Tractatus Logico-Philosophicus by Ludwig Wittgenstein

Tractatus

Logico

Philosophicus

By

LUDWIG WITTGENSTEIN

With an Introduction by

BERTRAND RUSSELL, F.R.S.

LONDON

KEGAN PAUL, TRENCH, TRUBNER & CO.

, LTD.

NEW YORK: HARCOURT, BRACE & COMPANY, INC.

1922PRINTED IN GREAT BRITAIN BY THE EDINBURGH PRESS,

9 AND 11 YOUNG STREET, EDINBURGH.NOTE

In rendering Mr Wittgenstein

s Tractatus Logico

Philosophicus avail



able for English readers, the somewhat unusual course has been adopted

of printing the original side by side with the translation. Such a method

of presentation seemed desirable both on account of the obvious diffiffifficul



ties raised by the vocabulary and in view of the peculiar literary character

of the whole. As a result, a certain latitude has been possible in passages

to which objection might otherwise be taken as over

literal.

The proofs of the translation and the version of the original which

appeared in the fifinal number of Ostwald

s Annalen der Naturphilosophie

(1921) have been very carefully revised by the author himself; and the

Editor further desires to express his indebtedness to Mr F. P. Ramsey, of

Trinity College, Cambridge, for assistance both with the translation and

in the preparation of the book for the press.

  1. K. O.INTRODUCTION

By BERTRAND RUSSELL

Mr Wittgenstein

s Tractatus Logico

Philosophicus, whether or

not it prove to give the ultimate truth on the matters with which it deals,

certainly deserves, by its breadth and scope and profundity, to be con



sidered an important event in the philosophical world. Starting from the

principles of Symbolism and the relations which are necessary between

words and things in any language, it applies the result of this inquiry to

various departments of traditional philosophy, showing in each case how

traditional philosophy and traditional solutions arise out of ignorance of

the principles of Symbolism and out of misuse of language.

The logical structure of propositions and the nature of logical in



ference are fifirst dealt with. Thence we pass successively to Theory of

Knowledge, Principles of Physics, Ethics, and fifinally the Mystical (das

Mystische)

.

In order to understand Mr Wittgenstein

s book, it is necessary to re



alize what is the problem with which he is concerned. In the part of his

theory which deals with Symbolism he is concerned with the conditions

which would have to be fulfifilled by a logically perfect language. There are

various problems as regards language. First, there is the problem what

actually occurs in our minds when we use language with the intention

of meaning something by it; this problem belongs to psychology

. Sec



ondly, there is the problem as to what is the relation subsisting between

thoughts, words, or sentences, and that which they refer to or mean; this

problem belongs to epistemology

. Thirdly, there is the problem of using

sentences so as to convey truth rather than falsehood; this belongs to

the special sciences dealing with the subject

matter of the sentences in

question. Fourthly, there is the question: what relation must one fact

(such as a sentence) have to another in order to be capable of being a

symbol for that other? This last is a logical question, and is the one with

which Mr Wittgenstein is concerned. He is concerned with the conditions

for accurate Symbolism, i.e. for Symbolism in which a sentence

means

something quite defifinite. In practice, language is always more or less

vague, so that what we assert is never quite precise. Thus, logic has two

problems to deal with in regard to Symbolism: (1) the conditions for

sense rather than nonsense in combinations of symbols; (2) the condi



tions for uniqueness of meaning or reference in symbols or combinations

7INTRODUCTION

of symbols. A logically perfect language has rules of syntax which pre



vent nonsense, and has single symbols which always have a defifinite and

unique meaning

. Mr Wittgenstein is concerned with the conditions for a

logically perfect language

not that any language is logically perfect, or

that we believe ourselves capable, here and now, of constructing a logi



cally perfect language, but that the whole function of language is to have

meaning, and it only fulfifils this function in proportion as it approaches

to the ideal language which we postulate.

The essential business of language is to assert or deny facts. Given the

syntax of a language, the meaning of a sentence is determinate as soon

as the meaning of the component words is known. In order that a certain

sentence should assert a certain fact there must, however the language

may be constructed, be something in common between the structure of

the sentence and the structure of the fact. This is perhaps the most

fundamental thesis of Mr Wittgenstein

s theory

. That which has to be

in common between the sentence and the fact cannot, so he contends,

be itself in turn said in language. It can, in his phraseology, only be

shown, not said, for whatever we may say will still need to have the same

structure.

