1 The world is everything that is the case.

1.1 The world is the totality of facts, not of things.

1.11 The world is determined by the facts, and by these being all the

facts.

1.12 For the totality of facts determines both what is the case, and

also all that is not the case.

1.13 The facts in logical space are the world.

1.2 The world divides into facts.

1.21 Anyone can either be the case or not be the case, and everything

else remain the same.

2 What is the case, the fact, is the existence of atomic facts.

2.01 An atomic fact is a combination of objects (entities, things).

2.011 It is essential to a thing that it can be a constituent part of an

atomic fact.

2.012 In logic nothing is accidental: if a thing can occur in an atomic

fact the possibility of that atomic fact must already be prejudged

in the thing.

2.0121 It would, so to speak, appear as an accident, when to a thing

that could exist alone on its own account, subsequently a state

of affairs could be made to fit.

If things can occur in atomic facts, this possibility must al-

ready lie in them.

(A logical entity cannot be merely possible. Logic treats of

every possibility, and all possibilities are its facts.)

Just as we cannot think of spatial objects at all apart from

space, or temporal objects apart from time, so we cannot think

of any object apart from the possibility of its connexion with

other things.

If I can think of an object in the context of an atomic fact,

I cannot think of it apart from the possibility of this context.

2.0122 The thing is independent, in so far as it can occur in all pos-

sible circumstances, but this form of independence is a form of

connexion with the atomic fact, a form of dependence. (It is

impossible for words to occur in two different ways, alone and

in the proposition.)

2.0123 If I know an object, then I also know all the possibilities of its

occurrence in atomic facts.

(Every such possibility must lie in the nature of the object.)

A new possibility cannot subsequently be found.

2.01231 In order to know an object, I must know not its external but all

its internal qualities.

2.0124 If all objects are given, then thereby are all possible atomic facts

also given.

2.013 Every thing is, as it were, in a space of possible atomic facts. I

can think of this space as empty, but not of the thing without

the space.

2.0131 A spatial object must lie in infinite space. (A point in space is

a place for an argument.)

A speck in a visual field need not be red, but it must have a

colour; it has, so to speak, a colour space round it. A tone must

have a pitch, the object of the sense of touch a hardness, etc.

2.014 Objects contain the possibility of all states of affairs.

2.0141 The possibility of its occurrence in atomic facts is the form of

the object.

2.02 The object is simple.

2.0201 Every statement about complexes can be analysed into a state-

ment about their constituent parts, and into those propositions

which completely describe the complexes.

2.021 Objects form the substance of the world. Therefore they cannot

be compound.

2.0211 If the world had no substance, then whether a proposition had

sense would depend on whether another proposition was true.

2.0212 It would then be impossible to form a picture of the world (true

or false).

2.022 It is clear that however different from the real one an imagined

world may be, it must have something--a form--in common

with the real world.

2.023 This fixed form consists of the objects.

2.0231 The substance of the world can only determine a form and not

any material properties. For these are first presented by the

propositions--first formed by the configuration of the objects.

2.0232 Roughly speaking: objects are colourless.

2.0233 Two objects of the same logical form are--apart from their ex-

ternal properties--only differentiated from one another in that

they are different.

2.02331 Either a thing has properties which no other has, and then one

can distinguish it straight away from the others by a description

and refer to it; or, on the other hand, there are several things

which have the totality of their properties in common, and then

it is quite impossible to point to any one of them.

For if a thing is not distinguished by anything, I cannot dis-

tinguish it--for otherwise it would be distinguished.

2.024 Substance is what exists independently of what is the case.

2.025 It is form and content.

2.0251 Space, time and colour (colouredness) are forms of objects.

2.026 Only if there are objects can there be a fixed form of the world.

2.027 The fixed, the existent and the object are one.

2.0271 The object is the fixed, the existent; the configuration is the

changing, the variable.

2.0272 The configuration of the objects forms the atomic fact.

2.03 In the atomic fact objects hang one in another, like the members

of a chain.

2.031 In the atomic fact the objects are combined in a definite way.

2.032 The way in which objects hang together in the atomic fact is

the structure of the atomic fact.

2.033 The form is the possibility of the structure.

2.034 The structure of the fact consists of the structures of the atomic

facts.

2.04 The totality of existent atomic facts is the world.

2.05 Thetotalityofexistentatomicfactsalsodetermineswhichatom-

ic facts do not exist.

2.06 The existence and non-existence of atomic facts is the reality.

(The existence of atomic facts we also call a positive fact,

their non-existence a negative fact.)

2.061 Atomic facts are independent of one another.

2.062 From the existence or non-existence of an atomic fact we cannot

infer the existence or non-existence of another.

2.063 The total reality is the world.

2.1 We make to ourselves pictures of facts.

2.11 The picture presents the facts in logical space, the existence and

non-existence of atomic facts.

2.12 The picture is a model of reality.

2.13 To the objects correspond in the picture the elements of the

picture.

2.131 Theelementsofthepicturestand, inthepicture, fortheobjects.

2.14 The picture consists in the fact that its elements are combined

with one another in a definite way.

2.141 The picture is a fact.

2.15 That the elements of the picture are combined with one another

in a definite way, represents that the things are so combined

with one another.

This connexion of the elements of the picture is called its

structure, and the possibility of this structure is called the form

of representation of the picture.

2.151 The form of representation is the possibility that the things are

combined with one another as are the elements of the picture.

2.1511 Thus the picture is linked with reality; it reaches up to it.

2.1512 It is like a scale applied to reality.

2.15121 Only the outermost points of the dividing lines touch the object

to be measured.

2.1513 According to this view the representing relation which makes it

a picture, also belongs to the picture.

2.1514 The representing relation consists of the co-ordinations of the

elements of the picture and the things.

2.1515 These co-ordinations are as it were the feelers of its elements

with which the picture touches reality.

2.16 In order to be a picture a fact must have something in common

with what it pictures.

2.161 In the picture and the pictured there must be something identi-

cal in order that the one can be a picture of the other at all.

2.17 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.

2.171 The picture can represent every reality whose form it has.

The spatial picture, everything spatial, the coloured, every-

thing coloured, etc.

2.172 The picture, however, cannot represent its form of representa-

tion; it shows it forth.

2.173 The picture represents its object from without (its standpoint is

its form of representation), therefore the picture represents its

object rightly or falsely.

2.174 But the picture cannot place itself outside of its form of repre-

sentation.

2.18 What every picture, of whatever form, must have in common

with reality in order to be able to represent it at all--rightly or

falsely--is the logical form, that is, the form of reality.

2.181 If the form of representation is the logical form, then the picture

is called a logical picture.

2.182 Every picture is also a logical picture. (On the other hand, for

example, not every picture is spatial.)

2.19 The logical picture can depict the world.

2.2 The picture has the logical form of representation in common

with what it pictures.

2.201 The picture depicts reality by representing a possibility of the

existence and non-existence of atomic facts.

2.202 The picture represents a possible state of affairs in logical space.

2.203 The picture contains the possibility of the state of affairs which

it represents.

2.21 The picture agrees with reality or not; it is right or wrong, true

or false.

2.22 The picture represents what it represents, independently of its

truth or falsehood, through the form of representation.

2.221 What the picture represents is its sense.

2.222 In the agreement or disagreement of its sense with reality, its

truth or falsity consists.

2.223 In order to discover whether the picture is true or false we must

compare it with reality.

2.224 It cannot be discovered from the picture alone whether it is true

or false.

2.225 There is no picture which is a priori true.

3 The logical picture of the facts is the thought.

3.001 ‘An atomic fact is thinkable’--means: we can imagine it.

3.01 The totality of true thoughts is a picture of the world.

3.02 The thought contains the possibility of the state of affairs which

it thinks.

What is thinkable is also possible.

3.03 We cannot think anything unlogical, for otherwise we should

have to think unlogically.

3.031 It used to be said that God could create everything, except what

was contrary to the laws of logic. The truth is, we could not say

of an ‘unlogical’ world how it would look.

3.032 To present in language anything which ‘contradicts logic’ is as

impossible as in geometry to present by its co-ordinates a figure

which contradicts the laws of space; or to give the co-ordinates

of a point which does not exist.

3.0321 We could present spatially an atomic fact which contradicted

the laws of physics, but not one which contradicted the laws of

geometry.

3.04 An a priori true thought would be one whose possibility guar-

anteed its truth.

3.05 We could only know a priori that a thought is true if its truth

was to be recognized from the thought itself (without an object

of comparison).

3.1 In the proposition the thought is expressed perceptibly through

the senses.

3.11 We use the sensibly perceptible sign (sound or written sign, etc.)

of the proposition as a projection of the possible state of affairs.

The method of projection is the thinking of the sense of the

proposition.

3.12 The sign through which we express the thought I call the prop-

ositional sign. And the proposition is the propositional sign in

its projective relation to the world.

3.13 To the proposition belongs everything which belongs to the pro-

jection; but not what is projected.

Therefore the possibility of what is projected but not this

itself.

In the proposition, therefore, its sense is not yet contained,

but the possibility of expressing it.

(‘The content of the proposition’ means the content of the

significant proposition.)

In the proposition the form of its sense is contained, but not

its content.

3.14 The propositional sign consists in the fact that its elements, the

words, are combined in it in a definite way.

The propositional sign is a fact.

3.141 The proposition is not a mixture of words (just as the musical

theme is not a mixture of tones).

The proposition is articulate.

3.142 Only facts can express a sense, a class of names cannot.

3.143 That the propositional sign is a fact is concealed by the ordinary

form of expression, written or printed.

(For in the printed proposition, for example, the sign of a

proposition does not appear essentially different from a word.

Thus it was possible for Frege to call the proposition a com-

pounded name.)

3.1431 The essential nature of the propositional sign becomes very clear

when we imagine it made up of spatial objects (such as tables,

chairs, books) instead of written signs.

The mutual spatial position of these things then expresses

the sense of the proposition.

3.1432 We must not say, ‘The complex sign ‘aRb’ says ‘a stands in

relation R to b’’; but we must say, ‘That ‘a’ stands in a certain

relation to ‘b’ says that aRb’.

3.144 States of affairs can be described but not named.

(Names resemble points; propositions resemble arrows, they

have sense.)

3.2 In propositions thoughts can be so expressed that to the objects

of the thoughts correspond the elements of the propositional

sign.

3.201 These elements I call ‘simple signs’ and the proposition ‘com-

pletely analysed’.

3.202 The simple signs employed in propositions are called names.

3.203 The name means the object. The object is its meaning. (‘A’ is

the same sign as ‘A’.)

