Socrates, than to go round by the general
proposition, "all men are mortal." For the probability that Socrates is
mortal is greater, on our data, than the probability that all men are
mortal. This is obvious, because if all men are mortal, so is Socrates;
but if Socrates is mortal, it does not follow that all men are mortal.
Hence we shall reach the conclusion that Socrates is mortal, with a
greater approach to certainty if we make our argument purely inductive
than if we go by way of "all men are mortal" and then use deduction.

Many years ago there appeared, principally owing to the initiative of
Dr. F. C. S. Schiller of Oxford, a comic number of _Mind_. The idea was
extraordinarily good, not so the execution. A German friend of Dr.
Schiller was puzzled by the appearance of the advertisements, which were
doubtfully humorous. However, by a syllogistic process, he acquired
information which was new and useful to him, and thus incidentally
refuted Mill. Presumably he started from the title of the magazine
(_Mind!_), for a mark of exclamation seems nearly always in German to be
a sign of an intended joke (including of course the mark after the
politeness expressed in the first sentence of a private letter or a
public address). There would be, then, the following syllogism:

  This is a book of would-be jokes (i.e. everything in this book is a
          would-be joke);
  This advertisement is in this book;
  Therefore, this advertisement is a would-be joke.

Thus the syllogism may be almost as powerful an agent in the detection
of humour as M. Bergson's criterion, to be described in a future
chapter.[57]

FOOTNOTES:

[56] [The following passage is almost word for word the same as a
passage on pp. 123-5 of Mr. Russell's _Problems of Philosophy_, first
published in 1912, a year after Mr. R*ss*ll's death. It is easy hastily
to conclude that Mr. Russell was indebted to Mr. R*ss*ll to a greater
degree than is usually supposed. But an examination of the internal
evidence leads us to another conclusion. The two texts, it will be
found, differ only in the names of the German Emperor, the Crown Prince
and the other personages being replaced, in the book of 1912, by those
of Messrs. Brown, Jones, Smith, and Robinson. Now, Mr. Russell, in a new
edition of his _Problems_ issued near the beginning of the European war
and before the Russian revolution, substituted "the Emperor of Russia"
for "the Emperor of China" of the first edition. Hence it seems quite
likely that Mr. Russell, who has always shown a tendency to substitute
existents for nonentities, wrote Mr. R*ss*ll's notes.--
<sc>
 Ed.
</sc>
]

[57] [See Chapter XLII.--ED.]




CHAPTER XXII

THE MORTALITY OF SOCRATES


The mortality of Socrates is so often asserted in books on logic that it
may be as well briefly to consider what it means. The phrase "Socrates
is mortal" may be thus defined: "There is at least one instant _t_ such
that _t_ has not to Socrates the one-many relation R which is the
converse of the relation 'exists at,' and all instants following _t_
have not the relation R to Socrates, and there is at least one instant
_t´_ such that neither _t´_ nor any instant preceding _t´_ has the
relation R to Socrates."

This definition has many merits. In the first place, no assumption is
made that Socrates ever lived at all. In the second place, no assumption
is made that the instants of time form a continuous series. In the third
place, no assumption is made as to whether Socrates had a first or last
moment of his existence. If time be indeed a continuous series, then we
can easily deduce[58] that there must have been _either_ a first moment
of his non-existence _or_ a last one of his existence, but not both;
just as there seems to be either a greatest weight that a man can lift
or a least weight that he cannot lift, but not both.[59] This may be set
forth as follows: for the present we will not concern ourselves with
evidence for or against human immortality; I will merely try to present
some logical questions which persistently arise whenever we think of
eternal life. One of the greatest merits of modern logic is that it has
allowed us to give precision to such problems, while definitely
abandoning any pretensions of solving them; and I will now apply the
logico-analytical method to one of the problems of our knowledge of the
eternal world.[60]

We will start from the generally accepted proposition that all men are
mortal. Clearly, if we could know each individual man, and know that he
was mortal, that would not enable us to know that all men are mortal,
unless we knew, in addition, that those were all the men there are. But
we need not here assume any such knowledge of general propositions; and,
though most of us will admit that the proposition in question has great
intrinsic plausibility, it is not strictly necessary for our present
purpose to assume anything more than the still more probable proposition
"Socrates is mortal." This last proposition, quite apart from the fact
that we have a large amount of historical evidence for its truth, has
been repeated so often in books on logic that it has taken on the
respectable air of a platitude while preserving the character of an
exceedingly probable truth. The truth also results from the fact that it
is used as the conclusion of a syllogism. For it is a well-known fact
that syllogisms can only be regarded as forming part of a sound
education if the conclusions are obviously true. The use of a syllogism
of the form "All cats are ducks and all ducks are mice, therefore all
cats are mice," would introduce grave doubts into the University of
Oxford as to whether logic could any longer be considered as a valuable
mental training for what are amusingly called the "learned professions."

If, then, we divide all the instants of time, whether past, present, or
future, into two series--those instants at which Socrates was alive, and
those instants at which he was not alive--and leave out of
consideration, for the sake of greater simplicity, all those instants
before he lived, we see at once, by the simple application of Dedekind's
Axiom, that, if Socrates entered into eternal life after his death,
there must have been either a last moment of his earthly life _or_ a
first moment of his eternal life, but not both.

Logic alone can give us no information as to which of these cases
actually occurred, and we are thrown back on to a discussion of
empirical evidence. It is no unusual thing to read of people who thought
"that every moment would be their last." In this case it is quite
obvious that they consequently thought that eternity would have no
beginning.

Now here we must consider two things: (1) It is plainly unsafe to
conclude from what people think will happen to what will happen; (2)
even if we could so conclude, it would be unsafe to deduce that there
was a last moment in the life of Socrates: we could only make the guess
plausible, as we should be using the inductive method.

There are two other pieces of evidence that there is a last moment of
any earthly existence, which we may now briefly consider. That this was
so was held by Carlo Michaelstaedter; but since he apparently only
believed this because he wanted, by attributing a supposed ethical value
to that moment, to give support to his theory of suicide, we ought not
to give great weight to this evidence. Secondly, Thomas Hobbes objected
to the principle "that a quantity may grow less and less eternally, so
as at last to be equal to another quantity; or, which is all one, that
there is a last in eternity" as "void of sense." Now, the principle
meant is true, so that, although the other proposition mentioned by
Hobbes does not follow logically from the first, there is some evidence
that this other is true. In fact, that Hobbes thought that such-and-such
a proposition followed from another proposition which he wrongly
believed to be false, is far better evidence for the truth of
such-and-such a proposition than any we have for the truth of most of
our most cherished beliefs.

Thirdly, Leibniz, in a dialogue[61] written on his journey of 1676 to
visit Spinoza, raised the question whether the moment at which a man
dies may be regarded as both the last moment at which he is alive and
the first at which he is dead, as it must be by Aristotle's theory of
continuity. Agreement with this view violates the law of contradiction;
denial of it implies that two moments can be immediately adjacent. By
the denial, then, we are led to regard space and time as made up of
indivisible points and moments, and thus, since we can draw one and only
one parallel from any point in the diagonal of a square to a given side,
the diagonal will contain the same (infinite) number of points as that
side, and will therefore be equal to it. In this Leibniz repeated an
argument used by the ancient Arabs, Roger Bacon, and William of Occam.
This Leibniz considered to be a proof that a line cannot be an aggregate
of points. Indeed, their number would be "the number of all numbers" of
the greatest possible integer, which _is_ not.

It does not seem, further, that any light is thrown on the logical
question of human mortality or immortality by legal decisions. It would
appear that one can, legally speaking, be alive for any period less than
twenty-four hours after one is dead and be dead for any period less than
twenty-four hours before one's death. At least, according to _Salkeld_,
i. 44, it was "adjudged that if one be born the first of February at
eleven at night, and the last of January in the twenty-first year of his
age, at one of the clock in the morning, he makes his will of lands, and
dies, it is a good will, for he was then of age." In Sir Robert Howard's
case (_ibid._, ii. 625) it was held by Chief Justice Holt that "if A be
born on the third day of September; and on the second day of September
twenty-one years afterwards he make his will, this is a good will; for
the law will make no fraction of a day, and by consequence he was of
age." But it is hardly necessary to remark that in this way the problem
with which we are concerned is merely shifted and not solved. For the
question as to whether there is or is not a last moment of a man's life
is not answered by the decision that he dies legally twenty-four hours
before or after he dies in the usual sense of the word, and the problem
arises as to whether there is or is not a last moment of his legal
age.[62]

So assuming that there was a last moment of Socrates's earthly life, and
consequently no first moment of his eternal life, we see, further, that,
unless the possibility of infinite numbers is granted, it would be quite
possible for us logically to doubt the possibility of an eternal life
for Socrates on the same grounds as those which led Zeno to assert that
motion was impossible and that Achilles could never overtake the
Tortoise. If, on the other hand, it be admitted that eternity, at least
in the case of Socrates, had a beginning, these same arguments of Zeno
would lead any one who denies the possibility of infinite number to
conclude that Socrates, like the worm, can never die. Thus is it quite
plain that the difficulties about immortality which meet us at the very
outset of our inquiry can partly be solved only by the help of the
theory of infinite numbers and partly, it would seem, not at all.

There is another difficulty about immortality which is quite distinct
from this and is analogous to another argument of Zeno. If, indeed, all
the instants of time be divided, as before, into the two series of
instants at which Socrates was alive and instants at which he was not
alive, it follows at once that no instant of time is not accounted for.
At none of these instants, however, does Socrates die; obviously he
cannot die either when he is alive or when he is dead. Thus it would
appear that Socrates never died, and that we ought to re-define the term
"mortal" to mean "a human being who is alive at some moments and dead at
some." Consequently we must avoid the very tempting conclusion that,
because Socrates never died, he was therefore immortal.

