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129752
AN INTRODUCTION TO
WITTGENSTEIN'S
TRACTATUS
f
A reference-list of Harper Torchbooks, classified
by subjects, is printed at the end of this volume.
AN INTRODUCTION TO
WITTGENSTEIN'S
TRACTATUS
Second Edition, Revised
G. E. M. ANSCOMBE
HARPER TORCHBOOKS T The Academy Library
Harper & Row, Publishers, New York
AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
Copyright G. E. M. Anscombe 1959
Printed in the United States of America
This book was originally published in 1959 in the Philosophy divi-
sion, edited by Professor H. J. Paton, of the Hutchinson University
Library, with a second edition, revised, in 1963. It is here reprinted
by arrangement.
First HARPER TORCHBOOK edition published 1965 by
Harper & Row, Publishers, Incorporated
49 East 3 3rd Street
New York, N. Y. 10016
CONTENTS
Foreword by Professor H. J. Paton 9
Note on Second Edition 10
Introduction 11
Glossary 21
1 Elementary Propositions 25
2 The Theory of Descriptions 41
3 Negation: (1) The Logicians* Definition of 'not p' 51
4 Negation: (2) The Picture Theory 64
5 Consequences of the Picture Theory 79
6 Sign and Symbol 87
7 Wittgenstein, Frege and Ramsey 98
8 Operations 113
9 Formal Concepts and Formal Series 122
"The General Form of Proposition
9
10 132
11 Generality 138
12 Knowledge and Certainty 150
13 'Mysticism' and Solipsism 161
Further Reading 174
Index
ACKNOWLEDGMENTS
Most of what appears here was originally delivered in the form of
lectures at Oxford in the Michaelmas and Hilary Terms, 1957-8.
I am indebted, first, to the Oxford audiences who in successive years
provided me with the occasion for getting my ideas of the Tractatus
straight; and, secondly, to Professor Pat on, the Editor of this series,
who made many useful criticisms of my exposition. I am also much
indebted to Peter Geach for help afforded in frequent discussions
and in revising the book. In particular, I obtained some logical
information and references from him; for example, he supplied me
with the explanations given in Chapter 9 of Wittgenstein's criti-
cisms of Principia Mathematica, concerning the 'ancestral relation"
and the unacknowledged use of 'formal series' made by Russell and
Whitehead. Finally, I have had the advantage of reading through
Professor Erik Stenius' highly interesting book on the Tractatus
before its publication. It presents a very different account from my
own of some important points, and enabled me to sharpen my own
ideas by contrast.
I am grateful to the Rockefeller Foundation, which has suppor-
ted me during six years* work on Wittgenstein's MSS. ; this book was
written during my tenure of a Research Fellowship sponsored by
the Foundation.
Oxford G, E. M. ANSCOMBE
FOREWORD
An Introduction to Wittgenstein's Tractatus may seem to differ in
character from the books hitherto published in the present philo-
sophical series: it makes a freer use of technical expressions, and it
demands a greater effort from the general reader. This is inevitable
from the very nature of the subject; for Wittgenstein's thinking
sprang from the modern development of mathematical logic; and
he makes few concessions to the uninitiated. Indeed, it is for this
reason that an introduction to his work is so urgently required; and
with its aid the reader who has grasped some elementary techniques
of modern logic (which should not prove too difficult) may hope to
find Ms way about in what to many has been a closed field of
thought. The opening up of such a field is fully in accord with the
general aim of the series, even if the difficulties to be tackled may be
greater than usual. It might almost be said, in view of the influence
exercised by Wittgenstein, that without an introduction of this kind
the aim of the series would be imperfectly realized.
To this it may be objected that the Tractatus has now been
superseded. In one sense this is true; for its author has expressly
recognized that it contains grave mistakes. But the Tractatus by no
means loses its historical importance because of this frank ad-
mission. As a philosophical work begins to recede into history, an
attempt to expound its doctrine and estimate its significance may
become both more necessary and more easy. Wittgenstein himself
was of the opinion that his more recent works could be seen in the
right light only by contrast with, and against the background of,
his earlier way of thinking. Hence it is my hope that Miss
Anscombe's work may serve as an introduction not merely to the
Tractatus, but also indirectly to his philosophy as a whole.
H. J. PATON
NOTE ON SECOND EDITION
Apart from the correction of some obvious mistakes and misprints,
and some other slight improvements of the text, 1 have taken the
opportunity to add a paragraph to the end of Chapter 10 and to
change pages 166-7.
G. E. M. ANSCOM3E
INTRODUCTION
Ludwig Wittgenstein was born in Vienna in 1889, the eighth child
of a family largely Jewish by extraction, though not by persuasion.
His father had started the modern iron and steel industry in Austria;
he was a man of forcible character who was generally both much
feared and much respected. The children were brought up in an
atmosphere of extreme contempt for most kinds of low standard.
The whole generation had an unusual fire about them. All were
aesthetically and, in particular, musically talented to a high degree;
the father, however, though sharing such interests up to a point,
regarded them as suited only to be a side-line for his sons: the only
fit career for them was civil engineering. (It had to be concealed
from him that one of them as a child played the violin in St. Peter's
Church in Vienna.) The combination of family temperament and
the attitude of the parents who could not conceive that their
children might suffer miseries worth taking account of led to
at least one suicide among the sons. Of himself, Ludwig said: *I had
an unhappy childhood and a most miserable youth'; but again, in
connection with the work that a man was content to publish: *I had
luck: I was very well brought up' i.e. as far as concerned critical
standards.
Ludwig came to Manchester at the age of about nineteen to
study engineering; but by 1911 his interest had been caught by the
philosophy of mathematics, and he went to Cambridge to study
under Russell. From 1912 to 1917 he was engaged in writing the
work which is the subject of this book. The greater part of the writing
was done while he was on active service in the Austrian army.
As a boy of sixteen Wittgenstein had read Schopenhauer and
had been greatly impressed by Schopenhauer's theory of the 'world
as idea' (though not of the 'world as will'); Schopenhauer then
struck him as fundamentally right, if only a few adjustments and
11
12 INTRODUCTION
clarifications were made. It is very much a popular notion of Witt-
genstein that he was a latter-day Hume; but any connections be-
tween them are indirect, and he never read more than a few pages
of Hume. If we look for Wittgenstein's philosophical ancestry, we
should rather look to Schopenhauer; specifically, his 'solipsism',
his conception of 'the limit' and his ideas on value will be better
understood in the light of Schopenhauer than of any other philo-
sopher. It is one of the oddities of the present day that Schopenhauer
isoften vaguely associated with Nietzsche and even with Nazism,
and is thought to be some kind of immoralist, worshipper of power
and praiser of suicide; it is not the mythical Schopenhauer of
popular repute, but the actual Schopenhauer, that we should
remember in connection with Wittgenstein.
For the rest, Wittgenstein's philosophical influences are pretty
well confined to Frege and to Russell, who introduced him to
Frege's writings. His relative estimate of them comes out in the
acknowledgment he makes in the Preface to the Tractatus: 'I owe a
great part of the stimulation of my thoughts to the great works of
Frege and to the writings of my friend Mr. Bertrand RusselL'
Frege, whose first important work was published in 1879 and
who died in 1925, was a philosopher whose greatness, up to the
present day, was acknowledged only by a very few. Russell and
Wittgenstein were the most eminent of those who recognized it. He
was not a general philosopher and had no concern with either ethics
or theory of knowledge, but purely with logic and the foundations
of mathematics; here however his considerations, being always
concerned with fundamentals, are of the greatest general philo-
sophical importance. I devote a good deal of space to Frege in this
book for the following reason Wittgenstein's Tractatus has captured
:
the interest and excited the admiration of many, yet almost all that
has been published about it has been wildly irrelevant. If this has
had any one cause, that cause has been the neglect of Frege and of
the new direction that he gave to philosophy. In the Tractatus
Wittgenstein assumes, and does not try to stimulate, an interest in
the kind of questions that Frege wrote about; he also takes it for
granted that his readers will have read Frege.
Now, empiricist or idealist preconceptions, such as have been
INTRODUCTION 13
most common in philosophy for a long time, are a thorough im-
pediment to the understanding of either Frege or the Tractates. It is
best, indeed, if one wants to understand these authors, not to have
any philosophical preconceptions at all, but to be capable of being
naively struck by such questions as the following ones: If I say that
Russell is a clever philosopher, I mention Russell, and say some-
thing about him: now, is what I say about him something that I
mention, just as I mention him? If so, what is the connection between
these two mentioned things? If not, what account are we to give
of the words expressing what I say about him? have they any
reference to reality? Further, suppose we try to explain the reference
to reality by adverting to the truth of what is said, then what about
false statements? These say that such and such is the case just as
much as true statements do so that the saying-that done by true
statements cannot be explained by their truth. The investigations
prompted by these questions are more akin to ancient, than to more
1
modern, philosophy.
Again, if I say that the evening star is the same as the morning
star, is this statement about the object mentioned, or about the signs
used in speaking of it? If it is about the object, then why is it in-
formative to say this, but not informative to say that the evening
star is the same as the evening star? If it is about the signs, then,
once more, how can it be informative since we can make signs stand
for what we like by arbitrary definition?
This latter problem led Frege to formulate his famous distinc-
tion between the sense (Sinn) and the reference (Bedeutung) of
words The expressions 'the morning star' and 'the evening star', he
:
said, have the same reference for they refer to the same object,
namely the planet Venus. But they have different senses, and that is
why the statement of identity can be informative. This distinction
between 'sense' and 'reference', once formulated, was used by Frege
throughout his accounts of truth and falsehood, of objects and
concepts, of propositions and thoughts.
1
cf. Plato's Theaetetus 189A: 'In judging, one judges something; in
judging something, one judges something real; so in judging something
unreal one judges nothing; but judging nothing, one is not judging at all.'
Wittgenstein returned to the problem presented by this argument again
and again throughout his life.
14 INTRODUCTION
Above all, Frege's enquiries were in no way psychological; nor
had he any interest in private mental contents. If people speak of the
number 100, they may have quite different images: one may have a
mental picture of the letter C, another of ten rows of ten dots,
another of a collection of hens in a yard; and the same man may
have different images at different times, or again one may have no
image. None of this has the slightest bearing on what is meant when
we speak of the number 100. Nor can the history of the race or of
language, nor again the mental processes by which a man grasps that
10 x 10=100, be relevant to the question what it is that he grasps.
Russell, who discusses many of the same questions as Frege,
differs from him by introducing the notion of immediate experience,
and hence that of private mental contents, into his explanations of
meaning and his theory of judgment. For Russell was thoroughly
imbued with the traditions of British empiricism. Wittgenstein's
admirers have generally been like Russell in this, and have assumed
that Wittgenstein was too; therefore they have had assumptions
about what is fundamental in philosphical analysis which were quite
out of tune with the Tractatus.
We owe to Frege the notion of a 'truth-value' which is current at
the present day. The truth-value of a proposition is its truth or
falsehood as the case may be. Frege says: its truth-value is the
true if it is true and the false if it is false. This term is now very
familiar to any student of elementary logic, and may easily seem to
be no more than a conveniently short expression by means of which
circumlocution is avoided. In Frege, however, we find it arrived at
through hard work on the theory of meaning and truth: work which
also produced other allied and fruitful conceptions.
One of the most important of these is the profound comparison 1
between a predicate and the expression of an arithmetical function,
2
*- The notion of what is termed a
e.g. '(x) 'propositional function',
e.g. 'x is bald', is directly based on this comparison: here we get a
proposition if we replace the *x* by a (real) proper name, just as
'
from *(x) 2 we get an expression of definite value by replacing the
V by a definite number. This comparison is fundamental to all
modern logic.
1
See Chapter 7.
INTRODUCTION 15
Frege also gave us the modern conception of 'quantification',
which is so useful and in such general use in logic that we regard it
as we regard the wheel, forgetting its inventor. Quantification
essentially consists in reformulating 'Everything is heavy' as: 'For
all x, x is heavy'; and 'Something is heavy' as: 'For some x, x is
heavy' or 'There is an x such that x is heavy'. These are written in a
1
symbolic notation.
The general reader may wonder at first whether the interest of
such a device is not purely technical. It is easy to bring out that this
is not so; it is of great general interest in philosophy.
For example, this formulation supplies us with a perspicuous
refutation of the celebrated Ontological Argument of Descartes:
people have been generally agreed that, but not how, it is to be
refuted. According to the Ontological Argument the notion of God
involves that of existence, as that of a triangle involves the various
properties of a triangle; therefore, God exists. Let us concede the
premise. (There is even good ground for it in the fact that e.g. "There
used to be a God, but isn't any more* seems to conflict with the
concept 'God'.) Tlie premise should be stated as follows: Just as, //
anything is a triangle, it has those properties, so if anything is God,
it must possess eternal existence. This is fair; we must be permitted
to take seriously the argument about triangles which Descartes relies
on. But in the sense in which the conclusion 'God exists* is intended,
it means that there is a God. And that by no means follows from the
premise. For, quite generally, from: 'For all x, if ^x, then ^x', we
f
cannot infer: 'There is an x such that <x.* That is, interpreting *<x
as *x is God' and YX' as 'x has eternal existence', we cannot infer
'There is a God' from 'For all x, if x is God, x has eternal existence*.
We can very well grant that and still ask 'But is there such a being?'
We may well say: *It belongs to the concept of a phoenix never to
die, but eternally to renew its life in the flames'; but we cannot infer
from the concept that there lives such a creature.
Again, the following fallacious piece of reasoning is found in
Aristotle: 'All chains of means to ends 2 must terminate in a final
end. This final end will be the supreme good.* The first statement is
1
See Glossary and Chapter 11.
2
i.e. every case of doing a in order that b in order that c. . . .
16 INTRODUCTION
reasonable; the second assumes that the first has shewn that there
issome one end, the same for all chains of means to ends, in which
they all terminate: the fallacy is immediately avoided by writing:
For all x, if x is a chain of means to ends, there is a y such that y
is a final end and x terminates in y,
which is quite different from:
There is a y such that y is a final end, and for all x, if x is a chain
of means to ends, x terminates in y.
It is of general usefulness to be armed against all fallacies of this
type.
Again, a possible limitation on the sense in which a man is free
can be brought out by considering that:
At all times there is a possibility of my abstaining from smoking,
is quite different from and unluckily does not imply:
There is a possibility of my abstaining from smoking at all times.
The difference here is quite easily made out in ordinary language;
but it is also easily missed. In symbolic notation it cannot be missed,
for there is no ambiguous way of writing down what we are saying.
Thus this device of modern logic at least is an instrument for the
clarification of thought which is of use to anyone who engages in
reasoning. And without the development of this part of logic by
Frege and Russell, it is inconceivable that Wittgenstein should have
written the Tractatus,
Russell, studying the same range of topics as Frege, criticized
and rejected one of Frege's (at first sight) most plausible devices :
the distinction between the sense and the reference of phrases. At
the same time he used the new way of representing 'all' and 'some'
in analysing sentences containing definite descriptions (singular
descriptions preceded by the definite article). Frege had treated
1
such descriptions as proper names which, while they always had
a sense, might or might not have a reference. According to him
sentences containing definite descriptions without reference were
neither true nor false. This view is dependent on the validity of the
distinction between sense and reference, not merely as he first in-
troduced it in connection with identity, but in the very extended
application that he made of it; if that has to be rejected, a new
1
See Chapter 2.
INTRODUCTION 17
account of such sentences has to be found. This was given by
Russell in his Theory of Descriptions. Wittgenstein embraced the
logical analysis afforded by Russell's theory with admiration, and,
as we shall see, it exercised a great influence on the thought of the
Tractatus.
The notions of 'sense', 'reference* and 'meaning' as they occur
in these three authors, Frege, Russell and Wittgenstein, perhaps need
a summary account. Frege's 'sense' corresponds roughly to the
ordinary English 'meaning'. When we ask for the meaning of a
phrase, we are not asking to have an object indicated to us we want
:
a paraphrase with the same meaning or, as Frege would say, the
same sense. On the other hand, if I say 'When I spoke of "that fat
charlatan", I meant Smith', what I 'meant' is a man; Frege's
'reference' (Bedeutung) corresponds to what is 'meant' in this use of
of the word.
Wittgenstein follows Frege in this use of the words 'Bedeutung\
'bedeuterf.Generally, in the Tractatus, they ought not to be
rendered, as C. K. Ogden rendered them, by 'meaning' and 'mean',
1
but rather by 'reference' and 'stand for'. Wittgenstein's conception
of 'sense' may be called the same as Frege's, if we are careful to add
that Wittgenstein had different theses about it: for he held that
names had no sense but only reference, and propositions no
reference but only sense; and also that a proposition could not have
a sense without being either true or false. Further, he uses the
suggestion of 'direction' that is contained in the word 'sense* when
he speaks of positive and negative as opposite senses: we shall see
that he considered significant negatability to be of the essence of a
and a proposition and its negation as like
significant proposition,
arrows pointing in opposite directions. ('Sinn* is ordinary German
for 'direction' ; in English usage 'sense' occurs with that meaning in
mathematics,)
Russell uses only one notion 'meaning' and holds that the
meanings of words must always be objects that one is directly
1
English readers of the Tractatus may need to be warned that Ogden's
translation is notoriously very bad. Wittgenstein told me that he had not
checked the whole of this translation, but only answered a few questions
that were put to him about some passages, e.g. I think we can see
Wittgenstein's hand in the free but excellent rendering of 4.023.
18 INTRODUCTION
acquainted with. He also speaks of 'denoting': a 'denoting' expres-
sion is such an expression as 'Some ambassador', 'Any horse', 'The
(one and only) earth satellite'. It was the object of the Theory of
Descriptions to analyse such expressions away, and so 'denoting*
has no part in Russell's final explanation.
This, then, is the historical background of the Tractatus. It is a
book which is apt to captivate people's minds, while at the same time
seeming in many parts excessively obscure. Some people, once they
have looked into it, are prevented from throwing it away in despair
of penetrating its meaning by the impression they receive of great
light in certain areas. This Introduction is addressed primarily to
such readers as these. It is certainly not meant to be of any value to
someone who does not read or propose to read the Tractatus itself.
The Tractatus is not presented in an order of demonstration from
premises ; if we want to find the grounds for its contentions, we must
look in the middle and not at the beginning. It is divided into a set
of numbered* remarks in a decimal notation shewing what is of
greater and what of subsidiary importance: the more decimal
places in a number, the more subsidiary the remark it is attached to.
The main propositions are the ones numbered with the whole
numbers 1 1. These run:
1. The world is everything that is the case.
2. What is the case the fact is the existence of atomic facts. 1
3. The logical picture1 of the facts is the thought.
4. The thought is the significant proposition.
5. The proposition is a truth-function 8 of elementary propo-
4
sitions.
(The elementary proposition is a truth-function of itself.)
6. The general form of truth-function is [p, J, N(|)]. & This is the
general form of proposition.
7. What we cannot speak of, we must be silent about. 6
1
See Chapters 1 and 4.
1
See Chapter 4.
8
See Glossary and Chapter 3.
4
See Chapter 1.
*
See Chapter 10.
See Chapters 5 and 13.
INTRODUCTION 19
It is clear enough from this that the principal theme of the book
is the connection between language, or thought, and reality. The
main thesis about this is that sentences, or their mental counter-
parts,
are pictures of facts. Only we must not suppose that what is
pictured by
a proposition has to exist: as Wittgenstein wrote in
explaining himself to Russell in 1919, a fact is what corresponds to a
if it is true. The proposition is the same picture whether
proposition
it is true or false i.e. whether the fact it is a picture of is a fact, is
the case, or not. This should not make us ask *How, then, can a fact
not be a fact?' For, following Wittgenstein's explanation, it means:
The proposition is the same picture whether what corresponds to
it if it is true is the .case or not: it is a picture of that. And what
corresponds to it if it is true is the same, whether it is true or false.
The world is the totality of facts i.e. of the counterparts in reality
of true propositions. And nothing but picturable situations can be
stated in propositions. There is indeed much that is inexpressible
which we must not try to state, but must contemplate without
words.
In his Introduction Wittgenstein suggests that he may be under-
stood only by people who have had the same thoughts as he;
certainly he can only be understood by people who have been per-
plexed by the same problems. His own writing is extraordinarily
compressed, and it is necessary to ponder each word in order to
understand his sentences. When one does this, they often turn out
to be quite straightforward, and by no means so oracular or
aphoristic as they have been taken to be. But few authors make such
demands on the close attention and active co-operation of their
readers.
In my account, I have not followed the arrangement of the
Tractatus at all. That, I think, is something to do when one reads
the book for enjoyment after one has come to understand its main
ideas. I have chosen what seem to me to be the most important
themes and problems of the book. My first six chapters aim at giving
the reader some idea of the 'picture theory' of the proposition. I
devote a great deal of space to the topic of negation, for 'not',
which is so simple to use, is utterly mystifying to think about; no
theory of thought or judgment which does not give an account of it
20 INTRODUCTION
can hope to be adequate. It is thus one of the central topics of the
Tractatus.
Chapter 7 is mainly concerned with what becomes of the great
problem of Universals in Wittgenstein's theory, and Chapter 8 with
s
certain aspects of 'not', 'and', or\ etc., which are not covered in my
account of the picture theory. Chapters 9 and 10 deal with important
technical notions which are rather special to the Tractatus, and could
be omitted by a beginner who wanted first to familiarize himself
with its foundations: these chapters, we may say, treat of the upper
storeys of the edifice. But with Chapter 11, on the theory of gener-
ality, we are once more working on the foundations. The last two
chapters are about some general philosophical consequences which
Wittgenstein drew from his investigations into the philosophy of
logic.
The logic, a knowledge of which is necessary for an understand-
ing of the Tractatus, is very elementary; my own aim has been to
write in such a way that someone who was not already familiar with
itcould pick it up as he went along. In case the symbols and technical
terms of elementary modern logic should be unfamiliar to a reader,
I append a short glossary.
GLOSSARY
p, q, r These small letters are used to represent
propositions. (By Wittgenstein, only
for elementary propositions.)
a, b, c These small letters (from the beginning of
the alphabet) are used to represent proper
names of objects.
fa, ga, ^a, <Aa represent propositions containing the
name 'a'. Similarly
f(a,b), <(a,b) represent a proposition containing the
names *a' and *b': 'f and V
arc here
shewn to be 'two-place' predicates, or
'dyadic relational expressions'.
R, S These large letters are used to represent
relations, e.g. 'to the right of, 'larger
than', 'father of. And
aRb symbolizes a proposition asserting a
relation between a and b.
x, y, z These small letters (from the end of the
alphabet) mark (different) empty places
in propositions written in the forms
4
fa' or *f(a,b)', or 'aRb', from which a
proper name or names have been re-
moved; e.g. if we remove *a' and 'b' from
'aRb' we have *-R-', which yields a
proposition if we put names in the
blanks. To differentiate blanks, we put
*xRy' ; to shew they are to be filled up the
same way, we put 'xRx\
21
22 GLOSSARY
Variable Such a small letter as x, y, z, in the role
just described. Variables are chiefly used
in the construction of
Quantified propositions Propositions containing the notions 'all',
'some'. When we are speaking of *a!T
so-and-so, and any example of so-and-so
would be an object which could have a
proper name, the proposition is written
in the form:
GO* For all x, #x; i.e. 'Everything is #'.
For some x, ^x; i.e. "Something is or
"There is an x such that #x*.
Truth-value The truth or falsehood (as the case may
be) of a proposition.
Function Cannot be defined, but only illustrated: a
(Value, Argument) function, say ( )*, takes different values for
different arguments: e.g. the value of the
a
function ( ) for the argument 3 is 9,
since (3)*=9. The value of the power
function ( ) < ) for arguments 2 and 3 in
that order is 8: since 2 to the power
3 (2 s) is 8.
Truth-function A function (e.g. '- and -') whose argu-
ment^) (e.g. *p' and *q') and values
(e.g. 'p and q') are propositions, such that
the truth-value of its value is determined
by the truth-value(s) of its argument(s).
Truth-functional con- Signs used to express truth-functions, e.g.
nectives (one kind of 'not', 'and', 'or'. For the truth-value of
logical constants*) *not-p* is determined by the truth-value
of p, and the truth-value of *p and q* and
*p or q' is determined by the truth-values
of p and of q.
GLOSSARY 23
not.
and.
or; non-exclusive, i.e. *p v q' is true when
4
both p' and 'q' are true, as well as when
only one of them is.
if ... then . . . , defined as 'either not ...
or . . .'. Thus *pi>q* is true if 'p' is false
or 'q* true, regardless of any real con-
nection in their subject-matters. This
(minimum) sense of *if . then' occurs in . .
*If that is so, I'm a Dutchman', which if I
am known not to be a Dutchman is a
way of saying that 'that' is not so.
Material implication the 'if ... then' expressed by 'D'.
Truth-table 1 A table designed to show the relation
(or: Matrix) between the truth-value of a truth-func-
tion and the truth-value(s) of its argu-
4
ment(s). Thus the truth-tables for p and
q' and 'p or q' are:
p q P and q porq
T(rue) T T T
T F F T
F(alse) T F T
F F F
Tautology Any truth-function such that whatever
the truth-values of its arguments, its value
for those arguments is always true. Ex-
amples: p v ~p; (p.pDq)Dq.
Logical product Conjunction of all the propositions of a
given set, e.g. p. q. r.
Logical sum Disjunction of all the propositions of a
given set, e.g. p v q v r.
1
Invented (independently) by Wittgenstein and Post.
24 GLOSSARY
= In logical contexts this is used as the sign
of identity, not of equality in quantity;
*a=b* means that a is b.
N( ) Joint negation of the propositions put
between the brackets; used only by Witt-
genstein (see Chapter 10).
O, Q variable signs for an operation in Wittgen-
stein's sense ; these symbols are peculiar to
the Tractatus (see Chapter 9).
, 77 variables for expressions, not tied to any
one kind, as is x, which is a name variable,
or again n, which is a numerical variable:
used in informal exposition by Frege and
Wittgenstein.
^ sign for a special operation, used only in
the present book (see Chapter 10). Read
as 'Resh'.
1
ELEMENTARY PROPOSITIONS
Karl Popper has described the Tractatus in the following way:
'Wittgenstein tried to shew that all so-called philosophical or
metaphysical propositions were in fact non-propositions or
pseudo-propositions: that they were senseless or meaningless.
All genuine (or meaningful) propositions were truth-functions
of the elementary or atomic propositions which described
"atomic facts", i.e. facts which can in principle be ascertained by
observation. In other words, they were fully reducible to ele-
mentary or atomic propositions which were simple statements
describing possible states of affairs, and which could be in
principle established or rejected by observation. If we call a
statement an "observation statement" not only if it states an
actual observation but also if it states anything that may be ob-
served, we shall have to say that every genuine proposition must
be a truth-function of and therefore deducible from, observation
statements. All other apparent propositions will be, in fact,
1
nonsense; they will be meaningless pseudo-propositions.'
I cite this passage because it expresses the most common view of
the Tractatus. It only needs a small supplement to express that view
completely. For it is sufficiently well known that the Tractatus con-
tains a 'picture theory' of language, of which Popper here makes no
mention. The whole theory of propositions is, then, on this view, a
merely external combination of two theories: a 'picture theory*
of elementary propositions (viz. that they have meaning by being
1
British Philosophy in Mid-Century, Allen and Unwin, 1957 pp. 163-4.
:
25
26 AN INTRODUCnON TO WITTGENSTEIN'S TRACTATUS
'logical pictures* of elementary states of affairs), and the theory of
truth-functions as an account of non-elementary propositions; this
latter theory breaks down rather easily, because it is impossible to
regard generalized propositions that relate to an infinitely numerous
universe as truth-functions of elementary propositions.
Someone who, having read the Tractatus9 reads Popper's account
of it, must be struck by one thing: namely that there is a great deal
about 'observation' in Popper's account, and very little about it in
the Tractatus. According to Popper, the elementary propositions of
the Tractatus are simple observation statements. Now can we find
any support for this view in the Tractatus itself? I think that the
strongest support that we can find is at 3.263: "The references of
primitive signs can be made clear by elucidations. Elucidations are
propositions containing the primitive signs. Thus they can only be
understood, if one is acquainted with the references of these signs.*
I think we can take it that 'primitive signs* are the same thing as
'names', from the passage above, 3.261 : Two
signs, one a primitive
sign, and the other defined by means of primitive signs, cannot
signify in the same way. Names cannot be expounded by means of
definitions.' Here it is clear enough that 'names' are 'primitive signs';
and as we know from elsewhere that Wittgenstein did not regard
logical signs as primitive signs, or as having anything that they stand
for, we can also say that the only primitive signs for him are what he
calls 'names'.Names, then, can be made clear by elucidations, by
sentences containing them spoken to someone who is acquainted
with the objects that they stand for.
An obvious example of a name might seem to be the word 'red*
uttered in a sentence, perhaps 'Red patch here' in the presence of
someone who is contemplating the red patch and who may be
supposed to have acquaintance with the object designated by the
word 'red'. And 'red patch here' would seem to be a candidate for
being a simple or elementary observation statement such as Popper
refers to.This suggests that the elementary propositions are not
merely observation statements, but sense-datum statements; as,
indeed, they were taken to be both by many members of the Vienna
Circle and for many years in Cambridge discussions. And I think it
is quite possible that Wittgenstein had roughly this sort of thing
ELEMENTARY PROPOSITIONS 27
rather vaguely in mind. His speaking of 'acquaintance' (for that
certainly seems the best rendering of 'ken?ien' and its compounds
where they occur in the Tractates) very strongly suggests this; we
immediately think of Russell's distinction between 'knowledge by
acquaintance' and 'knowledge by description'.
|
do not believe that any other support for Popper's view of
elementary propositions is to be found in the Tractatus. And this
passage is a rather slender support.
In the first place, Wittgenstein does not state, or even suggest,
that the proposition which contains an elementary name and
'elucidates' that name for a person acquainted with its reference
must be an elementary proposition.
In the second place, the kind of example that comes most readily
to mind, 'This is a red patch', can be proved not to be an elementary
proposition according to the Tractatus. For at 6.3751 we find in
parenthesis: 'It is clear that the logical product of two elementary
propositions can be neither a tautology nor a contradiction. The
assertion that a point in the visual field is two different colours at the
same time is a contradiction.' It follows directly from this that "This
is a red patch' cannot be an elementary proposition.
Indeed, quite generally, if elementary propositions are simple
observation statements, it is very difficult to see how what Wittgen-
stein says here can possibly hold good of them; for, for any proposi-
tion which could reasonably be called a 'simple observation state-
ment', one could find another that would be incompatible with
it and be precisely analogous to it logically. Therefore, whatever
elementary propositions may be, they are not simple observation
statements and this accounts for the lack of reference to observation
;
in all the remarks concerning elementary propositions; which would
surely be very strange if Popper's interpretation were the correct one.
With this is connected the fact that there is hardly any epistemo-
logy in the Tractatus; and that Wittgenstein evidently did not think
that epistemology had any bearing on his subject-matter. We find
epistemology put in its place at 4.1121: 'Psychology is no nearer
related to philosophy than is any other natural science. The theory
of knowledge is the philosophy of psychology.'
A letter to Russell in 1919, written from the prison camp at Monte
28 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
Cassino, throws further light on this. Russell had asked: '. But a
. .
Gedanke [thought] is a Tatsache [fact] what are its constituents and
:
components, and what is their relation to those of the pictured
TatsacheT To this Wittgenstein replies: 'I don't know what the
constituents of a thought are but I know that it must have consti-
tuents which correspond to the words of language. Again the kind
of relation of the constituents of the thought and of the pictured fact
is irrelevant. It would be a matter of psychology to find out.' That
is to say, it would be a matter of empirical investigation to find out,
both what the constituents of a thought are and how they are
related to the 'objects' occurring in facts, that is to say, to the objects
designated by the 'names' in language.
That this is fantastically untrue is shewn by any serious investiga-
tion into episteinology, such as Wittgenstein made in Philosophical
Investigations. But it is fair to say that at the time when he wrote the
Tractatus, Wittgenstein pretended that epistemology had nothing to
do with the foundations of logic and the theory of meaning, with
which he was concerned. The passage about the 'elucidation' of
names, where he says that one must be 'acquainted' with their
objects, gives him the lie.
More positively, the grounds on which Wittgenstein holds that
there are elementary propositions and simple names shew that the
elementary propositions have not the role of simple observation
statements.At 5.5562 we find: 'If we know, on purely logical
grounds, that there must be elementary propositions, then this must
be known by anyone who understands propositions in their un-
analysed form.' But it is clear that he thought we did know this on
purely logical grounds. That is to say, the character of inference, and
of meaning itself, demands that there should be elementary proposi-
tions. And that there should be simple names and simple objects is
equally presented as a demand at 3.23: 'The demand for the possi-
bility of the simple signs is the demand for definiteness of sense.' We
shall see that he holds that an indefinite sense would not be a sense
at all; indeed in the Preface he put this forward, not just as one of
the most important contentions of the book, but as an epitome of its
whole meaning: 'Whatever can be said at all, can be said clearly;
and what we cannot speak of, we must be silent on.'
ELEMENTARY PROPOSITIONS 29
Again, the simple objects are presented as something demanded
by the nature of language at 2.021, 2.0211 "The objects form the
:
substance of the world. That is why they cannot be complex. If the
world had no substance, then one proposition's making sense would
depend on another one's being true.' But this is not the case: we can
devise propositions at will and know what they mean, without
ascertaining any facts. If one proposition's making sense always
depended on another one's being true, then it would be impossible
to do this impossible, as Wittgenstein puts it, to devise a picture
of the world (true or false) (2.0212); he means by this no more than
devising a proposition.
We get further (though, I should judge, unnecessary) confirma-
tion from an entry in the notebooks out of which he composed the
Tractatus, in which he remarks (23.5,15): 'It also seems certain that
we do not infer the existence of simple objects from the existence of
particular simple objects, but rather know them by description, as
it were as the end product of analysis, by means of a process
leading to them.' The thought of this entry in the notebooks is in
fact echoed in the Tractatus text at 4.221 'It is obvious that in
:
analysing propositions we must arrive at elementary propositions
consisting of names in immediate combination.' This view of names,
and hence of our knowledge of objects, is a more truthful one than
is suggested by the remark about 'elucidations'. And in the note-
books he exclaims at the fact that he is absolutely certain that there
are elementary propositions, atomic facts, and simple objects, even
though he cannot produce one single example.
If the elementary propositions of the Tractatus are not simple
observation statements, it seems necessary to find some other
account of them before we can grasp the doctrines of the book even
in vague outline. For an understanding of the notion of an ele-
mentary proposition will help us with its correlate, an atomic fact,
or elementary situation.
Wittgenstein opens the Tractatus by saying that the world is the
totality of facts (Tatsachen). He quickly introduces a new term
(translated 'atomic fact'): 'Sachverhalt\ Literally this word simply
means 'situation'. Etymologically it suggests 'hold of things' i.e. a
way things stand in relation to one another. Wittgenstein plays
30 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
heavily on this suggestion. It rapidly becomes clear that by a
'situation' he means an arrangement of objects, which objects, he
says, are 'simple'. The 'situation' is a concatenation of simple
objects, which 'hang in one another like the links of a chain' (2.03).
Hence the word was translated 'atomic fact'; for 'situation', not
carrying with it the special suggestion of 'Sachverhalt', would have
been obscure; and 'atomic fact' had been Russell's term for the
correlate of a true 'atomic' proposition.
Writing to Russell from Monte Cassino in 1919, Wittgenstein
explained Sachverhalt as what corresponds to an elementary pro-
position if it is true, and a Tatsache as what corresponds to the
logical product (i.e. the conjunction) of elementary propositions
when this product is true. 1 This explanation concerns the first intro-
duction of 'Tatsache' or 'fact'. At 2.06 he introduced the further
4
expression a negative fact': 'We also call the non-existence of
atomic facts a negative fact.'
That is to say, to the question 'What is a fact?' we must answer:
'It is nothing but the existence of atomic facts.' This is a thesis about
facts; not the assignment of a technical meaning to the word in
Wittgenstein's system. And to the question: 'Is there such a thing as
a negative fact?' we must answer: 'That is only the non-existence of
atomic facts.' Thus the notion of a fact is supposed to be explained
to us by means of that of an atomic fact, or elementary situation.
And that in turn is simply what corresponds to a true elementary
proposition. Thus an exploration of this notion is indispensable.
1
Some critics have objected to the translation 'atomic fact' because an
atomic fact is presumably a fact, and it is awkward to speak of 'non-
existent facts'; but Wittgenstein does speak of non-existent Sachverhalte
(2.06). This objection does not amount to much. But it is added that
Wittgenstein never speaks of 'possible facts' (Tatsachen). For what he
speaks of as possible, he uses another German word, Sachlage, which
means 'state of affairs'. Prof. Stenius suggests that this is the real non-
atomic parallel to Sachverhalt, and that Wittgenstein was simply wrong
in giving Russell parallel accounts of Sachverhalt and Tatsache. I find
suggestions that Wittgenstein gave an incorrect account of the Tractatus
in 1919 quite unacceptable. In German a 'possible fact' (mogliche Tatsache)
would be something that is perhaps a fact i.e. for all we know to the
contrary; this irrelevant reference to our knowledge would surely be what
ruled the phrase out. The difficulties we encounter here are really those of
the subject-matter itself, and not of Wittgenstein's terminology. Wittgen-
stein accepted the translation 'atomic fact'.
ELEMENTARY PROPOSITIONS 31
The following appear to be theses which hold for elementary
propositions :
(1) They are a class of mutually independent propositions.
(2) They are essentially positive.
(3) They are such that for each of them there are no two ways of
being true or false, but only one.
(4) They are such that there is in them no distinction between an
internal and an external negation.
(5) They are concatenations of names, which are absolutely
simple signs.
As for the reasons for holding that there are such propositions
as these, we know at least that, according to the Tractatus, they are
'purely logical'. About these purely logical grounds I will only say
here that the main one is this : we can draw conclusions from a false
proposition. This is the same fact as that we can invent or devise a
proposition and know what it means, without first discovering the
facts which hold in regard to its subject-matter. For to understand a
proposition is to know what is the case if it is true.