3.21 To the configuration of the simple signs in the propositional

sign corresponds the configuration of the objects in the state of

affairs.

3.22 In the proposition the name represents the object.

3.221 ObjectsIcanonlyname. Signsrepresentthem. Icanonlyspeak

of them. I cannot assert them. A proposition can only say how

a thing is, not what it is.

3.23 The postulate of the possibility of the simple signs is the postu-

late of the determinateness of the sense.

3.24 A proposition about a complex stands in internal relation to the

proposition about its constituent part.

A complex can only be given by its description, and this

will either be right or wrong. The proposition in which there

is mention of a complex, if this does not exist, becomes not

nonsense but simply false.

That a propositional element signifies a complex can be seen

from an indeterminateness in the propositions in which it oc-

curs. We know that everything is not yet determined by this

proposition. (The notation for generality contains a prototype.)

The combination of the symbols of a complex in a simple

symbol can be expressed by a definition.

3.25 There is one and only one complete analysis of the proposition.

3.251 The proposition expresses what it expresses in a definite and

clearly specifiable way: the proposition is articulate.

3.26 The name cannot be analysed further by any definition. It is a

primitive sign.

3.261 Every defined sign signifies via those signs by which it is defined,

and the definitions show the way.

Two signs, one a primitive sign, and one defined by primitive

signs, cannot signify in the same way. Names cannot be taken to

pieces by definition (nor any sign which alone and independently

has a meaning).

3.262 What does not get expressed in the sign is shown by its appli-

cation. What the signs conceal, their application declares.

3.263 The meanings of primitive signs can be explained by elucida-

tions. Elucidations are propositions which contain the primitive

signs. They can, therefore, only be understood when the mean-

ings of these signs are already known.

3.3 Only the proposition has sense; only in the context of a propo-

sition has a name meaning.

3.31 Every part of a proposition which characterizes its sense I call

an expression (a symbol).

(The proposition itself is an expression.)

Expressions are everything--essential for the sense of the

proposition--that propositions can have in common with one

another.

An expression characterizes a form and a content.

3.311 An expression presupposes the forms of all propositions in which

it can occur. It is the common characteristic mark of a class of

propositions.

3.312 Itisthereforerepresentedbythegeneralformofthepropositions

which it characterizes.

And in this form the expression is constant and everything

else variable.

3.313 An expression is thus presented by a variable, whose values are

the propositions which contain the expression.

(In the limiting case the variables become constants, the ex-

pression a proposition.)

I call such a variable a ‘propositional variable’.

3.314 Anexpressionhasmeaningonlyinaproposition. Everyvariable

can be conceived as a propositional variable.

(Including the variable name.)

3.315 If we change a constituent part of a proposition into a variable,

there is a class of propositions which are all the values of the re-

sulting variable proposition. This class in general still depends

onwhat, byarbitraryagreement, wemeanbypartsofthatprop-

osition. But if we change all those signs, whose meaning was ar-

bitrarily determined, into variables, there always remains such

a class. But this is now no longer dependent on any agreement;

it depends only on the nature of the proposition. It corresponds

to a logical form, to a logical prototype.

3.316 What values the propositional variable can assume is deter-

mined.

The determination of the values is the variable.

3.317 The determination of the values of the propositional variable

is done by indicating the propositions whose common mark the

variable is.

The determination is a description of these propositions.

The determination will therefore deal only with symbols not

with their meaning.

And only this is essential to the determination, that it is

only a description of symbols and asserts nothing about what is

symbolized.

The way in which we describe the propositions is not essen-

tial.

3.318 Iconceivetheproposition--likeFregeandRussell--asafunction

of the expressions contained in it.

3.32 The sign is the part of the symbol perceptible by the senses.

3.321 Two different symbols can therefore have the sign (the written

signorthesoundsign)incommon--theythensignifyindifferent

ways.

3.322 It can never indicate the common characteristic of two objects

that we symbolize them with the same signs but by different

methods of symbolizing. For the sign is arbitrary. We could

therefore equally well choose two different signs and where then

would be what was common in the symbolization.

3.323 In the language of everyday life it very often happens that the

same word signifies in two different ways--and therefore belongs

to two different symbols--or that two words, which signify in

different ways, are apparently applied in the same way in the

proposition.

Thus the word ‘is’ appears as the copula, as the sign of

equality, and as the expression of existence; ‘to exist’ as an

intransitiveverblike‘togo’; ‘identical’ asanadjective; wespeak

of something but also of the fact of something happening.

(In the proposition ‘Green is green’--where the first word is

a proper name and the last an adjective--these words have not

merely different meanings but they are different symbols.)

3.324 Thus there easily arise the most fundamental confusions (of

which the whole of philosophy is full).

3.325 In order to avoid these errors, we must employ a symbolism

which excludes them, by not applying the same sign in different

symbolsandbynotapplyingsignsinthesamewaywhichsignify

in different ways. A symbolism, that is to say, which obeys the

rules of logical grammar--of logical syntax.

(The logical symbolism of Frege and Russell is such a lan-

guage, which, however, does still not exclude all errors.)

3.326 In order to recognize the symbol in the sign we must consider

the significant use.

3.327 The sign determines a logical form only together with its logical

syntactic application.

3.328 If a sign is not necessary then it is meaningless. That is the

meaning of Occam’s razor.

(If everything in the symbolism works as though a sign had

meaning, then it has meaning.)

3.33 In logical syntax the meaning of a sign ought never to play a

role; it must admit of being established without mention being

thereby made of the meaning of a sign; it ought to presuppose

only the description of the expressions.

3.331 From this observation we get a further view--into Russell’s The-

ory of Types. Russell’serrorisshownbythefactthatindrawing

uphissymbolicruleshehastospeakofthemeaningofthesigns.

3.332 No proposition can say anything about itself, because the prop-

ositional sign cannot be contained in itself (that is the ‘whole

theory of types’).

3.333 A function cannot be its own argument, because the functional

sign already contains the prototype of its own argument and it

cannot contain itself.

If, for example, we suppose that the function F(fx) could

be its own argument, then there would be a proposition

‘F(F(fx))’, and in this the outer function F and the inner

function F must have different meanings; for the inner has

the form ϕ(fx), the outer the form ψ(ϕ(fx)). Common to

both functions is only the letter ‘F’, which by itself signifies

nothing.

This is at once clear, if instead of ‘F(F(u))’ we write ‘(∃ϕ) :

F(ϕu).ϕu = Fu’.

Herewith Russell’s paradox vanishes.

3.334 The rules of logical syntax must follow of themselves, if we only

know how every single sign signifies.

3.34 A proposition possesses essential and accidental features.

Accidental are the features which are due to a particular way

of producing the propositional sign. Essential are those which

alone enable the proposition to express its sense.

3.341 The essential in a proposition is therefore that which is common

to all propositions which can express the same sense.

And in the same way in general the essential in a symbol is

that which all symbols which can fulfil the same purpose have

in common.

3.3411 One could therefore say the real name is that which all symbols,

which signify an object, have in common. It would then follow,

step by step, that no sort of composition was essential for a

name.

3.342 In our notations there is indeed something arbitrary, but this

is not arbitrary, namely that if we have determined anything

arbitrarily, then something else must be the case. (This results

from the essence of the notation.)

3.3421 A particular method of symbolizing may be unimportant, but it

is always important that this is a possible method of symboliz-

ing. And this happens as a rule in philosophy: The single thing

proves over and over again to be unimportant, but the possibil-

ity of every single thing reveals something about the nature of

the world.

3.343 Definitions are rules for the translation of one language into

another. Every correct symbolism must be translatable into

every other according to such rules. It is this which all have in

common.

3.344 What signifies in the symbol is what is common to all those

symbols by which it can be replaced according to the rules of

logical syntax.

3.3441 We can, for example, express what is common to all notations

for the truth-functions as follows: It is common to them that

they all, for example, can be replaced by the notations of ‘∼p’

(‘not p’) and ‘p∨q’ (‘p or q’).

(Herewith is indicated the way in which a special possible

notation can give us general information.)

3.3442 Thesignofthecomplexisnotarbitrarilyresolvedintheanalysis,

in such a way that its resolution would be different in every

propositional structure.

3.4 The proposition determines a place in logical space: the exis-

tence of this logical place is guaranteed by the existence of the

constituent parts alone, by the existence of the significant prop-

osition.

3.41 The propositional sign and the logical co-ordinates: that is the

logical place.

3.411 The geometrical and the logical place agree in that each is the

possibility of an existence.

3.42 Although a proposition may only determine one place in logical

space, the whole logical space must already be given by it.

(Otherwise denial, the logical sum, the logical product, etc.,

would always introduce new elements--in co-ordination.)

(Thelogicalscaffoldingroundthepicturedeterminesthelog-

ical space. The proposition reaches through the whole logical

space.)

3.5 The applied, thought, propositional sign is the thought.

4 The thought is the significant proposition.

4.001 The totality of propositions is the language.

4.002 Man possesses the capacity of constructing languages, in which

every sense can be expressed, without having an idea how and

whateachwordmeans--justasonespeakswithoutknowinghow

the single sounds are produced.

Colloquial language is a part of the human organism and is

not less complicated than it.

From it it is humanly impossible to gather immediately the

logic of language.

Language disguises the thought; so that from the external

form of the clothes one cannot infer the form of the thought they

clothe, because the external form of the clothes is constructed

with quite another object than to let the form of the body be

recognized.

Thesilentadjustmentstounderstandcolloquiallanguageare

enormously complicated.

4.003 Most propositions and questions, that have been written about

philosophical matters, are not false, but senseless. We can-

not, therefore, answer questions 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.

(TheyareofthesamekindasthequestionwhethertheGood

is more or less identical than the Beautiful.)

And so it is not to be wondered at that the deepest problems

are really no problems.

4.0031 Allphilosophyis‘Critiqueoflanguage’ (butnotatallinMauth-

ner’s sense). Russell’s merit is to have shown that the apparent

logical form of the proposition need not be its real form.

4.01 The proposition is a picture of reality.

The proposition is a model of the reality as we think it is.

4.011 At the first glance the proposition--say as it stands printed on

paper--does not seem to be a picture of the reality of which it

treats. But nor does the musical score appear at first sight to

be a picture of a musical piece; nor does our phonetic spelling

(letters) seem to be a picture of our spoken language. And yet

these symbolisms prove to be pictures--even in the ordinary

sense of the word--of what they represent.

4.012 It is obvious that we perceive a proposition of the form aRb as

a picture. Here the sign is obviously a likeness of the signified.