The fifirst requisite of an ideal language would be that there should be

one name for every simple, and never the same name for two difffferent

simples. A name is a simple symbol in the sense that it has no parts

which are themselves symbols. In a logically perfect language nothing

that is not simple will have a simple symbol. The symbol for the whole

will be a

complex,

containing the symbols for the parts. In speaking

of a

complex

we are, as will appear later, sinning against the rules

of philosophical grammar, but this is unavoidable at the outset.

Most

propositions and questions that have been written about philosophical

matters are not false but senseless. We cannot, therefore, answer ques



tions of this kind at all, but only state their senselessness. Most questions

and propositions of the philosophers result from the fact that we do not

understand the logic of our language. They are of the same kind as the

question whether the Good is more or less identical than the Beautiful

(4.003)

. What is complex in the world is a fact. Facts which are not

compounded of other facts are what Mr Wittgenstein calls Sachverhalte,

whereas a fact which may consist of two or more facts is called a Tatsa



che: thus, for example,

Socrates is wise

is a Sachverhalt, as well as a

Tatsache, whereas

Socrates is wise and Plato is his pupil

is a Tatsache

but not a Sachverhalt.

8INTRODUCTION

He compares linguistic expression to projection in geometry

. A geo



metrical fifigure may be projected in many ways: each of these ways corre



sponds to a difffferent language, but the projective properties of the orig



inal fifigure remain unchanged whichever of these ways may be adopted.

These projective properties correspond to that which in his theory the

proposition and the fact must have in common, if the proposition is to

assert the fact.

In certain elementary ways this is, of course, obvious. It is impossible,

for example, to make a statement about two men (assuming for the mo



ment that the men may be treated as simples), without employing two

names, and if you are going to assert a relation between the two men it

will be necessary that the sentence in which you make the assertion shall

establish a relation between the two names. If we say

Plato loves Soc



rates,

the word

loves

which occurs between the word

Plato

and the

word

Socrates

establishes a certain relation between these two words,

and it is owing to this fact that our sentence is able to assert a relation

between the person

s name by the words

Plato

and

Socrates.

” “

We

must not say, the complex sign

aRb

says

a stands in a certain relation

R to b

; but we must say, that

a

stands in a certain relation to

b

says

that aRb

(3.1432)

.

Mr Wittgenstein begins his theory of Symbolism with the statement

(2.1):

We make to ourselves pictures of facts.

A picture, he says, is

a model of the reality, and to the objects in the reality correspond the

elements of the picture: the picture itself is a fact. The fact that things

have a certain relation to each other is represented by the fact that in

the picture its elements have a certain relation to one another.

In the

picture and the pictured there must be something identical in order that

the one can be a picture of the other at all. What the picture must

have in common with reality in order to be able to represent it after its

manner

rightly or falsely

is its form of representation

(2.161, 2.17)

.

We speak of a logical picture of a reality when we wish to imply only

so much resemblance as is essential to its being a picture in any sense,

that is to say, when we wish to imply no more than identity of logical

form. The logical picture of a fact, he says, is a Gedanke. A picture

can correspond or not correspond with the fact and be accordingly true

or false, but in both cases it shares the logical form with the fact. The

sense in which he speaks of pictures is illustrated by his statement:

The

gramophone record, the musical thought, the score, the waves of sound,

all stand to one another in that pictorial internal relation which holds

9INTRODUCTION

between language and the world. To all of them the logical structure is

common.

(Like the two youths, their two horses and their lilies in the

story

. They are all in a certain sense one)

(4.014)

. The possibility of a

proposition representing a fact rests upon the fact that in it objects are

represented by signs. The so

called logical

constants

are not represented

by signs, but are themselves present in the proposition as in the fact. The

proposition and the fact must exhibit the same logical

manifold,

and

this cannot be itself represented since it has to be in common between

the fact and the picture. Mr Wittgenstein maintains that everything

properly philosophical belongs to what can only be shown, to what is

in common between a fact and its logical picture. It results from this

view that nothing correct can be said in philosophy

. Every philosophical

proposition is bad grammar, and the best that we can hope to achieve

by philosophical discussion is to lead people to see that philosophical

discussion is a mistake.