The five theses which hold good of elementary propositions can
be found at or inferred from several places in the Tractatus.
(1) Elementary propositions are a class of mutually independent
propositions. This we have already seen stated in a restricted form at
6.3751 'It is clear that the logical product of two elementary proposi-
:
1
tions can neither be a tautology nor a contradiction.' Strictly, it may
be said that this might be true and the general mutual independence
false; but we need not delay over the suggestion. It is worth noticing
that the existence of a great class of mutually independent proposi-
tions is implicit in the common explanation of truth-functions and
truth-functional tautologies. For we are told that a complex proposi-
tion is a truth-function of the proposition(s) contained in it if its truth-
value is uniquely determined by the truth- value of the proposition(s)
in question; and it is a tautology if it is true for all combinations of
the truth-values of its components. If it is a function of several
propositions, it is impossible that its tautological truth should con-
sist in its truth for all the combinations unless its components have
1
And also 4.211 and 5.134.
32 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
some mutual independence. To take an example, the syllogism
some Europeans are Mohammedans,
'If all Europeans are white and
then some white men are Mohammedans' is a logical truth in which
three propositions occur; its being a logical truth is equivalent to
the logical impossibility of the case in which the first two component
propositions are true and the last false. A
truth-table will thus not
display the tautological character of the proposition; for if one
constructs a truth-table for *If p and q, then r', one has to show the
truth-value (namely falsehood) of this conditional for the case
where 'p* and 'q' are both true but V
is false; and it is not the
truth-table but the interpretation of *p', 'q*, and 'r' which shows
that in the syllogistic case the conditional cannot be false.
Nevertheless a truth-table containing inconsistent rows may
display the tautological character of a proposition, e.g. Aristotle
felt a difficulty about the following form of geometrical argument:
4 1
All triangles are either isosceles or scalene; all isosceles triangles
have the property <f>; all scalene triangles have the property <f>;
therefore all triangles have the property <.' We can see that his
difficulty consisted in the argument's not being formalizable in his
syllogistic calculus. What he needed was the truth-functional
A
calculus. Let x be a figure; then let p'='x is a triangle', 'q'=*x
is isosceles*, 'r'='x is scalene', and 's'='x has the property <f>\
Then the fact that (pDqvr. qDs. rDs)D(pDs) 2 is a tautology
of the truth-functional calculus would have supplied the missing
formalization. Now being a tautology means being true for all
combinations of the possible truth-values of the elements (means
being, as logicians say, a tautology of the given elements), and
the truth-table setting forth these combinations will include a row
in which both *q' and *r' are true, and another in which both *q' and
4 4
r' are false while 'p' is true. But if 'q'= x is isosceles', and *r*='x
is scalene*, these combinations will be impossible.
We may conclude from this that a complex proposition can be
shewn to be a logical truth from the fact that it is a tautology of its
component propositions, even though some of these are mutually
1
two equal sides.
i.e. possessed of at least
1
In English 'p implies that q or r, and q implies that s, and r implies
that s, all implies that p implies that s'.
ELEMENTARY PROPOSITIONS 33
inconsistent; and from the syllogistic example, that it cannot be
shewn not to be a logical truth from the fact that it is not a tautology
1
of its component propositions. Nevertheless, the type of tautology
in which some of the combinations of truth-possibilities are incon-
sistent must be regarded as degenerate. The fact that by our calculus
the complex proposition turns out 'true' if we assign an inconsistent
set of truth-values to its components does not help to demonstrate its
tautological character; we might rather strike out inconsistent rows
of the truth-table as not counting. But if all cases were like this, with
now one now another row of our truth-tables inconsistent, then the
formal truth of the truth-functional tautology would vanish.
Thus either the theory of truth-functions has no application, or
there is a class of mutually independent propositions. But we apply
the calculus of truth-functions every time we reason e.g. 'If p, then
1
q, but not q, therefore not p a thing which we constantly do in the
:
most diverse contexts of ordinary life. Here is the beginning of a
justification for Wittgenstein's saying: 'We know on purely logical
grounds that there must be elementary propositions* and 'everyone
knows this who understands propositions in their unanalysed form*.
At any rate everyone manifests an implicit knowledge that there is
a (very large) class of mutually independent propositions.
(2) Elementary propositions are essentially positive. This we can
infer from 4.25: 'If the elementary proposition is true, the atomic
fact exists; if it is false the atomic fact does not exist* together with
2.06: 'We also call the existence of atomic facts a positive, and their
non-existence a negative fact* : the elementary proposition therefore
issuch as to express something positive, namely the holding of an
elementary situation. This, of course, does not mean that the
occurrence of the sign of negation in a prepositional sign would
prove that it did not state an elementary proposition. Wittgenstein
warns us at 4.0621 'The occurrence of negation in a proposition is
:
1
According to Wittgenstein, this logical truth can be exhibited as a
tautology of a set of elementary propositions, though not as a tautology of
the propositions explicitly occurring in it; see Chapter 11. Von Wright has
shown a simple way of exhibiting it as a tautology if we assume men to
have any given finite number; see Logical Studies (Routledge & Kegan
Paul, 1957), Chapterl;itis exhibited as atautology of singular propositions
about men.
34 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
not enough to characterize its sense' i.e. to characterize it as negative
rather than positive in sense; as stating, if true, a negative fact.
Russell in his letters to Wittgenstein after receiving the text of
the Tractatus once asked whether the negations of elementary pro-
positions were themselves elementary propositions, and received the
indignant-sounding rejoinder: 'Of course not.*
(3) Elementary propositions are such that for them there are no
two ways of being true or false but only one.
This is clearest for falsehood. By 4.25 the falsehood of an elemen-
tary proposition is simply the non-existence of a single atomic
situation.
At 3.24 Wittgenstein says: *A complex can be given only by its
description, which will hold or not hold. The proposition in which
there is mention of a complex will not be meaningless when the
complex does not exist, but merely false. That a prepositional
element designates a complex can be seen from an indefiniteness in
the propositions in which it occurs.* One kind of indefiniteness in a
proposition might be that there was more than one way of its being
false: the complex might exist, but what was said of it might not
hold; or the complex might not exist.
We could imagine a proposition in which there was mention of
a complex, which had only one way of being true, though two ways
of being false. Let us suppose a proposition *^a' such that *a' is a
simple name, being such that there was only one way for to hold
<f> <f>
of anything. Then let us suppose a complex A, which exists if bRc.
A
Then '^A' will be false if exists but ^ does not hold of it, and also
if not bRc, so there are two ways for it to be false; but only one way
for it to be true, namely that bRc, so that A exists, and </>A.
*We know,' Wittgenstein goes on, *that not everything is settled
by such a proposition' that is to say, by a proposition in which
there is mention of a complex. In the example that I have imagined,
'everything would be settled' by the truth of the proposition, but
not everything by its falsehood. What he principally had in mind was
the sort of proposition where there is a variety of ways for the
proposition to be true. (This is in fact the most ordinary sort of
proposition, of which alone one can give examples; to illustrate
other sorts of proposition one has to use dummy names and dummy
ELEMENTARY PROPOSITIONS 35
predicates and stipulate their characters.) Take for example 'My
watch is on the table', which Wittgenstein considers in
lying
his notebooks. There are hundreds of different, more minutely
statable, and incompatible states of affairs which would make that
proposition true. The elementary proposition will have only one
state of affairs that will make it true: 'everything' will be settled by
it i.e. nothing be left open.
(4) Elementary propositions are such that there is in them no distinc-
tion between an internal and an external negation. This is in part the
same point as has already been made in connection with definiteness
of sense. We can say 'The King of France is bald' has as a negation
:
'The King of France is not bald'; I distinguish this internal negation
of the proposition from the external negation: 'Not: The King of
France is bald' we have already seen how these differ in sense. To
take another case: the proposition 'Everyone is wise* has an internal
negation, 'Everyone is not wise' (or: 'is unwise'), and another,
external, negation: 'Not everyone is wise.' Aristotle was rather
puzzled by this difference between 'Socrates is wise' and 'Everyone
is wise' if 'Socrates is wise' is untrue, then 'Socrates is not wise' is
:
true; but if 'Everyone is wise' is untrue, still it does not follow that
'Everyone is not wise', or 'is unwise', is true; the contradictory is the
different proposition that not everyone is wise.
It is true that we sometimes use 'Everyone is not .* in the sense
. .
'Not everyone is .' and hence it is convenient to use the term 'un-
. .
;
1
wise' to make our point. But, to adapt what Frege says, it should
not be supposed from this attachment of the negation to 'wise' that
'what is negated is the content, not of the whole sentence, but just of
this part. ... It is incorrect to say: "Because the negative syllable is
combined with part of the sentence, the sense of the whole sentence
is not negated." On the contrary; it is by combining the negative
syllable with a part of the sentence that we do negate ... the whole
sentence.' That is to say, the sentence 'Everyone is wise' is certainly
made out to be untrue by someone who says 'Everyone is unwise';
but this is still a different negation from that expressed by 'Not
everyone is wise'.
1
Negation, Philosophical Writings of Gottlob Frege, ed. Geach &
Black (Blackwell, 1952), p. 131.
36 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
1
I choose 'internal and 'external' merely as convenient labels to
attach to these negations. An elementary proposition will be one for
which no such difference, between an internal and an external
negation, can be found. The falsehood of the elementary proposition
never consists in anything but the non-existence of a single atomic
fact.
(5) Elementary propositions are concatenations of names. This we
find stated explicitly at 4.22: "The elementary proposition consists
of names. It is a connection, a concatenation, of names.' Names are
simple signs; this is not merely asserted, but argued for, in the
Tractatus, at 3.341 1 'So it could be said that the real name is what all
:
symbols that designate the object have in common. Then we could
get the result, in a number of steps, that no kind of composition was
essential to the name.* That is to say, any name will of course have
a certain physical complexity, but you could replace it by another,
with a different complexity, without detriment to its doing the job of
naming the object. Whereas you could not, for example, adequately
symbolize a relation without using a symbol whose complexity
enabled you to shew the difference between, say, aRb and bRa.
So far what is argued about names would seem to be perfectly
applicable to ordinary names, such as 'Wittgenstein', which are not
names in the sense of the Tractatus. 'Wittgenstein' is what he calls a
'simple symbol' at 3.24: 'The contraction of the symbol of a complex
into a simple symbol can be expressed by means of a definition.'
This 'definition' will be basically the same thing as the 'description'
which he speaks of as 'giving' the complex.
Now the physical complexity of the name 'Wittgenstein' ex-
presses nothing, as can be shewn in the way suggested. But if the
'real* name, or 'real' symbol, of the object called Wittgenstein, has
something about it that implies complexity, then the name can be
said not to be a logically simple sign, even though it appears as a
simple sign in the sentence. For, as it is put at 3.262: 'What does not
get expressed in the signs is shewn by their application. Their
application declares what the signs fail to bring out.' And the appli-
cation of the name 'Wittgenstein' brings out that a great many
things, and a great variety of things, have to be true in order for there
to be true statements in which the name occurs. The same would be
ELEMENTARY PROPOSTHONS 37
true of any sign which had the same function as this sign 'Wittgen-
stein'. But what is common to all the symbols with this function is
what is essential to the symbol, as has been said at 3.341 'In general :
what is essential about a symbol is what all symbols capable of ful-
filling the same function have in common.' Therefore a certain
complexity, which only comes out in the application, is essential to
the name 'Wittgenstein*.
Thus the true names of the Tractatus will be, not physically
simple signs, but ones lacking the sort of complexity that the name
'Wittgenstein' has; and it is clear that elementary propositions can
contain only such names, since if they contained names like 'Witt-
genstein' they could not have only one way of being true or false.
So much here for the simplicity of names; we must now discuss
'concatenation'. The metaphor of a chain should suggest an essential
feature of elementary propositions. As we have seen, what is ex-
pressed by calling an elementary proposition a concatenation is
expressed for elementary situations ('atomic facts') at 2.03: 'In the
atomic fact the objects hang in one another like the links of a chain.*
In a literal chain consisting of links
A B C D
there is no difference between A's being linked to the B end of the
chain B C D, and D's being linked to the C
end of the chain
A B C. I think this element in the analogy should be taken
seriously; in the elementary proposition there must be nothing
corresponding to bracketing.
Let us look at what Wittgenstein says about bracketing at
5.461-5.461 1 : 'The apparently unimportant fact that logical pseudo-
relations like v (or) and D (if . . ., then . .
.) as opposed to
real relations require brackets is significant.' This remark has
been criticized on the ground that a bracketless notation, such as
that invented by Lukasiewicz, is possible. In this notation we write
Cpq
instead of
38 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
and then the difference between
and
will be expressed by the difference between
C(Cpq)r
and
Cp(Cqr)
where, though I have put brackets in, these are only an aid to read-
ing and are not needed to resolve any ambiguity. Now this is of
course true; it is true because the collecting done by brackets is done
by the rule for reading an expression containing 'C'. Some method of
collecting is required, and that is the essential point. 'Logical opera-
tion signs are punctuation marks,' Wittgenstein says, and Lukasie-
wicz's notation, far from refuting Wittgenstein's remarks about
brackets, brings out what Wittgenstein meant, for in it the collecting
or punctuating normally done by brackets is done by the rule for
reading the logical operation-signs. Now if the chain metaphor is to
be taken seriously, this differentiation of meanings by punctuation
or collection must somehow be inapplicable to the elementary
proposition.
What I call 'collection' or 'punctuation' can occur in proposi-
tions other than those (overtly) containing truth-functional con-
nectives. Consider the sentence 'Every man loves some girl.* We may
regard this as splitting upinto three 'expressions': 'Every man',
'loves' and 'some girl'. It is useful here to adopt the metaphor of
structural formulae in chemistry for the structure of sentences. An
expression will then sometimes correspond to what chemists call a
'radical': that is, a group of atoms which cannot by itself forma
stable molecule, but which can in chemical transformations pass
from one compound into another without the break-up of its own
inner connection of atoms. And the fact that what compound you
have depends, not only on what radicals you have, but also on how
ELEMENTARY PROPOSITIONS 39
they are fitted together, would be a parallel to such a difference as
that between 'Socrates loves Plato' and Tlato loves Socrates', or
again 'Every man loves some girl' and 'Some girl loves every man'.
There is, however, a difference between two possible senses of 'Every
man loves some girl' which could be brought out by a difference of
bracketing. In
(Every man) (loves some girl)
we could take the bracketing as indicating that 'every man' is
supplied as an argument in
loves some girl
and the sense will be that the predicate 'loves some girl' is true of
every man; whereas in
(Every man loves) (some girl)
'some girl' is supplied as an argument in
Every man loves
and the sense will be that of some girl it is true that every man loves
her. The difference is of course the one usually brought out by the
order of quantifiers. 1 This difference is one that cannot be illustrated
by our chemical analogy. And it is a sort of possibility of difference
that has to be absent from the elementary proposition.
Contrast with this case 'Socrates loves Plato'. We can indeed
introduce two different bracketings: '(Socrates) (loves Plato)', which
1
asserts of Socrates that he loves Plato, and '(Socrates loves) (Plato) ,
which asserts of Plato that Socrates loves him. But in this case
Wittgenstein, following Frege, would say that there was absolutely
no difference of sense.
Now it seems plausible to say that the reason why we have an
ambiguity resoluble by brackets in the one case but not in the other
1
See Chapter 11, pp. 138-41.
40 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
is that, at any rate as compared with 'Every man' and 'some girl', the
expressions 'Socrates* and 'Plato' are simple. This sort of considera-
tion may lead us to divine behind our propositions a kind of propo-
sition to which the chemical analogy of radicals will apply perfectly;
unlike a proposition in which, though you have the same expressions
combined in the same way, it makes a difference by what stages you
conceive the proposition as built up. Such a proposition will be a
concatenation of really simple signs, which have indeed an acci-
dental complexity, but one irrelevant to their function as signs.
2
THE THEORY OF DESCRIPTIONS
Wittgenstein's 'picture theory' of the proposition is much influenced
by Russell's Theory of Descriptions. According to that theory,
definite descriptions such as 'the author of Waverley', and 'the
present King of France', and again indefinite descriptions like *a
man' as this phrase occurs in 'I met a man', or 'A man has been
here', are not the designating expressions they at first seem to be.
At first sight, one readily assumes that, if the sentences in which
descriptions occur are true, each description stands for an object,
and the rest of the sentence expresses what holds of the object. To
say this is to compare descriptions with (real) proper names; but at
the same time the way in which descriptions stand for objects must
be different from the way in which proper names stand for objects;
indeed, the consideration of this leads to a breakdown of the idea
that descriptions 'stand-for' at all.
This is most obvious for indefinite descriptions; but is also true
of definite descriptions. A proper name will stand for its object
because that object is called by that name but a description, if it
;
stands for its object, does so because the object satisfies it, which is
clearly quite a different relation.
Further If a proper name (i.e. what has the superficial grammar
:
of a proper name) has in fact no bearer in the use that is being made'
of it, then nothing has been ascribed to any object by sentences in
which it occurs and so nothing has been said, truly or falsely. But
;
if it has a bearer (i.e. if it has the use, and not merely the superficial
grammar, of a proper name) then the sentence is false if what is
predicated in it does not hold of that bearer. Now if a sentence like
'Some man has been on the Moon* is false, this is not because *has
been on the Moon' is false of some man though if it is true, it is
41
42 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
true because 'has been on the Moon' is true of some man. So, if we
persist in thinking that the sentence would be made true by the fact
that something holds of what the grammatical subject stands for, it
turns out that its falsehood would not consist in the same thing's
not holding of what the grammatical subject stands tor.
When we turn to definite descriptions, it is easier to retain the
comparison with proper names; hence Frege called definite descrip-
tions proper names. But the comparison breaks down in various
ways. The predicate occurring as part of a definite description must
be uniquely true of something, if the description is to be taken as
standing for anything; whereas a proper name stands for a bearer
to which it has been assigned, without its being guaranteed, concern-
ing any given unique description, that the bearer satisfies it. Hence
we can give truth-conditions for statements containing definite des-
criptions regardless of whether the descriptions are vacuous or not.
It has been said (in the first instance by Frege) that the occurrence
of a vacuous definite description in a sentence disqualifies that
sentence from making a true or false statement. But this is unplausible
except when the sentence is a simple one. A
vacuous definite des-
cription can occur in a clause within a sentence without so disquali-
fying the whole sentence, e.g. 'Either he has no children or his
first child's name is Hilary.' All this shews that the object, if there is
one, satisfying a definite description, is not so designated by it that
nothing could be truly or falsely said by a sentence containing the
description if that object did not exist; whereas if Scott had never
existed, the use of the word 'Scott' as the name of that famous
author never could have existed either.
As a logical doctrine, Russell's Theory of Descriptions makes the
contrast between definite descriptions and (ordinary) proper names
which these considerations seem to demand. When doing logic,
Russell always treats e.g. 'Scott' as a proper name, by contrast with
descriptions like 'the author of Waverley'. His theory of knowledge,
on the other hand, leads him to propound the less convincing part
of the theory that ordinary proper names, like 'Scott', are not the
:
real proper names at all. Agenuine proper name must have a bearer;
this is a harmless point of logic; it becomes less harmless if it is
rendered as: What a proper name stands for must exist. For this
THE THEORY OF DESCRIPTIONS 43
may lead us to the idea that the bearers of the only genuine proper
names are existents not subject to Cartesian doubt (Russell's objects
of immediate acquaintance sense-data, etc.); or are eternal and
changeless simples (Wittgenstein's 'objects')-
Russell says that a definite description has no meaning by itself,
but the whole sentence in which it occurs has a meaning. He means
simply that a definite description does not function like a name. He
puts the point in that obscure way because of his idea of what it is for a
word or phrase to 'have meaning', namely: a word has meaning if
it is a word with which one means an object; to mean an object one
must be acquainted with it; for a word or phrase to have meaning,
then, it is necessary for what we mean by it to exist. In accordance
with this theory of meaning Russell passes from the truism that in
order to understand a sentence we must know what each word
means, to the doctrine that in judging or supposing we must be
acquainted with each of the objects that the judgment or supposition
is really about.
Thus, on Russell's view, if a description had 'meaning by itself,
itwould follow that what it stood for had some sort of being. In the
Principles of Mathematics Russell had actually thought that there
were entities 'meant' by descriptions, and called these entities
'denoting concepts'; not only definite descriptions, but such
phrases as 'any number' in 'Any number is either odd or even', had
denoting concepts as their 'meanings'. But what a proposition
containing a description asserted would ordinarily be asserted not
0/the corresponding denoting concept, but of a term or complex of
terms somehow connected with the denoting concept; e.g. it is not
the denoting concept answering to the phrase 'any number' that is
said to be odd or even. Russell's Theory of Descriptions represents
an escape from this position; he adopted Frege's way of handling
'some' and 'all', indefinite descriptions and phrases like 'any
number', and further applied it, as Frege never did, to definite
1
descriptions as well.
Frege's enquiries had already given the notion of proper names
an importance for logic and the theory of truth which it had never
1
See Chapter 11. cf. also Philosophical Writings of Gottlob Frege
(Blackwell, 1952), pp. 13-14, 16-20, 35-8, 93.
44 AN INTRODUCTTON TO WITTGENSTEIN'S TRACTATUS
had before. In Russell's theory it retains that importance, and at the
same time comprises only logically simple signs: *A "simple"
symbol is one which has no parts that are symbols. Thus "Scott" is
a simple symbol because though it has parts (namely separate
letters), these parts are not symbols. On the other hand "the author
of Waverley" is not a simple symbol because the separate words
1
that compose the symbol are parts which are symbols.' This
account of the simplicity of proper names is correct; as also it is
correct to say that the way a proper name contributes to the mean-
ing of a sentence in which it occurs is simply that it stands for its
bearer.
Russell analyses e.g. 'The author of Waverley drank port' as:
'For some x, x wrote Waverley and for all y, y wrote Waverley only
if y==x, and x drank port'; such an analysis of sentences containing
definite descriptions and other 'denoting phrases' excludes these
from the class of signs that contribute to the meaning of sentences
in which they occur by standing for their bearers. The denoting
phrases disappear, and only the predicates (and proper names, if
any) used in their construction play a part in the result of the
analysis. In consequence, 'standing-for' is shewn to be attributable
only to simple signs. 'Where,' Wittgenstein asked in a later writing,
'does language hook on to the world?' One place will be here, where
the proper name stands for its bearer.
In Russell's doctrine, a simple sign's having a meaning consists
in its standing for something; its meaning simply is the thing for
which it stands; and 'simple signs' will include not only proper
names of 'individuals', but also signs for 'universals' i.e. for rela-
tions, and for non-relational properties (if there are any) and per-
haps signs for logical forms as well; and these other simple signs
will equally have as their meanings the non-individual 'things' they
are signs for. Now, after Philosophical Investigations, it is easy to
distinguish between the two different elements of this doctrine
which we have mentioned: (a) the doctrine of the simplicity of the
proper name, and of its contributing to the meaning of the sentence
precisely by standing for its bearer; and (b) the idea that the mean-
ing of a name just is its bearer, or the meaning of a simple sign like
1
Introduction to Mathematical Philosophy, p. 173.
THE THEORY OF DESCRIPTIONS 45
'red' just is the quality with which we have immediate acquaintance.
At the time when Russell wrote, these elements were inextricably
conflated into one theory.
One of the most noteworthy features of Russell's doctrine is his
1
rejection of the Fregean distinction between sense and reference.
This distinction is highly plausible and tempting; but as an instru-
ment in the theory of truth it leads to great difficulties, and not to
the solution of our problems. Frege held that all symbols i.e.
both 'proper names' and predicates had 'sense', that some 'proper
names' had reference as well, and that all predicates had reference,
the reference of a predicate being what he termed a concept. (But
he does not explain the distinction of sense from reference for simple
predicates. ) An unasserted sentence (e.g. one occurring as a clause
2
in another sentence) is for him a proper name; if it has reference, its
reference is a truth-value. This raises the problem how it comes
about that certain senses namely those of sentences containing no
vacuous proper names are guaranteed to have reference.
On Russell's theory this problem does not arise; in any fully
analysed sentence there will occur nothing but words whose mean-
ings are present to us and are real things; for those meanings will
just be the 'things' (including relations, properties, and logical
forms signified by logical words and logical schemata) for which
the words stand. In his theory, we may say, 'language reaches right
up to reality', which is something we want to shew it doing.
But there are great defects in the theory as Russell states it, even
if for the moment we allow him to identify the meaning of a name
with its bearer. For Russell held that judgment and supposing are
(different) relations in which the mind stands to a set of objects
including a relation R; if R relates the objects other than inR
this set, then the judgment or supposition is true, and if not it is
false.
1
His detailed criticisms were, however, partly based on misunderstand-
ing: he wrongly assimilated Frege's views to what he had held in the
Principles of Mathematics.
2
For at least some complex predicates the distinction is easily made
out: the two predicates killed Socrates* and 'killed the philosopher who
4
was executed by a draught of hemlock' have different senses but the same
reference.
46 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
This theory (a) 'does not make it impossible to judge a nonsense',
as Wittgenstein complains at 5.5422; (b) fails to distinguish effec-
tively between judging (or supposing) that aRb and that bRa; and
(c) fails to explain negative judgments. For if when I judge that is A
to the right of B I stand in the judging relation to A, B, and the
relation to the right of, what happens when I judge that A is not to
the right of B? Do I stand in the judging relation to A, B, to the right
of, and not! Similar questions arise for the other logical constants,
'if', 'and', and 'or'.
This difficulty lies behind Wittgenstein's remark (3.42): Al-
though the proposition can only determine a single region of logical
space, still the whole of logical space must be given by it. Otherwise
negation, the logical sum,
1
the logical product,'2 etc. would
keep on introducing new elements in co-ordination' (sc. with those
previously introduced).
Wittgenstein avoids these difficulties, while retaining the idea
that the meaning of a simple sign is its bearer, by giving a different
account of propositions, judgments, and logical constants. On the
other hand he accepts Russell's Theory of Descriptions in its purely
logical aspect in so far as it shews how the analysis of propositions
into the complete statement of their truth-conditions is to be carried
out for propositions containing definite descriptions. The statement
of truth-conditions for a proposition containing a definite descrip-
tion thus includes a statement that there is one and only one object
satisfying the description, i.e. a statement that for some x, </>x, and,
for all y, <f>y only if y=x.
Let us now suppose that we have a proposition 'A is corrupt'.
Here *A' appears in the argument place in the function 'x is corrupt'.
Now let us consider the negation: 'A is not corrupt.' Can we dis-
tinguish between an external and an internal negation here i.e.
between taking this as the negation of the result of substituting 'A*
for Vin *x is corrupt' (external negation), and taking it as the result
of substituting 'A' for *x' in *x is not corrupt' (internal negation)?
If we cannot, then 'A' is a proper name; if we can, it is not. One sort
of case where we can is where 'A' is, e.g. 'Some committee' or 'Any
institution'. For these cases the point is readily seen (cf. Chapter 1,
THE THEORY OF DESCRIPTIONS 47
4
p. 35). But where A' is a definite description, the distinction be-
tween the internal and external negation still holds.
The question whether there must be simple signs in Russell's
sense thus leads us on to the question whether there must be substi-
tutions in 'x is corrupt' for which there is no distinction between
internal and external negation. Now if 'A' is an ostensibly singular
term, the distinction could arise only because the expression 'A*
itself indicated certain truth-conditions.
A proper name never does this; either one has to be told ad hoc
what, and what kind of thing, it is a proper name of; or one may
glean this latter information from the predicates associated with
the name, or guess it from custom: for example 'John' is custom-
arily a masculine human name in English-speaking countries
though this does not mean that a man makes a mistake if he calls
his sow 'John'.
Our question is Must there be (at least the possibility of) proper
:
names? And this question can be reformulated thus: Is it impossible
that, for any given f, every proposition which is a value of fx should
indicate truth-conditions C distinct from the conditions for the
holding of the property f ? Now the statement of the truth-condi-
tionsC will run: 'There is an x such that <x, and, for all y, <y only
if y=x'. But this could not be true, unless some singular proposi-
tion of the form *<b' were true. It might indeed be known, without
our knowing any singular proposition of the form '<b'; but if we
claimed to know it we should be postulating, even if we did not
know, a proposition of the form *<b'.
Thus we have
(1) fA, where *A' is of the form 'The <*.
(2) There is an x such that <x, and, for all y, ^y only if y=x.
(3) #b.
Now what do we postulate about the sign *b' in this postulated
proposition? Necessarily, that there is for it no distinction between
(~$)b and >-(<f>ty. For if there were such a distinction, the propo-
sition 'There is not an x such that <x' would in turn be ambiguous
in its truth-conditions: it might require that every proposition got
by substituting an expression of the kind 'b' in '(~<)x' shall be
true, or that every proposition got by substituting an expression of
48 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
the kind *b* in *$x' shall be false. There would thus be two quite
different ways in which '(Ex)^x' might be false, if the only substitu-
tions for x in '<x' were expressions *b* such that *<b' had different
possible negations.
So much follows from the logical part of the Theory of Descrip-
tions. This, however, does not lead us to 'simples' ; for the theory in
its logical aspect has nothing to do with any theory of reduction to
simples. It only demands that such simple symbols as 'Parliament'
shall be possible if such propositions as 'The body making laws for
Great Britain is corrupt' are to make sense. The type of name that
ispostulated here is the type: name of a body corporate. A truth-
condition for 'The British legislative body is corrupt' is There is an
x such that x corporately makes laws for Great Britain'. The variable
'x' here ranges over bodies corporate.
If there were no such things as bodies corporate, there would be
no proper names of them either; so it looks as if in that case 'There
is an x such that x corporately makes laws for Great Britain' would
lose its meaning. But in a world where there were no such things, it
might still be possible to imagine them. For a body corporate, e.g.,
to pass a measure by a large majority means that men stand in cer-
tain relations to one another and do certain things. If there were no
bodies corporate, someone might yet imagine certain men standing
in these relations to one another, might give an imaginary proper
name to the complex so formed, and might construct predicates that
were to hold of the complex when the individuals standing in these
relations did certain things. The proposition 'There is an x such
that x corporately makes laws' would then after all not fail to have
meaning.
Let us suppose that someone in a world without bodies corporate
has imagined there being such things, and has constructed this
proposition; there are now two quite different ways in which it can
be false, though not the same two ways as we considered before. It
is false in his world, because there are no bodies corporate at all
there. But in a possible world where there were bodies corporate,
there would be no distinction between 'All substitutions for "x" in
"x corporately makes laws" are false' and 'All substitutions in "x does
not corporately make laws" are true' these propositions would both
;
THE THEORY OF DESCRIPTIONS 49
be true or both false together, and if they were true then 'There is
an x such that x corporately makes laws' would be false.
Let us call the falsehood of this proposition in the world where
there are no bodies corporate 'radical falsehood*: the proposition
is radically false because certain propositions about men are false
they do not stand in such relations or do such things.
The question arises: could there always be the possibility of
radical, as opposed to ordinary, falsehood? It is clear that a 'radical
falsehood' always depends on the possibility of an ordinary false-
hood. Wittgenstein's starting-point is: We can construct proposi-
tions at will, without enquiry into any facts at all, and know what is
the case if they are true. On the supposition that the question of
'radical falsehood' can always be raised, we should always have to
distinguish between possible kinds of falsehood of our statements.
But then we could never determine the sense of the falsehood of
a proposition, except on the supposition of the truth of some prior
proposition for otherwise we should each time have to consider the
;
possibility of a radical falsehood, which must be explained in terms
of the ordinary falsehood of a prior proposition. Then we might
indeed start from the truth of certain propositions; but without this
we could never know the sense of any. 'Whether one proposition
made sense would depend on whether another one was true; so we
should not be able to invent a picture of the world (true or false)'
(2.0211-2).
Thus if we can construct propositions at will and know what is
the case if they are true, without knowing what is true and what is
false, it follows that there must be propositions incapable of what I
have called 'radical' falsehood. That is to say, there must be names
of simples which can only be named, and not defined\yy a description
as Parliament is, and whose existence is guaranteed. 'The demand
for the possibility of the simple signs is the demand that sense shall
be determinate' (3.23).
Here the 'simple signs' are not the 'simple symbols' of the Theory
of Descriptions, in its purely logical aspect as presented by Russell.
Wittgenstein shares with Russell the idea that the meaning of a
name is its bearer: but in him this is not noticeably based on the
British empiricist epistemology that influences Russell. We have,
50 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
rather, a Frege-like argument: Unless names have bearers, there is
no truth or falsehood. But if you always distinguish a sense and a
reference in names, as you must for a name like 'Parliament', the
connection between sense and truth-value becomes obscure. For
then the sense of a name will present a reference if something satis-
fies the description in which that sense might be set forth, i.e. if
something true; now this truth must be expressible by com-
is
bining a name and a predicate; and unless names are somewhere
nailed to reality without the mediation of senses which hold true of
objects, that relation between sentences and reality which consti-
tutes their truth will in no way have been explained.
NEGATION: (1) THE LOGICIANS' DEFINITION
OF 'NOT P'
'Everyone is unwise' is a negation of 'everyone is wise', but it is not
what logicians call the negation of it; in logic books, when the sign
for 'not' is introduced, we are told that 'not p' is "the proposition
that is true when p is false and false when p is true'. 'Everyone is
unwise' is not certainly true if 'Everyone is wise' is false; hence it
is notthe negation of 'Everyone is wise'. This was the point already
noticed by Aristotle in the De Interpretation >.
Such a definition of 'not p' as is found in many logic books may
make us ask (rather in the manner of Frege) what right anyone has
to give such a definition. I can define something as the so-and-so,
only if I am justified in being sure, first that there is a so-and-so, and
second that there is only one. If I have no such assurance, it is not
certain that I am succeeding in defining anything. How, then, am I
assured that there is one, and only one, proposition that is true when
p is false, and false when p is true?
It might seem that we could say: It is evident that a proposition
has two truth- values, as a coin has two sides; and we might think of
truth-functions as like bets on the results of tossing coins. Thus if
one coin is tossed once, there are two possible bets, which can be
represented as follows:
Coin Bet 1 Bet 2
H W L
T L W
in a table in which H and T stand for heads and tails, and W and L
51
52 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
for win and lose. This could be considered to correspond to the
following truth-table:
P P P
T T F
F F T
The analogy of course holds for the whole range of truth-
A and B, we could set
functions. For example, if we have two coins,
forth one possible bet as follows:
A B
H H W
H T L
T H W
T T W
Here the bet that the coins, in a single toss of both together, will fall
so that either A is tails or B is heads, corresponds to the truth-
function
p q or alternatively
T T T T
T F F T
F T T F
F F T T
~'p v q, or alternatively, to ~q v p; which, depends
that is to say: to
on whether we assimilate 'Heads' or Tails' to True'.
The most striking thing about this analogy is that when we set
forth the table of bets on the toss of a coin, we put different symbols
in the columns that display the bets from those we put in the
columns of possible results of the toss, whereas in the truth-table we
used the same signs in all three columns. The question arises with
what right, or on what grounds, or again, with a view to expressing
what, we use the same symbols in all the columns of the truth-table,
NEGATION: (1) 53
which we should not think of doing in all the columns of the betting
tables.
Consider the explanations of propositions and truth-functions,
or logical constants, which are commonly found in logic books. It is
usual for us to be told: first, propositions are whatever can be either
true or false; second, propositions can be combined in certain
ways to form further propositions; and third, in examining
these combinations, i.e. in developing the truth-functional cal-
culus, we are not interested in the internal structure of the com-
bined propositions. 1
Such explanations raise certain questions: e.g. has the internal
structure of the propositions, which does not concern us when we
study truth-functions, anything to do with the property of being true
or false? Again, is the property of being true or false, which belongs
to the truth-functions, the very same property as the property of
being true or false that belongs to the propositions whose internal
structure does not interest us? And, finally, if that is so, is it to be
regarded as an ultimate fact that propositions combine to form fur-
ther propositions, much as metals combine to form alloys which
still display a good many of the properties of metals?
In short, is there not an impression as it were of logical chemistry
about these explanations? It ,is this conception that Wittgenstein
opposes in the Tractates at 6.1 1 1 'Theories that make a proposition
:
of logic appear substantial are always wrong. It might be thought,
for example, that the words "true" and "false" denote two properties
among other properties, and then it would look like a remarkable
fact that every proposition possesses one of these properties.
This now looks no more a matter of course than the proposition
"all roses are either red or yellow" would sound, even if it were
true.*
Logical calculi are sometimes described as essentially sets of
marks with rules for manipulating them. For example Lewis and
Langford (Symbolic Logic, p. 227) say : 'Whatever more it may be, the
matrix method at least is a kind of game which we play with recog-
nizable marks, according to certain rules.' They then make some
1
See e.g. Hilbert and Ackermann, Mathematical Logic, p. 3.
54 AN INTRODUCnON TO WITTGENSTEIN'S -mACTATUS
remarks concerning an extract for a table for plq (which might
have the logical interpretation pDq, but of course need not):
P q Plq
1 1 1
1
They rightly assert that such a table need not have 'any "logical"
significance'; p and q may be 'any kind of things'. What is required
on their view is that in some game or other there should be 'an
operation or move, plq, which according to the rules can be taken
when p has the property A, only if q also has the property A'; and
in that case the table will tell us that if p has the property A, and plq
is an allowable move, then q must have A. The logical interpretation
will then consist in taking the property A, expressed by the figure
T, to be truth, and the property expressed by the figure '0' to be
falsehood, and reading *plq' as 'pDq'.
The animus behind such a view as this is a desire to get rid of the
notion of 'logical truth' in the mysterious character that it assumes
to someone with an empiricist outlook. But the argument presented
by Lewis and Langford fails, and, by the way in which it fails, helps
us to see the importance of the fact that we use the same signs, *T*
and 'F', or Tand '0', in all the columns of a truth-table. For as
has been remarked by Geach 1
there is an inconsistency here in the
interpretation of the figure 1, if p and q may be 'any kind of things'.