It is very important carefully to distinguish between the two arguments
I have just set forth. The second argument proves quite rigidly that
Socrates and, indeed, anybody else, never dies, whether there is or is
not a last moment of his life on earth. The first argument proves that,
if there is a first moment of Socrates's eternal life, his life on earth
never ends. But we have seen that we cannot conclude that this unending
life proves that he never is or will be in a state of eternity.

FOOTNOTES:

[58] By "Dedekind's Axiom," _E. N._, p. 11.

[59] _M._, vol. xx., 1910, pp. 134-5.

[60] [Here, again, Mr. R*ss*ll's work seems to anticipate some of Mr.
Russell's later work, e.g. in _Our Knowledge of the External World as a
Field for Scientific Method in Philosophy_, Chicago and London, 1914,
pp. 3-4, 55-6, _et passim._--ED.]

[61] "Pacidius Philalethi" in Louis Couturat, _Opuscules et Fragments
inédits de Leibniz_, Paris, 1903, pp. 594-627, especially pp. 599, 601,
608, 611. Cf. [A. E. Taylor, Hastings' _Encyclopædia of Religion and
Ethics_, vol. iv., Part 2, Edinburgh, 1912, p. 96.--ED.]; Robert Latta,
_Leibniz: The Monadology and other Philosophical Writings_, Oxford,
1898, pp. 21 ff, 29 (note); Couturat, _La Logique de Leibniz d'après des
documents inédits_, Paris, 1901, pp. 130, 132; and Russell, _Ph. L._,
pp. 108-16, 243-9.

[62] [It may be remarked that, according to _The Times_ of December 20,
1917, Mr. Justice Sargant, in the Chancery Division, also held that "the
law did not recognize fractions of a day," and that Lord Blackburn, in
his decision (9 _App. Cas._, 371, 373) that a man born on the thirteenth
of May 1853 attained the age of twenty-one on the thirteenth of May 1874
"was not speaking strictly."--ED.]




CHAPTER XXIII

DENOTING


A concept _denotes_ when, if it occurs in a proposition, the proposition
is not about the concept, but _about_ a term connected in a certain
peculiar way with the concept. Some people often assert that man is
mortal, and yet we never see announced in _The Times_ that Man died on a
certain day at his villa residence "Camelot" at Upper Tooting,[63] nor
do we hear that Procrastination was again the butt of Mr. Plowden's
jokes at Marylebone Police Court last week.

That two phrases may have different _meanings_ and the same _denotation_
was discovered by Alice and Frege. Alice[64] observed that the road
which led to Tweedledum's house was that which led to the house of
Tweedledee; and Frege pointed out that the phrases "the house to which
the road that leads to Tweedledum's house leads" and "the house to which
the road that leads to Tweedledee's house leads" have different _Sinn_,
but the same _Bedeutung_.

FOOTNOTES:

[63] Cf. _P. M._, pp. 53-4.

[64] See Appendix M.




CHAPTER XXIV

THE


The word "the" implies existence and uniqueness; it is a mistake to talk
of "the son of So-and-So" if So-and-So has a fine family of ten
sons.[65] People who refer to "the Oxford Movement" imply that Oxford
only moved once; and those quaint people who say that "A is quite the
gentleman" imply both the doubtful proposition that there is only one
gentleman in the world, and the indubitably false proposition that he is
that man. Probably A is one of those persons who add to the confusion in
the use of the definite article by speaking of his wife as "the wife."

In a certain Children's Hymn Book one reads:

  The river vast and small.

Few would deny that there is not more than one such river, but
unfortunately it is doubtful if there is such a river at all. The case
is exactly the same with the ontological proof of the existence of the
most perfect being.[66]

According to the _Daily Mail_ of October 9, 1906, Judge Russell decided
against a claim brought by an agent against his company for appointing
another agent, the claim being on the ground that he was appointed as
"the" agent.

Most people admit that the number 2 can be added to the number 2 to give
the number 4, but this is a mistake. They concede, when they use _the_,
that there is only one number 2, and yet they imagine that, when they
consider it apart as the first term of our above sum, they can find
another to add to it, and thereby form the third term. The truth is that
"2 + 2 = 4" is a very misleading equation, and what we really mean by
that faultily abbreviated statement is more precisely: If _x_ and _y_
denote any things which form a class B, and _x´_ and _y´_ any other
things that form a class (A) which, like that of _x_ and _y_, is a
member of the class (which we call "2") of those classes which have a
one-one correspondence with B (so that any member of A corresponds to
one, and only one, member of B, and conversely), the class of all the
terms of A and B together is a member of that class of classes which,
analogously, we call "4." In this, for the sake of shortness, we have
introduced abbreviations which should not be used in a rigorous logical
statement.

FOOTNOTES:

[65] Cf. _Md._, N. S., vol. xiv., 1905, pp. 481, 484.

[66] Cf. _ibid._, p. 491, note.




CHAPTER XXV

NON-ENTITY


When people say that such-and-such a thing "is non-existent" they
usually mean that there is not any "thing" of the kind spoken of. Venn
meant this when he described[67] his encounter with what he imagined to
be a very ingenious tradesman: "I once had some strawberry plants
furnished me which the vendor admitted would not bear many berries. But
he assured me that this did not matter, since they made up in their size
what they lost in their number. (He gave me, in fact, the hyperbolic
formula, _xy = c_, to connect the number and magnitude.) When summer
came, _no_ fruit whatever appeared. I saw that it would be no use to
complain, because the man would urge that the size of the non-existent
berry was infinite, which I could not see my way to disprove. I had
forgotten to bar zero values of either variable."

It is to be regretted that this useful note was omitted in the second
edition of Venn's book; one can imagine that it might have protected Mr.
MacColl and Herr Meinong (who believed, unlike Alice in what may be
called her first theory,[68] in round squares and fabulous monsters)
against the dishonest practices of traders who were too ready with
promises. For the death-blow to this kind of trade was not given until
1905, when Mr. Russell published his article "On Denoting,"[69] and took
up the position of the White King in opposition to Alice's later
assertions.[70]

Venn's experience illustrates another characteristic of mathematical
logic. It is necessary, in order to make our arguments conclusive, to
devote great care to the elimination of difficulties which rarely occur.
The White Knight--who was like Boole in being a pioneer of mathematical
logic in this way, and yet seems to have held, like Boole, those
philosophical opinions which would base logic on psychology--recognized
the necessity of taking precautions against any unusual appearance of
mice on a horse's back.[71]

FOOTNOTES:

[67] _S. L._, 1881, p. 339, note.

[68] See Appendix N.

[69] _Md._, N. S., vol. xiv., October 1905, pp. 479-93.

[70] See Appendix N.

[71] See Appendix O.




CHAPTER XXVI

IS


_Is_ has four perfectly distinct meanings in English, besides misuses of
the word. Among the misuses, perhaps the most important are those
referred to by De Morgan:[72] "... We say 'murder _is_ death to the
perpetrator' where the copula is _brings_; 'two and two _are_ four,' the
copula being 'have the value of,' etc."

Schröder[73] quite satisfactorily pointed out the well-known distinction
between an _is_ where subject and predicate can be interchanged (such
as: "the class whose members are Shem, Ham and Japhet is the class of
the sons of Noah") and an _is_ or _are_ where they cannot (such as:
Englishmen are Britons), but failed to see[74] the more important
distinction (made by Peano) of is in the sense of "is a member of." If
Englishmen are Britons, and Britons are civilized people, it follows
that Englishmen are civilized people; but, though the _Harmsworth
Encyclopædia_ is a member of the class Book (of one or more volumes),
and this class is the member of a class A of which it is the only
member, yet the _Harmsworth Encyclopædia_ is not a member of A, for it
is not true that it is the whole class of books; and such a statement
would not even be made except possibly in the form of an advertisement.

The fourth meaning of _is_ is _exists_; it is in certain rare moods a
matter for regret that there are difficulties in the way of using one
word to denote four different things. For, if there were not, we might
prove the existence of any thing we please by making it the subject of a
proposition, and thereby earn the gratitude of theologians.

FOOTNOTES:

[72] _F. L._, p. 268.

[73] _A. d. L._, i. pp. 127 sqq.

[74] _Ibid._, vol. ii. pp. 461, 597.




CHAPTER XXVII

_AND_ AND _OR_


When, with Boole, alternatives (A, B) are considered as mutually
exclusive, logical addition may be described as the process of taking A
_and_ B or A _or_ B. It is a great and rare convenience to have two
terms for denoting the same thing: commonly, people denote several
things by the same term, and only the Germans have the privilege of
referring to, say, _continuity_ as _Stetigkeit_ or _Kontinuierlichkeit_.
But Jevons[75] quoted Milton, Shakespeare, and Darwin to prove that
alternatives are not exclusive, and so attained first to recognized
views by arguments which were plainly irrelevant.

Of course, _and_ is often used as the sign of logical addition: thus one
may speak of one's brothers _and_ sisters, without being understood to
mean the null-class (as should be the case), or pray for one's
"relations and friends," without being sure that one's prayer would be
answered,--as it certainly would if one meant to pray for the
null-class, this being the class indicated. And a word like _while_ is
often used for a logical addition, when exclusiveness of the
alternatives is almost implied. Thus, a reviewer in _Mind_,[76] noticing
the translation of Mach's _Popular Scientific Lectures_ into American,
said of the lectures that: "Most of them will be familiar ... to
epistemologists and experimental psychologists: while the remainder,
which deal with physical questions, are well worth reading." The reader
has the impression, probably given unintentionally, that Professor
Mach's epistemological and psychological lectures are not, in the
reviewer's opinion, worth reading.

FOOTNOTES:

[75] _Pure Logic_ ..., London, 1864, pp. 76-9. Cf. Venn, _S. L._, 2nd
ed., pp. 40-8.

[76] N. S., vol. iv. p. 261.