Philosophy is not one of the natural sciences.

(The word

philosophy

must mean something which stands above or

below, but not beside the natural sciences.

) The object of philosophy is

the logical clarifification of thoughts. Philosophy is not a theory but an

activity

. A philosophical work consists essentially of elucidations. The

result of philosophy is not a number of

philosophical propositions,

but

to make propositions clear. Philosophy should make clear and delimit

sharply the thoughts which otherwise are, as it were, opaque and blurred

(4.111 and 4.112)

. In accordance with this principle the things that have

to be said in leading the reader to understand Mr Wittgenstein

s theory

are all of them things which that theory itself condemns as meaningless.

With this proviso we will endeavour to convey the picture of the world

which seems to underlie his system.

The world consists of facts: facts cannot strictly speaking be defifined,

but we can explain what we mean by saying that facts are what make

propositions true, or false. Facts may contain parts which are facts or

may contain no such parts; for example:

Socrates was a wise Athe



nian,

consists of the two facts,

Socrates was wise,

and

Socrates was

an Athenian.

A fact which has no parts that are facts is called by

Mr Wittgenstein a Sachverhalt. This is the same thing that he calls an

atomic fact. An atomic fact, although it contains no parts that are facts,

nevertheless does contain parts. If we may regard

Socrates is wise

as

an atomic fact we perceive that it contains the constituents

Socrates

and

wise.

If an atomic fact is analysed as fully as possible (theoret



ical, not practical possibility is meant) the constituents fifinally reached

10INTRODUCTION

may be called

simples

or

objects.

It is not contended by Wittgenstein

that we can actually isolate the simple or have empirical knowledge of

  1. It is a logical necessity demanded by theory, like an electron. His

ground for maintaining that there must be simples is that every complex

presupposes a fact. It is not necessarily assumed that the complexity of

facts is fifinite; even if every fact consisted of an infifinite number of atomic

facts and if every atomic fact consisted of an infifinite number of objects

there would still be objects and atomic facts (4.2211)

. The assertion that

there is a certain complex reduces to the assertion that its constituents

are related in a certain way, which is the assertion of a fact: thus if

we give a name to the complex the name only has meaning in virtue

of the truth of a certain proposition, namely the proposition asserting

the relatedness of the constituents of the complex. Thus the naming of

complexes presupposes propositions, while propositions presupposes the

naming of simples. In this way the naming of simples is shown to be

what is logically fifirst in logic.

The world is fully described if all atomic facts are known, together

with the fact that these are all of them. The world is not described

by merely naming all the objects in it; it is necessary also to know the

atomic facts of which these objects are constituents. Given this total of

atomic facts, every true proposition, however complex, can theoretically

be inferred. A proposition (true or false) asserting an atomic fact is called

an atomic proposition. All atomic propositions are logically independent

of each other. No atomic proposition implies any other or is inconsistent

with any other. Thus the whole business of logical inference is concerned

with propositions which are not atomic. Such propositions may be called

molecular.

Wittgenstein

s theory of molecular propositions turns upon his theory

of the construction of truth

functions.

A truth

function of a proposition p is a proposition containing p and

such that its truth or falsehood depends only upon the truth or falsehood

of p, and similarly a truth

function of several propositions p, q, r. . . is

one containing p, q, r. . . and such that its truth or falsehood depends

only upon the truth or falsehood of p, q, r. . . It might seem at fifirst

sight as though there were other functions of propositions besides truth



functions; such, for example, would be

A believes p,

for in general A

will believe some true propositions and some false ones: unless he is an

exceptionally gifted individual, we cannot infer that p is true from the fact

that he believes it or that p is false from the fact that he does not believe

11INTRODUCTION

  1. Other apparent exceptions would be such as

p is a very complex

proposition

or

p is a proposition about Socrates.

Mr Wittgenstein

maintains, however, for reasons which will appear presently, that such

exceptions are only apparent, and that every function of a proposition is

really a truth

function. It follows that if we can defifine truth

functions

generally, we can obtain a general defifinition of all propositions in terms

of the original set of atomic propositions. This Wittgenstein proceeds to

  1.  

It has been shown by Dr Sheffffer (Trans. Am. Math. Soc.