As regards p and q, the figure Tis taken to stand for some property
A, but as regards plq it is taken to stand for the property of being an
allowable move in a certain game. This is inconsistent unless A is the
property of being an allowable move and that is not necessary.
That it is not necessary is clearly seen if we take a simple non-
logical interpretation of the table. Let the figures T and '0' connote
the presence and the absence of an hereditary property A, and let
'plq' mean 'offspring by p out of q'. Then the table will have no
reference to moves in any game. It will state that the trait A is
present in the offspring, when it is present in the sire, if and only if
Jfs and Ands, Analysis, Vol. 9, 1948-9. This and the succeeding two
1
paragraphs are adapted from this article.
NEGATION: (1) 55
also present in the dam: a good example of what Wittgenstein
it is
would call a 'substantial' piece of information. Note that here plq is
not a 'move in a game' at all any more than p and q are; plq is an
animal.
Now though, as this example shows, you need not interpret the
figures T and 4
0' to mean that moves in a game are respectively
allowed and forbidden, it is of course perfectly permissible to do so.
But in that case p and q cannot be 'any kind of things' but must be
moves in the game, like plq. The table will then be equivalent to the
following sentence 'If p is an allowable move, then q is an allowable
:
move if and only if plq is also an allowable move/ And here the sign
T does not belong to the terminology of any special game, like 'Kt*
in chess; it expresses what we may fittingly call a logical relation of
the move plq to the moves p and q, so that the significance of the
notation 'plq' is after all 'logical'.
Now 'If p is an allowable move, then q is an allowable move if
and only if plq is an allowable move' at least sounds like a sub-
stantial piece of information about a game; at any rate if one can
specify p, q, and plq independently as moves: p might be a diagonal
move of a certain piece on a squared board, q a move of another
piece parallel with an edge of the board, and *plq' the name given
(we won't ask why, for the moment) to a move by yet another piece
along one of the edges of the board. Then the information 'If you
can move this piece diagonally you can move this piece along the
edge of the board if, and only if, you can move this piece parallel
with one of the edges' might either be a rule of the game, or inferable
from the rules of the game.
If, however, we now do ask why moving this piece along the
edge of the board should be symbolized by a sign mentioning those
two other moves, then we can answer by saying that
P q
1 1 1
1
defines a possible allowability in terms of given allowabilities, and
'plq' is a notation in which this allowability is set forth: that, and
56 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
that alone, is the meaning of this notation. Then the substantial
information as far as this game is concerned is that there do exist
some moves whose allowability is so conditioned, and that the move
along the edge of the board is in fact one of them but that any move
;
describable as plq is allowable, if p is allowable, if and only if q is
1
also allowable, is not a 'substantial' piece of information.
Similarly, if we revert to the analogy of tossing coins, we set forth
part of the betting table we have already considered:
Coin A Coin B Bet: A tails or B heads
H H W
H T L
and it is not a substantial piece of information that this is a possible
bet on the result of a toss of two coins, each with two possible sides
to come uppermost. Nor is it informative to say that this is the bet
*A tails or B heads'; that is simply another way of writing what is
already written down in the WL column one could simply point to
;
the column and say: That's my bet.
And so Wittgenstein says that such a sign as
P q
T T T
FTP
T
F
F
F
F
F
or, assuming a standard convention for the p and q columns,
'(TFFF) (p,q)' is a propositional sign: we find this statement at
4.442. '(TFFF) (p,q)' is just another way of writing 'p.q'.
This is the explanation of the symbolism introduced at 5.5:
The Greek letter f is a variable whose value is a proposition; the
1
The reader must be careful, in reading this passage, to distinguish
between *plq* and plq. plq is a move in the game, 'plq' a notation for the
move.
NEGATION: (1) 57
dots after it indicate a set of such variables of unspecified length. The
dashes in the left-hand bracket indicate an absence of Ts in the
truthrtable, however long this may be: its length in any given case
willdepend on how many propositions are indicated in the right-
hand bracket. The point is that only the bottom row of the matrix
has T set against it: this is the case in which all the propositions
indicated in the right-hand bracket are false; this combination is
to stand at the bottom, whichever of various possible conventions is
adopted for arranging the other possible combinations of truth- values
in the matrix. Thus Wittgenstein's formula '( T) (f
1
)'
is the negation of all the propositions in the right-hand bracket
Now in the coin-tossing case, the 'substantial* information is that
the toss of a coin has two possibilities, heads and tails. And as we
have seen, this substantial fact has an analogue in the opening ex-
planations of logic books: that propositions are what can be true or
false. We might say that for coin-tossing purposes a coin has only
two possibilities of falling: if a coin e.g. stood on its edge when it
reached the floor, that wouldn't count as a toss. And similarly if
(for whatever reason) a sentence hasn't got a truth-value, it doesn't
count for making statements with or for operating the truth-
functional calculus with.
Frege allows such sentences if a sentence is a fiction, it has not
:
got a truth- value; and it is a fiction if it contains empty names. It can
still have a perfectly good 'sense', but not have a truth-value. In our
day, Mr. P. F. Strawson has also introduced a concept of a sentence's
having a sense, which is not sufficient to guarantee that if it is uttered
a statement is thereby made. We know its sense, if we know in what
circumstances it could be used to make a statement. Now, apart
from sentences containing fictitious proper names, Frege found his
view inconvenient; he regarded the possibility of constructing such
2
sentences as a defect in the language of (a priori) science. So when
1
This is used rather thanV because 'p' is generally used for an elemen-
tary proposition, and there is no requirement here that the value of 5 be
elementary.
2
In an empirical science, such as astronomy, the possibility cannot be
regarded as a defect of language, or legislated away. e.g. at one time it was
thought that there was an extra planet, which was called 'Vulcan*. On
Frege's view propositions about Vulcan could not have a truth-value.
58 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
in developing the foundations of mathematics he needed to use a
descriptive phrase which as ordinarily interpreted might have no
reference, he used an artificial reinterpretation to guarantee that it
had a reference; 1 and this artificiality, as Russell remarks, is an
objection to his procedure. Mr. Strawson's own suggestion has not
been sufficiently worked out for us to estimate its value.
It is well known that Russell and Wittgenstein were on the other
side of this fence; for Wittgenstein 'having a sense' was one and the
same thing with being true-or-false. We have already seen this at
3.24: 'The proposition in which a complex is mentioned does not
become nonsensical if the complex does not exist, but simply false.'
And we see it again at 4.063, where he develops an illustration of the
concept of truth by a black spot on white paper: black corresponds
to true and white to false. If you indicate a point on the surface, that
is like pointing to what Frege calls a 'thought', or the sense of a
sentence; and you are, of course, pointing to something that is in
fact either black or white. But, Wittgenstein says, the point at which
the illustration goes lame is this: you can indicate a point on a sheet
of paper without so much as having a notion of black and white;
what would correspond to this would be indicating a thought with-
out so much as having a notion of true and false: 'but to a proposi-
tion without a sense there corresponds nothing, for a proposition
doesn't designate an object with the properties called "true" and
"false" ;' as, say, the description of a point designates an object with
the properties called 'black' and 'white'. That is to say, unless the
proposition is already something true or false, he calls it something
'without sense'.
Again, at 4.064, we find Wittgenstein saying: 'Every proposition
must already have a sense; assertion cannot give it one.' Since this
is an attack upon Frege, it may well confuse a reader; for of course
Frege would agree that every (well-formed) sentence must already
have a sense! But Wittgenstein holds that what already has a sense
1
In school mathematics one is told that *- does not mean any number
when y 0; a reinterpretation in Frege's style might stipulate that when
X X
O, -is the number z such that zy=x, and when y=0, =x.
NEGATION: (1) 59
must already be true or false; he is attacking Frege's idea that in
judging we 'advance from a thought to a truth-value'.
1
Wittgenstein remained on this side of the fence all his life; for in
the very passage of Philosophical Investigations in which he attacked
the ideas about complexes which he expounded in the Tractatus, he
asked 'Am I really prepared in advance to say what, and how much,
:
has got to turn out untrue before I give up my proposition about
Moses asfalseT The kind of thing he has been considering has been
facts that, taken together, would tend to shew that there was no such
person as Moses; and he spoke of giving up the proposition about
Moses, not as neither true nor false, but as false.
It will be worth while to say a few things about the Frege-
Strawson side of the fence. First, Frege was sure that a well-formed
sentence whose names were not empty had a truth-value. But is it
not strange to be sure of that? Is it not as if there were a great metal
wall with holes in it, and we had some way of casting metal objects,
and were absolutely certain that each object that was properly cast
would fit into a hole in the wall one way up or the other (the well-
formed proposition or its negation is true) although no connection
had been shewn between the principles of casting objects and the
character of the metal wall? The fact that Frege's account makes
things look like this is a sure sign that he has gone wrong, like the
accounts of *true' and 'false' which make 'Every proposition is either
true or false' like 'Every rose is either red or yellow'. Frege's reply
to this would be that a sentence is only well-formed if the concepts
itemploys are sharply defined, and a concept is sharply defined if
it isdetermined for every object whether it falls under that concept
or not. The problem now assumes the form how does it come about
:
that we can form concepts for which this is determined, without any
reference to the facts?
Secondly, Frege actually said that the truth-conditions deter-
mine the sense of a proposition. He specifies the truth-conditions,
and therefore (since he is working in an a priori discipline) the
truth-value, which on his theory is also the reference, of any well-
formed formula in his system; he then adds, as if to anticipate the
1
See 'Sense and Reference', in Philosophical Writings of Gottlob
Frege, p. 65.
60 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
objection that he has only specified the reference and not the sense,
that the sense of such a formula is the sense of this that its truth-
:
conditions are fulfilled. And to this, mutatis mutandis, we may see a
correspondence in Mr. Strawson's 'knowing in what circumstances
the sentence could be used to make a statement'. The propositions
embodying the truth-conditions, or describing the circumstances in
which a sentence could be used to make a statement, must themselves
be either true or false, or require explanation in terms of further
truth-conditions, or further circumstances. In view of this, the Frege-
Strawson position on the possibility of sentences without truth- value
appears to be a waste of time: in such an account the concept of
'sense' is not divorced from those of truth and falsehood; it is merely
determined that when certain of the truth-conditions of a proposi-
tion are false we are to say that 'nothing either true or false has been
said'.
We have observed that the most striking difference between the
coin-tossing tables and the truth-tables is that in the former we use
different symbols in setting out the possible results of coin-tossing
and the possible bets on these results. Now let us suppose that we
have a coin with 'win' written on one side and 'lose' on the other,
so that we said that we bet on the coin's coming up 'win' or 'lose'
rather than 'heads' and 'tails'. Then a bet that the coin would 'win'
a bet, so to speak, in agreement with the coin on the one hand,
and a bet that the coin would lose, would be exactly comparable
to the two truth-functions of a single proposition:
P
T T F
F F T
The objection immediately arisesfthat while we have a good sense
for 'winning' and 'losing' in connection with a bet on whether the
coin will fall one side up or the other, there really is no sense in these
terms as applied to the sides of the coin themselves, except that we
happen to write this sign on one side and that on the other. The signs
are not really the same; any more than 'jam' is the same word in
English and in Latin. Now that may be so: but we have in fact
NEGATION: (1) 61
already encountered a parallel difficulty in connection with using
the word 'true' for the elementary propositions and for the truth-
functions. It may be intuitively obvious that there is no equivoca-
tion; and it is certainly extremely natural to give the explanations
found in the logic books and then simply get on with the calculus.
But our questions were reasonable ones; if there is no way of
answering them, and we have just to rely on our intuition, that is of
itself an important fact at any rate, it is important for philosophy.
The Tractatus, however, does attempt to give an answer to these
questions other thai* that the correctness of these two uses of the
words 'true' and 'false' is intuitively obvious. I opened by raising
the question: If we offer a definition of 'not p* as 'that proposition
which is true when p is false and false when p is true' how can this
be justified if we are not assured that there is such a proposition, and
only one? Now grounds are given for saying that there is only one
such proposition at 5.513 'It could be said that what is common to
:
What
all symbols that assert p as well as q, is the proposition "p.q."
is common to all symbols that either assert p or assert q, is the
proposition "p v q".
'And in this way it can be said: Two propositions are opposed
to one another, if they have nothing in common, and: Every propo-
sition has only one negative, because there is only one proposition
which is wholly outside it.*
In the first of these paragraphs, we must understand that it is the
1
propositions 'p.q' and *p v q that are being explained in terms of
what is common to a class of symbols. We
have already seen Witt-
genstein saying (3.341) that the essential thing about a symbol, or
the real symbol, is what all symbols that do the same job have in
common. If then there is anything that a set of propositions all say,
then what is common to that set of propositions he calls the real
symbol for the thing that they all say. So you might say that a set of
propositions, 'A is red*, *A is green', 'A is blue*, etc., have something
they all say, namely: *A is coloured.' And the 'real symbol' for 'A
is coloured' will be what is common to the propositions 'A is red%
A
4
is green', 'A is blue', etc.
Now any set of independent propositions has something that
they all say; what this is is brought out by writing out a truth-table
62 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
where all places but the bottom are marked T, the bottom place
having F in it opposite the row of F's of the matrix. This truth-table
specifies a proposition, which is made true by the truth of any one of
the components; hence I call it something that they all say. And
according to Wittgenstein's dictum, the 'real symbol' for this will be
what is common to them all. But the proposition in question is of
course the disjunction of them all: and hence the 'real symbol' for
e.g. p v q is what is common to all propositions* that either assert
p or assert q.
How the common thing about a set of symbols, and hence the
'real symbol', is to be described is a matter of which the Tractatus
gives an account in the range of entries under 3.31, that is to say,
3.31-3.318, and there is further matter relevant to the specification of
sets of propositions at 5.501. Let us assume that the account is
satisfactory. For our present purpose is to shew how Wittgenstein
proposes to justify our assurance that every proposition has only
one negative.
'Two propositions are opposed to one another if they have
nothing in common.' That is to say, if there is nothing that they
both say. In this sense 'The King of France is bald' and The King
of France is not bald' are not opposed to one another (if the latter
is the result of substituting 'The King of France' for 'x' in 'x is not
bald'), for there is something that they both say. This comes out in
the fact that 'The King of France is bald or the King of France is
not bald' may not hold, namely if there is no King of France; the
proposition therefore asserts something, viz. that some one of the
situations in which it would hold is actual; it asserts something be-
cause these situations are not an exhaustive list of all possible
situations; it excludes the situation in which it would not hold.
He goes on to say: 'Every proposition has only one negative, be-
cause there is only one proposition that lies wholly outside it.' Let
us test this by supposing that there might be two. Ex hypothesi, these
two must have different senses, i.e. it must be possible for one to
hold and the other not. Let us write them as not-p (1) and not-p (2).
Then the disjunction of p and not-p (1) could be false, if what held
were not-p (2). So the disjunction of p and not-p (1) does assert
something, and there is something asserted by both p and not-p (1),
NEGATION: (1) 63
namely that some one of the situations in which the disjunction
would hold is actual. It follows that there can be at most one propo-
sition that has 'nothing in common' with any given proposition.
There can of course be many propositional signs for this proposition,
but their sense will all be the same.
That there can be at most one proposition of this character for
any given proposition does not shew that there is one; and we must
next shew how this is made out in the Tractatus.
4
NEGATION: (2) THE PICTURE THEORY
We have been troubled by the procedure of the logic books in e.g.
placing the same signs, T and F, under the signs of the elementary
proposition in the truth-table and in the final column, and by the
justifications of this procedure, which consist in quasi-factual pro-
nouncements. Let us now consider remedying this procedure. I adopt
two new signs, P and N, which mean 'Positive' and 'Negative',
which I put under the signs for the elementary propositions:
P
P T F
N F T
The proposition which is an element in a truth-function is thus
introduced as having two senses, the positive and the negative, rather
than two truth- values, true and false.
Now there is actually some foundation for looking at it like this,
in the Tractatus itself. At 4.463 Wittgenstein writes: 'The proposi-
tion, the picture, the model, in the negative sense are like a solid
body, which restricts the free movements of another; in the positive
sense, like the space limited by solid substance, in which a body may
be placed.' Here at any rate a proposition, as well as a picture or
model, is conceived as something that can have both a positive and
a negative sense.
As far as concerns a picture, this is quite reasonable. It is in fact
connected with one of the objections that it is most natural to feel to
Wittgenstein's 'picture' theory of the position. A picture is not like
a proposition it doesn't say anything.
: A picture is not an assertion
64
NEGATION: (2) 65
that something like it is to be found somewhere in the world, where-
as in a proposition something is said to be the case. If we accept
Wittgenstein's dictum at 4.022 that 'A proposition shews how things
are if it is true. And it says that they are so\ we might say: 'Just this
shews the difference between a proposition and a picture; for while
a picture may be said to shew how things are, //there is something
it is a correct representation of, it certainly does not say that that is
how things are; the most that one could grant would be that we
could use the picture in saying how things are we could hold the
:
'
picture up and ourselves say: "This is how things are."
Now in fact this is Wittgenstein's point. For in order to be able
to do this in a quite straightforward sense, it is necessary that the
elements of the picture should be correlated with objects.
For example, here is a picture:
and if I have correlated the right-hand figure with a man A, and the
left-hand figure with a man B, then I can hold the picture up and say :
'This is how things are.' But I can just as well hold the picture up
and say: *This is how things aren't.'
If you could not do this with, say, the figures drawn or painted
on a piece of paper, once they had been correlated with actual people
or objects, then what was on the paper would not be a picture, but
a set of figures each of which was correlated with some object.
I may for example draw a figure here
66 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
and call it 'Plato'; and then draw another figure here
and call it 'Socrates' the two figures do not together constitute a
;
picture, because although of course there is a relation between them
they are, say, a certain distance apart on a single leaf this rela-
tion is non-significant. Whereas in the drawing of the two men
fencing the relation of the ink strokes constituting the drawing of
the first man and those constituting the drawing of the second man
was significant.
The isolated figures labelled 'Plato' and 'Socrates' each consist
of strokes in significant relationship, and hence it seems reasonable
to speak of correlating such figures, one with one person, another
with another. But if I just put a single stroke
and then another stroke
the sense that there was in saying 'correlate the first mark with one
person and the second with another' would vanish. If someone said
this, we should wait for something to be done with the strokes; we
might think that this announcement was a preparation for some-
thing; unless it is that it is not anything at all. Suppose I said: 'That
door stands for Dante and that table for Bertrand Russell'. My
audience would, if anything, look at me enquiringly and say: 'Well?'
And here 'Well?' means 'Do something to shew the point of
this'; and that means 'Let something else come into such a relation-
ship with this door, or again with this table, that the terms in relation
NEGATION: (2) 67
represent something.' We
could say: 'Only in the connections that
make up the picture can the elements of the picture stand for
objects.'
The picture-theory of the proposition is that the proposition in
the positive sense says: 'This is how things are' and in the negative
sense says: 'This is how things aren't' the 'this in both cases being
9
the same the comparison is a comparison with a picture of the 'this*
:
in question. It is because of the character of the 'this' that there is the
possibility of saying 'it's how things are' or 'it's how things aren't'.
And this character is in pictures, ordinary pictures, themselves all
that is required for the possibility to be actualized is that their
figures be correlated with objects. This begins to tell us why Witt-
genstein says at 2.182 that 'Every picture is also a logical picture';
and at 2.1514 that 'the picturing relation consists of the co-ordina-
tions of the elements of the picture and of the things'.
The quite straightforward possibility of doing this depends on
the correlations' having been made; now this correlation is in one
way quite external. The picture of two men fencing was intelligible
as a picture, without our making any correlations of the figures with
individual men. We might compare to this picture, without indi-
vidual correlations, what Wittgenstein at 3.24 calls the 'proto-
picture' occurring in the generality notation: the 'xRy', for example,
in '(Ex)(y)xRy\
What I have called the externality of the correlations between the
elements of a picture and actual objects is an important feature of
Wittgenstein's account. Giancarlo Colombo, s.j., the Italian trans-
lator of the Tractatus, commented on Wittgenstein's theory of the
'isomorphism', as it is called, between language and the world, that
itwas difficult to see why a described fact should not be regarded as
itself a description of the proposition that would normally be said
to describe it, rather than the other way round. And as far as con-
cerns the internal features of proposition and fact, this is a strong
point; for all the internal features are supposed to be identical in the
proposition (or describing fact) and the described fact.
But after having stated at 2.15 that the way the elements are
connected in the picture is the same as the way it sets forth the things
as being connected, Wittgenstein goes on to compare it to a ruler
68 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
which you set up against an object (2.1512-2.15121) and then says:
*
(2.1513) According to this conception, the picture must have in
addition the depicting relation which makes it into a picture' l and, ;
as we have already seen, this depicting relation consists of the
correlations with objects (2.1514).
Thus there are two distinct features belonging to a picture (in
the ordinary sense of 'picture') :
first, the relation between the ele-
ments of the picture; and second, the correlations of the elements in
the picture with things outside the picture; and as we have seen, the
first feature must belong to a picture before the second one can only ;
if significant relations hold among the elements of the picture can
they be correlated with objects outside so as to stand for them. The
correlating is not something that the picture itself does; it is some-
thing we do.
We see this at 5.4733, where Wittgenstein says: 'Frege says:
Every well-formed proposition must have a sense; and I say: Every
possible proposition is well-formed, and if it doesn't make sense,
this can only come of our not having supplied any reference for some
of its component parts.' What Wittgenstein means by 'Every
possible proposition is well-formed' is that the relations that must
hold between the elements if a sentence is to be a sentence at all
must be there also in any nonsensical sentence, if you could make this
have a perfectly good sense just by changing the kind of reference
that some part of the sentence had. Here it is 'we' who 'give' a sign
its reference.
This is why at 3.13 Wittgenstein says that *A proposition has in
iteverything that a projection has; but it hasn't got the projected
thing in it; so it has the possibility of the projected thing in it, but
not the very thing itself: And so the proposition does not yet contain
its sense; what it does contain is the possibility of expressing that
1
Ogden's rendering of this sentence: 'the representing relation also
. . .
belongs to the picture' can be misleading. There is evidence in Ramsey's
review of the Tractatus (since Ramsey helped with the translation) that it
was intended in an incorrect sense. Ramsey says that the elements 'are
co-ordinated with the objects by the representing relation which belongs
to.the picture' (Foundations of Mathematics, p. 27 J). This interpretation
throws Wittgenstein's quite straightforward idea into obscurity; the sen-
tence has no such obscurity for educated native speakers of German.
NEGATION: (2) 69
sense. ... It contains the/orm of its sense, but not its content.' The 1
reason why the proposition doesn't 'contain its sense' is that the
correlations are made by us; we mean the objects by the components
of the proposition in 'thinking its sense' : this is part of what is meant
at 3. 1 1 : 'The method of projection is the thinking of the sense of the
proposition.' It is we who 'use the sensibly perceptible signs as a
projection of a possible state of affairs'; we do this by using the
elements of the proposition to stand for the objects whose possible
configuration we are reproducing in the arrangement of the elements
of the proposition. This is what Wittgenstein means by calling the
proposition a picture. It is at any rate clear enough that we could
use a picture in this way.
Now, confining ourselves to pictures, it is also clear that if we
'think2 the sense of the picture' by correlating its elements with
actual objects, we can in fact think it in either of two ways: namely
either as depicting what is the case, or as depicting what isn't the
case. That is to say, there are two senses which we can 'think' in
connection with the picture. For it is the very same picture we hold
up if we wish to say that // holds or that it doesn't hold. Or again, if I
hold up a picture and say 'If I correlate the elements of this picture
with things, I can say something by holding it up', someone might
reply: There are two things you could assert in holding the picture
up : first the existence, and second the non-existence, of that situation
which is represented by the picture so soon as its elements are
correlated with objects. And the difference between the two is not
that the relations between the elements are taken to be different; on
the contrary, they are exactly the same.
It is clear that one must convey what situation one is saying does
not exist, and this will be conveyed precisely by the picture depicting
that situation. No other picture could be involved: you could not for
example make a picture of the situation's not existing. We must be
1
Wittgenstein's use of 'projection' is a metaphorical extension of the
mathematical use, which may be explained thus: "The drawing of straight
lines through every point of a given figure, so as to produce a new figure
each point of which corresponds to a point of the original figure.' The new
figure is also said to be a projection 0/the original one, which is projected
into it (cf. The Shorter Oxford Dictionary).
2
This is a Germanism which it seems necessary to retain in English.
70 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
careful not to confuse what is not the case with what is the case
instead of it; if you tried to make a picture of a situation's not
existing you would only make a picture of what did exist instead of
it. The only exception to this is when we have the convention that
not shewing something shews that the thing does not exist: as when
a map shews that no large river passes through Birmingham by not
shewing a river passing through Birmingham.
These, then, are the reasons for speaking of a picture as having
or rather being capable of being given a positive and a negative
sense. The two senses are integral to the picture, once the correla-
tions have been established. Certainly a picture whose 'sense' is
'thought' one way or the other, as I have described, is a preposi-
tional sign.
What is mysterious about negation is that something's not being
the case should be capable of being something that is the case; 1 and
it isa peculiarity of a picture of something's being the case that it
can be taken as presenting us with something that is the case by
being a picture of what is not the case. In his notebooks,. Wittgen-
stein speaks of logical constants as giving the method of projection
of the proto-picture in the proposition. I think this conception is not
discarded in the Tractatus.
In the course of his researches prior to writing the Tractatus,
Wittgenstein invented what he called the a-b notation. He proposed
to write a proposition like this:
apb
the a and b being what he called the 'two poles' of the propo-
sition. This notation has survived in the Tractatus at 6.1203, except
that he writes T and F instead of a and b. But we could represent the
prepositional sign that is a picture (of the most ordinary kind) in
the same way:
1
cf. Philosophical Investigations, 429: *The agreement, the harmony,
of thought and reality consists in this: if I say falsely that something is red,
then, for all that, it isn't red.' The problem is the ancient one of how a
false proposition makes sense.
This rendering of the picture become proposition would stress the
fact that it has acquired two 'poles', or senses in which it can be
thought, by having the drawn figures correlated with actual men.
Now the question is: does this conception give us what is
essential to propositions, so that it is at all plausible to say that all
propositions have this character?
I believe that the most that we can say is that the bi-polarity of
the picture, of the occurrence of one picture in two senses, has a
very striking analogy in the fact that if we have a proposition, and
insert a 'not' into it, then what is being denied is exactly what the
original proposition said. In negating a proposition we use the
prepositional sign to form another, and we tend to feel that both
say something: and hence want an account that would justify this
feeling. Both propositions mention exactly the same things in the
same relation to one another. The picture-proposition we have
imagined gives us a very clear idea of structures for which these
points hold. And it is also true that the non-existence of a con-
figuration of things is a clear and intelligible idea. What constitutes
the truth and falsehood of the picture-proposition; its opposed
positive and negative senses; its possession of these senses inde-
pendently of whether it is true or false (i.e. of which truth-value it
has) all this is extremely intelligible: and what is intelligible here is
precisely the logical character of the picture-proposition. But is it not
the logical character that marks a proposition as such and that we
want clarified? Something that seems to make this really clear might
rather convincingly be taken to shew the essential character of a
proposition.
So far as I can see, these are the real grounds for being struck
even to the point of conviction by this account. It adds to its per-
suasiveness that it was capable of being further, and beautifully,
thought out, and that it seemed to offer a solution to many problems,
71
72 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
and finally even give a 'way of seeing the world rightly'. There are
indeed serious difficulties about it; nevertheless, we shall understand
the Tractatus best if we let ourselves succumb to the attractiveness of
this idea, assume its correctness, and follow up its consequences
throughout the Tractatus.
Every picture-proposition has two senses, in one of which it is a
description of the existence, in the other of the non-existence, of a
configuration of objects; and it is that by being a projection. It is the
peculiarity of a projection that from it and the method of projection
you can tell what is projected; the latter need not physically exist,
though the points in space that it would occupy must. The idea of a
projection is therefore peculiarly apt for explaining the character of
a proposition as making sense independently of the facts: as intelli-
gible before you know whether it is true; as something concerning
which you can ask whether it is true, and know what you are asking
before you know the answer.
If this explanation can be made to stick it will make the character
of a proposition completely clear. For supposing TpF and TqF to
be picture-propositions, then someone who says TpF' will be saying
(let us suppose) that the situation pictured by 'p' exists; he can say
itdoes not exist by reversing the T and F poles of TpF' a pro-
cedure represented by Wittgenstein with: 'F-TpF-T'. The diagram
gives a picture-proposition, whose T pole is joined to a line joining
the two T poles, and also to a line joining the two F poles, of TpF'
and TqF', and whose F pole is joined to a line joining the T pole of
TpF' with the F pole of TqF', and again to a line joining the F pole
NEGATION: (2) 73
of TpF' with the T pole of TqF'. This proposition is true if the
situations pictured by 'p' and *q' both exist, or again if neither exists;
otherwise false. This way of writing the prepositional sign brings it
out that nothing is mentioned but the objects correlated with the
4
elements of p' and 'q' and no configurations introduced except those
set forth in *p' and 'q'.
The propositional sign written here in diagram form is the same
proposition as what we would most commonly write (p.q) v (~p.
~q). This brings out why Wittgenstein says: The structure of the
fact consists of the structures of the elementary situations' (2.034);
'The picture presents ... the existence and non-existence of ele-
mentary situations' (2.1 1); The truth-conditions determine the play
left to the facts by the proposition' (4.463);'A proposition may be
an incomplete picture of a particular state of affairs' (in this case,
say, the state of affairs that neither p nor q) 'but it is always a com-
plete picture' (5.156). For his idea was that definiteness of sense
consists in this: a proposition may indeed leave a great deal open,
but it is clear what it leaves open.
We must now return to the fact that a picture (in the ordinary
sense) becomes a proposition the moment we correlate its elements
with actual things. To shew this, I draw it shewing the T-F poles
that it gets directly the correlation is made:
We must always remember the condition for the possibility of the
correlation: namely that the arrangement of the ink strokes them-
selves is significant, is capable of picturing a situation if the correla-
tions are made. 1
1
There is something that I slur over here for purposes of a first rough
exposition and that is the difference between the significant relations of the
:
mere ink strokes, and the relations of the drawn members of the drawn
figures and of the drawn figures among themselves. This corresponds to
the difference between the significant relations between the sensibly per-
ceptible signs, and the relations between the symbols. See 3.326.
74 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
Now, that some arrangement of shapes on a surface is capable of
being a projection of another arrangement of shapes on a surface is
obvious from their both being arrangements of shapes on a surface.
Wittgenstein calls being spatial (or again being coloured) a 'form',
and expresses the point by saying that a picture can depict anything
whose form it shares: the spatial picture can depict the spatial, the
coloured picture anything coloured, etc.
By analogy with this conception he erects one of 'logical form*.
As children we sometimes amuse ourselves by drawing imaginary
maps. Given the method of projection a person could say what the
imaginary country's coasts would be like. But that is not to say that
the imaginary map is already a true or false map of any actual
coastline. But we might say it becomes a true or false map of the
world given the method of projection the moment we pin it to
any actual place by correlating some of its points with places on the
globe.
This is to suppose that we call it a true map if, correlating one
point on the drawn coastline with one point on the globe, and
another with another, the projection of the drawn coastline coin-
cides with an actually existent coastline; and otherwise we call it a
false map of that part of the globe. These facts, however, do not imply
that, supposing it to be a false map in the sense I have defined, there
is another method of projection for that map which will make it a
true map of a coastline, the same points on the map and on the
globe being correlated. After all, the originally selected points might
be in the middle of a great ocean, in which case no method of pro-
jection would make the imaginary map a true map of a coastline.
Thus we can consider the T and F poles of the picture-proposition
as giving two senses, positive and negative (as it were, the different
methods of projection), in which the picture-proposition can be
thought. Now although a map is a picture-proposition once a
method of projection and correlations have been established, it is
not so simply in virtue of its spatial form. For saying *It's not like
this' is using the map to say something true; but it is not giving a
correct map. If then the positive and negative senses are compared to
different methods of projection, then it is not the spatial form, but
something comparable to spatial form, that makes the map into a
NEGATION: (2) 75
picture-proposition when correlations are established. This Witt-
genstein calls 'logical form'.
It is obvious enough that a proposition divides up into parts. It
is also obvious that the division is not arbitrary. You cannot divide
'The cat is often drunk' into 'The cat is of and 'ten drunk', although
each part could be significant: as Wittgenstein would say, the
first expression's standing to the left of the second is not what
signifies in our sentence, and (3.314) 'An expression has reference
only in the context of a proposition'. (It was of course on this pro-
nouncement, and on that of Frege, repeated by Wittgenstein, that
*A name has reference only in the context of a proposition* that I
modelled my statement about pictures: 'It is only in the connections
that make up the picture that its elements stand for objects.')
Propositions thus have a feature that is very comparable to a
We call the possibility of the kind of connection
feature of pictures.
that sets up a proposition 'logical form', as the possibility of any
particular spatial arrangement can be called spatial form. And since
logical form is that through which a structure can have T and F
poles, and for something to be true or false is the very same thing as
for reality to be thus or otherwise, Wittgenstein calls 'logical form*
also 'the form of reality'. Thus he provides a distinctive new solution
to the old old problem of shewing the connection between thought
and reality. That the logical form is common to reality and the
propositions is a further analogue to the way the spatial form is
common to the spatial object and its spatial picture: *A picture can
depict any object whose form it has.*
The notion of logical form leads straight to that of logical space.
We can construct a spatial illustration a bit like Wittgenstein's black
spot on white paper, but one which 'goes lame* at a different point.
If you consider an island marked on the surface of a sphere, it is
clear that it defines not merely its own shape but the shape of the rest
of the surface. A proposition is to be compared to such an island, its
negation to the rest of the surface.
Let us say that you illustrate the concept of truth by painting the
island white and the rest of the surface black, to correspond to
calling proposition true and its negation false; if on the other hand
a
it is the negation that is true, the island is black and the rest of the
76 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
surface white. Obviously you could do this with a real globe; and any
map, real or imaginary, would divide the globe. Only as we saw, the
divisions would not necessarily correspond to any actual coastlines.
But the division made by the two senses of any proposition is a division
of truth from falsehood; each coastline partitions the whole earth's
surface, so each proposition 'reaches through the whole of the
logical space'. But it is a proposition precisely by making a division
of true from false. Now let us represent the proposition saying that
either this or that is true, by a new globe with both the corres-
ponding areas white; what corresponds to saying that either a
proposition or its negation is true is painting the whole surface of
the globe white in which case you have no map. And similarly for
painting the whole surface black, which would correspond to 'not
(p or not p)'. But it is clear than an all-white or all-black globe is not
a map.
So when as sometimes happens in old-fashioned philosophical
textbooks the laws of contradiction and excluded middle are laid
down as truth with which reason starts, this my be compared to the
admiration of the sailors for the Bellman's map in the Hunting of the
Snark:
'Other maps are such shapes, with their islands and capes !
But we've got our brave captain to thank'
(So the crew would protest) 'that he's bought us the best
A perfect and absolute blank !'
At 4.463 Wittgenstein uses a similar but double analogy. He
says that the proposition in the positive sense is like the space in
which a body can be placed; in the negative sense it is like a solid
body which prevents any body from being placed in the space it
occupies. Now since any proposition p divides the whole space, then
the positive proposition 'p or not p' leaves the whole space empty,
both the island indicated by p and the rest of the space; and its
negative 'not (p or not p)' blocks the whole space.
The point at which such analogies go lame is that e.g. a globe
shewing Australia as land and the rest as sea, and one shewing the
rest as land and what is now Australia as sea, have the shape of a
NEGATION: (2) 77
coastline in common; so that one is saying something about the
globe if one says that either this or that representation of it is true.
On the other hand, each of these two globes could be used to
depict what the other depicts, by changing the conventions for shew-
ing sea and land. This feature does illustrate what holds for propo-
sitions: 'p' could be used to say what '~p' says and vice versa. A
code by which one always meant the negative of what one said need
not break down. Hence, Wittgenstein says, 'though "p" and "~p"
have opposite senses, one and the same reality corresponds to them'
(4.0621): the reality is the coastline itself. It is important to remem-
ber that if 'p' and 'not p' were so substituted for one another, 'not'
would still mean 'not' and this is enough to shew that 'not' itself
:
has nothing corresponding to it in reality: its presence does not
determine the sense of the proposition.
We can now understand some of what Wittgenstein says about
tautology and contradiction. They are not 'pictures' (4.462), just as
all-white or all-black globes are not maps. And so they are not
'logical connections of signs' (4.466) : the relations between them are
non-significant i.e. depict nothing: the representing relations, like
two projections which between them fill a space, cancel one another
out.
The all-white globe, though, might be said to be a representation
of the whole world. It is because of the shape of the whole that the
two shapes, p together with not-p, combine to make the shape of
the whole. And this throws light on what Wittgenstein means when
he says that the logical propositions describe, or rather represent,
the framework of the world. 'It must shew something, that certain
combinations of symbols are tautologies.' But what is represented
here is not something that "we express by means of the signs', but
that 'speaks out on its own account' (6.124).
It seems sure that the Tractatus account is wrong. This is partly
because one cannot believe in the simple objects required by the
theory; partly because it leads to dogmatic and plainly false con-
clusions about the will, about modality and about generalizations
in infinite cases. But it is a powerful and beautiful theory: and there
is surely something right about it if one could dispense with
'simples* and draw the limits of its applicability.
78 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
It represents a high point of development of an historic line of
thought. The idea that the proposition is an interweaving of simple
names representing an interweaving of simple elements is to be
found in Plato's Theaeteius\ Aristotle thought about it a great deal,
and rejected it largely because something more than the elements was
required, something connecting them. And the idea that the com-
plexity of a proposition reflects a complexity in its object has every-
where been influential: it is part of what is expressed, for example,
in the idea of natural theologians that God, being *simple', is not
really describable or knowable however many propositions they
might construct about him.