CHAPTER XXVIII

THE CONVERSION OF RELATIONS


The "Conversion of Relations" does not mean what it might be supposed to
mean; it has nothing to do with what Kant called "the wholesome art of
persuasion." What concerns us here is the convertibility of a logical
relation. If A has a certain relation R to B, the relation of B to A,
which may be denoted by [vR], is called the _converse_ of R. As De
Morgan[77] remarked, this conversion may sometimes present difficulties.
The following is De Morgan's example:

"Teacher: 'Now, boys, Shem, Ham and Japheth were Noah's sons; who was
the father of Shem, Ham and Japheth?' No answer.

"Teacher: 'Boys, you know Mr. Smith, the carpenter, opposite; has he any
sons?'

"Boys: 'Oh! yes, sir! there's Bill and Ben.'

"Teacher: 'And who is the father of Bill and Ben Smith?'

"Boys: 'Why, Mr. Smith, to be sure.'

"Teacher: 'Well, then, once more, Shem, Ham and Japheth were _Noah's_
sons; who was the father of Shem, Ham and Japheth?'

"A long pause; at last a boy, indignant at what he thought the attempted
trick, cried out: 'It _couldn't_ have been Mr. Smith.' These boys had
never converted the relation of father and son...."

FOOTNOTES:

[77] _Trans. Camb. Phil. Soc._, vol. x., 1864, part ii., note on page
334.




CHAPTER XXIX

PREVIOUS PHILOSOPHICAL THEORIES OF MATHEMATICS


Mathematicians usually try to found mathematics on two principles:[78]
one is the principle of confusion between the sign and the thing
signified (they call this principle the foundation-stone of the formal
theory), and the other is the Principle of the Identity of Discernibles
(which they call the principle of the permanence of equivalent forms).

But the truth is that if we set sail on a voyage of discovery with Logic
alone at the helm, we must either throw such principles as "the identity
of those conceptions which have in common the properties that interest
us" and "the principle of permanence" overboard, or, if we do not like
to act in such a way to old companions with whom we are so familiar that
we can hardly feel contempt for them, at least recognize them clearly as
having no logical validity and merely as psychological principles, and
reduce them to the humble rank of stewards, to minister to our human
weaknesses on the voyage. And then, if we adopt the wise policy of
keeping our axioms down to the minimum number, we must refrain from
creating or thinking that we are creating new numbers to fill up gaps
among the older ones, and thence recognize that our rational numbers are
not particular cases of "real" numbers, and so on.

We thus get a world of conceptions which looks, and is, very different
from that which ordinary mathematicians think they see; and perhaps this
is the reason why some mathematicians of great eminence, such as Hilbert
and Poincaré, have produced such absurd discussions on the fundamental
principles of mathematics,[79] showing once more the truth of the not
quite original remark of Aunt Jane, who

  ... observed, the second time
  She tumbled off a 'bus:
  "The step is short from the sublime
  To the ridiculous."

In their readiness to consider many different things as one thing--to
consider, for example, the ratio 2:1 as the same thing as the cardinal
number 2--such mathematicians as Peacock, Hankel, and Schubert were
forestalled by the Pigeon, who thought that Alice and the Serpent were
the same creature, because both had long necks and ate eggs.[80] It is,
however, doubtful whether the Pigeon would have followed the example of
the mathematicians just mentioned so far as to embrace the creed of
nominalism and so to feel no difficulty in subtracting from zero--a
difficulty which was pointed out with great acuteness by the Hatter[81]
and modern mathematical logicians.

FOOTNOTES:

[78] These principles, after many attempts to state them by Peacock, the
Red and the White Queen (see Appendix P), Hankel, Schröder, and Schubert
had been made, were first precisely formulated by Frege in _Z. S._; cf.
also Chapter VII.

[79] See Couturat, _R. M. M._, vol. xiv., March, 1906, pp. 208-50, and
Russell, _ibid._, September, 1906, pp. 627-34.

[80] See Appendix P.

[81] See _ibid._




CHAPTER XXX

FINITE AND INFINITE


I was once shown a statement made by an eminent mathematician of
Cambridge from which one would conclude that this mathematician thought
that finite distances became infinite when they were great enough. In
one of those splendidly printed books, bound in blue, published by the
University Press, and sold at about a guinea as a guide to some advanced
branch of pure mathematics, one may read, even in the second edition
published in 1900, the words: "Representation [of a complex variable] on
a plane is obviously more effective for points at a finite distance from
the origin than for points at a very great distance."

Plainly some of the points at a very great distance are at a _finite_
distance, for the same author mentions that Neumann's sphere for
representing the positions of points on a plane "has the advantage ...
of exhibiting the uniqueness of _z_ = [infinity symbol] as a value of
the variable."




CHAPTER XXXI

THE MATHEMATICAL ATTAINMENTS OF TRISTRAM SHANDY


Tristram Shandy[82] said that his father was sometimes a gainer by
misfortune; for if the pleasure of haranguing about it was as ten, and
the misfortune itself only as five, he gained "half in half," and was
well off again as if the misfortune had never happened.

Suppose that the unit (arbitrary) of pleasure is denoted by A, Tristram
Shandy, by neglecting, in this ethical discussion, to introduce negative
quantities (Kant's pamphlet advocating this introduction into philosophy
was made subsequently[83]), apparently made 15A to result, and this can
hardly be maintained to be the half of 10A. It is possible, however,
that Tristram Shandy succeeded in proving the apparently paradoxical
equation

  15A = 5A

by remarking that the axiom "the whole is greater than the part" does
not always hold. This remark follows at once from what Mr. Russell[84]
has called "The Paradox of Tristram Shandy." This paradox is described
by Mr. Russell as follows:

"Tristram Shandy, as we know, took two years writing the history of the
first two days of his life, and lamented that, at this rate, material
would accumulate faster than he could deal with it, so that he could
never come to an end. Now I maintain that, if he had lived for ever,
and not wearied of his task, then, even if his life had continued as
eventfully as it began, no part of his biography would have remained
unwritten."

This paradox is strictly correlative to the well-known paradox of Zeno
about Achilles and the Tortoise.[85] "The Achilles proves that two
variables in a continuous series, which approach equality from the same
side, cannot ever have a common limit: the Tristram Shandy proves that
two variables which start from a common term, and proceed in the same
direction, but diverge more and more, may yet determine the same
limiting class (which, however, is not necessarily a segment, because
segments were defined as having terms beyond them). The Achilles assumes
that whole and part cannot be similar, and deduces a paradox; the other,
starting from a platitude, deduces that whole and part may be similar.
For common-sense, it must be confessed that it is a most unfortunate
state of things." And Mr. Russell considers that, in the face of proofs,
it ought to commit suicide in despair.

Now, I suggest the extremely unlikely possibility that Tristram Shandy,
by reflection on his own life and literary labours, was led to the
correct course of accepting the paradox which resulted from this
reflection and rejecting the Achilles. Thus, he concluded that an
infinite whole may be similar (or, in Cantor's terminology,
"equivalent") to a proper part of itself, and hence, by a confusion of
similarity with identity (or equivalence with equality) which he shares
with some subsequent philosophers,[86] that a whole may be equal to a
proper part of itself. If A is an infinite class, it is not difficult to
see that we can have

  10A = 5A.

In this way many have avoided an opinion which rests on no better
foundation than that formerly entertained by the inductive philosophers
of Central Africa, that all men are black.[87]

FOOTNOTES:

[82] Cf. a letter of De Morgan in Mrs. De Morgan's _Memoir of Augustus
De Morgan_, p. 324.

[83] Kant's tract was published in 1763, while _Tristram Shandy_ was
published in 1760.

[84] _P. M._, pp. 358-9 [Cf. _M._, vol. xxii., January 1912, p.
187.--ED.]

[85] Cf. _P. M._, pp. 350, 358-9; _M._, vol. xxii., 1912, p. 157.

[86] [Cf. for example, Cosmo Guastella, _Dell' infinito_, Palermo,
1912.--ED.]

[87] Cf. Russell, _P. M._, p. 360.




CHAPTER XXXII

THE HARDSHIPS OF A MAN WITH AN UNLIMITED INCOME


I once heard a man refer to his income as limited, in order to
illustrate the hardship of a class of men, of which he of course was
one, in having to pay a somewhat high income-tax. It is obvious that
this man spoke enviously, and consequently admitted the existence of
more fortunately placed individuals who had unlimited incomes. A little
reflection would have shown the man that he was not taking up a
paradoxical attitude. A "paradoxical attitude" is of course the
assertion of one or more propositions of which the truth cannot be
perceived by a philosopher--and particularly an idealist--and can be
perceived by a logician and occasionally, but not always, by a man of
common-sense. Such propositions are: "The cat is hungry," "Columbus
discovered America," and "A thing which is always at rest may move from
the position A to the different position B."

Now, if a man had an unlimited income, it is an immediate inference
that, however low income-tax might be, he would have to pay annually to
the Exchequer of his nation a sum equal in value to his whole income.
Further, if his income was derived from a capital invested at a finite
rate of interest (as is usual), the annual payments of income-tax would
each be equal in value to the man's whole capital. If, then, the man
with an unlimited income chose to be discontented, he would be sure of a
sympathetic audience among philosophers and business acquaintances; but
discontent could not last long, for the thought of the difficulties he
was putting in the way of the Chancellor of the Exchequer, who would
find the drawing up of his budget most puzzling, would be amusing.
Again, the discovery that, after paying an infinite income-tax, the
income would be quite undiminished, would obviously afford satisfaction,
though perhaps the satisfaction might be mixed with a slight uneasiness
as to any action the Commissioners of Income-Tax might take in view of
this fact.