4.013 And if we penetrate to the essence of this pictorial nature we

see that this is not disturbed by apparent irregularities (like the

use of ♯ and ♭ in the score).

For these irregularities also picture what they are to express;

only in another way.

4.014 The gramophone record, the musical thought, the score, the

wavesofsound, allstandtooneanotherinthatpictorialinternal

relation, which holds 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.)

4.0141 In the fact that there is a general rule by which the musician is

able to read the symphony out of the score, and that there is

a rule by which one could reconstruct the symphony from the

line on a gramophone record and from this again--by means

of the first rule--construct the score, herein lies the internal

similarity between these things which at first sight seem to be

entirely different. And the rule is the law of projection which

projects the symphony into the language of the musical score.

It is the rule of translation of this language into the language of

the gramophone record.

4.015 The possibility of all similes, of all the imagery of our language,

rests on the logic of representation.

4.016 In order to understand the essence of the proposition, consider

hieroglyphic writing, which pictures the facts it describes.

And from it came the alphabet without the essence of the

representation being lost.

4.02 This we see from the fact that we understand the sense of the

propositional sign, without having had it explained to us.

4.021 The proposition is a picture of reality, for I know the state of

affairs presented by it, if I understand the proposition. And I

understand the proposition, without its sense having been ex-

plained to me.

4.022 The proposition shows its sense.

The proposition shows how things stand, if it is true. And

it says, that they do so stand.

4.023 The proposition determines reality to this extent, that one only

needs to say ‘Yes’ or ‘No’ to it to make it agree with reality.

Itmustthereforebecompletelydescribedbytheproposition.

A proposition is the description of a fact.

As the description of an object describes it by its external

properties so propositions describe reality by its internal prop-

erties.

The proposition constructs a world with the help of a logical

scaffolding, and therefore one can actually see in the proposition

all the logical features possessed by reality if it is true. One can

draw conclusions from a false proposition.

4.024 To understand a proposition means to know what is the case, if

it is true.

(One can therefore understand it without knowing whether

it is true or not.)

One understands it if one understands its constituent parts.

4.025 The translation of one language into another is not a process

of translating each proposition of the one into a proposition of

the other, but only the constituent parts of propositions are

translated.

(And the dictionary does not only translate substantives but

also adverbs and conjunctions, etc., and it treats them all alike.)

4.026 The meanings of the simple signs (the words) must be explained

to us, if we are to understand them.

By means of propositions we explain ourselves.

4.027 It is essential to propositions, that they can communicate a new

sense to us.

4.03 A proposition must communicate a new sense with old words.

The proposition communicates to us a state of affairs, there-

fore it must be essentially connected with the state of affairs.

And the connexion is, in fact, that it is its logical picture.

The proposition only asserts something, in so far as it is a

picture.

4.031 In the proposition a state of affairs is, as it were, put together

for the sake of experiment.

One can say, instead of, This proposition has such and such

a sense, This proposition represents such and such a state of

affairs.

4.0311 One name stands for one thing, and another for another thing,

and they are connected together. And so the whole, like a living

picture, presents the atomic fact.

4.0312 The possibility of propositions is based upon the principle of the

representation of objects by signs.

My fundamental thought is that the ‘logical constants’ do

not represent. That the logic of the facts cannot be represented.

4.032 The proposition is a picture of its state of affairs, only in so far

as it is logically articulated.

(Even the proposition ‘ambulo’ is composite, for its stem

gives a different sense with another termination, or its termina-

tion with another stem.)

4.04 In the proposition there must be exactly as many things distin-

guishable as there are in the state of affairs, which it represents.

They must both possess the same logical (mathematical)

multiplicity (cf. Hertz’s Mechanics, on Dynamic Models).

4.041 This mathematical multiplicity naturally cannot in its turn be

represented. One cannot get outside it in the representation.

4.0411 Ifwetried, forexample, toexpresswhatisexpressedby‘(x).fx’

by putting an index before fx, like: ‘Gen. fx’, it would not do,

we should not know what was generalized. If we tried to show

it by an index g, like: ‘f(x )’ it would not do--we should not

g

know the scope of the generalization.

If we were to try it by introducing a mark in the argument

places, like ‘(G,G) . F(G,G)’, it would not do--we could not

determine the identity of the variables, etc.

All these ways of symbolizing are inadequate because they

have not the necessary mathematical multiplicity.

4.0412 For the same reason the idealist explanation of the seeing of spa-

tial relations through ‘spatial spectacles’ does not do, because

it cannot explain the multiplicity of these relations.

4.05 Reality is compared with the proposition.

4.06 Propositions can be true or false only by being pictures of the

reality.

4.061 If one does not observe that propositions have a sense inde-

pendent of the facts, one can easily believe that true and false

are two relations between signs and things signified with equal

rights.

One could then, for example, say that ‘p’ signifies in the true

‘∼p’

way what signifies in the false way, etc.

4.062 Can we not make ourselves understood by means of false propo-

sitions as hitherto with true ones, so long as we know that they

are meant to be false? No! For a proposition is true, if what

we assert by means of it is the case; and if by ‘p’ we mean ∼p,

and what we mean is the case, then ‘p’ in the new conception

is true and not false.

4.0621 That, however, the signs ‘p’ and ‘∼p’ can say the same thing is

∼

important, for it shows that the sign ‘ ’ corresponds to nothing

in reality.

That negation occurs in a proposition, is no characteristic of

its sense (∼∼p = p).

The propositions ‘p’ and ‘∼p’ have opposite senses, but to

them corresponds one and the same reality.

4.063 An illustration to explain the concept of truth. A black spot on

white paper; the form of the spot can be described by saying

of each point of the plane whether it is white or black. To the

fact that a point is black corresponds a positive fact; to the fact

that a point is white (not black), a negative fact. If I indicate

a point of the plane (a truth-value in Frege’s terminology), this

corresponds to the assumption proposed for judgment, etc. etc.

But to be able to say that a point is black or white, I must

firstknowunderwhatconditionsapointiscalledwhiteorblack;

in order to be able to say ‘p’ is true (or false) I must have

determined under what conditions I call ‘p’ true, and thereby I

determine the sense of the proposition.

The point at which the simile breaks down is this: we can

indicate a point on the paper, without knowing what white and

blackare; buttoapropositionwithoutasensecorrespondsnoth-

ing at all, for it signifies no thing (truth-value) whose properties

are called ‘false’ or ‘true’; the verb of the proposition is not ‘is

true’ or ‘is false’--as Frege thought--but that which ‘is true’

must already contain the verb.

4.064 Every proposition must already have a sense; assertion cannot

give it a sense, for what it asserts is the sense itself. And the

same holds of denial, etc.

4.0641 One could say, the denial is already related to the logical place

determined by the proposition that is denied.

The denying proposition determines a logical place other

than does the proposition denied.

The denying proposition determines a logical place, with the

help of the logical place of the proposition denied, by saying that

it lies outside the latter place.

That one can deny again the denied proposition, shows that

what is denied is already a proposition and not merely the pre-

liminary to a proposition.

4.1 Apropositionpresentstheexistenceandnon-existenceofatomic

facts.

4.11 The totality of true propositions is the total natural science (or

the totality of the natural sciences).

4.111 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.)

4.112 The object of philosophy is the logical clarification 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.

4.1121 Psychology is no nearer related to philosophy, than is any other

natural science.

The theory of knowledge is the philosophy of psychology.

Does not my study of sign-language correspond to the study

of thought processes which philosophers held to be so essential

to the philosophy of logic? Only they got entangled for the most

part in unessential psychological investigations, and there is an

analogous danger for my method.

4.1122 The Darwinian theory has no more to do with philosophy than

has any other hypothesis of natural science.

4.113 Philosophy limits the disputable sphere of natural science.

4.114 It should limit the thinkable and thereby the unthinkable.

It should limit the unthinkable from within through the

thinkable.

4.115 Itwillmeantheunspeakablebyclearlydisplayingthespeakable.

4.116 Everything that can be thought at all can be thought clearly.

Everything that can be said can be said clearly.

4.12 Propositions can represent the whole reality, but they cannot

represent what they must have in common with reality in order

to be able to represent it--the logical form.

To be able to represent the logical form, we should have to

be able to put ourselves with the propositions outside logic, that

is outside the world.

4.121 Propositionscannotrepresentthelogicalform: thismirrorsitself

in the propositions.

That which mirrors itself in language, language cannot rep-

resent.

That which expresses itself in language, we cannot express

by language.

The propositions show the logical form of reality.

They exhibit it.

4.1211 Thus a proposition ‘fa’ shows that in its sense the object a

occurs, two propositions ‘fa’ and ‘ga’ that they are both about

the same object.

If two propositions contradict one another, this is shown by

their structure; similarly if one follows from another, etc.

4.1212 What can be shown cannot be said.

4.1213 Now we understand our feeling that we are in possession of the

right logical conception, if only all is right in our symbolism.

4.122 We can speak in a certain sense of formal properties of objects

and atomic facts, or of properties of the structure of facts, and

in the same sense of formal relations and relations of structures.

(InsteadofpropertyofthestructureIalsosay‘internalprop-

erty’; instead of relation of structures ‘internal relation’.

I introduce these expressions in order to show the reason

for the confusion, very widespread among philosophers, between

internal relations and proper (external) relations.)

The holding of such internal properties and relations cannot,

however, be asserted by propositions, but it shows itself in the

propositions, which present the atomic facts and treat of the

objects in question.

4.1221 An internal property of a fact we also call a feature of this fact.

(In the sense in which we speak of facial features.)

4.123 A property is internal if it is unthinkable that its object does

not possess it.

(This blue colour and that stand in the internal relation of

brighter and darker eo ipso. It is unthinkable that these two

objects should not stand in this relation.)

(Here to the shifting use of the words ‘property’ and ‘rela-

tion’ there corresponds the shifting use of the word ‘object’.)

4.124 Theexistenceofaninternalpropertyofapossiblestateofaffairs

is not expressed by a proposition, but it expresses itself in the

proposition which presents that state of affairs, by an internal

property of this proposition.

It would be as senseless to ascribe a formal property to a

proposition as to deny it the formal property.

4.1241 One cannot distinguish forms from one another by saying that

one has this property but the other that: for this assumes that

there is a sense in asserting either property of either form.

4.125 The existence of an internal relation between possible states of

affairs expresses itself in language by an internal relation be-

tween the propositions presenting them.

4.1251 Here the disputed question ‘whether all relations are internal or

external’ disappears.

4.1252 Serieswhichareorderedbyinternal relationsIcallformalseries.

The series of numbers is ordered not by an external, but by

an internal relation.