Here it is worth remarking that the truth of the Tractatus theory
would be death to natural theology; not because of any jejune
positivism or any 'verificationism', but simply because of the picture
theory of the 'significant proposition'. For it is essential to this that
the picturing proposition has two poles, and in each sense it repre-
sents what may perfectly well be true. Which of them is true is just
what happens to be the case. But in natural theology this is an im-
permissible notion; its propositions are not supposed to be the ones
that happen to be true out of pairs of possibilities; nor are they
supposed to be logical or mathematical propositions either.
Wittgenstein used to say that the Tractatus was not all wrong: it
was not like a bag of junk professing to be a clock, but like a clock
that did not tell you the right time. It is noticeable that he sounds
like himself in the Tractatus whenever he writes about negation in
the Philosophical Investigations. And at one place the voice of the
author of the Tractatus is heard, like that of the drowned ghost in the
1
song: 'A description is a projection of a distribution in a space.'
1
Philosophical Investigations , p. 187.
5
CONSEQUENCES OF THE PICTURE THEORY
Convinced that he had penetrated the essential nature of truth,
falsehood and negation with his picture theory, Wittgenstein now
had a great programme to carry out. He had to shew how the vast
number of propositions that do not immediately appear to fit in
with his theory do in fact fit in with it. There was a residue that
would never fit in with it; these he dismissed as nonsensical: perhaps
simply nonsensical, perhaps attempts to say the inexpressible. The
following list gives us some idea of the greatness of the task. He had
to deal with:
Laws of inference, and, generally, logical truths.
Statements that one proposition implies another.
Generality i.e. propositions containing 'all* and 'some'.
Propositions giving logical classifications of terms and expres-
' *
sions e.g. "to the right of" is a relation', "a is to the right of b"
is a proposition'.
Propositions that are important in the foundation of mathematics
such as 'a is the successor of b'.
Statements about the possibility, impossibility, necessity, and
certainty of particular states of affairs.
Statements of identity.
Propositions apparently expressing functions of propositions,
such as 'it is good that p', or 'p is possible', *p is necessary' or again
1
'A believes p or 'A conceives p'; and perhaps even statements about
e.g. the beauty of pictures.
Propositions stating probabilities.
Propositions of mathematics.
79
80 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
Propositions stating laws of nature.
Propositions about space and time.
Egocentric propositions.
Propositions about the world as a whole, about God and the
1
meaning of life.
It would be wrong to suggest that Wittgenstein formed his views
on all these topics simply so as to fit in with the picture-theory. It
was rather in most cases that his views on them all did fit in with the
picture-theory; the fact that what seemed to him true views on them
did so fit in would seem an extra confirmation of the picture theory.
There is, however, one exception to this; his view, expressed at
6.37, that 'there is only logical necessity', and at 5.525, that the
possibility of a state of affairs is simply expressed by an expression's
being a significant proposition, appears to be a pure exigency of the
picture theory of the proposition. It is a very common dogma at the
present day that there is no sense of 'necessity' and 'possibility'
except 'logical necessity' and 'logical possibility'. It is possible that
this dogma, which is in part an effect of the influence of Hume, is
also a hangover from the time of the overwhelming influence of the
Tractatus.
With this is connected Wittgenstein's inference from the fact that
there is no logical connection between the will and the world (6.374)
(since what I intend does not have to come about), to the view that
1
'the world is independent of my will (6.373): the connection must
be a purely accidental one. This means that 'all happening and being
this way or that', and 'everything that is the case' is independent of
my will. If one should object to this that it is obvious that what
people intend has some bearing on what happens, he would reply
that that is just 'a phenomenon, only of interest to psychology*
(6.423).
Even here, though, the view has a connection with his ideas about
ethics. For the will as it appears in the world, the 'mere phenomenon,
only of interest to psychology', is what is spoken of at 5.631 'If I
:
were to write a book: "The world as I found it", I should also have
1
1 am not able to discuss Wittgenstein's treatment of all of these
topics in the compass of this book.
CONSEQUENCES OF THE PICTURE THEORY 81
to give an account of my body in it, and to say which members are
subject to my will and which not.' Now that such-and-such members
are subject to my will is a mere fact; if I were suddenly so paralysed
that nothing happened, the will would remain I should still have
willed; but this will is not merely an impotent thought of the thing's
happening, but is of good or evil; and that, apart from the mere
vulgar facts of what happens, is the interest of the will. But of that
'we cannot speak' because value lies outside the world and we can
only express what is in the world. Now that value lies outside the
world is not a mere consequence of Wittgenstein's picture theory
of language; had he only been concerned with the fact that 'good'
and 'evil' could not fit into the picture theory, he might have done as
many positivists did, and debunked value altogether.
Thus the part of his views which seems to be nothing but a
dogmatic consequence of the 'picture theory' is in fact his rejection
1
of modality. Any sense of 'may', 'can 'possible', other than that of
,
"logically possible', would be unamenable to explanation in terms of
the picture theory. And the assertion that something is logically
possible itself requires explanation. For the picture theory does not
permit any functions of propositions other than truth-functions.
Indeed, we should not regard Wittgenstein's theory of the proposi-
tion as a synthesis of a picture theory and the theory of truth-
functions; his picture theory and theory of truth-functions are one
and the same. Every genuine proposition picks out certain existences
and non-existences of states of affairs, as a range within which the
actual existences and non-existences of states of affairs are to fall.
Something with the appearance of a proposition, but which does
not do this, cannot really be saying anything: it is not a description
of any reality.
Possibility of a state of affairs is said at 5.525 to be expressed not
in a proposition (*p is possible' is not a picture of a state of affairs)
but in an expression's being a significant proposition thus the logical
:
possibility of p is one of those things that cannot be asserted, accord-
ing to the Tractatus, but that 'shew'. This explanation does not get
us much further forward. For an expression's being a significant
proposition cannot be a 'fact' either: at 5.5351 we find Wittgenstein
A
criticizing Russell for trying to symbolize the 'nonsense' "p" is a
82 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
proposition' by the senseless tautology 'pDp' so as to guarantee
that only propositions should be put in the argument-places of the
succeeding propositions. This, he says reasonably enough, is absurd,
because if that were not already assured, it could not possibly be
assured by the extra premise that pop, which would become not
false but nonsensical with the wrong sort of substitution for *p'.
*
The objection to "p" is a proposition' is a case of a quite
general objection to a whole range of similar formations: 'n is a
l *
number'; <f> is a function'; 'it is a (possible) fact that p'; "the king
of France" is a complex'; 'a is an object'. 'Object', 'fact', 'proposi-
tion','number', 'function', 'complex': all these Wittgenstein called
'formal concepts', saying (4.126): 'That anything falls under a formal
concept as its object cannot be expressed by a proposition, but is
shewn in the sign for the object itself. (The name shews that it
designates an object, the numeral that it designates a number, etc.)
Formal concepts cannot, like proper concepts, be presented by a
function.' Now at least for the examples '2 is a number', 'red is a
colour' the point is easily made that these propositions cannot ex-
press anything that might be false; there are not two possibilities,
that 2 is, and that it is not, a number; that red is, and that it is not, a
colour; of which the first happens to be actual in each case.
Carnap strongly objected to Wittgenstein's doctrine with its
corollary of the 'unsayables* that are 'shewn', which seemed to lead
on to the 'mysticism* of the Tractatus. 1 In order to avoid it, he pro-
posed to use the 'formal mode of speech' ; instead of saying 'red is
a property*, '2 is a number*, 'to the right of is a relation* we are to say
' ' '
"red" is a predicate',
"2" is a numeral', "to the right of" is a
relational expression', which were held not to involve the same
difficulties.
This (perhaps deliberately) failed to take account of Wit gen- i
stein's doctrine that the real symbol is what is common to all the
symbols that can do the same job. To say of the 'real symbol' for 'to
the right of the common feature that enables all expressions for
this in all languages to have this meaning that // is a relational
1
I once had occasion to remark to Wittgenstein that he was supposed
to have a mystical streak. 'Like a yellow streak,' he replied; and that is
pretty well how the Vienna Circle felt about certain things in the Tractatus.
CONSEQUENCES OF THE PICTURE THEORY 83
expression is not to say something that has the true-false poles. It is
clear that Wittgenstein's objection to propositions in which an
object is said to fall under a formal concept is not limited to the ones
like 'red is a property' which are in the 'material mode of speech*,
and so would not be removed by translation into the 'formal mode*.
'Predicate' would be just as much a 'formal concept* for him as
'property'.
Carnap was well aware of this, and flatly denied that there was
any difficulty about propositions in which an object was stated to
fall under a formal concept, so long as these propositions were
translated into the formal mode. At first sight this seems reasonable
'
enough. In "red" is a predicate' we are saying something about
the object named by the first word of the sentence; that object is
itself a word. What is said about it might not have been true: the
sentence therefore has the true-false poles, and Wittgenstein's
supposed difficulties about it are illusory.
It is an essential part of Carnap 's view that the convention of
forming the name of a word by writing it in quotes is wholly
arbitrary; there is no necessity for any systematic relation, any more
than the names of shapes like 'square' and 'round' need have shapes
corresponding to the shapes named; and 'red' as a word no more
' '
occurs in its name "red" than it does in 'predatory*.
Carnap's view is, however, radically defective. This was made
clear by a Czech logician, K. Reach. 1 He gives a table, of which I
reproduce a part here:
remarking that really, instead of 'table* one should say 'museum*, for
a table correlates names of things, whereas in a museum things and
1
Journal of Symbolic Logic, September 1938 'The Name Relation and
:
the Logical Antinomies.*
84 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
their names are exhibited together. The table consists of two rows.
In the upper row there are samples of various single symbols of a
language beneath each is a word arbitrarily chosen as a name of the
;
given symbol. Carnap calls such a correlation of simple symbols
with their arbitrary names a 'syntaktische Zuordnung\ When we say
'red is a colour-word, our first word is the name of an object (as it
1
so happens, of a word) but there is no essential connection between
;
this object and its name, other than that this is its name, any more
than there is any other connection between, say, a man and his
name; and so we may (as here) use as names of symbols, symbols
that have no systematic connection with them.
Reach demonstrates the defects of Carnap's syntaktische Zuord-
nung by taking it quite seriously, as follows: 'The purpose of the
sentence "Secol is the name of Semicolon" is to give information
about the meaning of Secol (i.e. [the word] "Semicolon")- Does this
sentence serve its purpose? Suppose somebody asks "What is the
meaning of Secol?" and he receives the answer "Secol is the name of
Semicolon." If the answer is to convey anything to the questioner,
itmust be understood; i.e. the questioner must know what Sco and
Secol [i.e. what the words "Secol" and "Semicolon"] stand for in the
sentence. That he knows the former [knows what Sco, i.e. the word
"Secol", stands for] is shewn by the form of his question; but the
meaning of his question is that he does not know the latter [he does
not know what the word "Semicolon", i.e. Secol, stands for]. Hence
the answer is incomprehensible to the questioner.'
Reach's work suggests the formulation of a very simple paradox,
which takes Carnap's view of the use of quotation marks seriously:
It is impossible to be told anyone's name by being told That man's
name is "Smith" '; for then his name is named, not used as a name,
in that statement, and so what I hear is the name of his name and
not his name; and I can only learn his name if I know what name this
name-of-a-name is a name of, just as I can only obey the order 'Fetch
a red one' if I know what colour the colour-word 'red' is a name of.
But from Reach's argument it is clear that I cannot informatively
*
be told that this name-of-a-name, i.e. "Smith" ', is the name of
the name 'Smith'; if I do not already understand this, I shall not
understand the statement that it is so. This, then, seems to be a
CONSEQUENCES OF THE PICTURE THEORY 85
rather clear case of 'what can be shewn' but 'cannot be [informa-
tively] said*.
Nevertheless, 'what shews* in this sense can be illuminatingly
said. We
have an (admittedly rather trivial) example of a proposition
*
lacking the true-false poles in "Someone** is not the name of
someone*. This is obviously true. But it does not have the bi-
polarity of Wittgenstein's 'significant propositions'. For what is it
that it denies to be the case? Evidently, that 'someone* is the name of
someone. But what would it be for 'someone' to be the name of
someone? Someone might christen his child 'Someone*. But when
'
we say "Someone" is not the name of someone', we are not intend-
ing to deny that anyone in the world has the odd name 'Someone'.
What then are we intending to deny? Only a piece of confusion.
But this sort of denial may well need emphasizing. Students, for
example, may believe what Professor Flew tells us in the Introduc-
tion to his collection Logic and Language, 1st Series, pp. 7-8:
namely that 'somebody* refers to a person, that it is part of the
'logic' of 'somebody', unlike 'nobody', to refer to somebody. If this
were so, then on being told that everybody hates somebody, we
could ask to be introduced to this universally hated person. When we
'
say "Somebody" does not refer to somebody', what we are intend-
ing to deny is what Professor Flew meant. But he did not really
mean anything (even if he felt as if he did).
Here a statement which appears quite correct is not a statement
with true-false poles. Its contradictory, when examined, peters out
into nothingness. We may infer from this that Wittgenstein's account
of propositions is inadequate, correct only within a restricted area.
'
For it hardly seems reasonable to prohibit the formula: "Some-
body" does not refer to somebody' or "Someone" is not the name
'
of someone' nor, of course, is this logical truth in any sharp sense
;
of 'logical truth'. It is, rather, an insight; the opposite of it is only
confusion and muddle (not contradiction).
The example of ' "Someone" is not the name of someone' is
particularly clear, because the true proposition is negative. Accord-
ing to Wittgenstein, however, since what our proposition denies
does not turn out to be anything, it itself is not a truth; for there
isn't anything which it says is not the case, as opposed to the equally
86 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
possible situation of its being the case. Therefore Wittgenstein
would either have looked for a more acceptable formulation (which
I think is impossible) or have said it was something which shewed
stared you in the face, at any rate once you had taken a good look
but could not be said. This partly accounts for the comical frequency
with which, in expounding the Tractatus, one is tempted to say
things and then say that they cannot be said.
At 4.1121 Wittgenstein says: 'Does not my study of sign-
language correspond to the study of thought processes which
philosophers have held to be so essential to the philosophy of logic?
Only they got involved for the most part in inessential psychological
investigations, and there is an analogous danger with my method.'
The development represented by Carnap and his school seems to be
a fulfilment of this expectation.
6
SIGN AND SYMBOL
As we have seen, if the possibility of a state of affairs is expressed,
not in a proposition, but in an expression's being a significant
proposition, then according to the Tractatus the very thing that it is
expressed in again cannot be expressed by a proposition. But we
are not yet in the realm of the 'inexpressible* according to the
Tractatus: for instead of speaking of an expression's being a
significant proposition, we could speak of the fact that "p" says
*
that p'. And we shall find out that this, taken one way, is a genuine
fact. To understand this we must examine two reputedly obscure
passages. The first is 5.541-5.5421 :
*
At first sight it appears as if there were another way [other than
as a truth-argument] in which one proposition could occur in an-
other. Especially in certain psychological forms of proposition,
like "A
believes that p is the case" or "A conceives p" etc. 1 Here it
appears superficially as if the proposition p stood in some kind of
relation to an object A. And these propositions have actually been
so taken in modern theory of knowledge (Russell, Moore, etc.)- It is
clear, however, that "A believes that p", "A conceives p", "A says
"
p" are of the form *p' says p". And here what is in question is not
a correlation of a fact to an object, but a correlation between facts
by means of a correlation between the objects in them. This also
shews that the mind the subject etc. as it is conceived in the
superficial psychology of the present day, is a chimera. For a com-
posite mind would no longer be a mind.*
*
The statement that *A believes that p' etc. are of the form "p'*
1
Russell mentions such forms of proposition (Principia Mathematics
1st Edition, Vol. I, p. 8) in order to explain truth-functions by contrast.
87
88 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
says p' has been variously taken to mean that Wittgenstein held it
was impossible to have a thought without uttering a sentence; or
that he held that a person was to be analysed as a complex.
For the first interpretation I can see no reason at all. Against
the second it seems to be an objection that it takes a theory that a
person is a complex as Wittgenstein's ground for saying that *A
*
believes p' is of the form "p" says that p'. Whereas it is evident
that he is arguing: You can't explain the mind as 'the judging
'
subject' in 'A judges p', because 'A judges p' is of the form "p"
says p'; so that way you will only reach a complex, and a composite
mind would not be a mind. Therefore Wittgenstein's statement that
*
'A believes p* is of the form "p" says p' cannot be based on any
Humean theory that a person is a complex.
*It is clear,' he says; and of course what was clear to him was
that for anything to be capable of representating the fact that p, it
must be as complex as the fact that p; but a thought that p, or a
belief or statement that p, must be potentially a representation of
the fact that p (and of course actually a representation of it, if it is
a fact that p). It is perhaps not quite right to say that *A judges p' is
*
of the form "p'* says that p' what he should have said was that the
;
business part of
4
A judges that p', the part that relates to something's
having as its content a potential representation of the fact that p,
'
was of the form "p" says that p': 'A believes p' or 'conceives p' or
*says p' must mean 'There occurs in A or is produced by A something
which is (capable of being) a picture of p\ We should here remember
the letter to Russell in which he said he did not know what the con-
stituents of thoughts were, but he was certain that a thought must
have constituents corresponding to the words of language.
'
Here, then, we are given "p" says that p' as a possible form of
proposition. If Wittgenstein has not been careless, it must fit his
general account of propositions that is, it must have true-false
poles. Now if a sentence is an arrangement of words, it would seem
to follow in accordance with the general principles of the Tractatus
that a way of designating a sentence must be (or be defined by) a
description of its arrangement of words ; though it is a reasonable
complaint for a reader to make that Wittgenstein might have been
more explicit than he is on this important point. The passage which
SIGN AND SYMBOL 89
comes nearest to stating it is as well known, and has been found as
obscure, as the one we have just considered. It comes at 3.1432,
and runs:
4
"The complex sign 'aRb' says that a stands in the relation R to
b." No, not that, but rather "That 'a' stands in a certain relation to
'b' says that aRb".'
This statement is really not particularly obscure. Consider what
relation the sign 'a' does actually stand in to the sign *b' in virtue
of which the whole sign so composed says that aRb. There are all
sorts of possibilities. For example, if I happened to write the 'a' in
blue and the 'b' in red, the question could arise whether it is in
C
virtue of the fact that 'a', 'b', and R' are written side by side (the
order being immaterial), with the 'a' blue and the *b' red, that the
sign so composed says that aRb. In fact, we know that even if I do
this, this is not the expressive feature of the sign, though of course
itmight be. The expressive feature is that the *a' stands to the left
4
and the 'b' to the right of the R'; for if I reversed that, putting 'b'
to the left and *a' to the right, then, according to our present con-
ventions, a different sense would be expressed. From this we can see
how we should take ' "p" says that p'. The expression schematically
*
represented by "p" % which in a concrete case would consist of an
actual proposition in quotation marks, is to be taken as a way of
describing the arrangement of signs that constitutes the proposition.
4
"p" says that p' thus admits of various interpretations; e.g.:
"That in "aRb" "a" is written in italics and "b" in Roman says
that aRb'
*
might be the way that we interpreted "aRb" says that aRb'.
And although it contains a true description of the prepositional sign
as here occurring, it is a false statement (though it could be a true
V
one) ; for it is not, as it happens, this fact, but the fact that stands
to the left and 'b' to the right of 'R', that says that aRb. The use of
italic and Roman letters is immaterial as far as concerns the expres-
sion of a relation.
If this is the sort of thing we are to understand, then the propo-
'
sition "p" says that p' is a genuine proposition, with true-false
poles, according to the conceptions of the Tractates; for its truth
or falsity depends on how the propositional sign *p* is understood
90 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
to be described. Of course, in order to be false, the description has
got to be of some feature of the prepositional sign that might have
been used to express p. So while some interpretation or other of
*
"p" says that p' must be true, its exact interpretation is something
that can be true or false.
If a man says perhaps wonderingly something of the form
*
"p" says p', he need not be thinking of the interpretation of the
part of his expression which is a quoted expression, but that does
not matter: for as Wittgenstein says at 4.002: 'Man possesses the
capacity of constructing languages in which any sense can be ex-
pressed without having an inkling what each word stands for, and
how. Just as we speak without knowing how the individual sounds
are produced. The tacit conventions for understanding ordinary
. . .
language are enormously complicated.' Someone who had given no
4
thought to how aRb' says that aRb would immediately know that
someone else had gone wrong who thought that it was the italics
that mattered, and that one could say that bRfl by writing 'aR6\
e
That is to say, in"p" says that p' what is being considered is
the prepositional sign, mental or physical; and it was of course
primarily of the physical sign that Wittgenstein was thinking. Signs
are after all what we actually hear or see; it is from them that we
gather the meaning of what is said or written; and some of the
variations in them embody variations of meaning. That is to say,
the kind of sensibly perceived difference that there is between *aRb'
and 'bRa' is that from which we gather, and by means of which we
express, a difference of sense.
Now if we consider the difference between
A and B are poetical
and
A and B are identical
we shall have gone far wrong if we think that the difference in sense
between the two propositions is expressed purely by the difference
of four letters, that in the one one thing is asserted of and B, and A
in the other another thing, the difference of words expressing a
difference just in what is asserted. For that difference of two words
SIGN AND SYMBOL 91
signifies much more than that; as comes out in the fact that if A and
B are poetical, A is poetical; whereas if A and B are identical, we
can't go on from this to say 'A is identical'. Thus Wittgenstein says :
'What does not get expressed in the signs, comes out in their applica-
tion: What the signs fail to express, their application declares.' By
'application' he did not mean 'role in life', 'use', 'practice of the use'
in the sense of Philosophical Investigations; he meant 'logico-
syntactic application* (i.e. that kind of difference between the
syntactical roles of words which concerns a logician). 'Only together
with its logico-syntactic application does a sign determine a logical
form' (3.327). And it was by the possession of a logical form that a
proposition was capable of expressing a sense.
But it is pretty well impossible to discern logical form in every-
day language. As an example of the difficulty, consider the difference
between Roman and Arabic numeration. MCMXLVffl is the same
number as 1948, but reading it is more complicated. For example,
the way of reading MCM is different from the way of reading VH,
though each is composed of three of the elements placed side by side.
This does not mean that the Roman system fails to express the same
number as the Arabic. It expresses it perfectly.
This illustrates Wittgenstein's view of the difference between
ordinary language and a good symbolic notation. In his Introduction
(p. 9) Russell said that Wittgenstein was 'concerned with the con-
ditions for a logically perfect language not that any language is
logically perfect, or that we believe ourselves capable, here and now,
of constructing a logically perfect language, but that the whole
function of language is to have meaning, and it only fulfils this
function in proportion as it approaches to the ideal language which
we postulate.' This statement of Russell's was plainly contrary to
the intentions of the Tractates, as is very easily shewn. At 5.5563
Wittgenstein says: 'All the sentences of our everyday language, just
as they are, are logically in perfect order.' Language could not
approximate to having meaning; any language, just qua language,
fulfils its purpose perfectly.
It is a mistake to suppose that the dictum 'Ordinary language is
all right' is an expression only of Wittgenstein's later views. He was
dialectically expounding, not opposing, his point of view at the time
92 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
of writing the Tractatus, in the following passage of Philosophical
Investigations:
*On the one hand it is clear that every sentence in our
language "is all right as it is". That is, that we are not striving
after an ideal : as if our ordinary vague sentences had not yet got
an irreproachable sense, and a perfect language had yet to be
constructed by us. On the other hand this seems clear Where there
:
is sense, there must be perfect order. And so there must be perfect
order even in the vaguest sentence.' 1
That is to say, the sentences of ordinary language no more fail to
express a sense than our Roman numeral fails to express a number.
The one expresses a sense, the other a number, perfectly. And so the
ideal order that characterizes language is there in every sentence of
ordinary language. But: 'Everyday language is a part of the human
organism and is just as complicated. It is humanly impossible to
gather the logic of language from it directly' (4.002). This, then, is
why, according to Wittgenstein, we study logic and construct logical
symbolisms in order to understand the 'logic of language', so as
:
to see how language mirrors reality.
We want in pursuit of the picture-theory to be able to say that the
expressive feature of language is that sigTis are combined in certain
ways. We
compared 'aRb', and 'bRa', saying that we have here a
sensible difference in which a difference of sense is expressed. That
is to say, this is a particular instance of a kind of difference which is
essential to any relational expression in any language: we have here
an example of 'what is common to all symbols that can do the job'.
But of course 'aRb' expresses something, as e.g. 'X-O' does not,
because the elements in 'aRb' are not just signs in the sense of
'marks', but are symbols, as those in 'X-O' are not. So the expressive
feature of 'aRb' is not just an order of elements, but is the fact that
a sign *a', which is a symbol, stands to the lefy and the sign 'b',
which is also a symbol, to the right, of the sign again a symbol
'R'.
1
Philosophical Investigations, Part I, 95.
SIGN AND SYMBOL 93
On the other hand, we have to remember the central point of the
picture theory which we have already explained: 'Only in the con-
text of a proposition has a name reference' ; 'Only in the context of
a proposition has an expression reference.' This prohibits us from
4
thinking that we can first somehow characterize 'a', R' and *b* as
symbolic signs, and then lay it down how we can build propositions
out of them. If 'a' is a symbolic sign only in the context of a propo-
sition, then the symbol 'a' will be properly presented, not by putting
itdown and saying it is a symbol of such and such a kind, but by
representing the whole class of the propositions in which it can
occur.
This we may do provisionally by taking a proposition in which
'a* occurs, and retaining V, while we substitute a variable (I will
4
use '') for all the rest of the proposition. Then the symbol a' is
rightly presented, not just by putting it down and saying it is a sign
of an object, but by a variable proposition
a
This Wittgenstein says quite generally for symbols, or 'expres-
sions', at 3.311-3.313: The expression presupposes the forms of all
propositions in which it can occur. It is the common characteristic
mark of a class of propositions. It is therefore presented by the
general form of the propositions of which it is characteristic. And
in this form the expression will be constant and everything else
variable. Thus the expression is presented by a variable: the pro-
positions which contain the expressions are values of this variable
I call such a variable a "prepositional variable".' Equally, of course,
6
itwould have been possible, considering a proposition such as aRb'
in which *a' occurs, to take *Rb* as the expression to be presented,
and to substitute a variable (I will use V) for the *a'; then the ex-
pression will be presented by the variable proposition
This account is perhaps inspired by Frege's Concept and Object.
Fregesaid:
94 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
'Language has means of presenting now one, now another
part of the sentence as the subject; one of the most familiar is
the distinction of active and passive forms. ... It need not then
surprise us that the same sentence may be conceived as an
assertion about a concept and also as an assertion about an
object; only we must observe that what is asserted is different.' 1
Frege was thinking at first of the fact that we can re-form propo-
sitions, as is shewn by his reference to active and passive forms.
Language shews now one, now another, part of the sentence as the
subject, by altering the sentence, so that now one part, now another,
appears as the grammatical subject, e.g. 'John murdered James',
'James was murdered by John'. And, also, 'The sun is red', 'Red is
a property of the sun*.
But when Frege says: 'The same sentence can be conceived as
an assertion about a concept and also about an object; only we must
observe that what is asserted is different', he has passed from con-
sidering a reformulation of 'The sun is red', like 'Redness is a
property of the sun', to considering the one sentence 'The sun is red'
in two ways. And these two ways are very well explained by Wittgen-
stein. Adopting his explanations we can take them as the alternatives
of regarding it as a value of a variable sentence:
red'
which takes 'The sun' as an argument, and
*
'The sun
which takes 'red' as argument. In the first, we shall therefore be re-
garding the sentence as 'about' the sun; in the second as 'about'
red for what we are 'taking as the subject' is what fills the argument-
place. Only, as Frege says, if we so regard the sentence as now an
assertion about a concept, now about an object, what is asserted is
different, though the sense of the whole analysis is in each case the
same.
1
Philosophical Writings ofGottlob Frege, ed. Geach and Black, p. 49.
SIGN AND SYMBOL 95
This last point was missed by Ramsey in his essay Universals.
He speaks of a theory which he rejects that in a proposition
*aRb' we can discern 'three closely related propositions; one asserts
that the relation R holds between the terms a and b, the second
asserts the possession by a of the complex property of "having R
to b", while the third asserts that b has the complex property that a
has R to it. These must be three different propositions because they
have three different sets of constituents, and yet they are not three
propositions, but one proposition, for they all say the same thing,
namely that a has R to b. So the theory of complex universals is
'
responsible for an incomprehensible trinity Ramsey's thought
is bedevilled at this point by the idea that you cannot analyse a
proposition in a variety of ways: that if you say that 'Socrates taught
Plato' ascribes something to Socrates, you cannot also say that it
ascribes something to Plato without making it out a different pro-
position.
Ramsey's essay, however, quite apart from its intrinsic interest,
is also very helpful for exegesis of the Tractatus theory of 'ex-
pressions'. For Wittgenstein tells us at 3.314 that every variable can
be conceived as a propositional variable even the variable name.
But how can this be? The variable proposition
x loves Socrates
has as values only those propositions in which a name is substituted
for x; but the propositional variable
f loves Socrates
indicated by Wittgenstein has as values all the propositions in which
'loves Socrates' occurs, e.g. 'Everyone loves Socrates', 'Anyone who
loves Plato loves Socrates', 'No one loves Socrates', 'Plato does not
love Socrates'. And similarly for all other variables, as variables
are usually understood. 'Plato has n sons' is a variable proposition
whose values are e.g. Tlato has 6 sons', 'Plato has 100 sons',
Tlato has no sons', but not 'Plato has stupid sons', or 'Plato has
96 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
as good sons as Socrates'. That is to say, its values are not all the
propositions in which the expression Tlato has .... sons' can
occur.
Here, following Ramsey, we may draw a distinction between a
wider and a narrower range of propositions which an expression can
be used to collect. Only there is no need to follow Ramsey in holding
that the expression 'Socrates' cannot be used to collect a range in-
cluding e.g. 'Socrates is wise and Plato is not'. For this opinion of
Ramsey's was based on his rejection of 'complex universals'; and
this in turn is based on his conviction that if you define e.g. *<x*
4
as aRx\ and then treat '<' as a predicate of b in '<b', you must be
denying that '<b' is a relational proposition which of course is
absurd, if you have defined '<x' as 'aRx'.
Now it may be that proponents of 'complex universals' were
confutable by this argument; but it does not follow that anyone who,
like Frege, picks a name out of a proposition and calls the rest of the
proposition a predicate can be dealt with in this way. Only Ramsey's
belief that one analysis of a proposition excludes all others enabled
him, just on the strength of this argument, to deny that 'Socrates*
could be used to collect just as varied a range of propositions as e.g.
*wise'. For he thinks that 'wise' can be used to collect such proposi-
tions as e.g. 'Neither Socrates nor Plato is wise' or 'Someone is wise'.
And he goes on to distinguish between a wider and a narrower range
of such propositions one, all the propositions in which 'wise' occurs,
:
and the other a narrower collection of simpler propositions, of the
form *x is wise'. We can adopt this distinction, just as he intends it
for 'wise', for 'Socrates' also.
Now the question arises: why is there no hint of this in Wittgen-
stein's text? Itwould certainly seem from what Wittgenstein says
that the 'class of propositions' of which an expression was 'the
common characteristic mark' was the whole class of propositions in
which the expression could occur. But *x loves Socrates' can only
be completed into a proposition by substituting a name for 'x';
therefore 'x loves Socrates' cannot give us the general form of the
whole class of propositions, in which the expression 'loves Socrates'
occurs. We seem to be forced to call Ramsey's distinction to our aid,
and say that the 'class of propositions* presented by "x loves Socrates'
SIGN AND SYMBOL 97
is a narrower class that can be discerned within the wider class of all
the propositions in which 'loves Socrates' can occur.
The answer to this puzzle lies in the theory which is integral
to the picture theory of the proposition that all propositions are
truth-functions of the elementary propositions. 1 On this theory,
the 'wider class' that we have been considering will be the class of all
the truth-functions of any set of propositions among which are
propositions containing the expression in question. It immediately
follows from this that any expression presupposes the most general
form of all propositions, as well as the special form of the propo-
sition in which it immediately occurs. But unless which is very
possible I have missed some essential feature of Wittgenstein's
2
idea, it must be admitted that his account is sketchy, unsatisfactory
and obscure.
1
In particular, general propositions such as 'Everyone loves Socrates'.
See Chapter 11.
It is possible that Wittgenstein was satisfied with 'the great works of
2
Frege' as far as concerns the general form of all propositions in which a
given predicate (such as 'loves Socrates' or 'is clever') occurs. Frege's
general form is 'M #03)* : this is the general form of second-level functions,
such as 'Everything is <f>\ 'Something is <', *<f> is what James is and John is
''
not'. The shews that # is a concept or first-level function, being the mark
of an argument-place.
7
WITTGENSTEIN, FREGE AND RAMSEY
We have inferred from Wittgenstein's remarks on *expressions'
which must include names that a name *a' can be represented by a
prepositional variable *()a' which is an informal and somewhat
uninformative version of the most general form of propositions
in which 'a' occurs. Thus Wittgenstein would not accept Frege's way
of distinguishing between object and concept that an object is
something complete in itself, whereas a concept is in need of com-
pletion is, as it were, something with a hole in it. For it looks as if
Wittgenstein will make out both that expression in a sentence which
designates an object, and that expression which remains over in the
sentence when we have picked out the expression designating the
object, to be something with, so to speak, a hole in it. And this con-
ception is the same as the picture theory, in terms of which we have
explained what Wittgenstein meant by Frege's dictum: 'Only in the
context of the proposition has a name reference.'
Ramsey, in his essay on 'Universals', took Wittgenstein to mean
that there was no difference between qualities and objects.
'Against Mr. Russell it might be asked how there can be such
objects as his universals, which contain the form of a proposition
and are incomplete. In a sense, it might be urged, all objects are
incomplete; they cannot occur hi facts except in conjunction with
other objects, and they contain the form of propositions of which
they are constituents. In what way do universals do this more
than anything else?'
Ramsey therefore suggests that it is mere prejudice to dis-
98
WITTGENSTEIN, FREGE AND RAMSEY 99
tinguish between individuals and qualities; there is no reason why
we should not speak of Socrates' attaching to <f>
as well as of <'s
attaching to Socrates in a proposition Socrates'.
'<
The distinction has a practical point, he says, in that if '<' stands
for e.g. 'either having R to a or having S to b\ we cannot put <f>=
Ra v Sb because we should not know whether the blanks in ( )Ra
and ( )Sb were to be filled with the same or different arguments.
Instead we must put <x=xRa vxSb; which explains not what is
meant by <f> by itself, but that followed by any symbol x it is short for
*xRa v xSb'. But if <f> were a simple property, there would be no
reason to say that '<' is asserted of Socrates rather than that 'Socra-
tes' is asserted of the reference of '</>'. And he takes this to be
Wittgenstein's doctrine, chiefly because he observed quite correctly
that Wittgenstein holds that both a name, and the remainder of a
sentence from which a name has been removed, are represented by
'prepositional variables'; moreover, Wittgenstein does not speak of
'concepts* or 'universals' as a kind of thing that is to be found in the
world: it is quite clear that for him there is nothing but objects in
configuration.
That Ramsey has mistaken Wittgenstein's intention is fairly
clear from Wittgenstein's calling 'function', like 'object', a formal
concept (see Chapter 9) and from his explanation at 4.24: 'Names are
simple symbols, I indicate them by single letters ("x", "y", "z").
The elementary proposition I write as a function of names in the
form "f(x)", "<(x,y)'V Now it must not be supposed from this that
Wittgenstein intends *<(x,y)' to represent an atomic fact consisting
of three objects. He has only just remarked (4.2211): 'Even if the
world is infinitely complex, so that every fact consists of infinitely
many atomic facts and every atomic fact is composed of infinitely
many objects, even so there must be objects and atomic facts.' So
when he writes *<(x 3 y)', nothing whatever is indicated about how
many names may be covered by the sign of the function there might,
;
on the hypothesis that he has just mentioned, be an infinite number.
Wittgenstein's doctrine, however, is not at all easy to understand;
for on the one hand he speaks of the elementary proposition as a
concatenation of names, as consisting of names in immediate com-
bination; and on the other hand he says at 5.47: 'Where there is
100 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
complexity, there is argument and function' therefore the element-
:
ary proposition too consists of argument and function.
These remarks considered together raise the problem: if the
elementary proposition consists of names in immediate connection
if it is just a concatenation of names then it is not reproduced, even
if it can be faithfully represented, by a formula consisting of some
letters for names and some letters for functions. And this is borne
out by many passages. Notably for example 3.143: The nature of
the propositional sign becomes very clear, if we imagine it as com-
posed of three-dimensional objects (say tables, chairs, books) in-
stead of written signs. Here the spatial lay-out of those things ex-
presses the sense of the proposition.' We are reminded of the models
of cars, buses and buildings set out in a law court to shew how an
accident took place, which made Wittgenstein say: "That's what a
proposition is!' And in the succeeding entry, which we have already
considered, he says: 'That "a" stands in a certain relation toi "b"
says that aRb.* Now the actual relation in which, in the propositonal
sign 'aRb' 'a' stands to *b' was, as we remarked, that 'a' stands to
the left, and 'b' to the right, of a further sign 'R'.Now
let 'R' be 'to
the left of*. In a waxwork display shewing the way people stood, the
fact that a man A stood to the left of a man B will be shewn by having
the wax figure that goes proxy for A in the display standing to the
left of the wax figure that goes proxy for B, and there will be no need
for any third object to signify the relation. At 4.0311 Wittgenstein
makes the comparison with the tableau vivant: 'One name stands for
one thing, another for another, they are connected together: that is
how the whole images the atomic fact like a tableau vivant.'
It is natural and reasonable to say of this idea: This is all very
well; but it is possible only when the picture-proposition shares a
'form' as Wittgenstein calls it, other than what he calls 'logical form',
with what it depicts. The waxwork show and the tableau vivant need
no figures going proxy for the spatial relations just because, being
three-dimensional models of three-dimensional situations, they can
reproduce the spatial relations instead of having something standing
for them. And the coloured picture can represent that a cloak is
red without having the cloak in one place and the redness in another,
just because it is a coloured picture representing something coloured,
WITTGENSTEIN, FREGE AND RAMSEY 101
so that it can simply shew the cloak as red. Further, even if the picture
were in black and white, and represented the colour of objects by
some conventionally agreed shading still, it has the advantage of
being able to shew the shading that means red, on the cloak and not
somewhere else.