A problem of a wholly different nature is connected with the possible
purchase by the man with an unlimited income of an enumerable infinity
of pairs of boots. If he wished to prove that he had an even number of
boots, it would be easy if right boots were distinguishable from left
ones, but if the man were a faddist of such a kind that he insisted that
his left boots should not be made in any way differently from his right
ones, it would not be possible for him to prove the theorem mentioned
unless he assumed what is known as "the multiplicative axiom." In fact
this axiom shows that it is legitimate to pick out an infinite
succession of members of an infinite class in an arbitrary way. In the
case of the pairs of boots, each pair contains two members, and if there
is no means of distinguishing between them, when we wish to pick out one
of them for each of the infinity of pairs, we cannot say which ones we
mean to pick out unless we assume, by means of the above axiom, that a
particularized member can always be found even with things of each of
which it can be said that, like Private James in the _Bab Ballads_,

  No characteristic trait had he
  Of any distinctive kind.

However, a solution of the puzzle was given by Dr. Dénes König of
Budapest. You first prove that there are points in space such that, if P
is one of them, not more than a finite number of pairs of boots are such
that each centre of mass of the two members of a pair is equidistant
from P. Taking a point P of this sort, select from each pair the boot
whose centre of mass is nearest P. (There may be a finite number of
pairs left over, but they can be dealt with arbitrarily.)

Another form of the problem is as follows. Every time the man bought a
pair of boots he also bought a pair of socks to go with it; he had an
enumerable infinity of pairs of each, and the problem is to prove that
he had as many boots as he had socks. In this case the boots, we will
suppose, can be divided into right and left, but the socks cannot. Thus
there are an enumerable infinity of boots, but the number of the socks
cannot be determined without admitting the axiom mentioned above. A
further difficulty might arise if the owner of the boots and socks lost
one leg in some accident, and told his butler to give away half his
socks. Naturally the butler would find great logical difficulties in so
doing, and it would seem to be an interesting ethical problem whether he
should be dismissed from his situation for failing to prove the
multiplicative axiom. Again, if the butler stole a pair of boots, the
millionaire would have as many pairs as before, but might have fewer
boots. There is as yet no evidence that the number of his boots is equal
to or greater than the number of pairs.




CHAPTER XXXIII

THE RELATIONS OF MAGNITUDE OF CARDINAL NUMBERS


The theorems of cardinal arithmetic are frequently used in ordinary
conversation. What is known as the Schröder-Bernstein theorem was used,
long before Bernstein or Schröder, by Edward Thurlow, afterward the
law-lord Lord Thurlow, when an undergraduate of Caius College,
Cambridge. Thurlow was rebuked for idleness by the Master, who said to
him: "Whenever I look out of the window, Mr. Thurlow, I see you crossing
the Court." The provost thus asserted a one-one correspondence between
the class A of his acts of looking out of the window and a part of the
class B of Thurlow's acts of crossing the Court. Thurlow asserted in
reply a one-one correspondence between B and a part of A: "Whenever
I cross the Court I see you looking out of the window." The
Schröder-Bernstein theorem, then, allows us to conclude that there is a
one-one correspondence between the classes A and B. That A and B were
finite classes is not the fault of the Master or Thurlow; nor is it
relevant logically.




CHAPTER XXXIV

THE UNKNOWABLE


According to Mr. S. N. Gupta,[88] the first thing that every student of
Hindu logic has to learn when he is said to begin the study of inference
is that "all H is S" is not always equivalent to "No H is not S." "The
latter proposition is an absurdity when S is _Kebalánvayi_, i.e. covers
the whole sphere of thought and existence.... 'Knowable' and 'Nameable'
are among the examples of _Kebalánvayi_ terms. If you say there is a
thing not-knowable, how do you know it? If you say there is a thing
not-nameable, you must point that out, i.e. somehow name it. Thus you
contradict yourself."

Mr. Herbert Spencer's doctrine of the "Unknowable" gives rise to some
amusing thoughts. To state that all knowledge of such and such a thing
is above a certain person's intelligence is not self-contradictory, but
merely rude: to state that all knowledge of a certain thing is above all
possible human intelligence is nonsense, in spite of its modest,
platitudinous appearance. For the statement seems to show that we do
know something of it, viz. that it is unknowable.

To the last (1900) edition of _First Principles_ was added a "Postscript
to Part I," in which the justice of this simple and well-known criticism
as to the contradiction involved in speaking of an "Unknowable," which
had been often made during the forty odd years in which the various
editions had been on the market, was grudgingly acknowledged as
follows:[89]

"It is doubtless true that saying what a thing is not, is, in some
measure, saying what it is;... Hence it cannot be denied that to affirm
of the Ultimate Reality that it is unknowable is, in a remote way, to
assert some knowledge of it, and therefore involves a contradiction."

The "Postscript" reminds one of the postscript to a certain Irishman's
letter. This Irishman, missing his razors after his return from a visit
to a friend, wrote to his friend, giving precise directions where to
look for the missing razors; but, before posting the letter, added a
postscript to the effect that he had found the razors.

One is tempted to inquire, analogously, what might be, in view of the
Postscript, the point of much of Spencer's Part I. It is, to use De
Morgan's[90] description of the arguments of some who maintain that we
can know nothing about infinity, of the same force as that of the man
who answered the question how long he had been deaf and dumb.

But the best part of the joke against Mr. Spencer is that he, as we
shall see in Chapter XXXVIII, was refuted by a fallacious argument, and
thus mistakenly asserted the validity of the refutation of remarks which
happen to be unsound.

The analogy of the contradiction of Burali-Forti with the contradiction
involved in the notion of an "unknowable" may be set forth as follows.
If A should say to B: "I know things which you never by any possibility
can know," he may be speaking the truth. In the same way, [Greek: ô] may
be said, without contradiction, to transcend all the _finite_ integers.
But if some one else, C, should say: "There are some things which no
human being can ever know anything about," he is talking nonsense.[91]
And in the same way if we succeeded in imagining a number which
transcends _all_ numbers, we have succeeded in imagining the absurdity
of a number which transcends itself.

All the paradoxes of logic (or "the theory of aggregates") are
analogous to the difficulty arising from a man's statement: "I am
lying."[92] In fact, if this is true, it is false, and _vice versa_. If
such a statement is spread out a little, it becomes an amusing hoax or
an epigram. Thus, one may present to a friend a card bearing on both
sides the words: "The statement on the other side of this card is
false"; while the first of the epigrams derived from this principle
seems to have been written by a Greek satirist:[93]

  Lerians are bad; not _some_ bad and some _not_;
  But all; there's not a Lerian in the lot,
  Save Procles, that you could a good man call;--
  And Procles--is a Lerian after all.

This is the original of a well-known epigram by Porson, who remarked
that all Germans are ignorant of Greek metres,

  All, save only Hermann;--
  And Hermann's a German.

FOOTNOTES:

[88] _Md._, N. S., vol. iv., 1895, p. 168.

[89] _First Principles_, 6th ed., 1900, pp. 107-10. The first edition
was published in 1862.

[90] Note on p. 6 of his paper: "On Infinity; and on the Sign of
Equality," _Trans. Camb. Phil. Soc._, vol. xi., part i., pp. 1-45 (read
May 16, 1864).

[91] The assertion of the finitude of a man's mind appears to be
nonsense; both because, if we say that the mind of man is limited we
tacitly postulate an "unknowable," and because, even if the human mind
were finite, there is no more reason against its conceiving the infinite
than there is for a mind to be blue in order to conceive a pair of blue
eyes (cf. De Morgan, _loc. cit._).

[92] Russell, _R. M. M._, vol. xiv., September 1906, pp. 632-3, 640-4.

[93] _The Greek Anthology_, by Lord Neaves (Ancient Classics for English
Readers), Edinburgh and London, 1897, p. 194.




CHAPTER XXXV

MR. SPENCER, THE ATHANASIAN CREED AND THE ARTICLES


When, in what I believe is misleadingly known as "The Athanasian Creed,"
people say "The Father incomprehensible," and so on, they are not
falling into the same error as Mr. Spencer, for the Latin equivalent for
"incomprehensible" is merely "_immensus_," and Bishop Hilsey translated
it more correctly as "immeasurable."[94] It is a regrettable fact that
Dr. Blunt,[95] in his mistaken modesty, has added a note to this passage
that: "Yet it is true that a meaning not intended in the Creed has
developed itself through this change of language, for the Nature of God
is as far beyond the grasp of the mind as it is beyond the possibility
of being contained within local bounds."

Mr. Spencer seems no happier when we compare his statements with those
in the Anglican Articles of Religion. There God is never referred to as
infinite. It is true that His power and goodness are so referred to; but
this deficiency was presumably brought about intentionally, so that
faith might gain in meaning as time went on.

FOOTNOTES:

[94] _A. C. P._, p. 217.

[95] _Ibid._, p. 218.




CHAPTER XXXVI

THE HUMOUR OF MATHEMATICIANS


Brahmagupta's problem[96] appears to be the earliest instance of a kind
of joke which has been much used by mathematicians. For the sake of
giving a certain picturesqueness to the data of problems, and so to
excite that sort of interest which is partly expressed by a smile,
mathematicians have got into the habit of talking, for example, of
monkeys in the form of geometrical points climbing up massless ropes.
Professor P. Stäckel[97] truly remarked that physiological
mechanics--the mechanics of bones, muscles, and so on--is wholly
different from this. There was once a lecturer on mathematics at
Cambridge who used yearly to propound to his pupils a problem in rigid
dynamics which related to the motion of a garden roller supposed to be
without mass or friction, when a heavy and perfectly rough insect walked
round the interior of it in the direction of normal rolling.

Hitherto this has been the only mathematical outlet for the humour of
mathematicians; and those who really had the interests of mathematics at
heart saw with alarm the growing tendency towards scholasticism in
mathematical jokes. Fortunately the discovery of logic by some
mathematicians has removed this danger. Still to many mathematicians
logic is still unknown, and to them--to Professor A. Schoenflies for
example--modern mathematics, owing to its alliance with logic, appears
to be sinking into scholasticism. It is true that the word
"scholasticism" is not used by Professor Schoenflies in any
intentionally precise signification, but merely as a vague epithet of
disapproval, as the word "socialism" is used by the ordinary philistine,
and this would certainly serve as a sufficient excuse. But no excuse is
needed: these opinions are themselves a source of mathematical jokes.