Similarly the series of propositions ‘aRb’,

“(∃x) : aRx.xRb’,

“(∃x,y) : aRx.aRy .yRb’, etc.

(If b stands in one of these relations to a, I call b a successor

of a.)

4.126 In the sense in which we speak of formal properties we can now

speak also of formal concepts.

(I introduce this expression in order to make clear the confu-

sionofformalconceptswithproperconceptswhichrunsthrough

the whole of the old logic.)

That anything falls under a formal concept as an object be-

longingtoit, cannotbeexpressedbyaproposition. Butitshows

itself in the sign of this object itself. (The name shows that it

signifies an object, the numerical sign that it signifies a number,

etc.)

Formal concepts cannot, like proper concepts, be presented

by a function.

For their characteristics, the formal properties, are not ex-

pressed by the functions.

The expression of a formal property is a feature of certain

symbols.

The sign that signifies the characteristics of a formal con-

cept is, therefore, a characteristic feature of all symbols, whose

meanings fall under the concept.

The expression of the formal concept is therefore a proposi-

tional variable in which only this characteristic feature is con-

stant.

4.127 The propositional variable signifies the formal concept, and its

values signify the objects which fall under this concept.

4.1271 Every variable is the sign of a formal concept.

For every variable presents a constant form, which all its

values possess, and which can be conceived as a formal property

of these values.

4.1272 Sothevariablename‘x’ isthepropersignofthepseudo-concept

object.

Wherever the word ‘object’ (‘thing’, ‘entity’, etc.) is rightly

used, it is expressed in logical symbolism by the variable name.

For example in the proposition ‘there are two objects which

...’, by ‘(∃x,y) ...’.

Wherever it is used otherwise, i.e. as a proper concept word,

there arise senseless pseudo-propositions.

Soonecannot, e.g.say‘Thereareobjects’ asonesays‘There

are books’. Nor ‘There are 100 objects’ or ‘There are ℵ ob-

jects’. And it is senseless to speak of the number of all objects.

The same holds of the words ‘Complex’, ‘Fact’, ‘Function’,

‘Number’, etc.

They all signify formal concepts and are presented in logical

symbolism by variables, not by functions or classes (as Frege

and Russell thought).

Expressions like ‘1 is a number’, ‘there is only one number

nought’, and all like them are senseless.

(It is as senseless to say, ‘there is only one 1’ as it would be

to say: 2 + 2 is at 3 o’clock equal to 4.)

4.12721 The formal concept is already given with an object, which falls

under it. One cannot, therefore, introduce both, the objects

which fall under a formal concept and the formal concept it-

self, as primitive ideas. One cannot, therefore, e.g. introduce

(as Russell does) the concept of function and also special func-

tions as primitive ideas; or the concept of number and definite

numbers.

4.1273 If we want to express in logical symbolism the general propo-

sition ‘b is a successor of a’ we need for this an expression for

the general term of the formal series: aRb, (∃x) : aRx . xRb,

(∃x,y) : aRx.xRy.yRb, ... The general term of a formal series

can only be expressed by a variable, for the concept symbolized

by ‘term of this formal series’ is a formal concept. (This Frege

and Russell overlooked; the way in which they express general

propositions like the above is, therefore, false; it contains a vi-

cious circle.)

We can determine the general term of the formal series by

giving its first term and the general form of the operation, which

generates the following term out of the preceding proposition.

4.1274 Thequestionabouttheexistenceofaformalconceptissenseless.

For no proposition can answer such a question.

(For example, one cannot ask: ‘Are there unanalysable sub-

ject-predicate propositions?’)

4.128 The logical forms are anumerical.

Therefore there are in logic no pre-eminent numbers, and

therefore there is no philosophical monism or dualism, etc.

4.2 The sense of a proposition is its agreement and disagreement

with the possibilities of the existence and non-existence of the

atomic facts.

4.21 The simplest proposition, the elementary proposition, asserts

the existence of an atomic fact.

4.211 It is a sign of an elementary proposition, that no elementary

proposition can contradict it.

4.22 The elementary proposition consists of names. It is a connexion,

a concatenation, of names.

4.221 It is obvious that in the analysis of propositions we must come

toelementarypropositions, whichconsistofnamesinimmediate

combination.

The question arises here, how the propositional connexion

comes to be.

4.2211 Eveniftheworldisinfinitelycomplex, sothateveryfactconsists

of an infinite number of atomic facts and every atomic fact is

composed of an infinite number of objects, even then there must

be objects and atomic facts.

4.23 The name occurs in the proposition only in the context of the

elementary proposition.

4.24 The names are the simple symbols, I indicate them by single

letters (x, y, z).

The elementary proposition I write as function of the names,

in the form ‘fx’, ‘ϕ(x,y)’, etc.

Or I indicate it by the letters p, q, r.

4.241 If I use two signs with one and the same meaning, I express this

by putting between them the sign ‘=’.

‘a = b’ means then, that the sign ‘a’ is replaceable by the

sign ‘b’.

(If I introduce by an equation a new sign ‘b’, by determining

that it shall replace a previously known sign ‘a’, I write the

equation--definition--(like Russell) in the form ‘a = b Def.’. A

definition is a symbolic rule.)

4.242 Expressions of the form ‘a = b’ are therefore only expedients

in presentation: They assert nothing about the meaning of the

signs ‘a’ and ‘b’.

4.243 Can we understand two names without knowing whether they

signify the same thing or two different things? Can we under-

stand a proposition in which two names occur, without knowing

if they mean the same or different things?

If I know the meaning of an English and a synonymous Ger-

man word, it is impossible for me not to know that they are

synonymous, it is impossible for me not to be able to translate

them into one another.

Expressions like ‘a = a’, or expressions deduced from these

are neither elementary propositions nor otherwise significant

signs. (This will be shown later.)

4.25 If the elementary proposition is true, the atomic fact exists; if it

is false the atomic fact does not exist.

4.26 The specification of all true elementary propositions describes

the world completely. The world is completely described by the

specificationofallelementarypropositionsplusthespecification,

which of them are true and which false.

4.27 With regard to the existence of n atomic facts there are K =

n

n

(cid:80) (cid:0)n(cid:1)

possibilities.

ν

ν=0

It is possible for all combinations of atomic facts to exist,

and the others not to exist.

4.28 To these combinations correspond the same number of possibili-

ties of the truth--and falsehood--of n elementary propositions.

4.3 The truth-possibilities of the elementary propositions mean the

possibilities of the existence and non-existence of the atomic

facts.

4.31 The truth-possibilities can be presented by schemata of the fol-

lowing kind (‘T’ means ‘true’, ‘F’ ‘false’. The rows of T’s and

F’s under the row of the elementary propositions mean their

truth-possibilities in an easily intelligible symbolism).

p q r p q p

T T T T T T

F T T F T F

T F T T F

T T F F F

F F T

F T F

T F F

F F F

4.4 A proposition is the expression of agreement and disagreement

with the truth-possibilities of the elementary propositions.

4.41 The truth-possibilities of the elementary propositions are the

conditions of the truth and falsehood of the propositions.

4.411 It seems probable even at first sight that the introduction of the

elementarypropositionsisfundamentalforthecomprehensionof

theotherkindsofpropositions. Indeedthecomprehensionofthe

general propositions depends palpably on that of the elementary

propositions.

4.42 With regard to the agreement and disagreement of a proposition

with the truth-possibilities of n elementary propositions there

(cid:80)Kn

(cid:0)Kn(cid:1)

are = L possibilities.

κ n

κ=0

4.43 Agreement with the truth-possibilities can be expressed by co-

ordinating with them in the schema the mark ‘T’ (true).

Absence of this mark means disagreement.

4.431 The expression of the agreement and disagreement with the

truth-possibilities of the elementary propositions expresses the

truth-conditions of the proposition.

The proposition is the expression of its truth-conditions.

(Frege has therefore quite rightly put them at the beginning,

as explaining the signs of his logical symbolism. Only Frege’s

explanation of the truth-concept is false: if ‘the true’ and ‘the

∼p,

false’ were real objects and the arguments in etc., then

∼p

the sense of would by no means be determined by Frege’s

determination.)

4.44 The sign which arises from the co-ordination of that mark ‘T’

with the truth-possibilities is a propositional sign.

4.441 It is clear that to the complex of the signs ‘F’ and ‘T’ no object

(orcomplexofobjects)corresponds; anymorethantohorizontal

and vertical lines or to brackets. There are no ‘logical objects’.

Something analogous holds of course for all signs, which ex-

press the same as the schemata of ‘T’ and ‘F’.

4.442 Thus e.g.

‘ p q

T T T

F T T

T F

F F T ’

is a propositional sign.

(Frege’sassertionsign‘⊢’ islogicallyaltogethermeaningless;

in Frege (and Russell) it only shows that these authors hold as

true the propositions marked in this way.

‘⊢’ belongs therefore to the propositions no more than does

the number of the proposition. A proposition cannot possibly

assert of itself that it is true.)

Ifthesequenceofthetruth-possibilitiesintheschemaisonce

for all determined by a rule of combination, then the last column

is by itself an expression of the truth-conditions. If we write this

columnasarowthepropositionalsignbecomes: ‘(TT-T)(p, q)’,

or more plainly: ‘(TTFT)(p, q)’.

(The number of places in the left-hand bracket is determined

by the number of terms in the right-hand bracket.)

4.45 For n elementary propositions there are L possible groups of

n

truth-conditions.

The groups of truth-conditions which belong to the truth-

possibilities of a number of elementary propositions can be or-

dered in a series.

4.46 Among the possible groups of truth-conditions there are two

extreme cases.

In the one case the proposition is true for all the truth-pos-

sibilities of the elementary propositions. We say that the truth-

conditions are tautological.

In the second case the proposition is false for all the truth-

possibilities. The truth-conditions are self-contradictory.

In the first case we call the proposition a tautology, in the

second case a contradiction.

4.461 The proposition shows what it says, the tautology and the con-

tradiction that they say nothing.

The tautology has no truth-conditions, for it is uncondition-

ally true; and the contradiction is on no condition true.

Tautology and contradiction are without sense.

(Like the point from which two arrows go out in opposite

directions.)

(I know, e.g. nothing about the weather, when I know that

it rains or does not rain.)

4.4611 Tautology and contradiction are, however, not senseless; they

are part of the symbolism, in the same way that ‘0’ is part of

the symbolism of Arithmetic.

4.462 Tautologyandcontradictionarenotpicturesofthereality. They

present no possible state of affairs. For the one allows every

possible state of affairs, the other none.

In the tautology the conditions of agreement with the world

--the presenting relations--cancel one another, so that it stands

in no presenting relation to reality.