This is exactly what does not happen in a proposition. In a sen-
tence saying that the man wore a red cloak the word for the cloak is
not printed in red to shew this. Even if we had some such conven-
tions and perhaps we can admit we have something of the sort in
4
the difference between aRb' and *bRa' they do not take us very
far.Even in this very favourable case, we need a special sign for
the relation itself. 1 And rightly so, because there is some material
content to relations like 'to the right of or 'bigger than' ; that is why
signs between which the same kind of relations hold can reproduce
them; but if you were quite generally to express relations between
things by relations between their signs, then you would need to have
as many different relations between signs as we in practice have
words to express relations.
This is in fact Wittgenstein's requirement for the fully analysed
sentences of a language. For the fully analysed elementary proposi-
tion is a concatenation of simple names; though not a mere list,
because the way they are combined is expressive.
This does not mean that function and argument would disappear
in the final analysis. If for the moment we may give 'a-b-c-d* as an
elementary proposition, then *a-b-c-( )' and 'a-('X' ')-d' would be
4
two different functions; which might be represented as *fx*, <(x,y)'
respectively; and the representations of 'a-b-c-d' as a value of these
two functions would be 'fd', *<(b,c)*. I write primes in the second
function to shew that it can be completed with different names in
the two empty argument-places. (')-(' >(' ")-("") would be a
formula, *a logical form a logical proto-picture', of an elementary
proposition.
Now we just do not know the composition of any elementary
proposition; that is why Wittgenstein never gives any such example.
But Ramsey writes as if, say, 'a-b' were a specifiable elementary
1
A
case in which no sign occurs for the relation itself is 'Caius Marci*
in Latin 'Caius (is the son) of Marcus'.
102 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
proposition, which Wittgenstein chooses to write as, say, *f(b)'. That
'
is quite to misunderstand Wittgenstein's use of the sign 'f in *f(b)' :
4
f(b)' symbolizes an elementary proposition, but not necessarily one
in whose sense (the atomic fact) only two objects occur. The point
can be put most briefly like this: to represent a name 'a' by *()a',
i.e.by the most general way for that name to occur in a proposition,
is not to represent a name as a function, but only to stress that the
name has reference only in the context of a proposition.
The idea of conceiving a proposition as a function of the ex-
pressions contained in it comes from Frege, and to understand it we
have to go back to his great essay Function and Concept and follow
the steps by which he formulated this conception.
First we introduce the notion of a numerical function i.e. what
is expressed by a numerical formula containing one or more 'indefi-
nitely indicating' letters; if the letter or letters are replaced by signs
for a definite number or numbers, the expression so obtained has a
definite numerical value: x 2 , x-f-y. The function could be
e.g.
fittingly expressed by a formula with an empty place in it: ( )*, By
an 'argument' we mean what is signified by the sign we put into the
empty place. 'We give the name "the value of a function for an
argument** to the result of completing the function with the argu-
ment.* Thus e.g. 4 is the value of the function ( ) 2 for the argument
2. But it is necessary, if there is more than one empty place, to dis-
tinguish between cases where the function can be completed by
putting different things, and cases where it must be completed by
putting the same thing, into the empty places. That is why we use
letters instead of empty places.
There are functions whose value is always the same, whatever the
argument, such as 2+x x; and there are pairs of functions whose
values are always the same for the same argument: for example
Following Frege, we now add to the signs -h, , etc., which
serve for constructing a functional expression, such signs as =, > ,
< , which occur in arithmetical statements. So we speak of the
function x 2 =l. The value of the function for a given argument is
signified by the result of substituting a definite numeral for the letter
1
1 follow Frege in speaking of two functions here. It is not usual.
WITTGENSTEIN, FREGE AND RAMSEY 103
x. But the result of substituting a definite numeral for x here has
not a numerical value, but is something true or false; hence the now
familiar idea of a 'truth-value' is derived from this conception of
Frege's.
The Value' of x*= 1 is 'true' for a definite argument, e.g. for 1 ;
to say this is the same thing as to say that 1 is a square root of 1,
or that 1 has the property that its square is 1, or that 1 falls
under the concept 'square root of 1'. 'We thus see,* Frege says, 'how
closely what is called a concept in logic is connected with what we
call a function.* This suggests an interesting definition of a propo-
sition as 'the result of completing a sign of a function by filling tip
an argument-place, when the value of the result is a truth-value'.
And so far, Wittgenstein is in agreement with Frege, and expresses
his agreement at 3.318: '1 conceive the proposition like Frege and
Russell as a function of the expressions it contains.'
To speak of conceiving the proposition as a function o/the ex-
pressions it contains is of course not inconsistent with denying, as
Frege does, that a proposition is a function; it is like speaking of 8
as a function of 2, say its cube. It is important to grasp this point,
that what is a function of something is not a function tout court ;
confusion on this point is often found. A
function for example is
sometimes explained as a variable magnitude. Now it is true that,
say, the volume of a gas is a variable magnitude (i.e. variable in
time) and is also a function of the pressure and temperature. But the
volume of a gas is not a function tout court, and therefore we do not
get here an example of a function that is a variable magnitude. To
say that the volume is a function of pressure and temperature is to
say that there is a function f such that V=f(p,t). For the volume to
be a function tout court would be represented by the nonsense
V=f( ).
We must now consider Frege's next step. He has defined a
function as what is signified by an expression with an empty place;
and he says : 'An object is anything that is not a function, so that the
expression for it does not contain an empty place.' It follows that
(unassorted) propositions designate objects, since they have no
empty places; and since Frege regards a proposition as one kind of
completed functional expression, and considers that a completed
104 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
S
functional expression (e.g. *2 *) is a designation of a value of the
function, it becomes natural to say that propositions designate
values. This might be a matter of terminology, to which it would be
unreasonable to object, granted that the conception of a proposition
as a completed functional expression recommends itself.
Frege now proceeds to construct a function
whose value is 'the true' when *the true' is its argument, and in all
other cases is the false. By taking *the true* as argument Frege means
putting a true proposition in place of the 'x' you can put a designa-
;
tion of anything there instead a false proposition or a definite
description of a numeral or an ordinary proper name: anything, in
short, that stands for anything, without having any empty places
in it. For example
is a possible result of completing this function, and the value of the
function when so completed is false, or, as Frege puts it,
: 2 is the
false. This way of speaking is of course a consequence of the dis-
tinction between sense and reference. If I use an expression which
stands for something, then in using it I am speaking of what it stands
for; and if I have another name, *B', for that thing, I can use the
first name, *A', and say that A is B. So since *
2' is a designation
of the truth-value: false, Frege can say that 2 is the false. We
must accept this sort of consequence if we accept the prima facie
plausible distinction between the sense and reference of expressions;
this constitutes an objection to the distinction.
The reason why Frege wished to construct such a function is that
he has no truck with attempts to stipulate ranges of significance in
the manner of Russell. 1 If a truth-value is an object, it can be an
argument; but he is not willing to specify 'propositions' as the range
of significant substitutions for 'x' in functions taking truth-values
1
See Chapter 9, pp. 123-4.
WITTGENSTEIN, FREGE AND RAMSEY 105
as arguments; and indeed the specification of ranges of significance
is a very dubious business.
Ordinarily, if we write down *5 >4' we wish to assert something;
but according to Frege's view, 4 5>4' is just an expression for a
truth-value, without any assertion. Therefore, he says, we need a
special sign hi order to be able to assert something, as opposed to
expressing a mere assumption the putting of a case without a
1-
1
It has sometimes perplexed readers of Wittgenstein that he refers,
both in the Tractatus (4.063), and in Philosophical Investigations, to 'the
Fregean Annahme\ as if 'Annahme* (assumption) had been a technical
term in Frege, as it was in Meinong. His reference is to this passage; and it
is evident that his attention was especially fixed on it by a passage in
Russell's account of Frege in the Principles of Mathematics, Appendix A,
477. Russell says: There are, we are told, three elements in judgment:
(1) the recognition of truth, (2) the Gedanke (the thought), (3) the truth-
value. Here the Gedanke is what I have called an unassorted proposition
or rather, what I have called by this name covers both the Gedanke alone
and the Gedanke together with its truth-value. It will be well to have names
for these two distinct notions; I shall call the Gedanke alone a prepositional
concept \ the truth-value of a Gedanke I shall call an assumption' And here
Russell has a footnote referring to the passage in Function and Concept,
and, saying: 'Frege, like Meinong, calls this an Annahme\ 'Formally, at
least,' he goes on, 'an assumption does not require that its content should
be a prepositional concept; whatever x may be, "the truth of x" is a
definite notion. This means the true if x is true, and if x is false or not a
proposition it means the false.'
What Russell refers to as *the truth of x* is of course Frege's function
x. Frege introduces a second function
whose value is the false for just those arguments for which the value of
x is the true. Thus, as Russell says, we do not have assertions and
negations there is not a negation sign, corresponding to the assertion
sign but we have assertions of the truth and falsity of 'thoughts*, or, as
Russell calls them, 'prepositional concepts*.
It is a peculiarity of Russell's account that he takes
5>4
to be something different from
5>4
C *
and calls 5 >4* the 'thought' and 5 >4*the 'assumption'; thus turning
Frege's quite innocent and untechnical expression 'a mere assumption*
into a technicality.
What Russell failed to notice was that if a proposition is substituted
106 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
simultaneous judgment as to whether it holds or not. So he puts a
vertical stroke at the left of the horizontal, e.g.
I
2 -{-3= 5
and this expresses the assertion that 2+3=5.
We must now examine Wittgenstein's main criticism of Frege.
At 4.431, he says: 'The proposition is the expression of its truth-
conditions*, and then remarks: 'Hence Frege was quite right to
premise the truth-conditions as defining the signs of his symbolism.'
The reference is presumably to the passage in the Grundgesetze
'
for x in x' there is no difference at all, for Frege, either in sense or in
reference, between the proposition by itself and 5 the proposition with the
horizontal stroke attached; moreover a 'thought is not a proposition, not
even an unassorted proposition, but is the sense of a proposition, and
hence there is the same Gedanke when we have a proposition and when
we have a proposition with the stroke attached. It is only when we substi-
"
tute the designation of something other than a truth-value for 'x' in x'
that there is any difference, either in sense or in reference, between the
designation by itself and the designation with the stroke attached. In that
case, the designation designates whatever it does designate the Moon or
the number 3 for example; and the designation with the stroke attached
designates a truth-value, in these cases the false.
Russell's remarks, which mistakenly give special prominence to Frege's
use of the word 'assumption', must be the source of Wittgenstein's refer-
ences to it. Further, it appears that Wittgenstein actually accepted Russell's
interpretation; for his comment on Frege at the end of 4.063 is not other-
wise intelligible: 'The proposition does not stand for any object (truth-
value) whose properties are called "true'* or "false"; the verb of the
proposition is not "is true" or "is false" as Frege thought but what "is
true" must already contain the verb.* Although in Begriffsschrift Frege said
that the verb of the proposition was 'is true' a view which he rejected
in Sense and Reference he never thought this of 'is false'. But if we were
to adopt Russell's interpretation of the passage in Function and Concept,
we should say that according to Frege there are three stages
(1) x
(2) the truth of x
or: the falsehood of x
and then (3) the final stage of assertion, which we might think of as a tick
put against whichever is right, the truth of x or the falsehood of x; and
such a view might easily be rendered as a view that the real verb in the
proposition that gets asserted i.e. in the 'assumption' is 'is true' or
'is false'.
WTTTGENSmN, FREGE AND RAMSEY 107
where Frege says that he has specified the reference, i.e. the truth-
value, of any well-formed proposition in his symbolism by specify-
ing the truth-conditions, and that the sense of the proposition is the
sense of: such-and-such truth-conditions are fulfilled. 'Only,' Witt-
genstein continues, 'the explanation of the concept of truth is wrong:
if "the true" and "the false" were really objects, and were the
arguments in ~p etc., then according to Frege's own specifications
the sense of ~p would by no means be specified.*
Frege has specified the truth-values of his propositions by specify-
ing the truth-conditions, because his propositions are logical truths:
it is the characteristic feature of logical truths (or again of logical
falsehoods) that their truth-values are determined by determining
their truth-conditions. But he has also said that the sense of his
propositions is the sense of this: that their truth-conditions are
fulfilled; and in this way he has ensured that his propositions are
scientifically perfect; he has guaranteed a sense and a reference for
them, and determined which truth-value they have.
Thus, if Frege has a negative proposition, *~p", its sense must
also be the sense of the fulfilment of its truth-conditions. But his
explanation of negation is this he introduces a new function
:
x,
whose value is the false for just those arguments for which the value
of x is the true, and conversely; and so in *~p* we have a pro-
position determined as expressing the result of completing with the
argument 'p' a function whose value for given arguments is given;
but where is the sense of '~p'? *^p' appears to be defined in effect
as that proposition whose reference is the true in certain circum-
stances and the false in others. But on Frege's own principles you do
not specify a sense by specifying a reference; and so, Wittgenstein
says, according to Frege's own principles, the sense of *~p' is not
determined. 1
The problems involved here are at bottom the same as those I
discussed in Chapter 3. As a criticism of Frege the point can be
summarized by saying: 'If truth-values are the references of propo-
sitions, then you do not specify a sense by specifying a truth-value.*
1
This criticism is quite independent of the misinterpretation of Frege's
theory (taken over by Wittgenstein from Russell) which was discussed
in
the last footnote.
108 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
Now this objection is quite decisive; but the essential difficulty
about negation, although it receives a special form in connection
with Frege's theory, is, as we have already seen, not generated just
by Frege's conceptions. We encountered it at the very outset, when
we examined the customary definition of *~p' as the proposition
that is true when p is false and false when p is true; and we have seen
how Wittgenstein's picture theory of the proposition guaranteed
the legitimacy of the customary definition by supplying the con-
ditions required for offering such a definition: namely that there is
not more than one such proposition and that there always is such a
proposition.
Furthermore, negation gives us good grounds for rejecting
Ramsey's suggestions, not just as interpretations of the Tractates,
but in themselves. For you can negate a function, but not an object:
this shews that even the simplest possible sign of a function is not the
same thing as a name. It may be asked why, in analysing 'Socrates
is not wise', we should not take the negation with 'Socrates' rather
than with 'wise' 'Socrates-is-not wise'. We can certainly speak of
'all the things that Socrates is nof\ and Frege would have said that
this phrase stood for a second-level concept, its role being to say,
concerning the reference of a predicate, that this is one of the things
Socrates is not. But though it can be treated as an 'expression', the
common characteristic mark of a class of propositions, 'Socrates is
not' is not on an equal footing with 'Socrates' they cannot be
treated as one another's contradictories, like 'red' and 'not red'.
The result of attaching 'Socrates is not' to the conjunction of predi-
cates 'wise and just' is quite different from the conjunction of the
results of attaching it to 'wise* and 'just'; for the name 'Socrates'
no such difference can arise. Accordingly, 'Socrates is not' is not an
allowable interpretation of a name variable, in the way that a
negative predicate is always an allowable interpretation of a predi-
cate variable.
It should be apparent, however, that Wittgenstein's views are
extremely Fregean. What, then, has become of Frege's 'concepts' in
Wittgenstein's theory? They seem to have disappeared entirely;
actually, however, instead of making concepts or universals into a
kind of objects, as Ramsey wished to, Wittgenstein made the gulf
WITTGENSTEIN, FREGE AND RAMSEY 109
between concepts and objects much greater than Frege ever made it.
So far as concerns the content of a functional expression, that will
consist in the objects covered by it. But in respect of having argument-
places, concepts go over entirely into logical forms. In the 'com-
pletely analysed proposition', which is 'a logical network sprinkled
with names', 1 the Fregean 'concept', the thing with holes in it, has
become simply the logical form. Thus there is no question of two
kinds of reference for expressions; one which is incomplete, having a
hole in it that awaits, say, an object to complete it; and another,
complete and capable of completing the incomplete, itself requiring
no completion.
An interesting consequence follows about, say, two propositions
expressing (completely) different facts A is red, and B is red. If these
: :
propositions were 'completely analysed*, so that we had elementary
propositions consisting of names in immediate connection, then
the question arises whether the objects- that would be named, in
place of our using the colour-word 'red' in the two cases, would be
different. I think Ramsey would have supposed that they would be
the same. And no doubt he would have pooh-poohed the feeling that
in that case these objects would have the character of universals
rather than 'individuals'; we don't think A is a 'universal* because it
can enter into a variety of facts, so why should we think this of red
or if red is composite, of the objects into which 'red' is 'analysed*?
This is perhaps a proper reply; yet it is difficult not to feel that an
object that can exist all over the world in different facts has rather
the character of a universal. It takes a little mental habituation to
think that existence in several facts is the only feature that counts,
so that since both Aand red can exist in several facts, we should not
be impressed by A's at least existing in only one place at a time, while
red can exist in so many.
1
1 take this expression from a late notebook of Wittgenstein's in which
he makes some comments on the theories of the Tractatus. In his pre-
Tractatus notebooks Wittgenstein says: 'Properties and relations are
objects too' (16.6.15). On my view, he no longer holds this in the
Tractatus. I think my view necessary (a) to reconcile the various passages
I have cited about functions and elementary propositions and (b) because
if Wittgenstein held that objects fell into such radically distinct categories
as functions and individuals, it is an incredible omission not to have made
this clear.
110 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
Anyhow, whatever the merit of Ramsey's view, he is incorrect in
ascribing it to Wittgenstein. Let us pretend once more that we can
make an actual model of an elementary proposition 'with the names
in immediate connection'; then for Wittgenstein the two facts: A is
red, and: B is red, would be analysed into (1) facts corresponding to
the descriptions of the complexes A and B, and (2) facts about the
elements of the complex A along with certain further elements, say
a, b, c, for A's redness, and exactly corresponding facts about the
elements of the complex B along with certain other elements, say
d, e, f, for B's redness. There is no need for a, b, c, to be the same
as d, e, f, respectively; for it is only the 'logical network' that is
'universal'.
We normally tend to assume that different occurrences (at least
of the same shade) of red differ only in that there are different things
that are red that no real difference other than this answers to the
two predications of the predicate 'red'. This has helped to form the
belief in universals; though there have been philosophers, e.g. among
the medievals, who have wished to speak of 'individualized forms'
'this whiteness' for example. The problem of 'universals' can in fact
be given the form: was Frege right to introduce two wholly different
kinds of 'reference' for words, namely 'objects' and 'concepts'? A
'concept' was the 'reference' of a predicate; now the characteristic
mark of a predicate is its possession of an argument-place or -places,
which could be filled with the names of now one, now another object;
hence a 'concept' is a 'universal'. In Wittgenstein's fully analysed
proposition, we have nothing but a set of argument-places filled with
names of objects; there remains no kind of expression that could be
regarded as standing for a concept.
The objects 'behind' a true predication of *red' would indeed be
of the same logical form in every case. We must remember that
the original seat of form is the objects themselves: 'If things can
occur in atomic facts, this must be something that is in things them-
selves If I can imagine an object in the nexus of an atomic fact, I
cannot imagine it outside the possibility of this nexus.' (2.01 21) And
:
that is why Wittgenstein says: 'The possibility of its occurrence in
atomic facts is the form of the object' (2.0141), and: The objects
form the substance of the world' (2.021); and so they are "form and
WITTGENSTEIN, FKEGE AND fcAMSEY 111
content' (2.025). Thus at 2.0231 we learn that the substance of the
world i.e. the objects can determine only a form, not any material
properties. For it needs propositions (as opposed to names) to repre-
sent material properties; such properties are 'only formed by the
configuration of the objects*. Red is a material property, and there-
fore formed by a configuration of objects and, as I have said, by
the same configuration of different objects in the different facts that
exist when different things are red. These different objects, having the
capacity to enter into configurations forming the material property
red, will be of the same logical form: that of objects whose con-
figurations yield colours. (Hence colour is a 'form of objects':
2-0251).
This, then, will be why he immediately goes on to say: Two
objects of the same logical form apart from their external proper-
ties are only distinct from one another in that they are different'
(2.0233). The only 'external properties' his simple objects can have,
of course, are those of actually occurring in certain facts.
Here Wittgenstein adds a remark, which may seem at first sight
to contradict the previous one: 'Either a thing has properties that no
other has, in which case one can mark it out from the others
through a description without more ado, and point to it; or on the
other hand there are several things with all their properties in
common, and then it is absolutely impossible to point to one of
them. For if nothing marks a thing out, I cannot mark it out if I
did, it would be marked out.' It is possible that he is here thinking
of what is involved in e.g. distinguishing between and identifying
particles of matter. It would be wrong to infer from this passage
that he thinks that there cannot be two things with all their proper-
ties in common: at 5.5302 he is explicit that it makes sense to say
that two objects have all their properties in common.
Frege's notion of concepts led him to the awkwardness of saying :
"The concept horse is not a concept'; for, in statements about 'the
concept horse\ the concept horse is not the reference of these words
since they are not words being used predicatively as words that stand
1
for a concept must be. Frege came to think that any such statement
1
1 am informed of this by Mr. M. A. E. Dummett, who has read some
of Frege's unpublished writings at Miinster.
112 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
was ill-formed; a concept must not occur except predicatively. That
is,we can speak of 'the animal that both the Derby winner for 1888
and the Derby winner for 1889 are', but this expression, like 'a
horse' itself, can occur only predicatively; we cannot say: 'the
reference of this expression is the concept horse'.
Wittgenstein would say the sign for a function shewed itself to
be the sign for a function; that something falls under a formal
concept like 'function' is for him something that cannot be said; and
Frege's difficulties about 'the concept horse" explain the point of this.
If you say that your expression 'the animal that both the Derby
winners are' has a concept as its reference, you at once lay yourself
open to the question 'what concept?', with only one possible answer:
'the concept horse' yet this 'is not a concept'. But the formal con-
cept is rightly represented by the type of variable used in 'There is a
:
such that both the Derby winners are
<f> & :the variable employed
expresses what Frege wanted to express by the phrase 'the concept',
and yet saw he could not properly express in this way.
8
OPERATIONS
We must now consider Wittgenstein's remarks on operations.
They have a special interest in connection with his rejection of the
Frege-Russell assertion sign, and Wittgenstein arrived at them in
grappling with the problem of the assertion sign as introduced by
Frege.
Although this sign, ')
', is still in use in symbolic logic, it has
not now the same meaning as it had for Russell and Frege; it now
means 'is a theorem', and so can occur (as it could not in Russell and
Frege) hypothetically: 'if p', i.e. *if p is a theorem
]
.'. Russell's
. .
use of the sign explicitly follows Frege's; for Frege, the assertion
sign symbolizes the difference between the thought of something's
being the case and the judgment that it is the case it can thus never
occur in an //clause. Frege has two arguments for its necessity, one
weak and the other strong.
The weak argument is from the necessity of a distinction between
entertaining an hypothesis (formulating the content of a judgment,
having a 'sense' before one's mind) and asserting a proposition. He
says that an actor on the stage, for example, is not asserting. At that
rate, it would be an inexcusable faux pas to make an actor write the
assertion sign before a proposition on a blackboard in a play This !
argument need not delay us.
The strong argument in the light of which we can understand
Wittgenstein on operations is that we must distinguish between the
occurrence of a proposition in a conditional: *if p, then q', or a dis-
junction: 'either p, or q', and its occurrence when we simply say
that p. The distinction is an obvious one, but quite difficult to
express; it is natural to say that we are distinguishing between the
113
114 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
occurrence of the proposition, unasserted, as a component of an
assertion, and its occurrence when it is itself asserted: and here the
distinction we are trying to make is certainly not a psychological one.
But we cannot say that *p', when it occurs by itself and when it
occurs in a disjunction, *p v q', has a different sense; for from 'p v q'
*
and ~q* we can infer *p', and the proposition that stands by itself as
the conclusion must be the very same proposition as occurred as a
disjunct.
Russell 1 uses this point in his explanation of the 'non-psycho-
which is what according to him
logical' sense of ''being asserted*,
must accrue to a proposition (besides what it has just qua propo-
sition) when it is used as a premise to prove something, or is (rightly)
inferred as a conclusion from a premise. 'When we say therefore, we
state a relation which can only hold between asserted propositions,
and which thus differs from implication.'
1
Being asserted (in this 'logical sense) is, for both Russell and
Frege, something that cannot possibly attach to a proposition
unless it is true. But it is more than its being true; for in the dis-
junction 'p or q' it may be that one or the other proposition is true,
but neither is being asserted. In Frege's terminology, we might say
that if an unasserted proposition is true, it is (in fact) a designation
of the true; but in the asserted proposition the true is actually being
presented to us as such, it is not just that some designation of it
occurs in our discourse.
"
Wittgenstein says curtly: Trege's assertion sign "| is logically
quite meaningless: in Frege (and in Russell) it only indicates that
these authors hold the propositions so marked to be true' (4.442).
We must therefore enquire how he deals with the problem raised by
Russell, about therefore; and also with the difference between 'p' and
*q* by themselves and in *p v q* or 'pDq'.
The first point, about therefore, is dealt with at 6.1263-4: 'It is
clear in advance that the logical proof of a significant proposition
and proof in logic [i.e. proof of a proposition of logic from another
proposition of logic] must be quite different things. The significant
proposition says something, and its proof shews that things are as
it says; in logic every proposition is the form of a
proof. Every
1
Principles of Mathematics, 38.
OPERATIONS 115
proposition of logic is a symbolic representation of a modus ponens.
(And the modus ponens cannot be expressed by a proposition.)*
That is to say, Wittgenstein takes the tautology
(p.pDq) Dq
to be just another symbolic representation of the form of argument
called modus ponens , viz. :
p
P
pDq
Now 'representation* is the term Wittgenstein uses of a picture: what
a picture represents is its sense. So here he is saying that the impli-
cation '(p, and p implies q) implies q', is as it were a picture or
proposition with the modus ponens as its sense.
Russell says 'When we say therefore, we state a relation that can
:
only hold between asserted propositions.' He means, among other
things, that therefore is something we are wrong to say, unless the
premises are true and the conclusion too. This idea finds an echo in
a statement by Frege in his essay on negation: 'One cannot infer
anything from a false thought.' But that is not true. What is true
and, of course, what Frege was referring to is that one cannot
prove anything from false premises; one can criticize a proof by
saying that the premises are false or doubtful. But it is wrong to say
that 'therefore* is being misused hi a correct argument from false
premises. To be sure, 'therefore' is the utterance of someone who is
asserting one or more propositions that precede it, and one that
follows it, and he is in error if he asserts what is false; he is not
however committing a further error in using 'therefore' just because
his premises are false and his conclusion, accordingly, perhaps false.
If these are his errors, they do not import a further mistake into his
'therefore*.
We must of course distinguish between the way a proposition
occurs when used to assert what it means, and the way it occurs
when e.g. it is merely a subordinate clause in a proposition that is
116 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
asserted (a clause, moreover, that may be false, though the whole
proposition is true); it is a mistake, though it is natural, to describe
this difference as a difference between a 'logically asserted' and a
'logically unasserted' proposition. We have here a necessary dis-
tinction, wrongly made. 'Assertion' has only a psychological sense.
We might indeed perhaps accept 'logically asserted* and 'logically
unasserted* as technical descriptions of different ways in which
propositions may occur; but it can be no part of the requirements
for being 'logically asserted* that a 'logically asserted' proposition be
true, as both Frege and Russell thought; and once that feature of
'logically asserted' propositions is removed, the terms become a
mere pair of labels, and cease even to have an air of being ex-
planatory. 'Logical assertion' is no longer an extra feature attaching
to a proposition, or added to its sense; we have no idea what it is;
we only know when to call propositions 'asserted' in this sense.
But this is not the end of the matter: the difference, if it has only
been labelled, demands both to be made clearer and to be explained.
First, although you cannot prove anything unless you know some-
thing, you can construct the modus ponens that would be a proof if
you found out that its premises are true. 7/ these premises are true,
this conclusion is true' is then a description of this modus ponens;
which I suppose is what Wittgenstein meant by his remark (6.1264).
The premises that you construct may be quite hypothetical; or again,
you may know one premise and make an hypothesis of the other.
This is the reason why Aristotle rightly says a conclusion is reached
in just the same way in a 'demonstrative' and a 'dialectical' syllo-
gism: if you say 'suppose p, and suppose q, then r'; or if, being
given 'p', you say 'suppose q, then r' ; you are just as much inferring,
:
and in essentially the same way, as if you are given 'p' and 'q' as true
and say 'therefore r'.
This, Wittgenstein would say, is because 'the structures of (the)
propositions stand in internal relations to one another' (5.2). For
at 5. 1 3 1 he has said 'If the truth of one proposition follows from the
:
truth of others, this is expressed by relations in which the forms of
those propositions stand to one another. .
. These relations are
.
internal and exist simultaneously with, and through, the existence of
the propositions.*
OPERATIONS 117
Wittgenstein goes on (at 5.21) to say that we can 'emphasize
these internal relations in our form of expression, by representing
one proposition as the result of an operation that produces it out of
others (the bases of the operation)'. This is perhaps best explained in
a simple, but not quite familiar, example. Take a relation and its
converse, e.g. 'husband of* and 'wife of, and consider the two
propositions 'a is husband of b', 'a is wife of b*. We now introduce
:
an operation, called 'conversion', the sign of which is *Cnv* placed
before a relative term; thus, instead of writing e.g. *bRa\ we write
'aCnvRb'. Then 'aCnv(husband of)b* emphasizes the internal re-
lations of two propositions 'a is husband of b' and 'a is wife of b'
by exhibiting the second proposition as the result of an operation
upon the first (of course, an operation that could only be performed
on propositions of this relational form).
An operation must not be assumed to be necessarily an in-
ferential operation. In our present case, indeed, since 'husband of is
an asymmetrical relation, the two propositions are incompatible. An
operation upon a given proposition as base may produce one that is
compatible or incompatible with the proposition operated on; the
only thing it does not produce is something equivalent to the
proposition operated on. An operation is what has to happen to a
proposition in order to turn it into a different one (cf. 5.23). And 'the
operation is the expression of a relation between the structures of its
result and of its base' (5.22).
In this example we can also understand clearly enough what is
meant by saying: 'The occurrence of an operation does not char-
acterize the sense of a proposition. For the operation does not
assert anything, only its result does, and this depends on the bases of
the operation' (5.25). It is very clear in this instance that 'Cnv' is not
a distinguishing mark of the sense of a proposition, as 'not' might
easily be thought to be; for you might think you can pick out a
special class of negative propositions, but you would not be tempted
to think that you can pick out a special class of relations that are
converses; every relation has a converse, and is thus the converse
of its converse, and can be written 'CnvR' for some suitable inter-
1
pretation of R'.
Having grasped the general notion of an 'operation', we can now
118 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
proceed to the next step, which is taken at 5.234: 'The truth-
functions of the elementary propositions are the results of operations
with the elementary propositions as bases. (I call these operations
truth-operations).*
To say this is to make a radical distinction between a truth-
function and an ordinary function like *<x\ For as we have seen,
Wittgenstein says that *the occurrence of an operation is not a dis-
tinguishing mark of the sense of a proposition for the operation
:
does not assert anything, only its result does*. Now a function of
names is certainly a distinguishing mark of the proposition in which
it occurs: such a function certainly expresses something, marks out
a form and a content. But the sign of an operation not merely
stands for nothing has no reference it does not even mark out a
form: it only marks the difference between forms.
Consider the following propositions which have *p' and 'q' as
bases of truth-functional operations:
pvq
p .
q
Each of these can be written differently, e.g. we have the same
pair of propositions in
For the first of the pair, we have here two versions: in one the pro-
position reached is shewn as the result of disjunction performed on
*p' and *q'; in the other as the result of negation performed on the
result of conjoining the results of negation performed on 'p' and
'q'; this is sufficient to shew that the mere occurrence of disjunction,
or conjunction, or negation, is not a distinguishing mark of a pro-
position. On the other hand, if you perform a different operation on
the very same base, you get a difference of sense.
Let us now compare Wittgenstein's position with those taken up
by Frege and Russell. For Russell, a truth-function is one kind
among the functions that take propositions as arguments. Frege
OPERATIONS 119
places no such restrictions on what can be an argument; as we have
seen, he constructs a function whose value is the true if a designation
of the true occupies the argument-place, and is otherwise the false;
and another function whose value is the false for those arguments
for which the value of the previous function is the true, and vice
versa. He finally constructs a third function:
whose value is to be the false if we put a true proposition into the cy*
argument-place, and any designation which is not of the true (i.e.
either a false proposition, or any designation other than a propo-
sition) into the *x' argument-place; in all other cases the value of
the function is to be the true. This is material implication: 'either
not p or q'.
Thus both for Frege and for Russell a truth-function is the same
kind of thing as a function with an empty place for a name; but there
a certain difference between Frege and Russell. For Frege,
is still
the empty place is a place for a name\ propositions are counted
among names, for he takes propositions (whose components all
have reference) as names of truth-values. For Russell, propositions
are just the range of significant substitutions for the variable in this
kind of function; and he would not call a proposition a name.
Further, he does not construct a function and stipulate what its
values are to be for various arguments; he simply defines each of the
truth-functions as 'that function which with argument p or argu-
ments p and q (these being propositions) is the proposition that .
.', .
filling upthe that clause with some statement about the truth or
falsehood of the arguments) like *p is not true* or *p and q are both
true*.
Both Frege and Russell, however, would hold that the truth-
functional connectives themselves express functions. For Wittgen-
stein it is otherwise. We saw earlier that to say: *A proposition is a
function of the expressions it contains* is not incompatible with
saying: 'A proposition is not a function.* To say that a proposition
120 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
is a function of the expressions it contains is to say that it is the result
of completing them with one another. To say it is not a function is to
say that it is not itself something with an empty place awaiting com-
pletion. In this sense of course a proposition that is a truth-function
of others is not a function, on any view. But on Wittgenstein's view
we must go further: a truth-function of propositions is not a
function of those propositions', for it is the result of an operation, not
a result of completing one expression with another; and a truth-
argument is not the argument of a function, but the base on which
an operation is performed. At 5.25-5.251 it is said explicitly: 'Opera-
tion and function must not be confused. A function cannot be its
own argument, but the result of an operation can be the base of that
very operation.' For example, we cannot put 'x is a man' into the
argument-place in *x is a man' itself we get the nonsense 'x is a man
is a man'; but we can write *~p' for 'p' in '~p' itself, and the
result *~ ~-p' makes perfect sense. Similarly, any operation may be
iterated, any number of times. Sometimes an operation cancels out
when it is iterated: *~ ~p' reduces to *p', and *aCnv(CnvR)b' to
'aRb' (see 5.254). A genuine function never behaves in this way.
With this we come to the question: What is the 'occurrence' of a
proposition 'in' a complex proposition, which we have seen to be
wrongly characterized as the occurrence of an unasserted in an
asserted proposition? Wittgenstein's answer is that in the complex
proposition its component proposition has the role only of a truth"
argument: i.e. it is the base of a truth-operation.
We were inclined to argue: 'In "~p" the sense of "p" must
occur, but it is not being asserted, so we must distinguish between
the sense of "p" and the assertion; here "assertion" has a non-
psychological import, and signifies something that we add to the
sense of "p" when we assert that p.* But Wittgenstein says 'the sense
:
of a truth-function of p is a function of the sense of p* (5.2341) ; here he
is certainly making an arithmetical comparison as 2 is a function of
:
4, namely its square root, so the sense of '~p' is a certain function
of the sense of 'p*, and the sense of *p' only occurs in that of '~p'
in the way in which 4 occurs in 2.
The difference, then, that we first wanted to call the difference
between an asserted and an unasserted proposition because we
OPERATIONS 121
wished to say that the sense of 'p' must occur in *~p' or in *p v q'
is rightly described as a difference between the occurrence of the
sense, and the occurrence, not of the sense, but of a certain function
of the sense.
The argument by which we reached the view that 'assertion' was
an extra feature which somehow gets added to the sense ' "p"
must mean p, in "~p" and "p v q", which nevertheless do not
'
assert that p' could be compared to arguing: "7" must mean 7
in "I had 7-3 apples'*; so we must distinguish, even in empirical
propositions where numerals occur, between the use of a numeral
to designate a number of things, which we will call its positive use,
and uses where it has the same sense but does not designate a
number of things. Some extra feature therefore attaches to the use
1
of the numeral in "I had 7 apples' ,
but not in "I had 7-3 apples".'
By this argument, we might propose to symbolize that 'extra feature'
by prefixing the sign 'P' for 'positive' to certain occurrences of
numerals, and think it necessary to write: 'I had P7 apples' and
'I had P (7-3) apples'.
9
FORMAL CONCEPTS AND FORMAL SERIES
At the end of his life, as we saw, Frege came to think that if some-
thing is a concept, we cannot correctly say that it is a concept i.e.
predicate the term 'concept' of it because an expression for a
concept can significantly occur only in the place of a predicate, not
as a subject of the predicate 'concept'. This doctrine was what
Wittgenstein expressed by saying: 'Something's falling under a
formal concept, as an object 1 belonging to it, cannot be expressed by
a proposition, but is rather shewn by the sign for that object' (4. 1 26) ;
e.g. if something falls under the 'formal' concept concept or property,
this is shown by the predicative character of the sign we use for that
'something'; and again, a variable relating to properties will have
to be one that we take as having one or more argument-places.
1 This
use of the term 'object' must not mislead us into thinking we
have some evidence for Wittgenstein's counting properties and relations
as objects, of different 'type* from the objects they attach to (contrary to
the view stated in Chapter 7). It is the same use as Wittgenstein adopts,
but apologizes for, at 4.123 'A property is internal if its not belonging to
:
its object is inconceivable. (This blue and that blue eo ipso stand in the
internal relation of brighter and darker. It is inconceivable for these
objects not to stand in this relation.) (To the shift in the use of the words
"property" and "relation" there answers here a shift in the use of the word
"object").* Just as internal properties and relations are not properly
speaking properties and relations, so neither are shades of blue objects in
the proper sense.