, Vol. XIV.

pp

. 481

488) that all truth

functions of a given set of propositions can be

constructed out of either of the two functions

not

p or not

q

or

not

p

and not

q

.

Wittgenstein makes use of the latter, assuming a knowledge

of Dr Sheffffer

s work. The manner in which other truth

functions are

constructed out of

not

p and not

q

is easy to see.

Not

p and not

p

is

equivalent to

not

p,

hence we obtain a defifinition of negation in terms

of our primitive function: hence we can defifine

p or q,

since this is

the negation of

not

p and not

q,

i.e. of our primitive function. The

development of other truth

functions out of

not

p

and

p or q

is given

in detail at the beginning of Principia Mathematica. This gives all that is

wanted when the propositions which are arguments to our truth

function

are given by enumeration. Wittgenstein, however, by a very interesting

analysis succeeds in extending the process to general propositions, i.e. to

cases where the propositions which are arguments to our truth

function

are not given by enumeration but are given as all those satisfying some

condition. For example, let fx be a propositional function (i.e. a function

whose values are propositions), such as

x is human

”—

then the various

values of fx form a set of propositions. We may extend the idea

not

p

and not

q

so as to apply to simultaneous denial of all the propositions

which are values of fx. In this way we arrive at the proposition which

is ordinarily represented in mathematical logic by the words

fx is false

for all values of x.

The negation of this would be the proposition

there

is at least one x for which fx is true

which is represented by

(x)

.

fx.

If we had started with not

fx instead of fx we should have arrived at

the proposition

fx is true for all values of x

which is represented by

(x)

.

fx.

Wittgenstein

s method of dealing with general propositions [i.e.

(x)

.

fx

and

(x)

.

fx

] diffffers from previous methods by the fact that

the generality comes only in specifying the set of propositions concerned,

and when this has been done the building up of truth

functions proceeds

exactly as it would in the case of a fifinite number of enumerated arguments

p, q, r . . . .

12INTRODUCTION

Mr Wittgenstein

s explanation of his symbolism at this point is not

quite fully given in the text. The symbol he uses is (p, ξ, N(ξ))

. The

following is the explanation of this symbol:

p stands for all atomic propositions.

ξ stands for any set of propositions.

N(ξ) stands for the negation of all the proposi



tions making up ξ

.

The whole symbol (p, ξ, N(ξ)) means whatever can be obtained by

taking any selection of atomic propositions, negating them all, then tak



ing any selection of the set of propositions now obtained, together with

any of the originals

and so on indefifinitely

. This is, he says, the general

truth

function and also the general form of proposition. What is meant

is somewhat less complicated than it sounds. The symbol is intended to

describe a process by the help of which, given the atomic propositions,

all others can be manufactured. The process depends upon:

(a) Sheffffer

s proof that all truth

functions can be obtained out of

simultaneous negation, i.e. out of

not

p and not

q

;

(b) Mr Wittgenstein

s theory of the derivation of general propositions

from conjunctions and disjunctions;

(c) The assertion that a proposition can only occur in another prop



osition as argument to a truth

function. Given these three foundations,

it follows that all propositions which are not atomic can be derived from

such as are, by a uniform process, and it is this process which is indicated

by Mr Wittgenstein

s symbol.

From this uniform method of construction we arrive at an amazing

simplifification of the theory of inference, as well as a defifinition of the sort

of propositions that belong to logic. The method of generation which has

just been described, enables Wittgenstein to say that all propositions can

be constructed in the above manner from atomic propositions, and in this

way the totality of propositions is defifined.

(The apparent exceptions

which we mentioned above are dealt with in a manner which we shall

consider later.

) Wittgenstein is enabled to assert that propositions are

all that follows from the totality of atomic propositions (together with

the fact that it is the totality of them); that a proposition is always a

truth

function of atomic propositions; and that if p follows from q the

meaning of p is contained in the meaning of q, from which of course it

results that nothing can be deduced from an atomic proposition. All the

13INTRODUCTION

propositions of logic, he maintains, are tautologies, such, for example, as

p or not p

.

The fact that nothing can be deduced from an atomic proposition

has interesting applications, for example, to causality

. There cannot, in

Wittgenstein

s logic, be any such thing as a causal nexus.

The events

of the future,

he says,

cannot be inferred from those of the present.