4.463 The truth-conditions determine the range, which is left to the

facts by the proposition.

(The proposition, the picture, the model, are in a negative

sense like a solid body, which restricts the free movement of

another: in a positive sense, like the space limited by solid sub-

stance, in which a body may be placed.)

Tautology leaves to reality the whole infinite logical space;

contradiction fills the whole logical space and leaves no point to

reality. Neither of them, therefore, can in any way determine

reality.

4.464 The truth of tautology is certain, of propositions possible, of

contradiction impossible. (Certain, possible, impossible: here

we have an indication of that gradation which we need in the

theory of probability.)

4.465 The logical product of a tautology and a proposition says the

same as the proposition. Therefore that product is identical

with the proposition. For the essence of the symbol cannot be

altered without altering its sense.

4.466 To a definite logical combination of signs corresponds a definite

logical combination of their meanings; every arbitrary combina-

tion only corresponds to the unconnected signs.

That is, propositions which are true for every state of affairs

cannot be combinations of signs at all, for otherwise there could

only correspond to them definite combinations of objects.

(And to no logical combination corresponds no combination

of the objects.)

Tautology and contradiction are the limiting cases of the

combinations of symbols, namely their dissolution.

4.4661 Of course the signs are also combined with one another in the

tautology and contradiction, i.e. they stand in relations to one

another, but these relations are meaningless, unessential to the

symbol.

4.5 Now it appears to be possible to give the most general form of

proposition; i.e. to give a description of the propositions of some

one sign language, so that every possible sense can be expressed

by a symbol, which falls under the description, and so that every

symbol which falls under the description can express a sense, if

the meanings of the names are chosen accordingly.

It is clear that in the description of the most general form

of proposition only what is essential to it may be described--

otherwise it would not be the most general form.

That there is a general form is proved by the fact that there

cannotbeapropositionwhoseformcouldnothavebeenforeseen

(i.e. constructed). The general form of proposition is: Such and

such is the case.

4.51 Suppose all elementary propositions were given me: then we

can simply ask: what propositions I can build out of them. And

these are all propositions and so are they limited.

4.52 The propositions are everything which follows from the totality

of all elementary propositions (of course also from the fact that

it is the totality of them all). (So, in some sense, one could

say, that all propositions are generalizations of the elementary

propositions.)

4.53 The general propositional form is a variable.

5 Propositions are truth-functions of elementary propositions.

(An elementary proposition is a truth-function of itself.)

5.01 The elementary propositions are the truth-arguments of propo-

sitions.

5.02 It is natural to confuse the arguments of functions with the

indices of names. For I recognize the meaning of the sign con-

taining it from the argument just as much as from the index.

In Russell’s ‘+ ’, for example, ‘c’ is an index which indicates

c

that the whole sign is the addition sign for cardinal numbers.

But this way of symbolizing depends on arbitrary agreement,

and one could choose a simple sign instead of ‘+ ’: but in ‘∼p’

c

‘p’ is not an index but an argument; the sense of ‘∼p’ cannot

be understood, unless the sense of ‘p’ has previously been un-

derstood. (In the name Julius Caesar, Julius is an index. The

index is always part of a description of the object to whose name

we attach it, e.g. The Caesar of the Julian gens.)

Theconfusionofargumentandindexis, ifIamnotmistaken,

at the root of Frege’s theory of the meaning of propositions and

functions. For Frege the propositions of logic were names and

their arguments the indices of these names.

5.1 The truth-functions can be ordered in series.

That is the foundation of the theory of probability.

5.101 The truth-functions of every number of elementary propositions

can be written in a schema of the following kind:

(TTTT)(p,q) Tautology (if p then p, and if q then q) [p⊃p.q ⊃q]

(FTTT)(p,q) in words: Not both p and q. [∼(p.q)]

(TFTT)(p,q) ’’ ’’ If q then p. [q ⊃p]

(TTFT)(p,q) ’’ ’’ If p then q. [p⊃q]

(TTTF)(p,q) ’’ ’’ p or q. [p∨q]

(FFTT)(p,q) ’’ ’’ Not q. [∼q]

(FTFT)(p,q) ’’ ’’ Not p. [∼p]

(FTTF)(p,q) ’’ ’’ p or q, but not both. [p.∼q :∨:q.∼p]

(TFFT)(p,q) ’’ ’’ If p, then q; and if q, then p. [p≡q]

(TFTF)(p,q) ’’ ’’ p

(TTFF)(p,q) ’’ ’’ q

(FFFT)(p,q) ’’ ’’ Neither p nor q. [∼p.∼q or p|q]

(FFTF)(p,q) ’’ ’’ p and not q. [p.∼q]

(FTFF)(p,q) ’’ ’’ q and not p. [q.∼p]

(TFFF)(p,q) ’’ ’’ p and q. [p.q]

(FFFF)(p,q) Contradiction (p and not p; and q and not q.) [p.∼p.q.∼q]

Those truth-possibilities of its truth-arguments, which verify

the proposition, I shall call its truth-grounds.

5.11 If the truth-grounds which are common to a number of proposi-

tions are all also truth-grounds of some one proposition, we say

that the truth of this proposition follows from the truth of those

propositions.

5.12 In particular the truth of a proposition p follows from that of a

proposition q, if all the truth-grounds of the second are truth-

grounds of the first.

5.121 The truth-grounds of q are contained in those of p; p follows

from q.

5.122 If p follows from q, the sense of ‘p’ is contained in that of ‘q’.

5.123 Ifagodcreatesaworldinwhichcertainpropositionsaretrue, he

creates thereby also a world in which all propositions consequent

on them are true. And similarly he could not create a world in

which the proposition ‘p’ is true without creating all its objects.

5.124 A proposition asserts every proposition which follows from it.

5.1241 ‘p.q’ is one of the propositions which assert ‘p’ and at the same

time one of the propositions which assert ‘q’.

Two propositions are opposed to one another if there is no

significant proposition which asserts them both.

Every proposition which contradicts another, denies it.

5.13 That the truth of one proposition follows from the truth of other

propositions, we perceive from the structure of the propositions.

5.131 If the truth of one proposition follows from the truth of others,

this expresses itself in relations in which the forms of these prop-

ositions stand to one another, and we do not need to put them

in these relations first by connecting them with one another in

a proposition; for these relations are internal, and exist as soon

as, and by the very fact that, the propositions exist.

5.1311 When we conclude from p∨q and ∼p to q the relation between

the forms of the propositions ‘p∨q’ and ‘∼p’ is here concealed

by the method of symbolizing. But if we write, e.g. instead of

‘p∨q’ ‘p|q.|.p|q’ and instead of ‘∼p’ ‘p|p’ (p|q = neither

p nor q), then the inner connexion becomes obvious.

(The fact that we can infer fa from (x).fx shows that gen-

erality is present also in the symbol ‘(x).fx’.

5.132 If p follows from q, I can conclude from q to p; infer p from q.

The method of inference is to be understood from the two

propositions alone.

Only they themselves can justify the inference.

Laws of inference, which--as in Frege and Russell--are to

justify the conclusions, are senseless and would be superfluous.

5.133 All inference takes place a priori.

5.134 From an elementary proposition no other can be inferred.

5.135 In no way can an inference be made from the existence of one

state of affairs to the existence of another entirely different from

it.

5.136 There is no causal nexus which justifies such an inference.

5.1361 The events of the future cannot be inferred from those of the

present.

Superstition is the belief in the causal nexus.

5.1362 The freedom of the will consists in the fact that future actions

cannot be known now. We could only know them if causality

were an inner necessity, like that of logical deduction.--The

connexion of knowledge and what is known is that of logical

necessity.

(‘A knows that p is the case’ is senseless if p is a tautology.)

5.1363 If from the fact that a proposition is obvious to us it does not

follow that it is true, then obviousness is no justification for our

belief in its truth.

5.14 If a proposition follows from another, then the latter says more

than the former, the former less than the latter.

5.141 If p follows from q and q from p then they are one and the same

proposition.

5.142 A tautology follows from all propositions: it says nothing.

5.143 Contradiction is something shared by propositions, which no

proposition has in common with another. Tautology is that

which is shared by all propositions, which have nothing in com-

mon with one another.

Contradiction vanishes so to speak outside, tautology inside

all propositions.

Contradiction is the external limit of the propositions, tau-

tology their substanceless centre.

5.15 If T is the number of the truth-grounds of the proposition ‘r’,

r

T the number of those truth-grounds of the proposition ‘s’

rs

which are at the same time truth-grounds of ‘r’, then we call

the ratio T : T the measure of the probability which the prop-

rs r

osition ‘r’ gives to the proposition ‘s’.

5.151 Suppose in a schema like that above in No. 5.101 T is the num-

r

ber of the ‘T’’s in the proposition r, T the number of those

rs

‘T’’s in the proposition s, which stand in the same columns as

‘T’’s of the proposition r; then the proposition r gives to the

proposition s the probability T : T .

rs r

5.1511 There is no special object peculiar to probability propositions.

5.152 Propositions which have no truth-arguments in common with

one another we call independent.

Independent propositions (e.g. any two elementary proposi-

tions) give to one another the probability 1.

If p follows from q, the proposition q gives to the proposition

p the probability 1. The certainty of logical conclusion is a

limiting case of probability.

(Application to tautology and contradiction.)

5.153 A proposition is in itself neither probable nor improbable. An

event occurs or does not occur, there is no middle course.

5.154 In an urn there are equal numbers of white and black balls (and

no others). I draw one ball after another and put them back in

the urn. Then I can determine by the experiment that the num-

bers of the black and white balls which are drawn approximate

as the drawing continues.

So this is not a mathematical fact.

If then, I say, It is equally probable that I should draw a

white and a black ball, this means, All the circumstances known

to me (including the natural laws hypothetically assumed) give

to the occurrence of the one event no more probability than to

the occurrence of the other. That is they give--as can easily be

understood from the above explanations--to each the probabil-

ity 1.

WhatIcanverifybytheexperimentisthattheoccurrenceof

the two events is independent of the circumstances with which

I have no closer acquaintance.

5.155 The unit of the probability proposition is: The circumstances--

with which I am not further acquainted--give to the occurrence

of a definite event such and such a degree of probability.

5.156 Probability is a generalization.

Itinvolvesageneraldescriptionofapropositionalform. Only

in default of certainty do we need probability.

If we are not completely acquainted with a fact, but know

something about its form.

(A proposition can, indeed, be an incomplete picture of a

certain state of affairs, but it is always a complete picture.)

The probability proposition is, as it were, an extract from

other propositions.