FOOTNOTES:

[96] See Chapter XII.

[97] _Encykl. der math. Wiss._, vol. iv., part i., p. 474.




CHAPTER XXXVII

THE PARADOXES OF LOGIC


We have already[98] referred to the contempt shown by some
mathematicians for exact thought, which they condemn under the name
of "scholasticism." An example of this is given by Schoenflies in
the second part of his publication usually known as the _Bericht
über Mengenlehre_.[99] Here[100] a battle-cry in italics--

  "_Against all resignation, but also against all scholasticism!_"--

found utterance. Later on, Schoenflies[101] became bolder and adopted a
more personal battle-cry, also in italics, and with a whole line to
itself:

  "_For Cantorism but against Russellism!_"

"Cantorism" means the theory of transfinite aggregates and numbers
erected for the most part by Georg Cantor. Shortly speaking, the great
sin of "Russellism" is to have gone too far in the chain of logical
deduction for many mathematicians, who were perhaps, like
Schoenflies,[102] blinded by their rather uncritical love of
mathematics. Thus it comes about that Schoenflies[103] denounces
Russellism as "scholastic and unhealthy." This queer blend of qualities
would surely arouse the curiosity of the most _blasé_ as to what strange
thing Russellism must be.[104]

Schoenflies[105] said that some mathematicians attributed to the logical
paradoxes which have given Russell so much trouble to clear up,
"especially to those that are artificially constructed, a significance
that they do not have." Yet no grounds were given for this assertion,
from which it might be concluded that the rigid examination of any
concept was unimportant. The paradoxes are simply the necessary results
of certain logical views which are currently held, which views do not,
except when they are examined rather closely, appear to contain any
difficulty. The contradiction is not felt, as it happens, by people who
confine their attention to the first few number-classes of Cantor, and
this seems to have given rise to the opinion, which it is a little
surprising to find that some still hold, that cases not usually met
with, though falling under the same concept as those usually met with,
are of little importance. One might just as well maintain that
continuous but not differentiable functions are unimportant because they
are artificially constructed--a term which I suppose means that they do
not present themselves when unasked for. Rather should we say that it is
by the discovery and investigation of such cases that the concept in
question can alone be judged, and the validity of certain theorems--if
they are valid--conclusively proved. That this has been done, chiefly by
the work of Russell, is simply a fact; that this work has been and is
misunderstood by many[106] is regrettable for this reason, among others,
that it proves that, at the present time, as in the days in which
_Gulliver's Travels_ were written, some mathematicians are bad
reasoners.[107]

Nearly all mathematicians agreed that the way to solve these paradoxes
was simply not to mention them; but there was some divergence of opinion
as to how they were to be unmentioned. It was clearly unsatisfactory
merely not to mention them. Thus Poincaré was apparently of opinion that
the best way of avoiding such awkward subjects was to mention that they
were not to be mentioned. But[108] "one might as well, in talking to a
man with a long nose, say: 'When I speak of noses, I except such as are
inordinately long,' which would not be a very successful effort to avoid
a painful topic."

Schoenflies, in his paper of 1911 mentioned above, adopted the
convenient plan of referring these logical difficulties at the root of
mathematics to a department of knowledge which he called "philosophy."
He said[109] of the theory of aggregates that though "born of the
acuteness of the mathematical spirit, it has gradually fallen into
philosophical ways, and has lost to some extent the compelling force
which dwells in the mathematical process of conclusion."

The majority of mathematicians have followed Schoenflies rather than
Poincaré, and have thus adopted tactics rather like those of the March
Hare and the Gryphon,[110] who promptly changed the subject when Alice
raised awkward questions. Indeed, the process of the first of these
creatures of a child's dream is rather preferable to that of
Schoenflies. The March Hare refused to discuss the subject because he
was bored when difficulties arose. Schoenflies would not say that he was
bored--he professed interest in philosophical matters, but simply called
the logical continuation of a subject by another name when he did not
wish to discuss the continuation, and thus implied that he had discussed
the whole subject. Further, Schoenflies would not apparently admit that
the one method of logic could be applied to the solution of both
mathematical and philosophical problems, in so far as these problems are
soluble at all; but the March Hare, shortly before the remark we have
just quoted, rightly showed great astonishment that butter did not help
to cure both hunger and watches that would not go.[111] The judgment of
Schoenflies by which certain apparently mathematical questions were
condemned as "philosophical," rested on grounds as flimsy as those in
the Dreyfus Case, or the Trial in _Wonderland_.[112]

FOOTNOTES:

[98] Chapters VII and XXXVI.

[99] _Die Entwickelung der Lehre von den Punktmannigfaltigkeiten._
Bericht, erstattet der deutschen Mathematiker-Vereinigung, Leipzig,
1908.

[100] _Ibid._, p. 7. The battle-cry is: "_Gegen jede Resignation, aber
auch gegen jede Scholastik!_"

[101] "Ueber die Stellung der Definition in der Axiomatik," _Jahresber,
der deutsch. Math.-Ver._, vol. xx., 1911, pp. 222-5. The battle-cry is
on p. 256 and is: "Für den Cantorismus aber gegen den Russellismus!"

[102] _Ibid._, p. 251. "Es ist also," he exclaims with the eloquence of
emotion and the emotion of eloquence, "nicht die Geringschätzung der
Philosophie, die mich dabei treibt, sondern die Liebe zur
Mathematik;..."

[103] "Ueber die Stellung der Definition in der Axiomatik," _Jahresber,
der deutsch. Math.-Ver._, vol. xx., 1911, p. 251.

[104] [Cf. for this, _M._, vol. xxii., January 1912, pp. 149-58.--ED.]

[105] _Bericht_, 1908, p. 76, note; cf. p. 72.

[106] E.g. in F. Hausdorff's review of Russell's _Principles_ of 1903 in
the _Vierteljahrsschr. für wiss. Philos. und Soziologie_.

[107] [Cf. _M._, vol. xxv., 1915, pp. 333-8.--ED.]

[108] Russell, _A. J. M._, vol. xxx., 1908, p. 226.

[109] _Loc. cit._, p. 222.

[110] See Appendix Q.

[111] See Appendix R.

[112] See Appendix S.




CHAPTER XXXVIII

MODERN LOGIC AND SOME PHILOSOPHICAL ARGUMENTS


The most noteworthy reformation of recent years in logic is the
discovery and development by Mr. Bertrand Russell of the fact that the
paradoxes--of Burali-Forti, Russell, König, Richard, and others--which
have appeared of late years in the mathematical theory of aggregates and
have just been referred to, are of an entirely _logical_ nature, and
that their avoidance requires us to take account of a principle which
has been hitherto unrecognized, and which renders invalid several
well-known arguments in refutation of scepticism, agnosticism, and the
statement of a man that he asserts nothing.

Dr. Whitehead and Mr. Russell say:[113] "The principle which enables us
to avoid illegitimate totalities may be stated as follows: 'Whatever
involves _all_ of a collection must not be one of the collection,' or
conversely: 'If, provided a certain collection had a total, it would
have members only definable in terms of that total, then the said
collection has no total.' We shall call this the 'vicious-circle
principle,' because it enables us to avoid the vicious circles involved
in the assumption of illegitimate totalities. Arguments which are
condemned by the vicious-circle principle will be called 'vicious-circle
fallacies.' Such arguments, in certain circumstances, may lead to
contradictions, but it often happens that the conclusions to which they
lead are in fact true, though the arguments are fallacious. Take, for
example, the law of excluded middle in the form 'all propositions are
true or false.' If from this law we argue that, because the law of
excluded middle is a proposition, therefore the law of excluded middle
is true or false, we incur a vicious-circle fallacy. 'All propositions'
must be in some way limited before it becomes a legitimate totality, and
any limitation which makes it legitimate must make any statement about
the totality fall outside the totality. Similarly the imaginary sceptic
who asserts that he knows nothing and is refuted by being asked if he
knows that he knows nothing, has asserted nonsense, and has been
fallaciously refuted by an argument which involves a vicious-circle
fallacy. In order that the sceptic's assertion may become significant it
is necessary to place some limitation upon the things of which he is
asserting his ignorance; the proposition that he is ignorant of every
member of this collection must not itself be one of the collection.
Hence any significant scepticism is not open to the above form of
refutation."

In fact, the world of things falls into various sets of things of the
same "type." For every propositional function [Greek: ph](_x_) there is
a range of values of _x_ for which [Greek: ph](_x_) has a signification
as a true or a false proposition. Until this theory was brought forward,
there were occasionally discussions as to whether an object which did
not belong to the range of a certain propositional function possessed
the corresponding property or not. Thus, Jevons, in early days,[114] was
of opinion that virtue is neither black nor not-black because it is not
coloured, but rather later[115] he admitted that virtue is not
triangular.[116]

FOOTNOTES:

[113] _Pa. Ma._, p. 40.

[114] _S. o. S._ pp. 36-7.

[115] _E. L. L._, pp. 120-1.

[116] [It may perhaps be added that, some years after Mr. R*ss*ll's
death, Dr. Whitehead stated, in an address delivered in 1916 and
reprinted in his book on _The Organisation of Thought_ (London, 1917, p.
120), that "the specific heat of virtue is 0.003 is, I should imagine,
not a proposition at all, so that it is neither true nor
false...."--ED.]