Superstition is the belief in the causal nexus.

That the sun will rise

to

morrow is a hypothesis. We do not in fact know whether it will rise,

since there is no compulsion according to which one thing must happen

because another happens.

Let us now take up another subject

that of names. In Wittgenstein

s

theoretical logical language, names are only given to simples. We do

not give two names to one thing, or one name to two things. There

is no way whatever, according to him, by which we can describe the

totality of things that can be named, in other words, the totality of

what there is in the world. In order to be able to do this we should

have to know of some property which must belong to every thing by

a logical necessity

. It has been sought to fifind such a property in self



identity, but the conception of identity is subjected by Wittgenstein to a

destructive criticism from which there seems no escape. The defifinition of

identity by means of the identity of indiscernibles is rejected, because the

identity of indiscernibles appears to be not a logically necessary principle.

According to this principle x is identical with y if every property of x is a

property of y, but it would, after all, be logically possible for two things

to have exactly the same properties. If this does not in fact happen

that is an accidental characteristic of the world, not a logically necessary

characteristic, and accidental characteristics of the world must, of course,

not be admitted into the structure of logic. Mr Wittgenstein accordingly

banishes identity and adopts the convention that difffferent letters are to

mean difffferent things. In practice, identity is needed as between a name

and a description or between two descriptions. It is needed for such

propositions as

Socrates is the philosopher who drank the hemlock,

or

The even prime is the next number after 1.

For such uses of identity it

is easy to provide on Wittgenstein

s system.

The rejection of identity removes one method of speaking of the to



tality of things, and it will be found that any other method that may be

suggested is equally fallacious: so, at least, Wittgenstein contends and, I

think, rightly

. This amounts to saying that

object

is a pseudo

concept.

To say

x is an object

is to say nothing

. It follows from this that we

14INTRODUCTION

cannot make such statements as

there are more than three objects in the

world,

or

there are an infifinite number of objects in the world.

Objects

can only be mentioned in connexion with some defifinite property

. We can

say

there are more than three objects which are human,

or

there are

more than three objects which are red,

for in these statements the word

object can be replaced by a variable in the language of logic, the variable

being one which satisfifies in the fifirst case the function

x is human

; in

the second the function

x is red.

But when we attempt to say

there

are more than three objects,

this substitution of the variable for the

word

object

becomes impossible, and the proposition is therefore seen

to be meaningless.

We here touch one instance of Wittgenstein

s fundamental thesis,

that it is impossible to say anything about the world as a whole, and

that whatever can be said has to be about bounded portions of the world.

This view may have been originally suggested by notation, and if so, that

is much in its favour, for a good notation has a subtlety and suggestive



ness which at times make it seem almost like a live teacher. Notational

irregularities are often the fifirst sign of philosophical errors, and a perfect

notation would be a substitute for thought. But although notation may

have fifirst suggested to Mr Wittgenstein the limitation of logic to things

within the world as opposed to the world as a whole, yet the view, once

suggested, is seen to have much else to recommend it. Whether it is

ultimately true I do not, for my part, profess to know. In this Introduc



tion I am concerned to expound it, not to pronounce upon it. According

to this view we could only say things about the world as a whole if we

could get outside the world, if, that is to say, it ceased to be for us the

whole world. Our world may be bounded for some superior being who

can survey it from above, but for us, however fifinite it may be, it cannot

have a boundary, since it has nothing outside it. Wittgenstein uses, as

an analogy, the fifield of vision. Our fifield of vision does not, for us, have a

visual boundary, just because there is nothing outside it, and in like man



ner our logical world has no logical boundary because our logic knows of

nothing outside it. These considerations lead him to a somewhat curious

discussion of Solipsism. Logic, he says, fifills the world. The boundaries of

the world are also its boundaries. In logic, therefore, we cannot say, there

is this and this in the world, but not that, for to say so would apparently

presuppose that we exclude certain possibilities, and this cannot be the

case, since it would require that logic should go beyond the boundaries of

the world as if it could contemplate these boundaries from the other side

15INTRODUCTION

also. What we cannot think we cannot think, therefore we also cannot

say what we cannot think.

This, he says, gives the key to Solipsism. What Solipsism intends is

quite correct, but this cannot be said, it can only be shown. That the

world is my world appears in the fact that the boundaries of language

(the only language I understand) indicate the boundaries of my world.