5.2 The structures of propositions stand to one another in internal

relations.

5.21 We can bring out these internal relations in our manner of ex-

pression, by presenting a proposition as the result of an opera-

tion which produces it from other propositions (the bases of the

operation).

5.22 The operation is the expression of a relation between the struc-

tures of its result and its bases.

5.23 The operation is that which must happen to a proposition in

order to make another out of it.

5.231 And that will naturally depend on their formal properties, on

the internal similarity of their forms.

5.232 The internal relation which orders a series is equivalent to the

operation by which one term arises from another.

5.233 The first place in which an operation can occur is where a prop-

osition arises from another in a logically significant way; i.e.

where the logical construction of the proposition begins.

5.234 The truth-functions of elementary propositions, are results of

operations which have the elementary propositions as bases. (I

call these operations, truth-operations.)

5.2341 The sense of a truth-function of p is a function of the sense of p.

Denial, logical addition, logical multiplication, etc. etc., are

operations.

(Denial reverses the sense of a proposition.)

5.24 An operation shows itself in a variable; it shows how we can

proceed from one form of proposition to another.

It gives expression to the difference between the forms.

(And that which is common to the bases, and the result of

an operation, is the bases themselves.)

5.241 The operation does not characterize a form but only the differ-

ence between forms.

5.242 The same operation which makes ‘q’ from ‘p’, makes ‘r’ from

‘q’, and so on. This can only be expressed by the fact that

‘p’, ‘q’, ‘r’, etc., are variables which give general expression to

certain formal relations.

5.25 The occurrence of an operation does not characterize the sense

of a proposition.

For an operation does not assert anything; only its result

does, and this depends on the bases of the operation.

(Operation and function must not be confused with one an-

other.)

5.251 A function cannot be its own argument, but the result of an

operation can be its own basis.

5.252 Only in this way is the progress from term to term in a formal

series possible (from type to type in the hierarchy of Russell

and Whitehead). (Russell and Whitehead have not admitted

the possibility of this progress but have made use of it all the

same.)

5.2521 The repeated application of an operation to its own result I call

itssuccessiveapplication(‘O′O′O′a’ istheresultofthethreefold

successive application of ‘O′ξ’ to ‘a’).

In a similar sense I speak of the successive application of

several operations to a number of propositions.

5.2522 The general term of the formal series a,O′a,O′O′a, .... I write

thus: ‘[a, x, O′x]’. This expression in brackets is a variable. The

first term of the expression is the beginning of the formal series,

the second the form of an arbitrary term x of the series, and

the third the form of that term of the series which immediately

follows x.

5.2523 The concept of the successive application of an operation is

equivalent to the concept ‘and so on’.

5.253 One operation can reverse the effect of another. Operations can

cancel one another.

5.254 Operations can vanish (e.g. denial in ‘∼∼p’, ∼∼p = p).

5.3 All propositions are results of truth-operations on the elemen-

tary propositions.

The truth-operation is the way in which a truth-function

arises from elementary propositions.

According to the nature of truth-operations, in the same way

as out of elementary propositions arise their truth-functions,

from truth-functions arises a new one. Every truth-operation

creates from truth-functions of elementary propositions another

truth-function of elementary propositions, i.e. a proposition.

The result of every truth-operation on the results of truth-oper-

ations on elementary propositions is also the result of one truth-

operation on elementary propositions.

Every proposition is the result of truth-operations on ele-

mentary propositions.

5.31 The Schemata No. 4.31 are also significant, if ‘p’, ‘q’, ‘r’, etc.

are not elementary propositions.

And it is easy to see that the propositional sign in No. 4.442

expresses one truth-function of elementary propositions even

when ‘p’ and ‘q’ are truth-functions of elementary propositions.

5.32 All truth-functions are results of the successive application of a

finite number of truth-operations to elementary propositions.

5.4 Here it becomes clear that there are no such things as ‘logical

objects’ or‘logicalconstants’ (inthesenseofFregeandRussell).

5.41 For all those results of truth-operations on truth-functions are

identical, which are one and the same truth-function of elemen-

tary propositions.

5.42 That ∨, ⊃, etc., are not relations in the sense of right and left,

etc., is obvious.

Thepossibilityofcrosswisedefinitionofthelogical‘primitive

signs’ of Frege and Russell shows by itself that these are not

primitive signs and that they signify no relations.

And it is obvious that the ‘⊃’ which we define by means of

‘∼ ’ and ‘∨’ is identical with that by which we define ‘∨’ with

the help of ‘∼ ’, and that this ‘∨’ is the same as the first, and

so on.

5.43 That from a fact p an infinite number of others should follow,

∼∼p, ∼∼∼∼p,

namely etc., is indeed hardly to be believed, and

it is no less wonderful that the infinite number of propositions of

logic(ofmathematics)shouldfollowfromhalfadozen‘primitive

propositions’.

But all propositions of logic say the same thing. That is,

nothing.

5.44 Truth-functions are not material functions.

If e.g. an affirmation can be produced by repeated denial, is

the denial--in any sense--contained in the affirmation?

Does ‘∼∼p’ deny ∼p, or does it affirm p; or both?

‘∼∼p’

The proposition does not treat of denial as an object,

but the possibility of denial is already prejudged in affirmation.

‘∼ ‘∼∼p’

And if there was an object called ’, then would

have to say something other than ‘p’. For the one proposition

∼

would then treat of , the other would not.

5.441 This disappearance of the apparent logical constants also occurs

if ‘∼(∃x).∼fx’ says the same as ‘(x).fx’, or ‘(∃x).fx.x = a’

the same as ‘fa’.

5.442 If a proposition is given to us then the results of all truth-

operations which have it as their basis are given with it.

5.45 Iftherearelogicalprimitivesignsacorrectlogicmustmakeclear

their position relative to one another and justify their existence.

The construction of logic out of its primitive signs must become

clear.

5.451 If logic has primitive ideas these must be independent of one an-

other. If a primitive idea is introduced it must be introduced in

all contexts in which it occurs at all. One cannot therefore intro-

duce it for one context and then again for another. For example,

if denial is introduced, we must understand it in propositions of

theform‘∼p’, justasinpropositionslike‘∼(p∨q)’, ‘(∃x).∼fx’

and others. We may not first introduce it for one class of cases

and then for another, for it would then remain doubtful whether

its meaning in the two cases was the same, and there would be

no reason to use the same way of symbolizing in the two cases.

(In short, what Frege (‘Grundgesetze der Arithmetik’) has

saidaboutthe introductionofsignsbydefinitionsholds, mutatis

mutandis, for the introduction of primitive signs also.)

5.452 The introduction of a new expedient in the symbolism of logic

must always be an event full of consequences. No new symbol

may be introduced in logic in brackets or in the margin--with,

so to speak, an entirely innocent face.

(Thus in the ‘Principia Mathematica’ of Russell and White-

head thereoccurdefinitions and primitive propositions in words.

Why suddenly words here? This would need a justification.

There was none, and can be none for the process is actually

not allowed.)

But if the introduction of a new expedient has proved nec-

essary in one place, we must immediately ask: Where is this

expedient always to be used? Its position in logic must be made

clear.

5.453 All numbers in logic must be capable of justification.

Or rather it must become plain that there are no numbers

in logic.

There are no pre-eminent numbers.

5.454 In logic there is no side by side, there can be no classification.

In logic there cannot be a more general and a more special.

5.4541 The solution of logical problems must be simple for they set the

standard of simplicity.

Men have always thought that there must be a sphere of

questions whose answers--a priori--are symmetrical and united

into a closed regular structure.

A sphere in which the proposition, simplex sigillum veri, is

valid.

5.46 When we have rightly introduced the logical signs, the sense of

all their combinations has been already introduced with them:

therefore not only ‘p ∨ q’ but also ‘∼(p ∨ ∼q)’, etc. etc. We

should then already have introduced the effect of all possible

combinations of brackets; and it would then have become clear

thatthepropergeneralprimitive signsarenot‘p∨q’, ‘(∃x).fx’,

etc., but the most general form of their combinations.

5.461 The apparently unimportant fact that the apparent relations

like ∨ and ⊃ need brackets--unlike real relations is of great

importance.

Theuseofbracketswiththeseapparentprimitivesignsshows

that these are not the real primitive signs; and nobody of course

would believe that the brackets have meaning by themselves.

5.4611 Logical operation signs are punctuations.

5.47 It is clear that everything which can be said beforehand about

the form of all propositions at all can be said on one occasion.

Foralllogicaloperationsarealreadycontainedintheelemen-

tary proposition. For ‘fa’ says the same as ‘(∃x).fx.x = a’.

Where there is composition, there is argument and function,

and where these are, all logical constants already are.

One could say: the one logical constant is that which all

propositions, according to their nature, have in common with

one another.

That however is the general form of proposition.

5.471 The general form of proposition is the essence of proposition.

5.4711 To give the essence of proposition means to give the essence of

all description, therefore the essence of the world.

5.472 The description of the most general propositional form is the

description of the one and only general primitive sign in logic.

5.473 Logic must take care of itself.

A possible sign must also be able to signify. Everything

which is possible in logic is also permitted. (‘Socrates is identi-

cal’ means nothing because there is no property which is called

‘identical’. The proposition is senseless because we have not

made some arbitrary determination, not because the symbol is

in itself unpermissible.)

In a certain sense we cannot make mistakes in logic.

5.4731 Self-evidence, of which Russell has said so much, can only be

discardedinlogicbylanguageitselfpreventingeverylogicalmis-

take. That logic is a priori consists in the fact that we cannot

think illogically.

5.4732 We cannot give a sign the wrong sense.

5.47321 Occam’srazoris,ofcourse,notanarbitraryrulenoronejustified

byitspracticalsuccess. Itsimplysaysthatunnecessary elements

in a symbolism mean nothing.

Signs which serve one purpose are logically equivalent, signs

which serve no purpose are logically meaningless.

5.4733 Frege says: Every legitimately constructed proposition must

have a sense; and I say: Every possible proposition is legit-

imately constructed, and if it has no sense this can only be

because we have given no meaning to some of its constituent

parts.

(Even if we believe that we have done so.)

Thus ‘Socrates is identical’ says nothing, because we have

given no meaning to the word ‘identical’ as adjective. For when

it occurs as the sign of equality it symbolizes in an entirely dif-

ferent way--the symbolizing relation is another--therefore the

symbol is in the two cases entirely different; the two symbols

have the sign in common with one another only by accident.

5.474 The number of necessary fundamental operations depends only

on our notation.