CHAPTER XXXIX

THE HIERARCHY OF JOKES


Jokes may be divided into various types. Thus a joke or class of jokes
can only be the subject of a joke of higher order. Otherwise we would
get the same vicious-circle fallacy which gives rise to so many
paradoxes in logic and mathematics. A certain Oxford scholar succeeded,
to his own satisfaction, in reducing all jokes to primitive types,
consisting of thirty-seven proto-Aryan jokes. When any proposition was
propounded to him, he would reflect and afterwards pronounce on the
question as to whether the proposition was a joke or not. If he decided,
by his theory, that it was a joke, he would solemnly say: "There _is_
that joke." If this narration is accepted as a joke, since it cannot be
reduced to one of the proto-Aryan jokes under pain of leading us to
commit a vicious-circle fallacy, we must conclude that there is at least
one joke which is not proto-Aryan; and, in fact, is of a higher type.
There is no great difficulty in forming a hierarchy of jokes of various
types. Thus a joke of the fourth type (or order) is as follows: A joke
of the first order was told to a Scotchman, who, as we would expect, was
unable to see it.[117] The person (A) who told this joke told the story
of how the joke was received to another Scotchman thereby making a joke
about a joke of the first order, and thus making a joke of the second
order. A remarked on this joke that no joke could penetrate the head of
the Scotchman to whom the joke of the first order was told, even if it
were fired into his head with a gun. The Scotchman, after severe
thought, replied: "But ye couldn't do that, ye know!" A repeated the
whole story, which constituted a joke of the third order, to a third
Scotchman. This last Scotchman again, after prolonged thought, replied:
"He had ye there!" This whole story is a joke of the fourth order.

Most known jokes are of the first order, for the simple reason that the
majority of people find that the slightest mental effort effectually
destroys any perception of humour. It seems to me that a joke becomes
more pleasurable in proportion as logical faculties are brought into
play by it; and hence that logical power is allied, or possibly
identical, with the power of grasping more subtle jokes. The jokes which
amuse the frequenters of music-halls, Conservatives, and M. Bergson--and
which usually deal with accidents, physical defects, mothers-in-law,
foreigners, or over-ripe cheese--are usually jokes of the first order.
Jokes of the second, and even of the third, order appeal to ordinary
well-educated people; jokes of higher order require either special
ability or a sound logical training on the part of the hearer if the
joke is to be appreciated; while jokes of transfinite order presumably
only excite the inaudible laughter of the gods.

FOOTNOTES:

[117] [It may be that, like certain remarks about cheese and
mothers-in-law (see below), the statement that Scotchmen cannot see
jokes is a joke of the first order.--ED.]




CHAPTER XL

THE EVIDENCE OF GEOMETRICAL PROPOSITIONS


It has often been maintained that the twentieth proposition of the first
book of Euclid--that two sides of a triangle are together greater than
the third side--is evident even to asses. This does not, however, seem
to me generally true. I once asked a coastguardsman the distance from A
to B; he replied: "Eight miles." On further inquiry I elicited the fact
that the distance from A to C was two miles and the distance from C to B
was twenty-two miles. Now the paths from A to B and from C to B were by
sea; while the path from A to C was by land. Hence if the path by land
was rugged and the distance along the road was two miles, it would
appear that the coastguardsman believed that not only could one side of
a triangle be greater than the other two, but that one straight side of
a triangle might be greater than one straight side and any curvilinear
side of the same triangle. The only escape from part of this astonishing
creed would be by assuming that the distance of two miles from A to C
was measured "as the crow flies," while the road A to C was so hilly
that a pedestrian would traverse more than fourteen miles when
proceeding from A to C. Then indeed the coastguardsman could maintain
the true proposition that there is at least one triangle ABC, with the
side AC curvilinear, such that the sum of the lengths of AB and AC is
greater than the length of BC, and only deny the twentieth proposition
of the first book of Euclid.

Reasoning with the coastguardsman only had the effect of his adducing
the authority of one Captain Jones in support of the accuracy of his
data. Possibly Captain Jones held strange views as to the influence of
temperature or other physical circumstances, or even the nature of space
itself, on the lengths of lines in the neighbourhood of the triangle
ABC.




CHAPTER XLI

ABSOLUTE AND RELATIVE POSITION


Some people maintain that position in space or time must be relative
because, if we try to determine the position of a body A, if bodies B,
C, D with respect to which the position of A could be determined were
not present, we should be trying to determine something about A without
having our senses affected by other things. These people seem to me to
be like the cautious guest who refused to say anything about his host's
port-wine until he had tasted red ink.

"Wherein, then," says Mr. Russell,[118] "lies the plausibility of the
notion that all points are exactly alike? This notion is, I believe, a
psychological illusion, due to the fact that we cannot remember a point
so as to know it when we meet again. Among simultaneously presented
points it is easy to distinguish; but though we are perpetually moving,
and thus being brought among new points, we are quite unable to detect
this fact by our senses, and we recognize places only by the objects
they contain. But this seems to be a mere blindness on our part--there
is no difficulty, so far as I can see, in supposing an immediate
difference between points, as between colours, but a difference which
our senses are not constructed to be aware of. Let us take an analogy:
Suppose a man with a very bad memory for faces; he would be able to
know, at any moment, whether he saw one face or many, but he would not
be aware whether he had seen any of the faces before. Thus he might be
led to define people by the rooms in which he saw them, and to suppose
it self-contradictory that new people should come to his lectures, or
that old people should cease to do so. In the latter point at least it
will be admitted by lecturers that he would be mistaken. And as with
faces, so with points--inability to recognize them must be attributed,
not to the absence of individuality, but merely to our incapacity."

Another form of this tendency is shown by Kronecker, Borel, Poincaré,
and many other mathematicians, who refuse mere logical determination of
a conception and require that it be actually described in a finite
number of terms. These eminent mathematicians were anticipated by the
empirical philosopher who would not pronounce that the "law of thought"
that A is either in the place B or not is true until he had looked to
make sure. This philosopher was of the same school as J. S. Mill and
Buckle, who seem to have maintained implicitly not only that, in view of
the fact that the breadth of a geometrical line depends upon the
material out of which it is constructed, or upon which it is drawn, that
there ought to be a paste-board geometry, a stone geometry, and so
on;[119] but also that the foundations of logic are inductive in their
nature.[120] "We cannot," says Mill,[121] "conceive a round square, not
merely because no such object has ever presented itself in our
experience, for that would not be enough. Neither, for anything we know,
are the two ideas in themselves incompatible. To conceive a body all
black and yet white would only be to conceive two different sensations
as produced in us simultaneously by the same object--a conception
familiar to our experience--and we should probably be as well able to
conceive a round square as a hard square, or a heavy square, if it were
not that in our uniform experience, at the instant when a thing begins
to be round, it ceases to be square, so that the beginning of the one
impression is inseparably associated with the departure or cessation of
the other. Thus our inability to form a conception always arises from
our being compelled to form another contradictory to it."

FOOTNOTES:

[118] _Md._, N. S., vol. x., July, 1901, pp. 313-14.

[119] J. B. Stallo, _The Concepts and Theories of Modern Physics_, 4th
ed., London, 1900, pp. 217-27.

[120] _Ibid._, pp. 140-4.

[121] _Examination of the Philosophy of Sir William Hamilton_, vol. i.
p. 88, Amer. ed.




CHAPTER XLII

LAUGHTER


[It seemed advisable to give here[122] some views on laughter, most of
which were also held by Mr. R*ss*ll, though no written expression of his
views has yet been found. In a review[123] of M. Bergson's book on
_Laughter_,[124] Mr. Russell has remarked:

"It has long been recognized by publishers that everybody desires to be
a perfect lady or gentleman (as the case may be); to this fact we owe
the constant stream of etiquette-books. But if there is one thing which
people desire even more, it is to have a faultless sense of humour. Yet
so far as I know, there is no book called 'Jokes without Tears, by Mr.
McQuedy.' This extraordinary lacuna has now been filled. Those to whom
laughter has hitherto been an unintelligible vagary, in which one must
join, though one could never tell when it would break out, need only
study M. Bergson's book to acquire the finest flower of Parisian wit. By
observing a very simple formula they will know infallibly what is funny
and what is not; if they sometimes surprise their unlearned friends,
they have only to mention their authority in order to silence doubt.
'The attitudes, gestures and movements of the human body,' says M.
Bergson, 'are laughable in exact proportion as that body reminds us of a
mere machine.' When an elderly gentleman slips on a piece of orange-peel
and falls, we laugh, because his body follows the laws of dynamics
instead of a human purpose. When a man falls from a scaffolding and
breaks his neck on the pavement, we presumably laugh even more, since
the movement is even more completely mechanical. When the clown makes a
bad joke for the first time, we keep our countenance, but at the fifth
repetition we smile, and at the tenth we roar with laughter, because we
begin to feel him a mere automaton. We laugh at Molière's misers,
misanthropists and hypocrites, because they are mere types mechanically
dominated by a master impulse. Presumably we laugh at Balzac's
characters for the same reason; and presumably we never smile at
Falstaff, because he is individual throughout."

The review concludes with the reflection that "it would seem to be
impossible to find any such formula as M. Bergson seeks. Every formula
treats what is living as if it were mechanical, and is therefore by his
own rule a fitting object of laughter." Now, this undoubtedly true
conclusion has been obtained, as is readily seen, by a vicious-circle
fallacy which Mr. R*ss*ll would hardly have committed.--ED.]

FOOTNOTES:

[122] From a remark on p. 47 above, it is evident that Mr. R*ss*ll
intended to write some such chapter as this.

[123] _The Professor's Guide to Laughter, The Cambridge Review_, vol.
xxxii., 1912, pp. 193-4.

[124] _Laughter, an Essay on the Meaning of the Comic_, English
translation by C. Brereton and F. Rothwell, London, 1911.