The metaphysical subject does not belong to the world but is a boundary

of the world.

We must take up next the question of molecular propositions which

are at fifirst sight not truth

functions, of the propositions that they con



tain, such, for example, as

A believes p

.

Wittgenstein introduces this subject in the statement of his position,

namely, that all molecular functions are truth

functions. He says (5.54):

In the general propositional form, propositions occur in a proposition

only as bases of truth

operations.

At fifirst sight, he goes on to explain,

it seems as if a proposition could also occur in other ways, e.

g

.

A believes

p

.

Here it seems superfificially as if the proposition p stood in a sort of

relation to the object A.

But it is clear that

A believes that p,

’ ‘

A

thinks p,

’ ‘

A says p

are of the form

p says p

; and here we have no co



ordination of a fact and an object, but a co

ordination of facts by means

of a co

ordination of their objects

(5.542)

.

What Mr Wittgenstein says here is said so shortly that its point is

not likely to be clear to those who have not in mind the controversies

with which he is concerned. The theory with which he is disagreeing

will be found in my articles on the nature of truth and falsehood in

Philosophical Essays and Proceedings of the Aristotelian Society, 1906

  1. The problem at issue is the problem of the logical form of belief,

i.e. what is the schema representing what occurs when a man believes.

Of course, the problem applies not only to belief, but also to a host of

other mental phenomena which may be called propositional attitudes:

doubting, considering, desiring, etc. In all these cases it seems natural

to express the phenomenon in the form

A doubts p,

” “

A desires p,

etc.

,

which makes it appear as though we were dealing with a relation between

a person and a proposition. This cannot, of course, be the ultimate

analysis, since persons are fifictions and so are propositions, except in

the sense in which they are facts on their own account. A proposition,

considered as a fact on its own account, may be a set of words which a

man says over to himself, or a complex image, or train of images passing

through his mind, or a set of incipient bodily movements. It may be

16INTRODUCTION

any one of innumerable difffferent things. The proposition as a fact on its

own account, for example the actual set of words the man pronounces

to himself, is not relevant to logic. What is relevant to logic is that

common element among all these facts, which enables him, as we say, to

mean the fact which the proposition asserts. To psychology, of course,

more is relevant; for a symbol does not mean what it symbolizes in virtue

of a logical relation alone, but in virtue also of a psychological relation of

intention, or association, or what

not. The psychological part of meaning,

however, does not concern the logician. What does concern him in this

problem of belief is the logical schema. It is clear that, when a person

believes a proposition, the person, considered as a metaphysical subject,

does not have to be assumed in order to explain what is happening

. What

has to be explained is the relation between the set of words which is the

proposition considered as a fact on its own account, and the

objective

fact which makes the proposition true or false. This reduces ultimately to

the question of the meaning of propositions, that is to say, the meaning of

propositions is the only non

psychological portion of the problem involved

in the analysis of belief. This problem is simply one of a relation of

two facts, namely, the relation between the series of words used by the

believer and the fact which makes these words true or false. The series of

words is a fact just as much as what makes it true or false is a fact. The

relation between these two facts is not unanalysable, since the meaning

of a proposition results from the meaning of its constituent words. The

meaning of the series of words which is a proposition is a function of the

meanings of the separate words. Accordingly, the proposition as a whole

does not really enter into what has to be explained in explaining the

meaning of a proposition. It would perhaps help to suggest the point of

view which I am trying to indicate, to say that in the cases we have been

considering the proposition occurs as a fact, not as a proposition. Such

a statement, however, must not be taken too literally

. The real point

is that in believing, desiring, etc.

, what is logically fundamental is the

relation of a proposition considered as a fact, to the fact which makes it

true or false, and that this relation of two facts is reducible to a relation

of their constituents. Thus the proposition does not occur at all in the

same sense in which it occurs in a truth

function.

There are some respects, in which, as it seems to me, Mr Wittgen



stein

s theory stands in need of greater technical development. This ap



plies in particular to his theory of number (6.02 ffff.

) which, as it stands, is

only capable of dealing with fifinite numbers. No logic can be considered

17INTRODUCTION

adequate until it has been shown to be capable of dealing with transfifinite

numbers. I do not think there is anything in Mr Wittgenstein

s system

to make it impossible for him to fifill this lacuna.