5.475 Itisonlyaquestionofconstructingasystemofsignsofadefinite

number of dimensions--of a definite mathematical multiplicity.

5.476 It is clear that we are not concerned here with a number of

primitive ideas which must be signified but with the expression

of a rule.

5.5 Every truth-function is a result of the successive application of

the operation (- - - - -T)(ξ,....) to elementary propositions.

This operation denies all the propositions in the right-hand

bracket and I call it the negation of these propositions.

5.501 An expression in brackets whose terms are propositions I indi-

cate--if the order of the terms in the bracket is indifferent--by

a sign of the form ‘(ξ)’. ‘ξ’ is a variable whose values are the

termsoftheexpressioninbrackets, andthelineoverthevariable

indicates that it stands for all its values in the bracket.

(Thus if ξ has the 3 values P, Q, R, then (ξ) = (P, Q, R).)

The values of the variables must be determined.

The determination is the description of the propositions

which the variable stands for.

Howthedescriptionofthetermsoftheexpressioninbrackets

takes place is unessential.

We may distinguish 3 kinds of description: 1. Direct enu-

meration. In this case we can place simply its constant values

instead of the variable. 2. Giving a function fx, whose values for

all values of x are the propositions to be described. 3. Giving

a formal law, according to which those propositions are con-

structed. In this case the terms of the expression in brackets are

all the terms of a formal series.

5.502 Therefore I write instead of ‘(- - - - -T)(ξ,....)’, ‘N(ξ)’.

N(ξ) is the negation of all the values of the propositional

variable ξ.

5.503 As it is obviously easy to express how propositions can be con-

structed by means of this operation and how propositions are

not to be constructed by means of it, this must be capable of

exact expression.

5.51 If ξ has only one value, then N(ξ) = ∼p (not p), if it has two

values then N(ξ) = ∼p.∼q (neither p nor q).

5.511 How can the all-embracing logic which mirrors the world use

such special catches and manipulations? Only because all these

areconnectedintoaninfinitely finenetwork, tothegreatmirror.

5.512 ‘∼p’ is true if ‘p’ is false. Therefore in the true proposition

‘∼p’ ‘p’ is a false proposition. How then can the stroke ‘∼ ’

bring it into agreement with reality?

Thatwhichdeniesin‘∼p’ ishowevernot‘∼

’,butthatwhich

all signs of this notation, which deny p, have in common.

‘∼p’, ‘∼∼∼p’,

Hence the common rule according to which

‘∼p∨∼p’, ‘∼p.∼p’,

etc. etc. (to infinity) are constructed. And

this which is common to them all mirrors denial.

5.513 We could say: What is common to all symbols, which assert

both p and q, is the proposition ‘p.q’. What is common to all

symbols, which assert either p or q, is the proposition ‘p∨q’.

And similarly we can say: Two propositions are opposed

to one another when they have nothing in common with one

another; and every proposition has only one negative, because

there is only one proposition which lies altogether outside it.

Thus even in Russell’s notation it is evident that ‘q : p∨∼p’

says the same as ‘q’; that ‘p∨∼p’ says nothing.

5.514 If a notation is fixed, there is in it a rule according to which

all the propositions denying p are constructed, a rule according

to which all the propositions asserting p are constructed, a rule

according to which all the propositions asserting p or q are con-

structed, and so on. These rules are equivalent to the symbols

and in them their sense is mirrored.

5.515 It must be recognized in our symbols that what is connected by

‘∨’, ‘.’, etc., must be propositions.

And this is the case, for the symbols ‘p’ and ‘q’ presuppose

‘∨’, ‘∼ ’, etc. If the sign ‘p’ in ‘p ∨ q’ does not stand for a

complex sign, then by itself it cannot have sense; but then also

the signs ‘p∨p’, ‘p.p’, etc. which have the same sense as ‘p’

have no sense. If, however, ‘p∨p’ has no sense, then also ‘p∨q’

can have no sense.

5.5151 Must the sign of the negative proposition be constructed by

means of the sign of the positive? Why should one not be able

to express the negative proposition by means of a negative fact?

(Like: if ‘a’ does not stand in a certain relation to ‘b’, it could

express that aRb is not the case.)

But here also the negative proposition is indirectly con-

structed with the positive.

The positive proposition must presuppose the existence of

the negative proposition and conversely.

5.52 If the values of ξ are the total values of a function fx for all

values of x, then N(ξ) = ∼(∃x).fx.

5.521 I separate the concept all from the truth-function.

Frege and Russell have introduced generality in connexion

with the logical product or the logical sum. Then it would be

difficult to understand the propositions ‘(∃x).fx’ and ‘(x).fx’

in which both ideas lie concealed.

5.522 That which is peculiar to the ‘symbolism of generality’ is firstly,

that it refers to a logical prototype, and secondly, that it makes

constants prominent.

5.523 The generality symbol occurs as an argument.

5.524 If the objects are given, therewith are all objects also given.

If the elementary propositions are given, then therewith all

elementary propositions are also given.

5.525 It is not correct to render the proposition ‘(∃x).fx’--as Russell

does--in words ‘fx is possible’.

Certainty, possibility or impossibility of a state of affairs are

notexpressedbyapropositionbutbythefactthatanexpression

is a tautology, a significant proposition or a contradiction.

That precedent to which one would always appeal, must be

present in the symbol itself.

5.526 One can describe the world completely by completely general-

ized propositions, i.e. without from the outset co-ordinating any

name with a definite object.

In order then to arrive at the customary way of expression

we need simply say after an expression ‘there is one and only

one x, which ....’: and this x is a.

5.5261 A completely generalized proposition is like every other proposi-

tion composite. (This is shown by the fact that in ‘(∃x,ϕ).ϕx’

we must mention ‘ϕ’ and ‘x’ separately. Both stand indepen-

dentlyinsignifyingrelationstotheworldasintheungeneralized

proposition.)

A characteristic of a composite symbol: it has something in

common with other symbols.

5.5262 The truth or falsehood of every proposition alters something

in the general structure of the world. And the range which is

allowed to its structure by the totality of elementary proposi-

tions is exactly that which the completely general propositions

delimit.

(If an elementary proposition is true, then, at any rate, there

is one more elementary proposition true.)

5.53 Identity of the object I express by identity of the sign and not by

meansofasignofidentity. Differenceoftheobjectsbydifference

of the signs.

5.5301 That identity is not a relation between objects is obvious. This

becomes very clear if, for example, one considers the proposition

‘(x) : fx. ⊃ .x = a’. What this proposition says is simply that

only a satisfies the function f, and not that only such things

satisfy the function f which have a certain relation to a.

One could of course say that in fact only a has this relation

to a, but in order to express this we should need the sign of

identity itself.

5.5302 Russell’s definition of ‘=’ won’t do; because according to it one

cannot say that two objects have all their properties in com-

mon. (Even if this proposition is never true, it is nevertheless

significant.)

5.5303 Roughly speaking: to say of two things that they are identical is

nonsense, and to say of one thing that it is identical with itself

is to say nothing.

5.531 I write therefore not ‘f(a,b).a = b’, but ‘f(a,a)’ (or ‘f(b,b)’).

And not ‘f(a,b).∼a = b’, but ‘f(a,b)’.

5.532 Andanalogously: not‘(∃x,y).f(x,y).x = y’,but‘(∃x).f(x,x)’;

and not ‘(∃x,y).f(x,y).∼x = y’, but ‘(∃x,y).f(x,y)’.

(Therefore instead of Russell’s ‘(∃x,y) . f(x,y)’: ‘(∃x,y) .

f(x,y).∨.(∃x).f(x,x)’.)

5.5321 Insteadof‘(x) : fx ⊃ x = a’ wethereforewritee.g.‘(∃x).fx. ⊃

.fa : ∼(∃x,y).fx.fy’.

And the proposition ‘only one x satisfies f()’ reads: ‘(∃x).

fx : ∼(∃x,y).fx.fy’.

5.533 The identity sign is therefore not an essential constituent of log-

ical notation.

5.534 And we see that apparent propositions like: ‘a = a’, ‘a = b.b =

c. ⊃ a = c’, ‘(x).x = x’, ‘(∃x).x = a’, etc. cannot be written

in a correct logical notation at all.

5.535 So all problems disappear which are connected with such pseu-

do-propositions.

Thisistheplacetosolvealltheproblemswhicharisethrough

Russell’s ‘Axiom of Infinity’.

What the axiom of infinity is meant to say would be ex-

pressed in language by the fact that there is an infinite number

of names with different meanings.

5.5351 There are certain cases in which one is tempted to use expres-

sions of the form ‘a = a’ or ‘p ⊃ p’ and of that kind. And

indeed this takes place when one would like to speak of the

archetype Proposition, Thing, etc. So Russell in the Principles

of Mathematics hasrenderedthenonsense‘pisaproposition’ in

symbols by ‘p ⊃ p’ and has put it as hypothesis before certain

propositions to show that their places for arguments could only

be occupied by propositions.

(It is nonsense to place the hypothesis p ⊃ p before a propo-

sition in order to ensure that its arguments have the right form,

because the hypothesis for a non-proposition as argument be-

comes not false but meaningless, and because the proposition

itself becomes senseless for arguments of the wrong kind, and

thereforeitsurvivesthewrongargumentsnobetterandnoworse

than the senseless hypothesis attached for this purpose.)

5.5352 Similarly it was proposed to express ‘There are no things’ by

‘∼(∃x).x = x’. But even if this were a proposition--would it

not be true if indeed ‘There were things’, but these were not

identical with themselves?

5.54 In the general propositional form, propositions occur in a prop-

osition only as bases of the truth-operations.

5.541 At first sight it appears as if there were also a different way in

which one proposition could occur in another.

Especially in certain propositional forms of psychology, like

‘A thinks, that p is the case’, or ‘A thinks p’, etc.

Here it appears superficially as if the proposition p stood to

the object A in a kind of relation.

(And in modern epistemology (Russell, Moore, etc.) those

propositions have been conceived in this way.)

5.542 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.

5.5421 This shows that there is no such thing as the soul--the subject,

etc.--as it is conceived in contemporary superficial psychology.

A composite soul would not be a soul any longer.

5.5422 The correct explanation of the form of the proposition ‘A judges

p’ mustshowthatitisimpossibletojudgeanonsense. (Russell’s

theory does not satisfy this condition.)

5.5423 To perceive a complex means to perceive that its constituents

are combined in such and such a way.

This perhaps explains that the figure

can be seen in two ways as a cube; and all similar phenomena.

For we really see two different facts.

(If I fix my eyes first on the corners a and only glance at b,

a appears in front and b behind, and vice versa.)