It may be asked: Why then did Wittgenstein resort to this misleading
terminology? The answer, I think, is that the terminology of objects* falling
under concepts is less loaded with philosophical doctrine, and more of a
familiar way of speaking, in German than in English. Mr. Michael
Dummett tells me that at Minister railway station he saw a notice begin-
ning 'All objects that fallundertheconcept^/7^-/w^cg-e'(^//e Gegenstande,
:
die unter den Begriff Handgepack fallen).
122
FORMAL CONCEPTS AND FORMAL SERIES 123
In Wittgenstein, as we saw (Chapter 5, pp. 82-3), the notion of a
'formal* concept, a concept that cannot be properly expressed by a
predicate or general term, but only by the way we apply the corres-
ponding sort of sign, is extended much more widely than this. Not
only 'concept*, 'function', 'object', but also 'number', 'fact', 'com-
plex', are formal concepts; and, in opposition to Carnap, Wittgen-
stein would maintain that such linguistic concepts as 'name', 'predi-
cate', 'proposition', 'relational expression' are also formal concepts.
In none of these cases can it be informatively said of something that
it falls under the concept; the only proper way of expressing a formal
concept is Frege held for the concepts 'object', 'concept',
(as
'function') the use of a special style of variable; and what makes a
style of variable special is not (say) belonging to a special alphabet,
but something that comes out in the use of the variable. If any
proposition '<A* contains a symbol 'A' for something falling under
a formal concept, then we may always introduce the appropriate
*
style of variable into the two blanks of '(E )<f>
or 'For some .
<f>
Thus: 'Socrates is snubnosed and bald* 'For some x, x is
'.
snubnosed and x is bald.' 'Socrates is bald and Plato is not*
'For some f, Socrates is f and Plato is not f.' Ten men mowed the
meadow' 'For some n, n men mowed the meadow.* Thus: 'Along
with an object falling under a formal concept, that concept is itself
already given' (4.12721); the concept object is given by using
'Socrates', the concept property by using 'bald', the concept number
by using 'ten'; and in each case the formal concept is to be sym-
bolically expressed by a style of variable (4.1272).
As regards the formal concepts that he himself recognized,
Frege so constructed his logical notation that inappropriate substi-
tutions for the corresponding variables gave a visibly ill-formed
expression; he had no need to appeal to what the signs were supposed
to stand for, but only to formal rules. Russell abandoned this ideal
for a symbolic language, and his system requires at critical points
that he should tell us in English how the interpretation of his signs
is to restrict their use; Wittgenstein sharply criticizes him for this
at 3.33-.331 and again at 5.452. But Frege's own system does not,
on Wittgenstein's view, satisfy the ideal; if *<( )* is a predicate, the
empty place in it can according to Frege be filled up with any
124 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
'proper name*, and for Frege 'proper names' include ordinary
proper names, clauses in sentences, definite descriptions, and
numerals. The fact that we should not ordinarily attach any sense
at all to '(a rose is a rose) is white' or '7 is white' did not worry
Frege; the concept white, he says, can be handled in logic only if
we can stipulate, as regards any possible subject for the predicate
'white', what would be the condition for the predicate's holding; and
any 'proper name' is such a possible subject. It was this sort of
paradox that Russell sought to avoid by appealing to 'ranges of
significance'; but Wittgenstein thought this remedy worse than the
disease, because logic cannot rest on vague intuitions expressed in
the vernacular about what is 'significant', but 'must take care of
itself' (5.473).
Wittgenstein's own remedy was to give a wholly new account of
the formal concepts 'proposition' and 'number', which should of
itself show the mistake of treating clauses and numerals as proper
names of objects. This account brings in the notion of a formal
series. The doctrine underlying this notion is that operations, in the
sense of the word explained in the last chapter, can be iterated 'the
result of an operation can be the base of that very operation*
(5.251). For example if we double a number, we can double the
result. If we take *O' to be an arbitrary operation, then starting from
a base 'a' we get the formal series 'a, Oa, OOa, OOOa, n and
. . .
;
Wittgenstein represents an arbitrary term of this series by *[a, x, Ox]*,
where the first of the three expressions within the square brackets
represents the first term of the series, and the other two represent the
way of getting from each term on to the next. In important cases,
the variable expressing a formal concept will relate to the terms of a
formal series; Wittgenstein holds that this is so for the formal con-
cept 'number' and (as we shall see in the next chapter) for the formal
concept 'proposition' also.
1
1 omit the apostrophe that Wittgenstein puts after symbols of
opera-
tion to indicate that he is speaking of the result of the operation: this is a
\estige of the apostrophe used by Russell, who writes 'RV for 'the R
of a'.
In Russell this is significant, because *-Ra' is the predicate 'is an R
of a',
whereas *RV is a definite description. In Wittgenstein it is superfluous and
therefore meaningless, since operations are in any case sharply distin-
guished from relations. Nor is it used very consistently in the printed
version of the Tractatus.
FORMAL CONCEPTS AND FORMAL SERIES 125
One might thus well think that for the concept 'number' it would
have been enough for Wittgenstein to say as he does at 6.022-.03 :
"The concept "number" is the variable number. . .The general
.
form of the whole number is [0, +1]' so long as this was supple-
mented by some account of 0' and of the special operation + T. In
4 *
fact Wittgenstein goes about it in quite a different way. At 6.02 he
gives the following definitions:
This explains the meaning of a zero exponent of the operator 'fl*
and also the meaning of an exponent of the form *n+l' given the
meaning of the exponent 'n'. 1 He then defines the ordinary nu-
merals in terms of and +1, as follows:
1=0+1; 2=0-1-1+1; 3=0+1+1+1; etc.
This enables us to interpret the use of any ordinary numeral as
an exponent; e.g. Q 3x=QQQx. And a number is always 'the ex-
ponent of an operation' (6.021); sentences where numerals appear
to have other uses must be translatable into sentences where they
occur as exponents e.g. '2+2=4' into 'fl2fi 2 x=,Q 4x' (cf. 6.241).
The formal concept 'term of such-and-such a formal series' just is
the concept 'result of applying such-and-such an operation an
arbitrary number of times to such-and-such a base', and a number is
an exponent of any such operation; it would thus involve a vicious
circle to treat numbers as just one special case of formal series. The
informal way of writing the general term of a formal series, '[x,,
?]', (whose informal character is shown by the use of '', just as
in Frege), is thus replaced by:
1
] (6.02) .
1
He uses here a capital omega, instead of a Roman O as in other
O', where the big O is the sign of the operation and the
4
places, because
exponent is 0, is disagreeably unperspicuous. The use of a Greek v, by
assimilation to the use of '', is pointless.
126 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
Wittgenstein's reason for introducing numbers only here, when
he has already often used the conception of a formal series, is that
you can explain what is meant by 'an arbitrary term of a formal
series' quite clearly without explicit mention of numbers: 'First we
have a; then Oa the result of performing a certain operation on a;
then OOa the result of performing it on Oa then OOOa and so on
; ; ;
"an arbitrary term" means "some term or other reached in this way".'
To give a concrete example: We might explain 'ancestor in the
male line' by saying: 'There's my father, and my father's father, and
my father's father's father, and so on.' "The concept of the successive
9
application of an operation is equivalent to the concept and so on
(5.2523). We
have the concept of an arbitrary term of a formal series
when we understand 'and so on' in connection with the series; e.g.
for the series of relations: father, father's father, father's father's
father . . .,* we know what its general term is when we know what
*and so on' (or the row of dots I have just written) means in con-
nection with the series. But if we ask, as regards some term of such a
series, which term it is, which performance of the generating opera-
tion the term results from, the interrogative 'which?' is really an
ordinal interrogative (Latin quotusl) requiring an ordinal numeral
as an answer.
For Frege and Russell, (natural) number was not a formal con-
cept, but a genuine concept that applied to some but not all objects
(Frege) or to some but not all classes of classes (Russell); those
objects, or classes, to which the concept number applied were picked
out from others of their logical type as being and the successors
of 0. The relation successor of was in turn defined by means of the
relation immediate successor of; plainly these two are related in
the same way as ancestor (in the male line) of and father of the
one relation is, as Russell says, the ancestral of the other in each
case. This brings us to the Frege-Russell account (independently
devised by each of them in essentially the same form) of what it is
for one relation to be the ancestral of another. 2 For simplicity's
lf
The series of ancestors is not a formal series, of course; but the
series of relations thus involved is one.
1
The relevant passages can be found in Frege's Foundations of Arith-
metic, 79-80, and in Russell and Whitehead's Principia Mathematica,
Vol. I, Part II, Section E,
FORMAL CONCEPTS AND FORMAL SERIES 127
sake, I shall merely explain how ancestor would be defined in
terms of parent; the generalization of this account can readily be
supplied.
We first define the notion of a hereditary property: viz. a
property which, if it belongs to one of a man's parents, belongs also
to him. We then define a is an Ancestor of b' to mean:
6
4
a is a parent of some human being, say x, all of whose hereditary
properties belong to b.'
Let us for the moment treat this as an arbitrary verbal stipu-
lation of what the defined term Ancestor' is to mean; we must now
*
enquire whether it is true that, on this definition, a is an Ancestor
of b if and only if a is in the ordinary sense an ancestor of b; if so,
we have an adequate definition of ordinary ancestorship which does
not introduce the 'and so on' brought into our ordinary explanations
of the term (cf. the last paragraph but two). And it is quite easy to
show intuitively that this equivalence between 'Ancestor' and 'an-
cestor' does hold.
A. Suppose a is an ancestor of b. Then either (1) a is a parent of
b, or (2) a is a parent of an ancestor of b.
(1) If a is a parent of b, then b himself fulfils the conditions
of being a human being, x, whose parent is a and whose here-
ditary properties all belong to b. So a is an Ancestor of b by our
definition.
(2) If a is a parent of some human being x, who is b's ancestor,
then the hereditary properties of x will all descend, through a finite
number of generations, to b; so once again a will be a parent of
some human being x whose hereditary properties all belong to b
i.e. will be an Ancestor of b.
Hence, if a is an ancestor of b, a is an Ancestor of b.
B. Suppose a is an Ancestor of b. Then there is some human
being, x, whose parent is a and whose hereditary properties all
belong to b. But the property of having a as an ancestor is itself a
hereditary property, since any human being, one of whose parents
has a as an ancestor, himself has a as an ancestor; hence, since this
hereditary property belongs to x, and all hereditary properties of x
belong to b, this property belongs to b i.e. b has a as an ancestor.
Hence, if a is an Ancestor of b, a is an ancestor of b.
128 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
The italicized assertion in proof B may well make the reader
suspect a vicious circle if we are attempting a definition of ancestor-
;
ship, how can we without circularity, in a proof that the definition is
adequate, bring in properties that are themselves defined in terms of
ancestorship? In an informal argument to show that a formal
definition fits our ordinary idea of ancestorship, this might, to be
sure, not seem to matter very much; for we are anyhow supposed to
know informally what 'ancestor' means, and it is 'ancestor', not
'Ancestor' the formally defined term, that is the word used in speci-
fying the questionable 'hereditary property*. This threat of circu-
larity is, however, not found only in informal arguments; in Prin-
cipia Mathematica some of the theorems and proofs formally and
explicitly assume that the properties that are hereditary with respect
to a relation R include some that are themselves defined in terms of
that ancestral relation which corresponds to R
as Ancestor cor-
1
responds to parent. Wittgenstein's accusation of having run into a
vicious circle about the ancestral relation (4.1273) was a peculiarly
vicious blow against Russell, who had elaborately contrived the
system of Principia in order to avoid circles of this sort (see op. cit.,
Vol. I, Introduction, Ch. II). Russell came to be seriously worried
by the accusation, and in the second edition of Principia he added as
an appendix a new chapter of proofs designed to avoid the vicious
circle.
Wittgenstein himself did not need the analysis of ancestral
relations in order to give an account of the number-concept; but
such relations are logically important in their own right, and sup-
plied him with an application for his notion of a formal series. It is
clear that the series of propositions
aRb; (Ex) aRx.xRb; (Ey) (Ex) aRx.xRy.yRb;
and so on
is a formal series in Wittgenstein's sense, as he
says (4.1273). He does
*op. cit., Vol. *90.163, *90.164, and proof of *90.31. (To make
I,
this strictly correct, one would have to modify slightly the definition of
'Ancestor' so as to count also as his own Ancestor anyone who either was
or had a parent.)
FORMAL CONCEPTS AND FORMAL SERIES 129
not, however, tell us what operation must be applied to each term to
yield the next one; and if we use 'x', 'y', etc. as variables, the opera-
tion is not perspicuously displayed. For here 'what the signs conceal,
their use reveals'; it is the case, though the style of variables here
used conceals it, that we have to think of variables as themselves
forming an indefinitely long series, a formal series, in which each
variable is derived from its predecessor; otherwise our capacities
of expression would run out as soon as we had used all the letters of
the alphabet. 'The same operation turns the variable "p" into the
variable "q", "q" into "r", and so on. This can only be explained by
these variables' giving general expression to certain formal relations'
(5.242). Wittgenstein's idea of an unlimited stock of variables, given
once for all by a formal rule, is taken for granted in modern dis-
cussions of logical syntax.
The series may be exhibited more clearly as a formal series by
7
using 'x, x', x", x" , .' instead of 'x, y, z,
. . 'as variables, and
writing the terms of the series as follows:
(Ex') (Ex) a=x.xRx'.x'=b
(Ex") (Ex') (Ex) a=x.xRx'.x'Rx".x"=b
7
(Ex'") (Ex") (Ex ) (Ex)a=x.xRx'.x'Rx".x''Rx"'.x"'=b
and so on. 1
Indeed, it would be quite easy to put into words a formal rule
for deriving each successive line from the last line. If we use *s
(Hebrew Resh) as a symbol of the operation successively applied,
the general term of this formal series will be written in Wittgen-
stein's style as:
hP,
where 'p' is '(Ex') (Ex) a=x.xRx'.x'=b'. And to say that b
1
Remember that *a=x' means that a is the same thing as x; and V=b',
that x' is the same thing as b.
130 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
is a successor of a with respect to the relation will beR
to assert the logical sum of this formal series of propositions
i.e. to assert that some proposition or other in the series is
true. 1
Wittgenstein used his doctrine of formal series to make a further
important criticism of Russell and Whitehead: that they kept on
making tacit use of formal series whose component propositions
were, on their own theory, propositions of different 'logical type'
and as such could not form a single series (5.252). One example of
this fallacy occurs in their use of the Axiom of Reducibility. This
axiom may be stated as follows: For any property of individuals that
*
is specified in terms of quantification over properties of individuals
there is another property that applies to the same individuals and is not
specified in terms of such quantification?
Now at *20.112 Russell requires, not this axiom, but a parallel
axiom got by substituting 'properties of individuals' throughout for
'individuals'', yet he simply refers back to the Axiom of Reducibility
for individuals (in fact he merely gives its number, *12.1). Clearly he
would in strictness need a new Axiom for properties of individuals,
and another for properties of properties of individuals, and so on
i.e. a formal series of axioms, each constructed in a uniform way
from its predecessor; and the only legitimate way of supplying what
is required would be to give the generating operation of this series.
But Russell's Theory of Types explicitly rules out the possibility of
1
In his paper 'On Derivability', Journal of Symbolic Logic; Vol. II,
No. 3 (September 1937), Quine shews that a certain notion, practically
the same as that of a formal series of expressions, enables us to define
ancestral relations without any such quantifications as 'every property
that .'or 'some one of the relations
. . . .' at all. This
.
definition, which
may be called a fulfilment of Wittgenstein's intentions, accordingly avoids
any risk of a vicious circle, such as might arise if some of the properties or
relations covered by the quantifications employed had themselves to be
specified in terms of the ancestral relation.
a
e.g. the property of having all the properties of a great general; or
again, the property of having all the vices of Charles I and none of his
virtues.
s
The property of being a great general applies to the same individuals
as the property of having all the properties o/a great general; but is not
itself specified, as that property is, in terms of
quantification over proper-
ties of individuals.
FORMAL CONCEPTS AND FORMAL SERIES 131
such an operation, since each successive proposition in the series
would be of higher 'type' than its predecessor, and not groupable
with it in a single series. 1 Unlike the criticism about ancestral relations,
this criticism was ignored by Russell in preparing the second edition
of Principia Mathematica.
1
Wittgenstein's objection that the Axiom of Reducibility would not be
a logical truth anyhow (6.1233) is independent, and not here relevant. For
a clear modern account of this problem, see Quine's From a Logical Point
of View, Chapter VI, 6.
10
'THE GENERAL FORM OF PROPOSITION'
It is now possible to explain the formula [, f N(|)] which Wittgen-
,
stein gives as 'the general form of truth-functions*. Russell's explana-
tion given in the Introduction is useless and should be disregarded.
Anyone with a taste for exegesis for its own sake will be able to work
out how Russell came to give this explanation, which was probably
prompted by an answer given by Wittgenstein to a question of his:
'What is f here?'
The formula is a particular example of the 'general term of a
formal series', with a set of propositions as the first term and joint
negation as the generating operation. We have seen that the presence
of in a formula is the mark of informal exposition; hence, once we
can use numbers, this formula can be given as
B, Nm N + n 1
(p>]
1
p is the whole collection of elementary propositions; the formula
therefore presupposes that if the number of elementary propositions
were finite, we could say exactly what, say, the hundred and third
term of the series of truth-functions (whose general term this
formula is) would be.
Let us consider the case of only two elementary propositions, p
and q, and shew how the totality of the truth-functions of p and q can
be generated by 'successive applications' of the operation N(). The
1
The stroke over the variable indicates that the variable stands for an
arbitrarily stipulated list of propositions which are the values of the
variable (see 5.501). These are a mere list, and hence not a single propo-
sition except in the degenerate case where the list has only one item. The
application of the operation N(jp to the values of the variable, on the
other hand, always results in a single proposition.
132
'THE GENERAL FORM OF PROPOSITION' 133
general term of the formal series of truth-functions having p andqas
truth-arguments will be
[P,q, N-foq), JP+Hp.q)]
We can easily state the second, third, fourth, etc. up to the seven-
teenth term of this series; it has only seventeen terms; for there are
fourteen distinct truth-functions of p and q, plus tautology and
contradiction. The first term, *p,q*, is of course not a truth-function
and not a proposition; it is just a list of the bases of the operation.
The second term, and first truth-function, in the series is N(p,q)
(= neither p nor q); and the second truth-function is the result of
applying the operation to the result of the first application of it : it is
therefore N(N(p,q)), i.e. p v q.
If we applied the operation once more to this result, we should
only get back to the first truth-function, since ~
^P^p. So we
apply the operation to the two results obtained so far, and obtain
N(N(p,q), N(N(p,q))), which, being the joint denial of a pair of
contradictories, is contradiction. The next result will be obtained by
applying the operation to this last-obtained result, i.e. negating it,
and is tautology.
If we now apply the operation to the last two results as values of
f, the result is once again contradiction, which we have already got,
so we discard it. And if we apply the operation to the third result,
contradiction, together with the first, we get tautology conjoined
with the negation of the first result, which is the same as the second
result; similarly if we apply the operation to the third result, contra-
diction, together with the second, we get tautology conjoined with
the negation of the second, which is the same as the first result.
Similarly it is fruitless to apply the operation to the fourth result,
tautology, in any combination, since this will always yield con-
tradiction. And if we now proceed to make the first four results,
first in such threes as are possible, and then all at once, the subject
of the operation, we shall reach no new result.
For our next truth-function we therefore bring down the first of
the bases, combining it with the result of the first application, and
write N((N(p,q)),p), i.e. "~(~p. ~q). ~p, i.e. p v q. ~p, i.e. q. The next
application, N(N((N(p,q)),p), negates the previous one and yields
134 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
~q; no more can be done by a reapplication of the operation to the
result so obtained, which will only give us q again. We therefore try
bringing our base, p, into combination with the result of the second
application of N(f). This was N(N(p,q)), and we write
N((N(N(p,q))),p)
which gives us ~(p v q). ~p, i.e. ^. ~q. ~p. But that is the same as
~P*~3 which we have already had; so it adds nothing, and neither
will its negation.
At the next step we take as values of the result of the third
application,which was contradiction, and our first base, p; now
N(contradiction, p) is (tautology. ~p), which is the same as ~p.
The result of the next application will of course be the negation of
this, i.e. p.
We next take as values of the result of the fourth application,
tautology, and p; but this gives us (contradiction. ~p), which is still
contradiction, and the negation of it will be tautology.
When we have gone as far as we can with our first base and
previous results, we bring down our second base, q, and combine it
with the result of the first application; this will yield p, and an
application of N() to this its negation, ~p; having reached these
already, we discard them.
We go on in this way until no new application of the operation to
results hitherto reached, together with our second base, yields any
proposition not already obtained. We go on taking as values of any
results not hitherto taken together of course we are not confined
to only one or two values for at a time, but use all possible numbers
in a systematic order. It is clear that we can have here a series with a
definite order, if we suppose that we are given the original bases in a
definite order. This of course is the precise reason why Wittgenstein
*
says at 5.242 that "p", "q*' and "r" must be variables which give
general expression to certain formal relations*. The use in symbolism
of the variable prepositional signs p, q, r must bring these formal
relations out; and they would be brought out most clearly if we
wrote p', p", p'", eta
That Wittgenstein's account makes sense for any finite set of
bases is clear enough. What was needed was that the description 'the
*THE GENERAL FORM OF PROPOSTITON' 135
th
result of the n application of the operation N() to these bases*
should be an absolutely precise determination of a proposition for
any number n up to the total number of truth-functions that there
are for this set of bases. It is clear that it is such a precise determina-
tion of a proposition.
Here we come to the defect in the Tractatus which Wittgenstein
described, later, as a failure to distinguish between the 'dots of lazi-
ness', as when we represent the alphabet by writing just
C
A,B,C, . . .*,
and the indispensable dots used to represent an infinite series, as in
*
1,2,3,4,. .'. It is easy to see how he came not to make this dis-
.
tinction, if we consider the following diagram:
pqrstv
TTTTTTTTTTTT
FTTTTTTTTTTT
XFXXXXXXXXXX
.
.
F
F
FFXXXXXXX.
XXFXXX .F . .
F
F
FXFXXX
XFFXXX F
F
FFFXXX
XXXFXX F
F
FXXFXX F
XXFFFFFF
FXFFFFFF F
F
XFFFFFFF
FFFFFFFF F
X
(In the left-hand column, the X's and F's alternate; in the second
column they alternate hi pairs, in the third in fours; and so on.)
136 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
This table represents a quite definite truth-function of a set of
propositions of unspecified number; this truth-function has the
truth-value false for all combinations of truth-values except the final
one; it does not matter how many propositions there are, because
the bottom line of arguments is all Fs anyway. This truth-table
defines, for an arbitrary number of bases, our operation N(|) -joint
negation of all the propositions in the set; and the liberal use of dots
in it may well seem not to matter the rule for constructing the
table is plain. If, therefore, the repeated application of this operation
to the given bases will in any finite case generate all their truth-
functions, and if it is possible to specify a set of propositions other-
wise than by enumeration, then it is very natural to say: what does
it matter that the number in the set is not known, what does it
matter even if it is infinite?
There would indeed be a serious objection if operating thus upon
the set of propositions did not generate a simple infinite series (a
progression) but one that was e.g. like the set of odd numbers in
natural order followed by the set of even numbers in natural order.
That series can of course be rearranged as the progression 1 ,2,3,4, . . .
;
but it might be that without ceasing to be a formal series our series
could not so be rearranged. In that case the expression 'the nth term
of the series' would never, for any finite n, get you into the part of
the series that began after you had started on a second infinite
series. Hence, if in the generation of a series of truth-functions by
repeatedly performing the operation N(|) upon the set of elementary
propositions (in the way I have described) there would have to be a
series of generations which you could only begin after you had gone
through the process of bringing down each new member of the set
in turn, right to the end of the infinite set; then Wittgenstein's idea
would be worthless.
But this difficulty can in fact never arise. You can perform the
new operations on the terms up to the nth term, which are made
st
possible by the introduction of the n-fl term, as soon as you have
introduced the n-fl st term; and there is never an infinity that you
have to finish before you can get on, because you quickly generate
propositions that either are tautologies or contradictions, or are
identical with propositions you have already generated, and you do
'THE GENERAL FORM OF PROPOSITION' 137
not have to proceed further with those. You clear up as you go
along.
On the other hand, the claim that is being made, in offering this
as the general term of the series of truth-functions of an infinite set
of elementary propositions, is apparently in conflict with the well-
known theorem that the truth-functions of an infinite set of elemen-
tary propositions form a non-denumerable set. This is so, because
the number of different assignments of truth-values to n propositions
is 2n The number of different assignments of truth-values to X
.
propositions (i.e. to a denumerably infinite set of propositions) is
SI
therefore 2" But this has been proved by Cantor to be greater
.
than S ;
that is to say, you could not find a one-one correlation
Sg
between a set whose number was 2 and a set whose number was
X And the truth-functions of S
.
propositions must be at least as
many as the possible ways of assigning truth-values to them. There-
fore an account which correlates the series of truth-functions of an
infinite set of elementary propositions with the series of natural
numbers, as Wittgenstein's does, must be wrong.
It seems likely enough, indeed, that Wittgenstein objected to
Cantor's result even at this date, and would not have accepted a
Cantorian device for specifying an infinite subset of the elementary
propositions such that a truth-function of it could not be generated
by his formula. For though he came to think his idea wrong, it was
certainly not through any conversion to Cantor that this happened.
On the contrary: whether or no he already objected to Cantor at the
time when he wrote the Tractatus, he certainly did so later.
However, the theory of the Tractatus, promising though it
looked at the time, has been clearly and cogently refuted in another
way. If all truths of logic are tautological truth-functions of ele-
mentary propositions, then there is in principle a decision procedure
for them all. But it was proved by Church in the 1930's that multiple
quantification theory has no decision procedure; that is, that there
cannot be a method by which one could settle, concerning any
well-formed formula of that theory, whether it was a theorem or
not.
11
GENERALITY
Frege's invention of the quantifier-notation must be reckoned among
the greatest benefits conferred on philosophy by logic. The fallacies
which are excluded by the insight it gives have been committed over
and over again by the greatest philosophers. No one should now be
able to get away with transitions like that from 'Every boy loves
some girl' to 'Some girl is loved by every boy*. In this down-to-earth
example, the fallacy sounds silly and impossible to commit; in
abstract contexts, it and similar fallacies (involving the notions, not
just of 'some* and 'all*, but of one of these combined with 'necessary*,
or 'possible') have proved very difficult to avoid.
A recent example of this sort of fallacy is afforded by Professor
Ayer; he argues from the fact that it is not possible, and a fortiori
1
not necessary, that every identification or recognition (of a person,
shape, quality, etc.) should in fact be checked, to the innocuousness
of the notion of an uncheckable identification. An argument running
'It is not necessary that every identification is checkable; ergo^ it is
possible that some identification is uncheckable' has all the appear-
ance of formal validity *Not necessarily (every S is P); ergo>
possibly (some S is not P)*. But in fact it is an illicit transition from:
(1) It is possible that it is not possible that every identification
should be checked
to
1
TTie problem of knowledge, pp. 60-1. The passage concerns Wittgen-
stein's objection (in Philosophical Investigations) to 'private' ostensive
definition. Professor Ayer seems to accept a kind of checkability as
necessary to the notion of an identification; but in reply to the objection
that 'private* checks are not checks, he retorts that in any case checks
always have to come to an end somewhere.
138
GENERALITY 139
(2) It is possible that there should be some identification that it
is not possible to check.
It is one of the uses of the quantifier-notation tomake this clear. Let
'M' represent 'possibly'; (1) and (2) then come out as:
(1) M~M(x) (x is an identification DX is checked)
(2) M(Ex) (x is an identification. ~M (x is checked))
or (equivalently): M~(x) (x is an identification DM
(x is checked))
The quantifier-signs now in use, *(x)<x' for 'Everything is <* and
1
'(Ex)^x' for 'Something is <f>\ were given us by Russell and White-
head; but the former is a variation of Frege's generality notation,
and the latter can be defined in terms of it, so the real inventor was
Frege.
Often enough a logical symbolism simply puts some new sign in
place of a word or phrase; this may be helpful. But what the quanti-
fier-notation does is quite different. At first sight 'Everyone is clever*
looks to be just such a sentence as 'Socrates is clever*. It becomes
clear that it is not, as soon as we consider negation: if 'Socrates is
clever' is untrue, then 'Socrates is not clever* is true; but, as Aristotle
remarked, the same does not hold for 'Everyone is clever'. Frege's
genius consisted in inventing a notation in which a formula of a
different layout is employed for universal propositions; and not just
of a different layout, but of the right layout.
This was surely partly what prompted Wittgenstein to say at
3.323: 'In ordinary language it is enormously common for ... two
words, which signify in different ways, to be applied in the sentence
in ways that are outwardly the same. ... In this way there easily
arise the most fundamental confusions (with which the whole of
philosophy is filled).
'In order to avoid these errors, we must use a symbolism which
excludes them A symbolism, then, that follows logical grammar
logical syntax'. And again, it will have been this that inspired the
'feeling' that he speaks of at 4.1213, the 'feeling that we are in pos-
session of the right logical conception if only all is right in our
symbolism'.
At 4.041 1 Wittgenstein dilates on the excellence of the symbolism
1
Russell and Whitehead actually use an inverted *E*; I am following
Hilbert and Ackermann.
140 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
'(x)fx*. He brings this out by considering alternative ways of ex-
pressing what we use this symbolism to express. We might try
putting 'Gen.fx'; but 'this would not tell us what was generalized'.
That is, it would be ambiguous as between what we should now
write as *(x)fx' and '(f)fx*. If we try to make good this defect by
writing the sign for generality as a subscript to the x, thus: *f(xg )*, it
would not do: 'we should not know the scope of the generality-
still
sign*: That is, *<xg v */rXg' would be ambiguous as between '(x)<x
v (x)ix' and '(x>x v ^x*. Finally, if we thought of writing the
generality sign itself in the argument-place: (G,G)f(G,G) 'we should
not be able to determine the identity of the variables'. That is to say,
the expression
would be ambiguous as between what we should now write as
and
In particular, we could not distinguish between these cases :
which holds for any relation <f>,
and
which means that the relation is symmetrical.
<
These difficulties could of course be got over by supplementary
conventions, corresponding to the 'enormously complicated tacit
conventions' which Wittgenstein mentions at 4.002 as needed for
the understanding of ordinary language. Think of the English
sentence 'If you can eat any fish, you can eat any fish*, which sounds
like a tautology, but is, on the contrary, a false judgment. Any
GENERALITY 141
native English-speaker will understand that sentence: few could
explain how it works. And again e.g. 'You can fool some of the
people all of the time' is ambiguous; the ambiguity is resolved, in
some complicated way, by the context. It is clear that the Fregean
quantifier-notation is far more perspicuous than any that has to be
backed up with complicated conventions.
Turning now to Wittgenstein's special treatment of generality,
we shall find it helpful to place the opening entry, 5.52, in juxta-
position with 5.51, the immediately preceding entry of the same
numerical level in the book:
5.51 'If
: has only one value, then N(|)= '-^p (not p); if it has
two values, then N(|)= ~p.~-q (neither p nor q).'
5.52: 'If the values of are all the values of a given function fx
for all values of x, then N(f) will be the same as ~(Ex)fx.'
Russell's account in the Introduction, then, is quite correct:
'Wittgenstein's method of dealing with general propositions [i.e.
"(x)fx" and "(Ex)fx"] differs from previous methods by the fact that
the generality comes only in specifying the set of propositions con-
cerned, and when this has been done the building up of truth-
functions proceeds exactly as it would in the case of a finite number
of enumerated arguments p, q, r, . . .* Wittgenstein emphasizes the
difference by saying: '/ separate the concept all from the truth-
function' and goes on to accuse Frege and Russell of not having
done this: Trege and Russell introduced generality in connection
with the logical product [p.q.r. ] or the logical sum [p v q
vrv ]. This made it difficult to understand the propositions
"(Ex)fx" and "(x)fx", which cover both ideas.'
Now there is no ground in their texts for a direct accusation that
either Frege or Russell 'introduced generality in connection with
the logical product or the logical sum*. We must therefore see in this
remark Wittgenstein's comment on their way of introducing gen-
erality: a claim that this is what it amounts to. So we must examine
how they actually introduce generality.
Frege introduced his generality notation in this way in Function
and Concept: he constructs the sign
142 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
in which what he has done is (to quote from his Begriffsschriff) to
replace an argument, say 'London' in 'London is a capital city', 'with
a German letter, and insert a concavity in the content stroke, and make
this same German letter stand over the concavity'.
The sign so constructed signifies the thought that 'The function
is a fact whatever we take its argument to be.* Or, as he puts it in
Function and Concept, the sign
is 'to mean the true when the fimction/(*) always has the true as its
value, whatever the argument may be*. Certainly there is nothing
here about a logical product. So what is Wittgenstein's argument?
It is based on his own view: the truth of such a proposition as
s
-
(x)fx' (to use the signs
'fa.fb.fc.fd
all the names
'
now usual) is the truth of the logical product
where the dots cover up our failure to write down
there are as arguments in the function fx. Therefore
:
when Frege explains his symbol, by stating what is judged to be the
case in the judgment that it symbolizes, he is in fact introducing 'all'
in connection with the logical product.
-
Frege does not employ an existential quantifier like *(Ex)' in
constructing the symbol for judgments of the form 'Some ;
he
simply uses negation together with his universal quantifier, just as
'
we can define *(Ex)fx' as *~(x)~fx*; but the same point would hold
for the explanation of particular judgments: their truth according
-
to Wittgenstein consists in the truth of a logical sum (fa v fb
v fc v fd ) and hence what they say is that that logical sum is true.
So someone who explains them by explaining what they say is 'intro-
ducing generality hi connection with the logical sum'.
Russell's explanations are not relevantly different from Frege's.
'This,' Wittgenstein says, 'made it difficult to understand "(x)fx"
and "(Ex)fx", which cover both ideas.' By 'both ideas' he means
both the idea of generality on the one hand, and that of the logical
product (in the case of universal propositions), or the logical sum (in
the case of particular propositions), on the other. The reason why
*it became difficult to understand* these propositions was that their
pictorial character was obscured. Their pictorial character consists
GENERALITY 143
in their being truth-functions of a set of propositions. But the
notation also covers the way of specifying the set a truth-function of
which is being asserted, viz. giving a function all of whose values
are the set in question. 'The function's being a fact whatever we
take the argument to be' explains generality in terms of the truth
of the generalized proposition. With such an explanation, how are
we to understand the inference from (x)^x to <a; i.e. from some-
thing's holding of a function, to something's holding of an object?
As Ramsey says, Wittgenstein's view 'explains how "fa" can be
inferred from "For all x, fx", and "There is an x such that fx" from
"fa". The alternative theory that "There is an x such that fx" should
be regarded as an atomic proposition of the form "F(f)" ("f has
application") leaves this entirely obscure; it gives no intelligible
connection between a being red and red having application, but
abandoning any hope of explaining this relation is content merely
to label it "necessary".' 1
Wittgenstein goes on to make further comments on the generality
notation. It has, he says, two peculiarities: it points to a logical
proto-picture, and it emphasizes constants. Ramsey explains the
second point to us. 'Let us consider when and why an expression
occurs, as it were, as an isolated unit. "aRb" does not naturally
divide into "a" and "Rb", and we want to know why anyone should
so divide it and isolate the expression "Rb". The answer is that if it
were a matter of this proposition alone, there would be no point in
dividing it in this way, but that the importance of expressions arises,
as Wittgenstein points out, just in connection with generalization.
Itis not"aRb" but "(x)xRb" which makes "Rb" prominent. In writing
(x)xRb we use the expression "Rb" to collect together the set of
propositions xRb which we want to assert to be true; and it is here
that the expression "Rb" is really essential because it is this which is
common to this set of propositions.' 2
Wittgenstein does not explicitly say that the importance of 'ex-
pressions' arises in connection with generalization: rather he uses
the notion of an expression to form his theory of generality. For him
expressions explain generality: an expression, by being 'the common
1
The Foundations of Mathematics, pp. 153-4.
1
Ramsey, ibid., pp. 123-4.
144 AN iNi-RODUcnoN TO WITTGENSTEIN'S TRACTATUS
characteristic mark of a class of propositions', gives us that class
the class of them all. But the class in question is clearly that narrower
range of propositions in which an expression occurs, which Ramsey
found it necessary to distinguish.
At 4.12721 Wittgenstein tells us: 'The formal concept is already
given with an object that falls under it.' That is, if we have been
given fa, the formal concept presented by the name-variable x is
already given: the 'proto-picture' 'fx' is given. Here I assume that T
is a constant; thus this proto-picture is not the 'logical proto-picture'
that is obtained by turning all the constants, into which a propo-
sition divides up, into variables, as was described at 3.315: 'If we
change one component of a proposition into a variable, then there is
a class of propositions which are all values of the resulting variable
proposition. This class in general still depends on what we, by
arbitrary convention, mean by parts of that proposition. But if
we change all those signs whose reference has been arbitrarily deter-
mined into variables, there is still always such a class. This, however,
now no longer depends on convention; it depends only on the nature
of the proposition. It corresponds to a logical form a logical proto-
picture' *(x)x moves slower than light', for example, lays emphasis on
:
'moves slower than light' as an expression which collects together a
class of propositions, and points to a 'logical proto-picture' xRy,
where (taking R as variable) all the constants have been turned into
variables.
This paves the way for Wittgenstein's next remarks: 'The sign
of generality appears as an argument. Once objects are given, that
of itself gives us all objects. Once elementary propositions are given,
that is enough for all elementary propositions to be given* (5.523-4).