More interesting than such questions of comparative detail is Mr

Wittgenstein

s attitude towards the mystical. His attitude upon this

grows naturally out of his doctrine in pure logic, according to which the

logical proposition is a picture (true or false) of the fact, and has in

common with the fact a certain structure. It is this common structure

which makes it capable of being a picture of the fact, but the structure

cannot itself be put into words, since it is a structure of words, as well

as of the facts to which they refer. Everything, therefore, which is in



volved in the very idea of the expressiveness of language must remain

incapable of being expressed in language, and is, therefore, inexpressible

in a perfectly precise sense. This inexpressible contains, according to Mr

Wittgenstein, the whole of logic and philosophy

. The right method of

teaching philosophy, he says, would be to confifine oneself to propositions

of the sciences, stated with all possible clearness and exactness, leaving

philosophical assertions to the learner, and proving to him, whenever he

made them, that they are meaningless. It is true that the fate of Socrates

might befall a man who attempted this method of teaching, but we are

not to be deterred by that fear, if it is the only right method. It is not

this that causes some hesitation in accepting Mr Wittgenstein

s position,

in spite of the very powerful arguments which he brings to its support.

What causes hesitation is the fact that, after all, Mr Wittgenstein man



ages to say a good deal about what cannot be said, thus suggesting to

the sceptical reader that possibly there may be some loophole through

a hierarchy of languages, or by some other exit. The whole subject of

ethics, for example, is placed by Mr Wittgenstein in the mystical, in



expressible region. Nevertheless he is capable of conveying his ethical

opinions. His defence would be that what he calls the mystical can be

shown, although it cannot be said. It may be that this defence is ade



quate, but, for my part, I confess that it leaves me with a certain sense

of intellectual discomfort.

There is one purely logical problem in regard to which these diffiffifficulties

are peculiarly acute. I mean the problem of generality

. In the theory

of generality it is necessary to consider all propositions of the form fx

where fx is a given propositional function. This belongs to the part of

logic which can be expressed, according to Mr Wittgenstein

s system.

But the totality of possible values of x which might seem to be involved

18INTRODUCTION

in the totality of propositions of the form fx is not admitted by Mr

Wittgenstein among the things that can be spoken of, for this is no other

than the totality of things in the world, and thus involves the attempt

to conceive the world as a whole;

the feeling of the world as a bounded

whole is the mystical

; hence the totality of the values of x is mystical

(6.45)

. This is expressly argued when Mr Wittgenstein denies that we

can make propositions as to how many things there are in the world, as

for example, that there are more than three.

These diffiffifficulties suggest to my mind some such possibility as this:

that every language has, as Mr Wittgenstein says, a structure concern



ing which, in the language, nothing can be said, but that there may be

another language dealing with the structure of the fifirst language, and

having itself a new structure, and that to this hierarchy of languages

there may be no limit. Mr Wittgenstein would of course reply that his

whole theory is applicable unchanged to the totality of such languages.

The only retort would be to deny that there is any such totality

. The

totalities concerning which Mr Wittgenstein holds that it is impossible

to speak logically are nevertheless thought by him to exist, and are the

subject

matter of his mysticism. The totality resulting from our hier



archy would be not merely logically inexpressible, but a fifiction, a mere

delusion, and in this way the supposed sphere of the mystical would be

abolished. Such an hypothesis is very diffiffifficult, and I can see objections

to it which at the moment I do not know how to answer. Yet I do not

see how any easier hypothesis can escape from Mr Wittgenstein

s con



clusions. Even if this very diffiffifficult hypothesis should prove tenable, it

would leave untouched a very large part of Mr Wittgenstein

s theory,

though possibly not the part upon which he himself would wish to lay

most stress. As one with a long experience of the diffiffifficulties of logic

and of the deceptiveness of theories which seem irrefutable, I fifind myself

unable to be sure of the rightness of a theory, merely on the ground that

I cannot see any point on which it is wrong

. But to have constructed

a theory of logic which is not at any point obviously wrong is to have

achieved a work of extraordinary diffiffifficulty and importance. This merit,

in my opinion, belongs to Mr Wittgenstein

s book, and makes it one

which no serious philosopher can afffford to neglect.

Bertrand Russell.

 

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