CHAPTER XLIII

"GEDANKENEXPERIMENTE" AND EVOLUTIONARY ETHICS


The "Gedankenexperimente," upon which so much weight has been laid by
Mach[125] and Heymans,[126] had already been investigated by the White
Queen,[127] who, however, seems to have perceived that the results of
such experiments are not always logically valid. The psychological
founding of logic appears to be not without analogy with the surprising
method of advocates of evolutionary ethics, who expect to discover what
_is_ good by inquiring what cannibals have _thought_ good. I sometimes
feel inclined to apply the historical method to the multiplication
table. I should make a statistical inquiry among school-children, before
their pristine wisdom had been biassed by teachers. I should put down
their answers as to what 6 times 9 amounts to, I should work out the
average of their answers to six places of decimals, and should then
decide that, at the present stage of human development, this average is
the value of 6 times 9.

FOOTNOTES:

[125] See, e.g., _E. u. I._, pp. 183-200.

[126] _G. u. E._, vol. i.

[127] See Appendix T.




APPENDIXES


A. LOGIC AND THE PRINCIPLE OF IDENTITY.

_T. L. G._, p. 45: "'Contrariwise," continued Tweedledee, "if it was so,
it might be; and if it were so, it would be: but as it isn't, it ain't.
That's logic."

       *       *       *       *       *

_S. B._, p. 159: The Professor said: "The day is the same length as
anything that is the same length as _it_."

       *       *       *       *       *

_S. B._, p. 161: Bruno observed that, when the Other Professor lost
himself, he should shout: "He'd be sure to hear hisself, 'cause he
couldn't be far off."


B. SYNTHESIS OF CONTRADICTORIES.

_T. L. G._, p. 71: "'What a beautiful belt you've got on!' Alice
suddenly remarked.... 'At least,' she corrected herself on second
thoughts, 'a beautiful cravat, I should have said--no, a belt, I mean--I
beg your pardon!' she added in dismay, for Humpty-Dumpty looked
thoroughly offended, and she began to wish she hadn't chosen that
subject. 'If only I knew,' she thought to herself, 'which was neck and
which was waist!'"


C. EMPIRICAL PHILOSOPHERS AND MATHEMATICS.

_T. L. G._, p. 79: "'... Now if you had the two eyes on the same side of
the nose, for instance--or the mouth at the top--that would be _some_
help.'

"'It wouldn't look nice,' Alice objected. But Humpty-Dumpty only shut
his eyes and said: 'Wait till you've tried.'"

       *       *       *       *       *

_T. L. G._, p. 72: "'And if you take one from three hundred and
sixty-five, what remains?'

"'Three hundred and sixty-four, of course.'

"Humpty-Dumpty looked doubtful. 'I'd rather see that done on paper,'
he said."


D. NOMINAL DEFINITION.

_T. L. G._, p. 73: "'When _I_ used a word,' Humpty-Dumpty said in rather
a scornful tone, 'it means just what I choose it to mean--neither more
nor less.'

"'The question is,' said Alice, 'whether you _can_ make words mean
different things.'

"'The question is,' said Humpty-Dumpty, 'which is to be master--that's
all.'"


E. CONFORMITY OF A PARADOXICAL LOGIC WITH COMMON-SENSE.

_T. L. G._, p. 100:

  "But I was thinking of a plan
  To dye one's whiskers green,
  And always use so large a fan
  That they could not be seen."
                                   (Verse from White Knight's song.)


F. IDEALISTS AND THE LAWS OF LOGIC.

_T. L. G._, p. 52-3: Tweedledee exclaimed: "'... if he [the Red King]
left off dreaming about you [Alice], where do you suppose you'd be?'

"'Where I am now, of course,' said Alice.

"'Not you!' Tweedledee retorted contemptuously. 'You'd be nowhere. Why,
you're only a sort of thing in his dream!'

"'If that there King was to wake,' added Tweedledum, 'you'd go
out--bang!--just like a candle!'

"'I shouldn't!' Alice exclaimed indignantly. 'Besides, if _I'm_ only a
sort of thing in his dream, what are _you_, I should like to know?'

"'Ditto,' said Tweedledum...; 'you know very well you're not real.'

"'I _am_ real!' said Alice, and began to cry."

       *       *       *       *       *

_T. L. G._, p. 97: "'How _can_ you go on talking so quietly, head
downwards?' Alice asked, as she dragged him out by the feet, and laid
him in a heap on the bank.

"The Knight looked surprised at the question. 'What does it matter where
my body happens to be?' he said. 'My mind goes on working all the same.
In fact, the more head downwards I am, the more I keep inventing new
things.'"

       *       *       *       *       *

_T. L. G._, p. 98: "'... Everybody that hears me sing--either it brings
the _tears_ into their eyes, or else----'

"'Or else what?' said Alice, for the Knight had made a sudden pause.

"'Or else it doesn't, you know.'"


G. DISTINCTION BETWEEN SIGN AND SIGNIFICATION.

_T. L. G._, pp. 98-9: "'The name of the song is called "_Haddocks'
Eyes_."'

"'Oh, that's the name of the song, is it?' Alice said, trying to feel
interested.

"'No, you don't understand,' the Knight said looking a little vexed.
'That's what the name is _called_. The name really _is_ "_The Aged Aged
Man_."'

"'Then I ought to have said "That's what the _song_ is called"?' Alice
corrected herself.

"'No, you oughtn't: that's another thing. The _song_ is called "_Ways
and Means_": but that's only what it's _called_, you know!'

"'Well, what _is_ the song, then?' said Alice, who was by this time
completely bewildered.

"'I was coming to that,' the Knight said. 'The song really _is
"A-sitting on a Gate_"....'"


H. NOMINALISM.

_A. A. W._, p. 70: "'Then you should say what you mean,' the March Hare
went on.

"'I do,' Alice hastily replied; 'at least--at least I mean what I
say--that's the same thing, you know.'

"'Not the same thing a bit!' said the Hatter. 'Why, you might just as
well say that "I see what I eat" is the same thing as "I eat what I
see."'

"'You might just as well say,' added the March Hare, 'that "I like what
I get" is the same thing as "I get what I like"!'

"'You might just as well say,' added the Dormouse, which seemed to be
talking in its sleep, 'that "I breathe when I sleep" is the same as "I
sleep when I breathe"!'

"'It _is_ the same thing with you,' said the Hatter; and here the
conversation dropped,..."


I. UTILITY OF SYMBOLIC LOGIC.

_A. A. W._, p. 92: "'I quite agree with you,' said the Duchess, 'and the
moral of that is--"Be what you would seem to be"--or if you'd like it
put more simply--"Never imagine yourself not to be otherwise than what
it might appear to others that what you were or might have been was not
otherwise than what you had been would have appeared to them to be
otherwise."'

"'I think I should understand that better,' Alice said very politely,
'if I had it written down: but I can't quite follow it as you say it.'

"'That's nothing to what I could say if I chose,' the Duchess replied,
in a pleased tone."


J. MISTAKE AS TO THE NATURE OF CRITICISM.

_T. L. G._, p. 105: "'She's in that state of mind,' said the White
Queen, 'that she wants to deny _something_--only she doesn't know what
to deny.'

"'A nasty, vicious temper,' the White Queen remarked; and then there was
an uncomfortable silence for a minute or two."


K. A CRITERION OF TRUTH.

_H. S._, p. 3:

  "Just the place for a Snark! I have said it twice:
  That alone should encourage the crew.
  Just the place for a Snark! I have said it thrice:
  What I tell you three times is true."

       *       *       *       *       *

_H. S._, p. 50:

  "'Tis the note of the Jubjub! Keep count. I entreat;
  You will find I have told it you twice.
  'Tis the song of the Jubjub! The proof is complete,
  If only I've stated it thrice."


L. UNIVERSAL AND PARTICULAR PROPOSITIONS.

_T. L. G._, p. 40: The Gnat had told Alice that the Bread-and-butterfly
lives on weak tea with cream in it; so:

"'Supposing it couldn't find any?' she suggested.

"'Then it would die, of course.'

"'But that must happen very often,' Alice remarked thoughtfully.

"'It always happens,' said the Gnat."


M. DENOTING.

_T. L. G._, p. 43: Tweedledum and Tweedledee were, in many respects,
indistinguishable, and Alice, walking along the road, noticed that
"whenever the road divided there were sure to be two finger-posts
pointing the same way, one marked 'TO TWEEDLEDUM'S HOUSE' and the other
'TO THE HOUSE OF TWEEDLEDEE.'

"'I do believe,' said Alice at last, 'that they live in the same
house!...'"


N. NON-ENTITY.

_T. L. G._, p. 87: "'I always thought they [human children] were
fabulous monsters!' said the Unicorn....

"'Do you know [said Alice], I always thought Unicorns were fabulous
monsters, too! I never saw one alive before!'

"'Well, now that we _have_ seen each other,' said the Unicorn, 'if
you'll believe in me, I'll believe in you. Is that a bargain?'"

       *       *       *       *       *

_T. L. G._, pp. 80-1: "'I see nobody on the road,' said Alice.

"'I only wish _I_ had such eyes,' the [White] King remarked in a fretful
tone. 'To be able to see Nobody! And at that distance, too! Why, it's as
much as _I_ can do to see real people by this light!'"

       *       *       *       *       *

_A. A. W._, p. 17: "And she [Alice] tried to fancy what the flame of a
candle looks like after the candle is blown out, for she could not
remember ever having seen such a thing."

       *       *       *       *       *

_A. A. W._, p. 68: "... This time it [the Cheshire Cat] vanished quite
slowly, beginning with the end of the tail, and ending with the grin,
which remained some time after the rest of it had gone.

"'Well! I've often seen a cat without a grin,' thought Alice; 'but a
grin without a cat! It's the most curious thing I ever saw in all my
life!'"

       *       *       *       *       *

_A. A. W._, p. 77: "... The Dormouse went on,...; 'and they drew all
manner of things--everything that begins with an M.'

"'Why with an M?' said Alice.

"'Why not?' said the March Hare.

"Alice was silent.