When he says 'the sign of generality occurs as an argument' he is
referring to the 'x' in '(x>x': we have passed from the form '^a' to
the construction of the form *^ everything' which we can do because
the expression *<( )' collects all propositions of the same form as
'$a': it determines a certain range of propositions. '(x)<x' is then
just the proposition which is a certain truth-function of those
propositions: we saw just why this should be so good a notation at
the opening of the present chapter. Thus it is that we are formally
given 'all objects' and therewith the possibility of all their con-
'
GENERALITY 145
nections, which form the elementary situations ; and thus it is
that we are given 'all elementary propositions*, and therewith all
possible propositions, i.e. all possibilities of being the case or not
the case.
Wittgenstein's view has the following strong advantage. If we
introduce V
as a truth-functional connective, then unless we adopt
Wittgenstein's view we need a new account of it in such propositions
as '(x)^>x v i/fx' e.g. 'All roses are either red or yellow', for here it
does not conjoin clauses to which a truth-value can be assigned.
'If logic has primitive notions,' he says at 5.451, 'they must be inde-
dependent of one another. If a primitive notion is introduced, it must
be introduced for all the contexts in which it occurs at all. Thus we
cannot introduce it first for one context, and then introduce it all
over again for another. For example: If negation has been intro-
duced, we must understand it in propositions of the form "~p"
just as in propositions like "~(p v q)", "(Ex) fx", etc. We
cannot
introduce it first for the one class of cases, and then for the other,
for then it would be doubtful whether it meant the same in both
cases, and there would be no ground for using the same kind of con-
nective in both cases. In a word, what Frege said (in the Grundgesetze
der Arithmetik) about the introduction of signs by means of defi-
nitions holds, mutatis mutandis, for primitive signs.' Russell and
Whitehead did introduce *~' and V
all over again for uses with
quantifiers (see Sections *9 and *10 of Principia Mathematical.
Modern logicians mostly introduce them with a merely truth-
functional explanation, and then go on using them 'with innocent
1
faces' in the predicate calculus.
The concept 'all* is all-pervasive in the Tractatus. 'The world is
everything that is the case the totality of facts determined by the
facts and by their being all the facts* (1-1.11). And at 4.51-2 we
find: 'Suppose all elementary propositions were given me: then we
can simply ask what propositions I can form from them. And these
are all propositions: that is how they are limited. Propositions are:
All that follows from the totality of elementary propositions (and of
course from its being the totality of them all). Thus it might be said
1
cf. Tractatus, 5.452. For an example, see Quine, Methods of Logic,
Section 16.
146 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
that in a certain sense all propositions are generalizations of the
elementary propositions.'
We have to think of the case in which the world is finite, remem-
bering that Wittgenstein did not think there was any essential dif-
ference between the finite and the infinite case. If we want to say not
merely that such-and-such things are green, but that everything (in
a box, for example) is green, this can be expressed by saying: 'There
are an x and a y in the box, x and y are green, and there are not an
x and a y and a z in the box.' (Here I am using Wittgenstein's pro-
posed convention about identity: 'Identity of the object I express by
identity of the sign, not by a sign of identity: difference of the object
1
by difference of the sign' (5.53).) It is clear that 'Everything in the box
is green' does not follow just from 'The objects a and b, which are
1
in the box, are green it must also be the case that a and b are all the
:
objects in the box.
Similarly, that such-and-such an independent possibility is not
the case follows from all the facts, together with the fact that these
are all the facts. (As Professor Stenius has pointed out to me, at this
stage (1.1 1) Wittgenstein means by 'the facts' only 'positive facts':
he has not yet introduced the expression 'a negative fact* for the
'non-existence of atomic facts', but has only brought in 'facts' which
are stated to consist in the existence of atomic facts.) And so Witt-
genstein says: 'the totality of facts determines both what is the case
and also all that is not the case' (1.12); and this is so whether the
world is finite or infinite.
How this is so is seen clearly in the finite case; yet in the finite
case Wittgenstein's doctrine appears to have a rather inconvenient
consequence which Ramsey drew. Ramsey argues that 'There are an
x and a y such that x^y' is the logical sum of the propositions
x^y, which are tautologies if x and y have different values, con-
tradictions if they have the same value. Hence it is itself a tautology
1
He has sometimes been taken to demand this convention. This is a
misunderstanding; he merely puts it forward as a possible one. The fact
that (however inconvenient) it is possible shews that identity is not a
genuine function. His view, then, does not require the abandonment of the
sign of identity if it should be convenient to use it. But it does exclude
uses of it which make a genuine function of it: as for example, in the
attempt to express *a exists' by Tor some x, x is identical with a*.
GENERALITY 147
if any one of the set is a tautology, but otherwise a contradiction.
That is, it is a tautology if x and y can take different values (i.e. if
there are two individuals), but otherwise a contradiction. 1 He
concludes that the series "There is one individual. . There are at
. .
least 2 individuals ---- There are at least n individuals .' begins by
. .
being tautologous; but somewhere it begins to be contradictory, and
the position of the last tautologous term shews the number of
individuals.
Now Wittgenstein rejected propositions of the form "There is
not an x such that x^x*, jeering at it with the remark: 'Would this
not be true if "there were things'* but these were not identical with
themselves?* (5.5352). He did not regard identity as a genuine
function. But the point can be made without using identity. Let us
suppose, for the sake of simplicity, that there are only two objects,
a and b, and one function, f. Then the possibilities fa, ~fa, fb,
~fb, will be all the possibilities that there are. Suppose that we
write these possibilities down as follows :
(1) There are an x and a y and a such that <f>x an
<f>
(2) There are an x and a y and a such that <f>x and
(f>
(3) There are an x and a y and a such that ~<x an
<f>
(4) There are an x and a y and a such that ~<f>x and
<j>
These are all the possibilities; therefore, the 'complete description
of the world in a completely generalized proposition*, of which
Wittgenstein speaks at 5.526, will be given by one of them.
Here we have 'described the world* without any preliminary
correlation of a name with a particular object. And then, in order to
arrive at the usual way of putting it, in which names are used, we
need only add e.g. 'x is a, and y is b, and is f. But that this is a
<f>
complete description, i.e. is all the facts there are, can only be ex-
pressed if we can add such propositions as 'and there are not an x
and a y and a z, and there are not a $ and a such that .'. Hence
. .
it is required at 5.526 that we shouldhave propositions stating: "There
is one and only one x such that .
.', which would have to be of the
.
form: 'There is an x and there are not an x and a y, such that .'. . .
This surely means that the 'complete description in completely
1
Ramsey, The Foundations of Mathematics, pp. 59-60.
148 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
generalized propositions' would, in the finite case, consist partly of
existential propositions employing more variables than there are
names of distinct objects. But if what is false can be true, then the
completely generalized propositions will allow more play to the
facts than the totality of elementary propositions. Yet at 5.5262
Wittgenstein denies this: 'The truth or falsehood of every propo-
sition alters something about the general structure of the world. And
the play which is allowed to its structure by the totality of elementary
propositions is just that which is limited by the completely general
propositions.*
This conclusion can only be avoided by adopting Ramsey's sug-
gestion and saying that the series of propositions
v ~<x
v <~^4x.<y v ~<y.z v z
etc., would go over, in the finite case, from being tautologies to being
contradictions at the point at which the number of different name
variables employed exceeded the number of different names. And it
is really only in the finite case that Wittgenstein's theory can be ex-
pounded with much clarity.
I find this conclusion unsatisfactory: in the infinite case, Witt-
genstein's theory can hardly be explained at all: we have to take the
finite case and say that he saw no important difference between it
and the infinite case; while in the finite case the view seems to lead to
a sudden transition of an existential proposition from tautology to
contradiction.
Did not Wittgenstein resist any attempt to symbolize "The
universe is not empty* in "There are things', regarding these as
'pseudo-propositions' attempting to say what shews! At 5.535 he
says that what the 'Axiom of Infinity' (which says that there is an
infinite number of objects) is supposed to mean 'would be expressed
in language by there being infinitely many names with different
references'.
If (Ex)fx is the logical sum: 'fa v fb v fc . . .' then (Ex)fx v ~fx
is the logicalsum of the singular tautologies: fa v ~fa v f b v ~fb
vfcv ~fc...; and so (Ex,$$x v ~^x will be the logical sum of all
the singular tautologies: fa v ~fa vfb v ~fb ga v ~ga vgb v
. . .
GENERALITY 149
. . . etc. In these propositions, then, we can see how 'the existence of
things' is something 'shewn' and not said. But if Wittgenstein allows
(Ex)<^x. (Ex,y)<x.<y as a way of saying that only one thing has <f>,
as he does at 5.5321 , it is difficult to see how he could avoid a way of
admitting formulae which say 'There are only n things and func- m
tions* without using either 'thing' or "function* as a function.
12
KNOWLEDGE AND CERTAINTY
Probably the best-known thesis of the Tractates is that 'meta-
physical' statements are nonsensical, and that the only sayable
things are propositions of natural science (6.53). Now
natural
science is surely the sphere of the empirically discoverable; and the
'empirically discoverable' is the same as *what can be verified by the
senses'. The passage therefore suggests the following quick and easy
way of dealing with 'metaphysical' propositions what sense-observa-
:
tions would verify and what falsify them? If none, then they are
senseless. This was the method of criticism adopted by the Vienna
Circle and in this country by Professor A. J. Ayer.
There are certain difficulties about ascribing this doctrine to the
Tractatus. There is nothing about sensible verification there. If
Wittgenstein means to suggest that we can test a proposition for
significance by seeing if we can state the sense-observations that
would verify it, then it is surprising that he does not say so. Nor is
a reference to sensible verifiability immediately implicit in the
identification of 'what can be said' with 'the propositions of natural
science*; for the totality of natural science has been defined earlier
in the book (4. 11) as the totality of true propositions. Nowhere have
we any suggestion of a general method for criticizing sentences,
according to which we may say: 'What observations would verify
(or falsify) that? If none, then it does not mean anything.' Such a
general method for criticizing sentences would obviously need a
preliminary justification; and it is difficult to see how the Tractatus,
for example, can be taken as such a preliminary justification, when
it says nothing about sensible observation.
150
KNOWLEDGE AND CERTAINTY 151
The general method that Wittgenstein does suggest is that of
'shewing that a man has supplied no meaning [or perhaps: "no
reference"] for certain signs in his sentences'. I can illustrate the
method from Wittgenstein's later way of discussing problems. He
once greeted me with the question: 'Why do people say that it was
natural to think that the sun went round the earth rather than that
the earth turned on its axis?* I replied: *I suppose, because it looked
as if the sun went round the earth.* 'Well,' he asked, 'what would it
have looked like if it had looked as if the earth turned on its axis?*
This question brought it out that I had hitherto given no relevant
meaning to 'it looks as if* in 'it looks as if the sun goes round the
earth'. My reply was to hold out my hands with the palms upward,
and raise them from my knees in a circular sweep, at the same time
leaning backwards and assuming a dizzy expression. 'Exactly!' he
said. In another case, I might have found that I could not supply
any meaning other than that suggested by a naive conception, which
could be destroyed by a question. The naive conception is really
thoughtlessness, but it may take the power of a Copernicus effect-
ively to call it in question.
Different philosophers have meant different things by 'meta-
physical'. Kant also attacked metaphysics: but Kant would not have
called 'Every rod has a length', or Time is one-dimensional and has
only one direction', metaphysical in the sense hi which he attacked
metaphysics; whereas for Wittgenstein they are so.
The criticism of sentences as expressing no real thought, accord-
ing to the principles of the Tractatus, could never be of any very
simple general form; each criticism would be ad hoc, and fall within
the subject-matter with which the sentence professed to deal. For
example, if someone says that time moves only in one direction, we
investigate this by asking him what processes he is comparing.
One frequently used tool in such enquiries is: 'What would it
be for it to be otherwise?' when, e.g. someone has said: 'Time has
only one direction.' Here we are asked for an intelligible description
of a state of affairs in which the asserted proposition let it be, say,
does not hold. As far as sensible
1
'the future comes after the past
verification is concerned, the asserted proposition and the alterna-
tive to it that is being asked for are, or may be, on the same level;
152 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
the relation of actual sense-experiences to each is not necessarily
being investigated. What is operative here is evidently not a sensible-
verification theory, but the picture theory of the significant des-
cription: both the proposition and its negation are supposed to
describe a possibility, otherwise the status of the proposition is
other than that of a significant description.
'Psychology is no more akin to philosophy than any other natural
science. Theory of knowledge is the philosophy of psychology'
(4.1121). In this passage Wittgenstein is trying to break the dic-
tatorial control over the restof philosophy that had long been
exercised by what is called theory of knowledge that is, by the
philosophy of sensation, perception, imagination, and, generally,
of 'experience*. He did not succeed. He and Frege avoided making
theory of knowledge the cardinal theory of philosophy simply by
cutting it dead; by doing none, and concentrating on the philosophy
of logic. But the influence of the Tractatus produced logical posi-
tivism, whose main doctrine is Verificationism'; and in that doctrine
theory of knowledge once more reigned supreme, and a prominent
position was given to the test for significance by asking for the ob-
servations that would verify a statement. (Further, in the period
between the Tractatus and the time when he began to write Phil-
osophical Investigations, Wittgenstein's own ideas were more closely
akin to those of the logical positivists than before or after.)
We can see how the Tractatus generated logical positivism,
although the two philosophies are incompatible, by studying
*
Moritz Schlick's essay, Meaning and Verification: Whenever we
ask about a sentence, "What does it mean?" what we expect is
instruction as to the circumstances in which the sentence is to be
used; we want a description of the conditions under which the sen-
tence will form a true proposition, and of those which will make it
false* Here Schlick seems to follow the Tractatus, except in the last
clause of his statement: the Tractatus says that I determine the
sense' of a proposition by 'determining in what circumstances I
call it true* (4.063). (It is implicit in this that the 'circumstances* in
question may hold or not hold; for it is an essential part of the
picture theory that a proposition which held in all circumstances
would not have 'sense' : it would lack TF poles.)
KNOWLEDGE AND CERTAINTY 153
Schlick calls the 'description of the conditions* under which a
word has application, or a sentence is true, the 'rules for the use' of
the word or sentence. These 'rules' will consist partly of 'ostensive
definitions', of which the simplest form will be a pointing gesture
combined with the pronouncing of the word; this can be done with
words like 'blue*. For words like 'immediate', 'chance', 'because*,
'again', Schlick says, the ostensive definition is of a more compli-
cated kind : 'in these cases we require the presence of certain complex
situations, and the meaning of the words is defined by the way we
use them in these different situations.* All rules for use 'ultimately
point to ostensive definitions*. 'This,' Schlick says, 'is the situation,
and nothing seems to me simpler or less questionable. It is this
situation and nothing else that we describe when we affirm that the
meaning of a proposition can be given only by giving the rules of its
verification in experience. (The addition "in experience" is really
1
superfluous, as no other kind of verification has been defined.)'
This shews us the transition from the Tractatus to 'verification-
ism* very clearly. What Schlick says leads immediately (a) to the
quick test for significance: 'What experience would verify this?* and
(b) to the maintenance of theory of knowledge as the cardinal
theory of philosophy.
In the Tractatus, the 'determination of the circumstances in
which I call a proposition true' must be a statement of its truth-
conditions. This is a completely different thing from a 'rule for the
use* of a sentence, if this takes the form of an 'ostensive definition*.
There could be no statement of the truth-conditions of an elementary
proposition, other than a restatement of it; and for all non-ele-
mentary propositions there can always be statements of truth-
conditions. If, then, Schlick is following the Tractatus, 'ostensive
definition' can only be relevant to the elementary proposition.
Further, Schlick insists that our 'rules for use' are 'arbitrary*; we
give what rules we like; all that is essential is that we give some. The
only arbitrariness in the Tractatus is in the assignment of names.
There is no arbitrariness about the fact that a certain type of
arrangement of names is capable of representing such-and-such a
1
Moritz Schlick, Meaning and Verification, reprinted in Feigl and
Sellars, Readings in Philosophical Analysis.
154 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
situation; it can do that only by reproducing in its own structure the
arrangement of objects in the situation, and we cannot make it do
so at will. Therefore, on the Tractatus view, there is no room for
criticizing a sentence on the ground that we have not stipulated what
situation it describes; but only on the ground that we have not
assigned a reference to some of the words in it. The utterance of a
sentence in a context in which it is true does not take the place of a
stipulation of truth-conditions; the most that it can do is to shew
someone the reference of the words; he will then understand the
prepositional sign, in its positive or negative sense, by meaning
the objects named in it. Then 'you have said something meaningless'
could only mean 'you have not assigned a reference to this expres-
sion', and never 'you have not shewn what observations would
establish the truth of this'.
On the Tractatus view, then, one could not ask what observations
would establish the truth of a proposition unless the 'structures' of
possible observation statements already stood in certain internal
relations to the 'structure' of the proposition. In the presence of
these internal relations, the question of meaningfulness cannot arise,
except in the form of a question about the reference of the indi-
vidual signs; if these signs are not given a reference, the proposition
could not be 'given* a sense, even by stipulating that its truth would
be established if and only if such-and-such observation statements
were verified. An alleged 'proposition' that was so 'given a sense'
would necessarily be, not a proposition, but the simple sign of a
complex; and then the sentences in which the 'proposition' occurred
would have to stand in internal relations to the 'observation state-
1
ments these internal relations would then supply us with the des-
;
cription of a complex, and the definition of a simple sign for that
complex; and the 'observation statements' would give the truth-
conditions of propositions in which that sign occurred. This doctrine
is quite different from Schlick's.
In Philosophical Investigations, where Wittgenstein makes an
extensive investigation of psychological concepts, his object was to
shew that it is not necessary to introduce the problems of epistem-
ology of i.e. of perception, imagination, and generally of 'experi-
encing' into the discussion of other problems of philosophy. That
KNOWLEDGE AND CERTAINTY 155
is to say, we can discuss e.g. the problems implicit in the expression
*the process of time*, without laying foundations by giving an
account of the ways in which we apprehend time memory, ex-
pectation, experience of succession, and so on.
Knowledge and certainty, however, are topics for the philosophy
of logic. In doing logic we are not indeed interested in what is the
case, or in what things are certainly known, or in the conditions for
certainty in practice. But logical theory must allow for the certainty
of propositions which are not logically necessary. Otherwise logic
would have no application. For 'It is clear in advance that the logical
proof of a significant proposition and proof in logic (i.e. proof of a
logical proposition) must be two quite different things. The sig-
nificant proposition asserts something, and its proof shows that it is
so' (6.1263, 6.1264). Thus the proof of a significant proposition is not
hypothetical. If its proof proves that it is the case, it is presupposed
that those propositions from which it is proved are known to be
true; for if they were uncertain, the conclusions would be equally
uncertain. The only 'certainty' would then be hypothetical that if
the premises are true the conclusion is ; but that is not what Wittgen-
stein calls a significant proposition; it is a proposition of logic, and
proof of it nothing but a 'mechanical expedient to facilitate the
recognition of it as a tautology' (6.1262). Thus, if we are to speak of
proving significant propositions, 'A knows p' cannot be an ideal
form of description without specifiable instances, nor one exemplified
only in 'knowledge' of tautologies.
It is easy to misunderstand certain remarks in the Tractatus
which have to do with this question and to suppose that Wittgen-
stein calls only tautologies certain. At 4.464 he says: 'The truth of
tautology is certain, that of a proposition is possible, and of contra-
diction impossible. (Certain, possible, impossible: here we have a
hint of that gradation which we need in probability theory.)' And at
5.525: 'Certainty, possibility or impossibility of a state of affairs are
expressed, not by a proposition, but by an expression's being a
tautology, a significant proposition or a contradiction.* It would be
natural at first sight to take these remarks as implying that certainty
belongs only to tautology. But the 'state of affairs' whose certainty is
expressed by an expression's being a tautology cannot be a state of
156 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
affairs described by a tautology; for Wittgenstein is insistent that
tautology describes no state of affairs is true for every possible
Again the 'significant proposition asserts
state of affairs (4.466).
something, and its proof shows that it is so' but there will be no
;
such proof if certainty belongs only to a tautology.
Now if we take the hint given by the parenthetical remark at
4.464 and examine the theory of probability as it is described by the
Tractates, we find that the first impression perhaps conveyed by
these propositions is mistaken, as it must be if Wittgenstein is
consistent.
The account of probability is closely connected with the view
that all the propositions are truth-functions of elementary propo-
sitions. At 5.15 we are told: 'If Tr is the number of truth-grounds of
the proposition "r", Tw the number of the truth-grounds of the
proposition "s" which are at the same time truth-grounds of "r",
then we call the ratio T^ Tr the measure of the probability given by
:
'
the proposition "r" to the proposition "s" (5.15). That is, if we
assume *p* and *q' to be elementary, since *p or q* has 3 possible
combinations of the truth-values of 'p' and 'q' which make it true, and
only 1 in common with 'p and q', the measure of the probability
given by 'p or q' to *p and q' is 1 :3.
This account of probability has been criticized as resting upon
the arbitrary dogma that all elementary propositions are equally
probable. 'Two elementary propositions give one another the
probability y (5.152). Now Wittgenstein also says: 'Propositions
which have no truth-arguments in common with one another, we
call independent of one another' (5.152). This is not an author's 'we'.
Turning it round we might say: 'When we speak of propositions as
independent of one another, what this really means is that they have
no truth-arguments in common, i.e. are truth-functions of quite
separate sets of elementary propositions.* With this we get some
light on what is meant by saying 'the application of logic decides
what elementary propositions there are' (5.557). That is to say: if
in the application of logic i.e. reasoning not 'in logic' but from
facts we (rightly) say 'even if this is so, that would not have to be
so, it is not even made probable, they have nothing to do with
another': then we have found propositions that are truth-functions
KNOWLEDGE AND CERTAINTY 157
of quite separate sets of elementary propositions. But he goes on to
say at this place: 'Logic cannot anticipate what resides in its applica-
tion* and 'Logic and its application must not overlap.' Thus the
question what are the elementary propositions does not belong to
logic at all.
These passages shew the doubtfulness of part of Wittgenstein's
criticism of the Tractatus in Philosophical Investigations. He jeers at
the idea that when I say "The broom is in the corner' I really mean
"The broomstick is in the corner and so is the brush and the broom-
stick is stuck in the brush.' But I shall recognize the negation of any
of those propositions as constituting an objection to The broom is
in the corner'; and that is all that the Tractatus theory requires. If I
understand a proposition, I shall know what more detailed state-
ments are inconsistent with it; these will then be more elementary
than it is.
To return to the probability theory: 'If p follows from q, then
the proposition "q" gives the proposition "p" the probability 1. The
certainty of the logical conclusion is a limiting case of probability*
(5.162). This can readily be seen from the Tractatus account of
probability together with its account of inference, according to
which what follows from a proposition is already stated by it
(5.14-.141).
Now, however, we are in a position to understand the propo-
sition: 'Certainty, possibility, or impossibility of a state of affairs
isexpressed not by a proposition, but by an expression's being a
tautology, a significant proposition, or a contradiction/ Since an
expression that is a tautology (or contradiction) does not answer to
any 'state of affairs', what expresses the certain (or impossible) 'state
of affairs* itself as opposed to expressing its certainty (or impossi-
9
bility), will not be the tautologous (or contradictory) expression, but
rather one of the propositions that occur as components of this
tautology (or contradiction). Moreover, in order to get 'a hint of
that gradation which we need in probability theory', 'possibility*
must here be taken as excluding both certainty and impossibility.
Take a case where *s* is a significant proposition and 'r' expresses
something we know. Then the 'state of affairs' expressed by V will
be certain if 'r.'-^s* is a contradiction (i.e. if 'ros' is a tautology); it
158 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
will be impossible if *r.s' is a contradiction (i.e. if 'rz> ~s' is a
tautology); it will be, relative to our knowledge, merely 'possible' if
'r.s* and 'r.< <$' are both significant propositions (each of them must
be either a significant proposition or a contradiction, if *r' and 's*
are both significant propositions).
This raises the question how we know that r; does the same
account apply as would apply to V if it were 'certain' that s, and
does this go on indefinitely, or do we come to a stop somewhere?
Wittgenstein's view is at this point obscure; but he refers to 'being
completely acquainted with a fact' (5.156), and presumably held that
here we do come to a stop.
Thus Wittgenstein offers an extraordinarily over-simplified
account of knowledge, which would presumably have to be filled
out with an account of 'acquaintance with facts'. 'A knows p', he
remarks at 5.1362, 'is senseless if p is a tautology. '(We should notice
that the word is 'senseless*, not 'nonsensical'; that is to say, the
knowledge that p, when 'p' is a tautology, is treated as he treats the
truth of 'p'.) But he has just said that the connection between know-
ledge and what is known is that of logical necessity. He is not
referring to the mere fact that *A knows p, but p is not true' is a
contradiction; but to his theory, which would be the foundation for
that fact, that the certainty of a state of affairs comes out in an
expression's being a tautology. That is to say, if A
knows p, then,
for some q, the fact that q is a fact that A
is 'acquainted* with, and
qDp is a tautology.
The remark: 'Certainty, possibility and impossibility of a state of
affairs are expressed, not by the proposition but . . .' stands as a
comment on 'It is incorrect to give "fx is possible" as the verbal
rendering of (Ex)(fx), as Russell does.' Russell held that necessity,
possibility (contingency) and impossibility belong, not to propo-
*
sitions, but to prepositional functions, such as *fx'. "fx" is neces-
sary', he says, means that all values of fx are true.
1
In the passage we have been considering, Wittgenstein discusses
not necessity, possibility and impossibility, but certainty, possibility
and impossibility. This might seem insignificant, from his saying
1
See e.g. the final paragraph of a chapter 'Prepositional Functions* in
Russell's Introduction to Mathematical Philosophy.
KNOWLEDGE AND CERTAINTY 159
'the truth of tautology is certain'; but, as we have seen, he cannot
hold that only the truth of tautology is certain. His objection to
Russell's account of necessity (and hence of logical impossibility) is
made elsewhere, at 6.1231 : 'The mark of logical propositions is not
general validity. For to be general only means: to be accidentally
valid for all things. An ungeneralized proposition can be tautologous
just as well as a generalized one.'
'That precedent,' Wittgenstein concludes 5.525, 'to which one
would always like to appeal, must reside in the very symbol itself.*
He evidently refers to a reason why it is especially tempting to
equate '(Ex)(fx)' and *fx is possible'. The most fundamental motive
for adopting Russell's views is that it would be one way of getting
rid of the puzzling character of 'necessary', 'possible' and 'im-
possible'; Wittgenstein has his own way of doing that. There
remains, however, the feeling that a case will guarantee possibility,
and thus give the assertion of possibility a sense, as nothing else
could; this is like the lawyer's feeling that the best way of showing a
procedure to be legal is to cite a precedent for it. So Russell thought
that *fx' is possible only if there is an actual case of an f
.
Now Wittgenstein acknowledges this desire for *a precedent*, but
says that this precedent resides in the symbol itself. The 'symbol
itself' will be the significant proposition. For 'in the proposition a
situation is as it were put together experimentally' (4.031). It is as if
the construction of small models of mechanisms were used to make
reports on what machines there were in some place, and one also
constructed hypothetical models, say in order to ask whether there
are any of these in that place. If the models are in clay and do not
move, one might want to know what makes them express possible
hypotheses. But if the models are themselves working mechanisms,
the 'precedent' to which one would want to appeal would be in the
models themselves. And so it is, Wittgenstein says, with significant
propositions.
No 'precedent' is to be found in tautology and contradiction;
Wittgenstein's remark has sole application to significant propo-
sitions. For 'sentences which are true for every state of affairs
cannot be connections of signs at all, for otherwise only particular
connections of objects will correspond to them. (And there isn't
160 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
any logical combination to which there corresponds no com-
bination of the objects.)' (4.466). To regard tautologies (logically
necessary propositions) as descriptions is as if one were to regard
the empty space where the mechanism was to go as itself a model
for all possible mechanisms. But the significant proposition is a
logical working model of the situation it asserts to exist.
13
'MYSTICISM 5
AND SOLIPSISM
When Russell received the MS. of the Tractatus from Wittgenstein,
then in an Italian prison-camp, after the First World War, he wrote
some comments and questions. Presumably he laid great stress on
the account of logical propositions as tautologies ; his letter is not
extant but we know he attached great importance to Wittgenstein's
researches on this subject from a footnote in his Introduction to
Mathematical Philosophy. In reply he received a letter from Wittgen-
stein saying:
'Now I'm afraid you haven't really got hold of my main
contention, to which the whole business of logical propositions is
only corollary. The main point is the theory of what can be
expressed (gesagt) by propositions i.e. by language (and, what
comes to the same, what can be thought) and what cannot be
expressed by propositions, but only shown (gezeigt); which, I
believe, is the cardinal problem of philosophy.
We have seen what 'can be said* according to this theory: that,
and that only, 'can be said* the negative of which is also a possi-
bility, so that which of the two possibilities is actual has to be dis-
covered by 'comparing the proposition with reality*. This notion is
rather vague, but it is clearly implied that the comparison does not
consist in mere thought: 'there is no picture that is true a priori".
That is to say, if a proposition has a negation which is a perfectly
good possibility, then it cannot be settled whether the proposition is
true or false just by considering what it means. The locus of direct
'comparison with reality* lies in the facts we are 'acquainted with*;
161
162 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
if so, then, on Wittgenstein's theory of inference, we can have no
knowledge of significant propositions that is not a restatement (or
an 'abstract' 5.156) of propositions known by 'acquaintance* to
be true. (This theory of knowledge has, to be sure, nothing but
clarity and simplicity to recommend it.) As for tautologies and
contradictions, they are devoid of 'sense' and 'say nothing'.
But an important part is played in the Tractatus by the things
which, though they cannot be 'said', are yet 'shewn' or 'displayed'.
That is to say: it would be right to call them 'true' if,perimpossibile9
they could be said; in fact they cannot be called true, since they
cannot be said, but 'can be shewn', or 'are exhibited', in the pro-
positions saying the various things that can be said.
Now the things that would be true if they could be said are
obviously important. Can we then draw a distinction between things
that are 'shewn', and things the opposite of which is 'shewn' be- ;
tween the things that would be true if they could be said, and those
that would be false if they could be said? It is impossible to speak
like this of attempted contradictions of what 'is shewn', as we have
'
already seen in a trivial case: whereas "Someone" is not the name
of someone' is intended to say something 'quite correct' (as Wittgen-
'
stein says of solipsism), we must say concerning
"Someone" is the
name of someone* that what such a proposition intends is not
merely not correct, but quite incoherent and confused; the demon-
stration that this is so completely destroys the idea that there is
anything at all behind the would-be statement. Nevertheless there
are utterances which at least sound like attempts to say the opposite
of the things that are 'quite correct' in this sense; and there will be
more error, or more darkness, in such attempts than in trying to say
the things that are 'shewn', even if they are really unsayable.
It would presumably be because of this that Wittgenstein
regards the sentences of the Tractatus as helpful, in spite of their
being strictly nonsensical according to the very doctrine that they
propound; someone who had used them like steps 'to climb out
beyond them' would be helped by them to 'see the world rightly'.
That is to say, he would see what 'is shewn', instead of being down
in a bog confusedly trying to propound and assert sometimes cases
of what is 'shewn \ sometimes would-be contradictions of these.
'MYSTICISM* AND SOLIPSISM 163
This idea of philosophic truth would explain one feature of
philosophy what a philosopher declares to be philosophically false
:
is supposed not to be possible or even really conceivable; the false
ideas which he conceives himself to be attacking must be presented
as chimaeras, as not really thinkable thoughts at all. Or, as Wittgen-
stein put it: An impossible thought is an impossible thought (5.61)
and that is why it is not possible to say what it is that cannot be
thought; it can only be forms of words or suggestions of the
imagination that are attacked. Aristotle rejecting separate forms,
Hume rejecting substance, exemplify the difficulty: if you want to
argue that something is a philosophical illusion, you cannot treat
it asa false hypothesis. Even if for purposes of argument you bring
it into contempt by treating it as an hypothesis, what you infer from
it is not a contradiction but an incoherence.
We must distinguish in the theory of the Tractates between
logical truths and the things that are 'shewn*; logical truths, whose
character we have already discussed, are the 'tautologies', and are
'sense-less' propositions (lacking TF poles), their negations being
'contradictions'; attempts to say what is 'shewn* produce 'non-
sensical* formations of words i.e. sentence-like formations whose
constituents turn out not to have any meaning in those forms of
sentences e.g. one uses a formal concept like Concept' as if it were
a proper concept (Chapter 7, pp. 111-1 2). Here the attempt to express
what one sees breaks down.
The connection between the tautologies, or sense-less propo-
sitions of logic, and the unsayable things that are 'shewn', is that the
tautologies shew the 'logic of the world'. But what they shew is not
what they are an attempt to say: for Wittgenstein does not regard
them as an attempt to say anything. They are, however, legitimate
constructions, introduced into the system of propositions as was
introduced into the system of numerals. Nor are they the only
propositions which 'shew' anything, or which shew 'the logic of the
world' on the contrary, every proposition at least does that.
:
7
Of all the things that are unsayably 'shewn ,
the most prominent
4
in the Tract atus is this 'logic of the world' or 'of the facts*. My most
fundamental thought is this logical constants are not proxies for
:
anything. The logic of the facts cannot have anything going proxy
164 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
for if (4.0342). Here he is contrasting logical constants with names,
which 'go proxy' for their objects: The possibility of sentences,' he
has just said, 'rests upon the principle of signs as going proxy for
objects* and what this principle in turn amounts to is the possi-
bility of logical picturing through one fact's having the same logical
form as another for only in the context of the proposition will a
sign go proxy for an object.
Sentences thus cannot represent, and nothing in them can stand
for, 'the logic of the facts': they can only reproduce it. An attempt
to say what it is that they so reproduce leads to stammering. This
view does seem to be true. I once bought toffees with the names of
the flavours, 'treacle', 'Devon cream* and so on printed on the papers,
and was momentarily startled to find one labelled 'fruit or nut*. It
cannot be 'fruit or nut', I said; it's fruit or it's nut! Any attempt to
say what the truth-functional constants like 'or' mean must fail; we
can only shew it.
Or again; if asked to explain the composition of the simplest
statement, we say that this word means, or refers to, such-and-such,
and this one means such-and-such, and together they mean that . . .
(or: someone who puts them together makes the statement that .) . .
and there follows just such another composition of signs as we
were trying to explain.
Again, if we try to explain the essence of a relational expression
to ourselves, we reproduce the relational form in our explanation.
For, as we have seen, we must make the distinction between *aRb'
and *bRa'; and if we do this by e.g. saying that in one the relation
goes from a to b, and in the other from b to a, we produce a sen-
tence which employs the essential relational form; for it reproduces
the distinction produced by exchanging the places of the terms.
All the logical devices the detailed twiddles and manipulations
of our language combine, Wittgenstein tells us at 5.511, into an
infinitely fine network,forming 'the great mirror' that is to say,
the mirror of language, whose logical character makes it reflect the
world and makes its individual sentences say that such-and-such is
the case. The simplest and most characteristic mark of this is that
we do not have to learn the meanings of all the sentences of our
language; given the understanding of the words, we understand and
'MYSTICISM' AND SOLIPSISM 165
construct sentences, and know what they mean without having it
explained to us.
It is essential to see that logic does not describe any facts: that
there are no logical facts. It was at one time natural to think that the
field of logic was the field of what was a priori true, i.e. true inde-
pendently of all existence. On this Wittgenstein says at 5.552: 'The
"experience" that we need to understand logic is not that something
is thus or thus, but that something is: but that is not an experience.
Logic precedes any experience that something is thus. It comes
before the How, not before the What.' According to the Tractatus
the 'what' is conveyed by the simple names, which cannot be taken
to pieces by definitions (3.261) and which name the 'substance of the
world' (2.0211). Thus even when a simple name is replaced by a
definite description, the description is merely 'about' the object, it
could not 'express' it (3.221).
From this, it is clear that Wittgenstein held that what was given
by experience was always facts; a grasp of the 'substance of the
world', which we shew we have in being able to describe the facts
we experience, is not given by any experience. For of any experience
it can be asked what it shews as being the case; since this can be
the case or not, indifferently, its being the case cannot possibly tell us
anything about logic. For everything logical about a (significant)
proposition is understood before it is known whether the proposition
is true or not.
But logic cannot be thought of as something quite independent
of the world either. For then 'How could we apply logic?' (5.5521).
That is to say, if logical truth were there without any world, then
when there was a world, how could it be said such-and-such cannot
:
be, because there is a logical fact that is inconsistent with it? It
could be said: If there would be such a thing as logic, even if there
were no world, then how can there be such a thing as logic, when
there is a world?* (i.e.: if logic comprised facts that the facts in the
world had to be consistent with, then logic would no longer be
logic, for it is logic that judges of the consistency of facts.) 'It has
been said that God can create everything but what is contrary to the
laws of logic. The point is, that we could not say what an "un-
logical" world would be like' (3.031). So the medieval philosopher
166 AN INTRODUCTION TO WITTGENSTEIN'S TRACT ATUS
says that God, to whom no 'word* is impossible, yet cannot change
the past, because 'change the past' is not a 'word'.
Thus when the Tractatus tells us that 'Logic is transcendental',
it does not mean that the propositions of logic state transcendental
truths; means that they, like all other propositions, shew some-
it
thing that pervades everything sayable and is itself unsayable. If it
were sayable, then failure to accord with it would have to be ex-
pressible too, and thus would be a possibility.
I will now consider the most notorious of the things that Wittgen-
stein says are 'shewn', but cannot be said: the truth of solipsism.
I arn a soiipsist if I think: *I am the only /: the world, including all
the people in it, is essentially an object of experience, and therefore
of my experience.' Wittgenstein says: what is intended here is right,
but it can't be said. (5.62).
It is a fairly natural thought that 'where there is consciousness,
there is an I*; but this raises immediate questions both about
'consciousness', and about the legitimacy of speaking of 'an I'. If
one considers examples of consciousness, as it is being approached
here, one does not think of how e.g. one can see that someone is
now conscious (he was, say, asleep before); but rather of contents
of consciousness, such as pain, images, the visual field; and what is
there to the consciousness except the pain, the image, or the visual
field itself?The essential thing is that these are being considered
'from inside'; all that there is from outside corresponding to these
words is manifestations of them in words and behaviour*
So it comes out that it is illegitimate to speak of 'an l\ 'From
inside' means only 'as / know things'; I describe those things
something, however, I cannot communicate or express: I try to,
by saying I speak 'from an inside point of view'. But there is no
other point of view. Suppose others too speak of the 'inside point
of view' ? That is my experience or my supposition of spoken words.
In a later writing Wittgenstein imagined a convention whereby
some individual, A, is used as a centre for the language of experience
and thought in the following way: Everyone says 'There is pain*, or
'It thinks*, when A is in pain, or is thinking; for other people the
locution would be, say, 'X is behaving as A behaves when there is
pain etc.* Now, he said, if I am the centre, this language has a pe-
'MYSTICISM* AND SOLIPSISM 167
culiar but quite inexpressible advantage over the languages in which
other people are taken as centres. It is inexpressible, because if I
try to express it in the language with myself as centre naturally my
statement, as coming from the person who is the centre, is specially
related to the language that has me as centre; and if I try to express
it in a language with someone else as centre, then the description, in
that language, of the alternative possible language with me as centre
gives it no special position hi comparison with any other alternative
language. But all these languages correspond, and one and the same
reality corresponds to them all and to the 'physical language'.
This passage, though written some years after the Tractatus,
appears very close to it in thought. For that 'advantage* was abso-
lute. Thus I am the centre, but this is inexpressible.
In the Tractatus Wittgenstein speaks of 'my language* (5.6) and
1
explains this as meaning 'the only language that I understand'
(5.62). Its limits 'stand for the limits of my world*. I cannot postulate
a language for talking about the relation of language, the world,
and the philosophical I, in which my world (the world given by
the limits of my language) would be one particular thing to talk
about. I can only say how things are in the world corresponding
to my language. But this manifests 'the 'all-comprehending world-
mirroring logic'.
That is why, having said at 5.6 'The limits of my language mean
the limits of my world', Wittgenstein gives as the first comment on
this pronouncement a number of remarks on logic: 'Logic fills the
world: the limits of the world are also its limits' (5.61). The argument
is: 'The limits of my language mean the limits of my world; but all
languages have one and the same logic, and its limits are those of
the world; therefore the limits of my world and of the world are one
and the same; therefore the world is my world.'
1
The emphasized definite article shews that *der Sprache' in 5.62 means
'that language* making a back reference which must be to 'my language'
in 5.6. In the first edition of this book I translated the parenthesis in
5.62 'the language that only I understand'. But Dr. C. Lewy has found a
copy of the first edition of the Tractatus with a correction by Wittgenstein
giving 'the only language that I understand'.
168 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
But the T
of this way of talking is not something that can be
found as a mind or soul, a subject of consciousness, one among
others; there is no such thing to be 'found' as the subject of con-
sciousness in this sense. All that can be found is what consciousness
is of, the contents of consciousness: 'I am my world' and The world
and life are one'. Hence this T, whose language has the special
position, is unique; the world described by this language is just the
real world Thoroughly thought out, solipsism coincides with pure
:
realism' (5.64).
It is not possible to understand this passage unless one has a
good deal of sympathy with solipsism. We should remember that
Wittgenstein had been much impressed by Schopenhauer as a boy;
many traces of this sympathy are to be found in the Tractatus.
Probably no one who reads the opening of The World as Will and
Idea: "The world is my idea', without any responsiveness, will be
able to enter into Wittgenstein's thought here.
Mrs. Ladd Franklin is reported to have written to Bertrand
Russell saying that she was a solipsist and could not understand why
everyone else was not too ! It is possible that the comic effect was
intentional, and the joke on her side. The necessity of solipsism is
very arguable; why should a solipsist not argue it with everyone
capable of arguing? Nothing would follow, even if two solipsists
exchanged views with mutual congratulation, about any cession by
either to the other of the unique position he conceives for himself. If
two people discuss Descartes' 'Cogito', they can agree that This is
an argument I can administer to myself alone,' and each can hold
the other would be incorrect to have disputed that; if snow fell in
appropriate sentences, one could dispute, agree and disagree with,
those sentences.
Further, it is very difficult to think of ways out of solipsism.
Indeed solipsism is often held to be irrefutable, but too absurd to
concern oneself with. In Wittgenstein's version, it is clear that the
T of solipsism is not used to refer to anything, body or soul; for in
respect of these it is plain that all men are alike. The T
refers to the
centre of life, or the point from which everything is seen.
It is difficult to get rid of such a conception once one has it. One
may well want to do so; e.g. one may feel that it makes the 'I* too
'MYSTICISM' AND SOLIPSISM 169
godlike. 'What is history to me? Mine is the first and only world!'
Wittgenstein wrote in his notebooks on this theme, and: 'There are
two godheads: God, and the L* (He did not follow Schopenhauer in
saying: The world is my will*; on the contrary: 'The world is inde-
pendent of my will and anything' s being as I want it is a "grace of
fate".' (6.373-4). 'Hence,' he wrote in his notebooks, 'the feeling of
being dependent on an alien will.')
It is not altogether easy to understand Wittgenstein's idea of 'the
limit'. It too is partly derivative from Schopenhauer, who wrote of
it as follows:
'Any one percipient being, with the object, constitutes the
whole world as idea just as fully as the existing millions would
do; but if this one were to disappear then the whole world as
idea would cease to be. These halves are therefore inseparable
even for thought, for each of the two has meaning and existence
only through and for the other, each appears with the other and
vanishes with it. They limit each other immediately.'
Again, Schopenhauer remarks on the fear of death that it is
really the fear of losing the present, and says that this is like being
afraid of slipping down the sides of the globe, as someone might do
who did not realize that the top is wherever he is.
The idea of the world as having limits which philosophy displays
to us appears over and over again in the Tractatus. It is perhaps best
known in the dictum of 6.45 : 'The view of the world sub specie aeterni
is the view of it as a limited whole. The feeling of the world as a
limited whole is the mystical feeling.' The world 'as a limited whole'
is notsuddenly introduced here as a new topic. We encounter the
world conceived as a whole as all that is the case and as limited
namely by being all that is the case at the very outset of the book;
the feeling of the world as a whole appears in the remark at 1.2:
'The world splits up into facts', for it is only of a whole that we
can say it splits up.
'Mysticism* is a rather odd name for what Wittgenstein is speak-
ing of; in popular language it suggests extraordinary and unusual
experiences, thoughts and visions peculiar to an extraordinary type
170 AN INTRODUCTION TO WITTGENSTEIN'S TRACTATUS
of individual; and no doubt it has been taken in that sense, and
written off, the more easily because Wittgenstein was himself well
known to be an extraordinary individual the very man to have
some mysticism about him. But Wittgenstein took the term over
from Russell, who used it in a special way, with reference to an
entirely ordinary feeling; one that is well expressed at 6.52: *We feel
that even if all possible scientific questions have been answered, still
the problems of life have not been touched at all.' And his further
comment on this is 'Of course there then just is no question left, and
:
just this is the answer.'
This comment can be taken in two ways: First, Wittgenstein
might be saying and this is what Professor Ayer, for example,
would make of his remarks that people who have wanted to say
what the meaning of life consisted in have had nothing in them but
a lot of nonsense. This cannot be the right interpretation; for he
speaks of people *to whom the meaning of life has become clear'.
But he says of them that they have not been able to say it. Now such
people have not failed for want of trying; they have usually said a
great deal. He means that they have failed to state what they wished
to state; that it was never possible to state it as it is possible to state
indifferent truth. He probably had Tolstoy especially in mind, whose
explanations of what he thought he understood are miserable
failures; but whose understanding is manifested, and whose preach-
ing comes through, in a story like Hadji Murad.
Wittgenstein's idea is probably made clearest at 6.41 'The mean-
:
ing of the world must lie outside the world. In the world everything
is as it is, everything happens as it does happen; there is no value
in it if there were any, it would have no value. If there is a value
that has value, it must lie outside all happening and outside being
this way or that. For all happening and being this way or that is
accidental. What makes it non-accidental cannot be found in the
world, for otherwise this thing would in its turn be accidental.' And:
*God does not reveal himself in the world' (6.432) i.e. it is not in
things' being this way as opposed to that that God is revealed.
This follows from the picture theory; a proposition and its
negation are both possible; which one is true is accidental. Why
then, having said that whatever is the case is accidental, does Witt-
'MYSTICISM' AND SOUPSBM 171
genstein speak of 'what makes it non-accidental'? To understand
this, we have to understand what he says about the will. The most
important remark that he makes here is "The facts all belong to the
:
task set, and not to the solution* (6.4321). 'Aufgabe\ which I
translate 'task set', is the German for a child's school exercise, or
piece of homework. Life is like a boy doing sums. (At the end of his
life he used the analogy still.) Now
the reason why the solution
cannot bring in any facts is that it is concerned with good and evil;
and the good or evil character of what is good or evil is non-
accidental; it therefore cannot consist in this happening rather than
that, for that is accidental.
In doing one's task, one receives certain laws, of the form "Thou
shalt . . .* This, Wittgenstein says, prompts the question: And sup-
pose I do not do it? 'But it is clear that ethics has nothing to do with
punishment or reward in the ordinary sense.' Still 'there must be
something right about that question. There must be a kind of ethical
reward and ethical punishment, but these must reside in the action
itself. (And it is also clear that the reward must be something
pleasant and the punishment something unpleasant)' (6.422).
Now what is 'the action itself'? Wittgenstein insists that 'the
world is independent of my will; there is no logical connection
between will and world' no logical connection between my will and
what actually happens at all. In so far as an event in the world can
be described as voluntary, and volition be studied, the will, and
therefore action, is 'a phenomenon, of interest only to psychology'.
Therefore 'action', in the ethical sense, is something independent of
what happens; and this is the bearer of good and evil. Thus the 'will
that is the bearer of the ethical' (6.423) belongs among the transcen-
dentals of the Tractatus, along with the mystical and the meaning of
life. The connection of will with the world is that 'the facts* belong to
the task one is set. If one has reached a solution, this is made to be
a solution, not by any alteration of the facts that may have taken
place any such alteration, even if one intended it, is accidental and
merely a 'grace of fate' but by an alteration 'hi the limits of the
world' (6.43).
It is this part of the Tractatus that seems to me most obviously
wrong. As Wittgenstein asks in Philosophical Investigation (644):
172 AN INlRODUCnON TO WITTGENSTEIN'S TRACTATUS
'Did not your intention [of which you are ashamed] reside also in
what you did? *What happens' includes 'actions', in the sense of
the word in which *good' and *bad' are predicated of actions. But
the philosophy of the Tractatus could not allow this to be so; hence
the chimerical *wiD' which effects nothing in the world, but only
alters the 'limits' of the world. In his notebooks Wittgenstein enter-
tained some more reasonable considerations, which are closely akin
to His thought in Philosophical Investigations', and then rejected
them, in the following revealing passage:
*The consideration of willing makes it look as if one part of
the world were closer to me than another (which would be in-
tolerable). But, of course, it is undeniable that in the popular
sense there are things that I do and other things not done by me.
In this way, then, the will would not confront the world as its
1
equivalent, which must be impossible.*
The true philosophical account of this matter has still to be
found; saying that the Tractatus is obviously wrong here, I do not
wish to suggest that I know what is right.
There is a strong impression made by the end of the Tractatus, as
if Wittgenstein saw the world looking at him with a face ; logic helped
to reveal the face. Now a face can look at you with a sad or happy,
grave or grim, good or evil expression, and with more or less ex-
pression. And so he speaks of the world 'waxing or waning as a
whole', i.e., in terms of my analogy, as having more or less ex-
pression, or a good or evil expression. The world thought of, not as
how things are, but as however they are seen as a whole is the
matter of logic; thought of as my life, it is the matter of ethics;
thought of as an object of contemplation, the matter of aesthetics:
1
The contrast with Schopenhauer, as well as the kinship, is interesting.
Schopenhauer thought that 'the world is my will' and is bad; and that the
only redemption for the will is 'to turn and freely deny itself'. Wittgenstein
thought that the world is good and independent of my will; good and bad
willing are attitudes to the world as a whole. The goodness of the world,
however, is not anything in how it is, but in its being at all; and lies outside
allbeing-this-way-or-that. The good will therefore will not be concerned
with how things are, and in that sense is like Schopenhauer's good will.
'MYSTICISM' AND SOLIPSISM 173
all these, then, are 'transcendental'. Many years later, in a 'lecture
on Ethics', Wittgenstein wrote: 'If I want to fix my mind on what I
mean by absolute or ethical value, it always happens that one par-
ticular experience presents itself to me which is therefore in a sense
1
my experience for excellence . . . the best way of describing it is to
9
say that when I have it I wonder at the existence of the world. The
identification of ethics and aesthetics (6.421) comes about in this
way: good and bad willing changes the world only as the object of
contemplation as a whole.
The man, however, who having been helped by the Tractatus
*sees the world rightly', i.e. sees what logic reveals as 'shewn*, will
not attempt to say it, since he knows it is unsayable. As for how
much advantage it is to him, Wittgenstein makes no great claim; in
the Introduction he said: "The whole meaning of the book could
perhaps be summed up as follows: What can be said at all can be
said clearly, and what cannot be spoken of we must be silent about.*
But his final judgment on the value of the book was this : 'It shews
how little has been done, when these problems have been solved.'
Wittgenstein's English is bad here. He means: *The experience which
1
I think of when I want to remind myself what I mean by excellence.''
FURTHER READING
WITTGENSTEIN, L. Philosophical Investigations; Blackweil, Oxford,
especially Part I, 1-116.
Remarks on the Foundations of Mathematics; Blackwell,
Oxford, Part L
Notebooks 1914-16; Blackwell, Oxford.
FREGE, G. Translations from the Philosophical Writings of Gottlob
Frege; Blackwell, Oxford.
The Foundations of Arithmetic; Blackwell, Oxford.
RUSSELL, B. The Principles of Mathematics; Allen & Unwin,
London, especially Chapters IH-V and VII.
The Problems of Philosophy; Home University Library.
Logic and Knowledge; Allen & Unwin, London, especially
*On Denoting' and "The Philosophy of Logical Atomism*.
Introduction to Mathematical Philosophy; Allen &, Unwin,
London.
RAMSEY, F. P. The Foundations of Mathematics; Routledge, London.
BASSON AND O'CONNOR. Introduction to Symbolic Logic (2nd
edition); University Tutorial Press, London.
QUINE, W. V. O. Methods of Logic; Routledge, London.
HTT.RERT AND ACKERMANN. Mathematical Logic; Chelsea Publishing
Co., New York.
VON WRIGHT, G. H. Logical Studies; Routledge, London, L
INDEX
Ethics, 80, 171 f.
Experience, 14, 25-7, 150 f.
Acquaintance, 26-7, 161-2
Aristotle, 15, 32, 116, 163
Assertion-sign, 113 f.
Axiom of Infinity, 148
Axiom of Reducibility, 130 Facts, 19, 28, 29-30, 123
Ayer, A. J., 138, 150, 170 atomic, 25, 29--30
negative, 30, 34, 146
Falsehood, 13, 34, 48-9, 53, 58,
74-7
Flew, A. G. N., 85
Cantor, G., 137
Form, logical, 75, 87 f., 91, 110 f.
Carnap, R., 82 f. Formal concepts, 82, 99, 123-4
Colombo, G., 67 Formal series, 124-31
Complexes, 34, 36-40, 123
Frege, 12-16, 35, 42, 43, 45, 50, 51,
Concatenation, 36-40
57, 58, 59, 68, 75, 92, 93-4, 97,
Contradiction, 77, 159-60, 163
98-112 passim, 113,114, 118-
Copernicus, 151
19, 122-4, 126, 138-42, 152
Function, 14, 98-112/?ars//n, 149
prepositional, 14
truth- (see Truth-function)
Denoting, 18
Descartes, 15,43, 168
Descriptions, Theory of, 17-18,
41-50
Geach, P., 54
Dummett, M. A. E., Ill ., 122 .
Generality, 26, 138-49
God, 15, 78, 80, 165-6, 169, 170
Elementary propositions, 18, 25-40 H
passim, 99 f.
Empiricism, 12-13,14 Hilbert (and Ackermann), 53, 139
Epistemology, 27-8, 152 Hintikka, J., 167
175
176 INDEX
Hume, 12, 80, 163 Negation, 17, 19, 35-6, 46-9, 51-78
Hunting of the Snark, 76 passim, 117
Nietzsche, 12
Numbers, 124 f.
Idealism, 12-13
Identity, 13, 146
Inexpressible, the, 81-6, 87, 161 f.
objects
Inference, 28, 114f., 157 122 f., 165
Observation (see Experience)
K Ogden, C. K., 17 ., 68 n.
Ontological argument, 15
Kant, 151 Operations, 113-21 passim
Ladd-Franklin, Mrs. C., 168
Lewis (and Langford), 53 Philosophical Investigations, 28, 44,
59, 70, 78, 91, 92, 138, 154,
Logic, 165 f.
Lukasiewicz, J., 37 157, 171-2
Picture, 100
logical, 18
M theory, 19, 25-6, 64-78 passim,
79-82, 93, 160, 170
Material implication, 23, 119 Plato, 13, 28
Meaning, 17, 28, 43-5 Popper, K., 25, 26
Metaphysics, 150f. Possibility, 80 f., 155, 157-60, 161
Modality (see Necessity and Principia Mathematica (Russell and
Possibility) Whitehead), 87, 126, 128 f., 139
Modus ponens, 115 Probability, 156 f.
Moore, G. E., 87 Projection, 68-9, 72
Moses, 59
'Mysticism', 82, 169 f.
N
Quantification, Quantifier, 15 f.,
Names, 26, 28-9, 36-40, 41-50, 165 22, 138 f.
Necessity, 80, 158-9 Quine, W. v. O M 130, 131, 145
INDEX 177
R Stenius, E., 30, 146
Strawson, P., 57-8
Ramsey, F. P., 68, 95 f., 98-112 Symbols, 61-2, 73, 87-97
passim^ 143 f.
Reach, K., 83 f.
Reference, 13, 45, 60, 68, 93, 107 f.
Relations
and judgment, 45-6 Tautology, 23, 31, 77, 115, 155 f.,
and propositions, 89, 95, 101, 164 159-60, 163
converse, 117 Tolstoy, 170
Russell, 12, 13, 16 f., 27, 34, 41-50, Truth, 13, 22, 29, 53-4, 57-61, 74,
80, 87, 91, 98, 104, 105-6, 161
113 f., 123, 124, 126 f., 141, -function, 22, 31, 33, 51-7, 60,
142, 158-9, 161, 170 118 f.
Schlick, M., 152 f. Variables, 22, 93 f., 123, 129, 144,
Schopenhauer, 11 f., 168 f. 148
Sense, 13, 17, 29, 45, 57-60, 64, 68, 'VerificationisnV, 78, 150 f.
107 f., 117f., 162-3 Vienna circle, 26, 82, 150
Signs, 63, 73, 87-97
logical, 45, 53-6, 118 f., 145,
163-4 W
simple, 44, 48
Solipsism, 12, 162 f. Will, 80, 169, 171 f.
Space, logical, 75 f. Wright, G. H. von, 33
Revised October, 1965
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Vol I TB/io86; Vol. II 78/1087 M s ENSLIN The Literature of the Christian Movement
TB/6
EMILE DURJCHETM et al Essays on Sociology and Philoso-
phy With Analyses of Durkheim's Life and Work JOHN GRAY- Archaeology and the Old Testament World.
Edited by Kurt H Wolff 78/1151 nlus. TB/127
LEON FESTINGER, HENRY W. RIECKEN & STANLEY SCHACH7ER. H H. ROWLEY- The Growth of the Old Testament
When Prophecy Fails: A Social and Psychological Ac- TB/107
count of a Modern Group that Predicted the Destruc- D. WINTON THOMAS, Ed Documents from Old Testament
tion of the World \
78/1132 Times TB/85
ALVIN w GOULDMER: Wildcat Strike A Study in Worker-
The Judaic Tradition
Management Relationships \ 78/1176
FRANCIS j. GRLTND Aristocracy in America: Social Class MARTIN BUBER Eclipse of God Studies in the Relation
in the Formative Years of the New Nation TB/IOOI Between Religion and Philosophy TB/H
KURT LEWIN: Field Theory in Social Science- Selected MARTIN BUBER Moses: The Revelation and the Cove-
Theoretical Papers Edited with a. Foreword by nant TB/27
Dorwm Cartwright 78/1135 MARTIN BUBER: Pointing the Way. Introduction by
R. M. MAC TVER- Social Causation 78/1153 Maurice S. Friedman TB/1O3
MARTIN BLTIER The Prophetic Faith TB/73 G. MOLLAT. The Popes at Avignon, 1305-1378 TB/joS
MARTIN BUBER Two Types of Faith- the interpenetration GORDON RUPP Luther's Progress to the Diet of Worms
o Judaism and Christianity TB/75 TB/120
ERNST LirowiG EHRIICH A Concise History of Israel
from the Earliest Times to the Destruction of the Christianity: The Protestant Tradition
Temple in A.D. 70 TB/HS
KARL BARTH: Church Dogmatics- A Selection TB/gj
MAURICE s FRIEDMAN Martin Buber The Life of Dia-
KARL BARTH. Dogmatics in Outline TB/56
logue TB/04
KARL BARTH The Word of God and the Word of Man
FLAVIUS JOSEPHVS The Great Roman-Jewish War, with
The Life of Josephus Introduction by l\':Uiam R TB/X3
Farmer RUDOLF BLLTMANN-, GERHARD EBELING et al Translating
TB/74.
T. j MEEK Hebrew Origins Theology into the Modern Age Historical, System-
TB/t>g
atic and Pastoral Reflections on Theology and the
Church in rhe Contemporary Situation Volume 2 of
Chnsi:anity General Journal for Theology and the Church, edited by
ROLAND H. BAINTON Christendom A Short History of Robert WFunk in association with Gerhard Ebeling
Chistianity y-nx. its Impact c*j Western Crvilizatioi TB/2J2
llluS Vol. I, TB '131, Vol. II, TB''l3i WINTHROP HUDSON- The Great Tradition of the American
Churches TB/g8
Christianity: Origins & Early Development SOREN KIERKEGAARD Edifying Discourses Edited with
an Introduction by Paul Holmer TB/32
AUGUSTINE. An Augustine Synthesis Edited by Erich
SORES KTERKEGAARD: The Journals of Kierkegaard.
Prz&vara TB/335
Edited mith an Introduction by Alexander Dru
ADOLF DEISSMANN- Paul. A Study :n Social and Religious
TB/15
TB/52
History
SOREN KIERKEGAARD. The Point of View for My Work as
EDIVMHJ GIBBON* The Triumph of Christendom in the
an Author A Report to History, S Preface by Benja-
Roman Empire (Ckavs XV-XX of "Decline and Fall,"
f min Nelson TB/88
J B. 5u-y edition) . IV.us. rzfj.6
SOREN KIERKEGAARD The Present Age. Translated and
MtzrRiCE GDCL-EI.: Jesus arid the Origins of Christianity
edited by Alexander Dru Introduction by Walter
Introduction in, C LesJie A!irfon
Volume I Prolc^omera to the Life cf Icsus rs/65
Volurrell The Life of Jesus rs/66 SOREN KIERKEGAARD Purity of Heart TB/4
EDGAR J GOODSPEED- A Life of JeSUS TB/Z SOREN KIERKEGAARD, Repetition. An Essay in Experi-
mental Psychology Translated with Introduction &
ADOL? HASNACK The Mission and Expansion of Christi-
Notes by Walter Lowrie 13/117
anity in the JVsf T,Vee Centuries Introduction by
lar^stz? Pe7:Jcn rs/gz
SOREN KIERKEGAARD Works of Love Some Christian
An Introdtict ion Reflections in the Form of Discourses TB/Z^I
s. K HARRISON -The Dead Sea Scrolls
TB/34 WALTER LOWRTE: Kierkegaard. A Life Vol. I rz/8 9
EDW2f HATCH The Influence of Greek Ideas on Christi- Vol. II TB/ 9 o
Introduction and Bibliography by Frederick PERRY MILLER & T. H. JOHNSON, Editors. The Puntans A
anity
C. C'-ar.i TB/28 Sourcebook of Their Writings Vol I 13/1093
ARTHUR DARBY NOCK Early Gentile Chnstiamt> and Its r Vol. II TB/1094
Hellenistic Background TB'III JAMES M ROsrNsoN, MARTIN NOTH et al The Bultmann
School of Biblical Interpretation New Directions?
ARTHUR DARBV N'OCK St. Paul TB/IO^
Volume i of Journal for Theology and the Church,
SULPICTCS S5VEKUS et a: The Western Fathers Being the
l.rcs of MerJir: of Tours, Ambrose, Augustine of
edited by Robert WFunk in association with Ger-
hard Ebeling TB/ija
H:vpc, Hanc r xtus cf Aries and Germanus of Auxerre
E.tjfeJ 2nd trxnslxted b-y F Hoars O F SCHLEIERMACHER The Christian Faith. Introduction
16/309
by Richard R Niebuhr Vol. I TB/IOS
F VAV DER MEER Augushne the Bishop: Church and
Vol n TB/109
Society st the Dawn of the Middle Ages TB/yCq
F SCHLETERMACHER On Religion: Speeches to Its Cul-
T
OHANVE5 WEISS Earliest Christianity A History of the tured Despisers Intro by Rudolf Otto
Pencd A D. 30-^53. Introduction and Bibliography TE/^6
PAUL TTLLICH Dynamics of Faith
by' Frederick C Grant Volume 1 TB/53 TB/42
Volume II TB/54 EVELYN UVDERHILL Worship TB/1O
G. VAN DER LEEUW Religion in Essence and Manifesta-
Christianity : The Middle Ages and The Refor- tion: A Study in Phenomenology. Appendices by
Hans H Penner Vol I TB/loo; Vol. II TB/loi
'CH4.N-.VES ECKHA.RT Meister Eckhart A Modem Trans- Christianity, The Roman and Eastern
lation byRB Blakney TB/S
Traditions
3Es:t)ERius ERASMUS Christian Humanism and the
Reformation Selected IVrjtiigs. Edited and trans- A ROBERT CAPONIGRI, Ed. Modern Catholic Thinkers I
lated by John C Olin TB/n66 God and Man TB/3O6
EriENNE G:LSON Dante and Philosophy TB/xoSg A. ROBERT CAPONIGRI, Ed Modem Catholic Thinkers II.
WILLIAM HAILES The Rise of Puritanism TR'ZZ The Church and the Political Order 13/307
JOHAN HL-IZINGA. Erasmus and the Age of Reforma- THOMAS CORBISHLEY s j. Roman Catholicism 13/112
tion /Tas TB/19 CHRISTOPHER DAWsoN The Historic Reality of Christian
A. e MCGUJTET ProiestanJ Thought Before Kant Pref- Culture TB/J05
ace by IcraslaT} Pelikan TB'QJ c p FEDOTOV. The Russian Religious Mind. Kievan
JOHN T MCNETLL Makers of the Christian Tradition Christianity, the tenth to the thirteenth centuries
From Alfred, the Crest to Schleiermacher TB/ISI TB/7o
G P FEDOTOV, EdA Treasury of Russian Spirituality c c GILUSPIE Genesis and Geology: The Decades be-
TB/303 fore Darwin 18/51
DAVID KNOWLES The English Mystical Tradition KYLE HASELDEN The Racial Problem in Christian Per-
TB/302 spective TB/XI6
GABRIEL MARCEL Being and Having. Introduction by WALTER KAUFMANN, Ed Religion from Tolstoy to
James Collins 16/310 Camus Basic Writings on Religious Truth and
GABRIEL MARCEL Homo Viator Introduction to a Msta- Morals Enlarged Edition 18/123
physic of Hope TB/397 JOHNTMCNEILL A History of the Cure of Souls 18/126
GUSTAVE WEIGEL, s J Catholic Theology in Dialogue
.
KENNETH B MURDOCK Literature and Theology in
TB/301 Colonial New England 18/99
H RICHARD NIEBUHR Christ and Culture TB/3
Oriental Religions. Far Eastern, Near Eastern H RICHARD NIEBUHR The Kingdom of God m America
TOR ANDRAE. Mohammed The Man and His Faith is/49
TB/62 RALPH BARTON PERRY- Puritanism and Democracy
EDWARD CONZE Buddhism Its Essence and Develop- 18/1x38
ment Foreword by Arthur Waley 73/58 PAUL PFUETZE: Self, Society, Existence Human Nature
EDWARD CONZE et al Editors Buddhist Texts Through
,
and Dialogue in the Thought of George Herbert Mead
the Ages TB/XIJ and Martin Buber 18/1059
ANANDA COOMARASWAMY Buddha and the Gospel of WAITER RAUSCHENBUSCH. Christianity and the Social
Buddhism lllus. iB/xxg Crisis + Edited by Robert D. Cross 18/3059
H G. CREEL Confucius and the Chinese Way TB/63 KURT SAMUELSSON Religion and Economic Action: A
FRANKLIN EDGERTON, Trans & Ed The Bhagavad Gita
-
Critique of Max Weber's The Protestant Ethic and the
TB/1I5 Spirit of Capitalism Trans, by E G. French, Ed
with Intro by D. C. Coleman
SWAMI NIKHILANANDA, Trans. & Ed. The Upanishads A
. 18/1131
One-Volume Abridgment 13/114 ERNST TROELTSCH- The Social Teaching of the Christian
HELLMUT WILHELM Change: Eight Lectures on the I Churches Vol I TB/JI, Vol. II 18/72
Ching TB/20X9
NATURAL SCIENCES
Philosophy of Religion
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NICOLAS BERDYAEV- The Beginning and the End TB/14 Biological Sciences
NICOLAS BERDYAEV Christian Existentialism A Berd- CHARLOTTE AUERBACH The Science of Genetics 3
yaev Anthology. Ed by Donald A Lowrie TV/I-JO 18/568
NICOLAS BERDYAEV: The Destiny of Man TB/6i MARSTON BATES The Natural History of Mosquitoes.
RUDOLF BULTMANN- History and Eschatology- The Pres- lllus 18/578
ence of Eternity TB/gx A. BELLAIRS: Reptiles Life History, Evolution, and
RUDOLF BULTMANN AND FIVE CRITICS Kerygma and Structure. lllus 18/520
Myth A Theological Debate TB/SO LUDWIG VON BERTALAVFFY: Modern Theories of Develop-
RUDOLF BULTMANN and KARL KUNDSiN: Form Criticism. ment An Introduction to Theoretical Biology 18/554
Two Essays on New Testament Research. Translated LUDWIG VON BERIALANFFY Problems of Life. An Evalua-
by Frederick C. Grant 18/96 tion of Modern Biological and Scientific Thought
MIRCEA ELIADE: The Sacred and the Profane TB/8x 18/521
LUDWIC FEUERBACH: The Essence of Christianity. In- HAROLD r BLUM Time's Arrow and Evolution TB/555
troduction by Karl Earth Foreword by H Richard JOHN TYLER BONNER The Ideas of Biology 2 lllus
Niebu.hr IB/XI is/570
ADOLF HARNACK What is Christianity? Introduction A. j. CAIN Animal Species and their Evolution lllus.
by Rudolf Bultmann TB/X/ TB/519
FRIEDRICH HEGEL On Christianity: Early Theological WALTER B CANNON: Bodily Changes in Pain, Hunger,
Writings. Edited by Richard Kroner and T. KnoxM Fear and Rage, lllus 18/562
13/79 w E LE GROS CLARK- The Antecedents of Man: An Intro-
KARL HEIM: Christian Faith and Natural Science TB/i6 duction to the Evolution of the Primates. lllus
IMMANUEL KANT Religion Within the Limits of Reason TB/559
Alone. 5 Introduction by Theodore M. Greene and w H DOWDESWELL Animal Ecology, lllus TB/543
John Silber TB/67 w. H. DOWDESWELL The Mechanism of Evolution. lllus.
JOHN MACQUARRIE An Existentialist Theology A Com- 18/527
parison of Heidegger and Bultmann Preface by R w GERARD. Unresting Cells lllus. 18/541
Rudolf Bultmann 18/125 DAVID LACK Darwin's Finches lllus. 1*8/544
PAUL RAMSEY, Ed Faith and Ethics- The Theology of E MORTON Molluscs An Introduction to their Form
j
H Richard Niebuhr 18/129 and Functions Illus 18/529
PIERRE TEILHARD DE CHARDiN. The Phenomenon of Man ADOLF PORTMANN: Animals as Social Beings lllus.
TB/83 18/572
o w. RICHARDS The Social Insects. lllus. 18/542
Religion, Culture & Society p. M. SHEPPARD Natural Selection and Heredity lllus
TB/52S
JOSEPH L BLAU, Ed.- Cornerstones of Religious Freedom
in America Selected Basic Documents, Court De- EDMUND w siNNOrr Cell and Psyche The Biology of
cisions and Public Statements. Revised and Enlarged Purpose 18/546
Edition TB/xx8 c. H. WADDLNGION How Animals Develop. lllus 18/553
o. c SUTTCW. Mathematics in Action Foreword by
Chemistry
James R Newman Illus 78/51.8
jR P TON- A Short History of Chemistry Illus
FREDERICK WAisMANN Introduction to Mathematical
T8/522
Thinking Foreword by Karl Menger 78/511
j. READ -
A Direct Entry to Organic Chemistry Illus
TB/523
j. READ Through Alchemy to Chemistry. Illus. 78/561 Philosophy of Science
R B BRAiTHtVArrE Scientific Explanation ^8/5x5
Communication Theory I BRONOWSKI Science and Human Values Revised and
1 R PIERCE Symbols, Signals and Noise. The Nature Enlarged Edition 13/505
aid Process of ComT-iumcation 76/574 ALBERT EINSTEIN et al Albert Einstein Philosopher-
Scienhst Edited by Paul A Schilpp Vol. I 13/502
Vol II 7B/503
Geography
WERNER HEISENBERG Physics and Philosophy The Reuo-
a. E COKEX This Great and Wide Sea: An Introduction l*tion in Modern Science
to Qccar.-ra<c".-j And Mxnie Biology. Illus TB''55i.
^8/549
JOHN MAYSARD KEYNEs A Treatise on Probability
F K. HARE The Restless Atmosphere 78/560
Introduction by N R. Hanson T*/557
KARL R. POPPER The Logic of Scientific Discovery
History of Science
TB/576
w DAMPER, Ed Readings in the Literature of Science STEPHEN TOULMIN. Foresight and Understanding. An
I"US TB/512 Enquiry into the Aims of Science Foreword by
A. HUNTER DVPSEE Science in the Federal Government Jacques Barzun 1-8/564
A
Histo-n; of Pohcies and Activities to 1040 TB '573 STEPHEN TOULMIN The Philosophy of Science- An ln~
AtEXANDRE KOYRE. From the Cosed World to the Infinite
r
Universe Copernicus. Kesler, Galileo, \ ewion, etc G j WHITROW The Natural Philosophy of Tune
TB/31
78/563
A G. VAN MEISSEN' From Atoinos to Atom. A History of
the Concept Atom 78/517
;
Physics and Cosmology
o NEUGEBA'JER The Exact Sciences in Antiquity TB 552
H T PLEDGE: Science S:rce 1500. A Short History of
DA.VID BOHM Causality and Chance m Modern Physics
Foreword by Louis de Broghs 78/536
\laihe*nztii:$, Physics, Chemistry and Biology Illus.
TB/506 p w BRIDGMA.V: The Nature of Thermodynamics
HANS THIRRINC Energy- for Man From IVindmills to 78/537
Xudear Powc* 73/556 p Y, BRTDGMAN A Sophisticate's Primer of Relativity
v>riL T AM LAW WH-iTE Essay or. Aionusrn: From Democ- TB/575
r.tus to ;<j6o TB/565 A. c CROMBIE, Ed Turning Point in Physics TB/535
A WOLF A History of Science Technology and Philoso- c v DURELL- Readable Relativity
Foreword by Freeman
phy in the i6th and 17th Centuries Illus Dyson
7
TB/530
Vol. I TB/joS; Vol II 73/509 ARTHUR EDDINGTON Space, Time and Gravitation- An
A VfQis A History of Science Technology, and Philoso- outline of the General Relativity Theory
73/510
phy in the Eighteenth Century. Illus GEORGE GAMOW Biography of Physics 2
Voi I TB, 539, Vol. II TB, 540 76/567
MAX JAMMER- Concepts of Force A Study in the Founda-
tion of Dynamics
\laihernatics 73/550
MAX JAMMER Concepts of Mass in Classical and Modern
H DAI EXTOar The Higher Arithmetic An Introduction
Physics rz / 5TL
ro the T'ico^ of lumbers 73/526
MAX JAMMER Concepts of Space The History of
H C FOROER GeoTietry An Irtroducnon TB,'548 Theories of Space in Physics Foreword
by Albert
GO77O3 FILEGE The Foundations of Arithmetic. A Einstein
^533
logiro-.Mj^c'n-firal En^ttiry 7B,'534 STEPHEN 70ULMIN &. JUNE GOODHELD The Fabric of the
3 AOR.VES- The Philosophy of Mathematics An Intro- Heavens: The Development of
.-iajfrcn
Astronomy and Dy-
78/547 namics Illus
TB/579
D i L:TT-_ZWOCD Skeleton Kev of Mathematics- A EDMUND WHnTAfcER History of the Theories of Aether
Sir*:?lc Account o+ Cjrj^.'ez Algebraic Problems and Electricity
78 -'525 Volume I The Classical Theories 18/531
CEOSCE E OWEN Fundamentals of Scientific Mathe- Volume II The Modern Theories
:
78/532
rr^tics TB '569 G. j WHITROW The Structure and Evolution of the Uni-
'A-:v_ARD VAN OSAI.\N QuiNE Mathematical Logic 73/558 verse An Introduction to Cosmology Illus. 78/504
33S4
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