"... [The Dormouse] went on: '--that begins with an M, such as
mouse-traps, and the moon, and memory, and muchness, you know you say
things are "much of a muchness"--did you ever see such a thing as a
drawing of a muchness?'

"'Really, now you ask me,' said Alice, very much confused, 'I don't
think----'

"'Then you shouldn't talk,' said the Hatter."


O. OBJECTS OF MATHEMATICAL LOGIC.

_T. L. G._, p. 93: "'I was wondering what the mouse-trap [fastened to
the White Knight's saddle] was for,' said Alice. 'It isn't very likely
there would be any mice on the horse's back.'

"'Not very likely, perhaps,' said the Knight, 'but, if they _do_ come, I
don't choose to have them running all about.'

"'You see,' he went on after a pause, 'it's as well to be provided for
_everything_. That's the reason the horse has all these anklets round
his feet.'

"'But what are they for?' Alice asked in a tone of great curiosity.

"'To guard against the bites of sharks,' the Knight replied."


P. THE PRINCIPLE OF PERMANENCE.

_T. L. G._, p. 106: "'Can you do Subtraction? [said the Red Queen] Take
nine from eight.'

"'Nine from eight I can't, you know,' Alice replied very readily 'but--'

"'She can't do Substraction,' said the White Queen."

       *       *       *       *       *

_A. A. W._, p. 56: [Said the Pigeon to Alice]: "'... No, no! You're a
serpent; and there's no use denying it. I suppose you'll be telling me
next that you never tasted an egg!'

"'I _have_ tasted eggs certainly,' said Alice, who was a very truthful
child; 'but little girls eat eggs quite as much as serpents do, you
know.'

"'I don't believe it,' said the Pigeon; 'but if they do, why then
they're a kind of serpent, that's all I can say.'

"This was such a new idea to Alice, that she was quite silent for a
minute or two, which gave the Pigeon the opportunity of adding, 'You're
looking for eggs, I know _that_ well enough; and what does it matter to
me whether you're a little girl or a serpent?'

"'It matters a good deal to _me_,' said Alice hastily;..."

       *       *       *       *       *

_A. A. W._, p. 75: "'But why [asked Alice] did they live at the bottom
of a well?'

"'Take some more tea,' the March Hare said to Alice, very earnestly.

"'I've had nothing yet,' Alice replied in an offended tone, 'so I can't
take more.'

"'You mean you can't take _less_,' said the Hatter: 'it's very easy to
take _more_ than nothing.'"


Q. MATHEMATICIANS' TREATMENT OF LOGIC.

_A. A. W._, p. 74: The Hatter had told of his quarrel with Time, and of
Time's refusal now to do anything he asked: "'... It's always six
o'clock now!'

"A bright idea came into Alice's head. 'Is that the reason so many tea
things are put out here?' she asked.

"'Yes, that's it,' said the Hatter, with a sigh: 'it's always tea time,
and we've no time to wash the things between whiles.'

"'Then you keep moving round, I suppose?' said Alice.

"'Exactly so,' said the Hatter: 'as the things get used up.'

"'But what happens when you come to the beginning again?' Alice ventured
to ask.

"'Suppose we change the subject,' the March Hare interrupted, yawning.
'I'm getting tired of this.'"

       *       *       *       *       *

_A. A. W._, p. 99: "'And how many hours a day did you do lessons?' said
Alice, in a hurry to change the subject.

"'Ten hours the first day,' said the Mock Turtle, 'nine the next, and so
on.'

"'What a curious plan!' exclaimed Alice.

"'That's the reason they're called lessons,' the Gryphon remarked,
'because they lessen from day to day.'

"This was quite a new idea to Alice, and she thought it over a little
before she made her next remark. 'Then the eleventh day must have been a
holiday.'

"'Of course it was,' said the Mock Turtle.

"'And how did you manage on the twelfth?' Alice went on eagerly.

"'That's enough about lessons,' the Gryphon interrupted in a very
decided tone...."


R. METHOD IN MATHEMATICS AND LOGIC.

_A. A. W._, p. 71: "'Two days wrong!' sighed the Hatter. 'I told you
butter wouldn't suit the works!' he added, looking angrily at the March
Hare.

"'It was the _best_ butter,' the March Hare meekly replied.

"'Yes, but some crumbs must have got in as well,' the Hatter grumbled;
'you shouldn't have put it in with the bread-knife.'

"The March Hare took the watch and looked at it gloomily: then he dipped
it into his cup of tea, and looked at it again: but he could think of
nothing better to say than his first remark, 'It was the _best_ butter,
you know.'"


S. VERDICT THAT LOGIC IS PHILOSOPHY.

_A. A. W._, pp. 119-23: "... 'Consider your verdict,' he [the King] said
to the jury, in a low trembling voice.

"'There's more evidence to come yet, please your Majesty,' said the
White Rabbit, jumping up in a great hurry: 'this paper has just been
picked up.'

"'What's in it?' said the Queen.

"'I haven't opened it yet,' said the White Rabbit, 'but it seems to be a
letter written by the prisoner to--to somebody.'

"'It must have been that,' said the King, 'unless it was written to
nobody, which isn't usual, you know.'

"'Who is it directed to?' said one of the jurymen.

"'It isn't directed at all,' said the White Rabbit, 'in fact there's
nothing written on the _outside_.' He unfolded the paper as he spoke,
and added, 'It isn't a letter, after all: it's a set of verses.'

"'Are they in the prisoner's handwriting?' asked another of the jurymen.

"'No they're not,' said the White Rabbit, 'and that's the queerest thing
about it.' (The jury all looked puzzled).

"'He must have imitated somebody else's hand,' said the King. (The jury
brightened up again.)

"'Please your Majesty,' said the Knave, 'I didn't write it, and they
can't prove that I did: there's no name signed at the end.'

"'If you didn't sign it, said the King, that only makes the matter
worse. You _must_ have meant some mischief, or else you'd have signed
your name like an honest man.'

"There was a general clapping of hands at this: it was the first really
clever thing the King had said that day.

"'That _proves_ his guilt, of course,' said the Queen, 'so, off
with----'

"'It doesn't prove anything of the sort!' said Alice. 'Why, you don't
even know what they're about!'

"'Read them,' said the King.

"The White Rabbit put on his spectacles. 'Where shall I begin, please
your Majesty?' he asked.

"'Begin at the beginning,' the King said very gravely, 'and go on till
you come to the end: then stop.'

"There was dead silence in the court, whilst the White Rabbit read out
these verses:

  "'_They told me you had been to her,
  And mentioned me to him;
  She gave me a good character,
  But said I could not swim._

  _He sent them word I had not gone
  (We know it to be true):
  If she should push the matter on,
  What would become of you?_

  _I gave her one, they gave him two,
  You gave us three or more;
  They all returned from him to you,
  Though they were mine before._

  _If I or she should chance to be
  Involved in this affair,
  He trusts to you to set them free
  Exactly as they were._

  _My notion was that you had been
  (Before she had this fit)
  An obstacle that came between
  Him, and ourselves, and it._

  _Don't let him know she liked them best,
  For this must ever be
  A secret kept from all the rest,
  Between yourself and me._'

"'That's the most important piece of evidence we've heard yet,' said the
King, rubbing his hands, 'so now let the jury----'

"'If any one of them can explain it,' said Alice (she had grown so large
in the last few minutes that she wasn't a bit afraid of interrupting
him), 'I'll give him sixpence. _I_ don't believe there's an atom of
meaning in it.'

"The jury all wrote down on their slates, 'She doesn't believe there's
an atom of meaning in it,' but none of them attempted to explain the
paper.

"'If there's no meaning in it,' said the King, 'that saves a world of
trouble, you know, as we needn't try to find any. And yet I don't know,'
he went on, spreading out the verses on his knee and looking at them
with one eye; 'I seem to see some meaning in them after all. "_-- said
I could not swim_"; you can't swim, can you?' he added, turning to the
Knave.

"The Knave shook his head sadly. 'Do I look like it?' he said. (Which he
certainly did _not_, being made entirely of cardboard.)

"'All right, so far,' said the King; and he went on muttering over the
verses to himself: ''_We know it to be true_'--that's the jury, of
course--'_If she should push the matter on_'--that must be the
Queen--'_What would become of you?_' What indeed!--'_I gave her one,
they gave him two!_' why, that must be what he did with the tarts, you
know----'

"'But it goes on, '_They all returned from him to you_,'' said Alice.

"'Why, there they are!' said the King, triumphantly pointing to the
tarts on the table. 'Nothing can be clearer than that. Then
again--'_Before she had this fit_'--you never had fits, my dear, I
think?' he said to the Queen.

"'Never!' said the Queen furiously, throwing an inkstand at the Lizard
as she spoke. (The unfortunate little Bill had left off writing on his
slate with one finger, as he found it made no mark; but he now hastily
began again, using the ink that was trickling down his face, as long as
it lasted.)

"'Then the words don't _fit_ you,' said the King, looking round the
court with a smile. There was a dead silence.

"'It's a pun!' the King added in an angry tone, and everybody laughed.

"'Let the jury consider their verdict,' the King said, for about the
twentieth time that day.

"'No, no!' said the Queen. 'Sentence first--verdict afterwards.'

"'Stuff and nonsense!' said Alice loudly. 'The idea of having the
sentence first!'

"'Hold your tongue!' said the Queen, turning purple...."


T. "GEDANKENEXPERIMENTE."

_T. L. G._, p. 61: "Alice laughed. 'There's no use trying,' she said:
'one _can't_ believe impossible things.'

"'I daresay you haven't had much practice,' said the [White] Queen.
'When I was your age, I always did it for half-an-hour a day. Why,
sometimes I've believed as many as six impossible things before
breakfast.'"


_Printed in Great Britain by_

UNWIN BROTHERS, LIMITED, THE GRESHAM PRESS, WOKING AND LONDON





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