5.55 Wemustnowanswerapriorithequestionastoallpossibleforms

of the elementary propositions.

The elementary proposition consists of names. Since we can-

notgivethenumberofnameswithdifferentmeanings,wecannot

give the composition of the elementary proposition.

5.551 Our fundamental principle is that every question which can be

decided at all by logic can be decided without further trouble.

(And if we get into a situation where we need to answer such

a problem by looking at the world, this shows that we are on a

fundamentally wrong track.)

5.552 The ‘experience’ which we need to understand logic is not that

such and such is the case, but that something is; but that is no

experience.

Logic precedes every experience--that something is so.

It is before the How, not before the What.

5.5521 And if this were not the case, how could we apply logic? We

could say: if there were a logic, even if there were no world, how

then could there be a logic, since there is a world?

5.553 Russell said that there were simple relations between different

numbers of things (individuals). But between what numbers?

And how should this be decided--by experience?

(There is no pre-eminent number.)

5.554 The enumeration of any special forms would be entirely arbi-

trary.

5.5541 It should be possible to decide a priori whether, for example, I

can get into a situation in which I need to symbolize with a sign

of a 27-termed relation.

5.5542 May we then ask this at all? Can we set out a sign form and

not know whether anything can correspond to it?

Has the question sense: what must be in order that some-

thing can be the case?

5.555 It is clear that we have a concept of the elementary proposition

apart from its special logical form.

Where,however,wecanbuildsymbolsaccordingtoasystem,

there this system is the logically important thing and not the

single symbols.

And how would it be possible that I should have to deal with

forms in logic which I can invent: but I must have to deal with

that which makes it possible for me to invent them.

5.556 Therecannotbeahierarchyoftheformsoftheelementaryprop-

ositions. Only that which we ourselves construct can we foresee.

5.5561 Empirical reality is limited by the totality of objects. The

boundary appears again in the totality of elementary propo-

sitions.

The hierarchies are and must be independent of reality.

5.5562 If we know on purely logical grounds, that there must be ele-

mentary propositions, then this must be known by everyone who

understands the propositions in their unanalysed form.

5.5563 All propositions of our colloquial language are actually, just as

they are, logically completely in order. That most simple thing

which we ought to give here is not a simile of truth but the

complete truth itself.

(Our problems are not abstract but perhaps the most con-

crete that there are.)

5.557 The application of logic decides what elementary propositions

there are.

What lies in the application logic cannot anticipate.

It is clear that logic may not collide with its application.

But logic must have contact with its application.

Therefore logic and its application may not overlap one an-

other.

5.5571 If I cannot give elementary propositions a priori then it must

lead to obvious nonsense to try to give them.

5.6 The limits of my language mean the limits of my world.

5.61 Logic fills the world: the limits of the world are also its limits.

We cannot therefore say in logic: This and this there is in

the world, that there is not.

For that would apparently presuppose that we exclude cer-

tain possibilities, and this cannot be the case since otherwise

logic must get outside the limits of the world: that is, if it could

consider these limits from the other side also.

What we cannot think, that we cannot think: we cannot

therefore say what we cannot think.

5.62 Thisremarkprovidesakeytothequestion, towhatextentsolip-

sism is a truth.

In fact what solipsism means, is quite correct, only it cannot

be said, but it shows itself.

That the world is my world, shows itself in the fact that the

limits of the language (the language which only I understand)

mean the limits of my world.

5.621 The world and life are one.

5.63 I am my world. (The microcosm.)

5.631 The thinking, presenting subject; there is no such thing.

If I wrote a book ‘The world as I found it’, I should also have

therein to report on my body and say which members obey my

will and which do not, etc. This then would be a method of

isolating the subject or rather of showing that in an important

sense there is no subject: that is to say, of it alone in this book

mention could not be made.

5.632 The subject does not belong to the world but it is a limit of the

world.

5.633 Where in the world is a metaphysical subject to be noted?

You say that this case is altogether like that of the eye and

the field of sight. But you do not really see the eye.

And from nothing in the field of sight can it be concluded

that it is seen from an eye.

5.6331 For the field of sight has not a form like this:

5.634 This is connected with the fact that no part of our experience is

also a priori.

Everything we see could also be otherwise.

Everything we can describe at all could also be otherwise.

There is no order of things a priori.

5.64 Here we see that solipsism strictly carried out coincides with

pure realism. The I in solipsism shrinks to an extensionless

point and there remains the reality co-ordinated with it.

5.641 There is therefore really a sense in which in philosophy we can

talk of a non-psychological I.

The I occurs in philosophy through the fact that the ‘world

is my world’.

The philosophical I is not the man, not the human body or

thehumansoulofwhichpsychologytreats, butthemetaphysical

subject, the limit--not a part of the world.

6 The general form of truth-function is: [p,ξ,N(ξ)].

This is the general form of proposition.

6.001 Thissaysnothingelsethanthateverypropositionistheresultof

successive applications of the operation N′(ξ) to the elementary

propositions.

6.002 Ifwearegiventhegeneralformofthewayinwhichaproposition

isconstructed, thentherebywearealsogiventhegeneralformof

the way in which by an operation out of one proposition another

can be created.

6.01 ThegeneralformoftheoperationΩ′(η)istherefore: [ξ,N(ξ)]′(η)

(= [η, ξ, N(ξ)]).

This is the most general form of transition from one propo-

sition to another.

6.02 And thus we come to numbers: I define

x = Ω0′x Def. and

Ω′Ων′x = Ων+1′x Def.

According, then, to these symbolic rules we write the series

x, Ω′x, Ω′Ω′x, Ω′Ω′Ω′x.....

as: Ω0′x,Ω0+1′x,Ω0+1+1′x,Ω0+1+1+1′x.....

Therefore I write in place of ‘[x,ξ,Ω′ξ]’,

“[Ω0′x,Ων′x,Ων+1′x]’.

And I define:

0+1 = 1 Def.

0+1+1 = 2 Def.

0+1+1+1 = 3 Def.

and so on.

6.021 A number is the exponent of an operation.

6.022 The concept number is nothing else than that which is common

to all numbers, the general form of number.

The concept number is the variable number.

And the concept of equality of numbers is the general form

of all special equalities of numbers.

6.03 The general form of the cardinal number is: [0,ξ,ξ +1].

6.031 The theory of classes is altogether superfluous in mathematics.

This is connected with the fact that the generality which we

need in mathematics is not the accidental one.

6.1 The propositions of logic are tautologies.

6.11 The propositions of logic therefore say nothing. (They are the

analytical propositions.)

6.111 Theories which make a proposition of logic appear substantial

are always false. One could e.g. believe that the words ‘true’

and ‘false’ signify two properties among other properties, and

then it would appear as a remarkable fact that every proposi-

tion possesses one of these properties. This now by no means

appears self-evident, no more so than the proposition ‘All roses

areeitheryelloworred’ wouldsoundevenifitweretrue. Indeed

our proposition now gets quite the character of a proposition of

natural science and this is a certain symptom of its being falsely

understood.

6.112 The correct explanation of logical propositions must give them

a peculiar position among all propositions.

6.113 It is the characteristic mark of logical propositions that one can

perceive in the symbol alone that they are true; and this fact

contains in itself the whole philosophy of logic. And so also it is

one of the most important facts that the truth or falsehood of

non-logical propositions can not be recognized from the propo-

sitions alone.

6.12 The fact that the propositions of logic are tautologies shows the

formal--logical--properties of language, of the world.

That its constituent parts connected together in this way

give a tautology characterizes the logic of its constituent parts.

In order that propositions connected together in a definite

way may give a tautology they must have definite properties of

structure. That they give a tautology when so connected shows

therefore that they possess these properties of structure.

6.1201 That e.g. the propositions ‘p’ and ‘∼p’ in the connexion ‘∼(p.

∼p)’

give a tautology shows that they contradict one another.

That the propositions ‘p ⊃ q’, ‘p’ and ‘q’ connected together

in the form ‘(p ⊃ q).(p) :⊃: (q)’ give a tautology shows that q

follows from p and p ⊃ q. That ‘(x).fx :⊃: fa’ is a tautology

shows that fa follows from (x).fx, etc. etc.

6.1202 Itisclearthatwecouldhaveusedforthispurposecontradictions

instead of tautologies.

6.1203 In order to recognize a tautology as such, we can, in cases in

which no sign of generality occurs in the tautology, make use

of the following intuitive method: I write instead of ‘p’, ‘q’,

‘r’, etc., ‘TpF’, ‘TqF’, ‘TrF’, etc. The truth-combinations I

express by brackets, e.g.:

and the co-ordination of the truth or falsity of the whole prop-

osition with the truth-combinations of the truth-arguments by

lines in the following way:

This sign, for example, would therefore present the proposi-

tion p ⊃ q. Now I will proceed to inquire whether such a prop-

∼(p.∼p)

osition as (The Law of Contradiction) is a tautology.

‘∼ξ’

The form is written in our notation

the form ‘ξ .η’ thus:--

∼(p.∼q)

Hence the proposition runs thus:--

Ifhereweput‘p’ insteadof‘q’ andexaminethecombination

of the outermost T and F with the innermost, it is seen that

the truth of the whole proposition is co-ordinated with all the

truth-combinations of its argument, its falsity with none of the

truth-combinations.

6.121 The propositions of logic demonstrate the logical properties of

propositions, by combining them into propositions which say

nothing.

This method could be called a zero-method. In a logical

proposition propositions are brought into equilibrium with one

another, andthestateofequilibriumthenshowshowtheseprop-

ositions must be logically constructed.

6.122 Whence it follows that we can get on without logical proposi-

tions, for we can recognize in an adequate notation the formal

properties of the propositions by mere inspection.

6.1221 If for example two propositions ‘p’ and ‘q’ give a tautology in

the connexion ‘p ⊃ q’, then it is clear that q follows from p.

E.g. that ‘q’ follows from ‘p ⊃ q .p’ we see from these two

propositions themselves, but we can also show it by combining

themto‘p ⊃ q.p :⊃: q’ andthenshowingthatthisisatautology.

6.1222 This throws light on the question why logical propositions can

no more be empirically established than they can be empirically

refuted. Not only must a proposition of logic be incapable of

being contradicted by any possible experience, but it must also

be incapable of being established by any such.

6.1223 Itnowbecomesclearwhyweoftenfeelasthough‘logicaltruths’

must be ‘postulated’ by us. We can in fact postulate them in so

far as we can postulate an adequate notation.

6.1224 It also becomes clear why logic has been called the theory of

forms and of inference.

6.123 It is clear that the laws of logic cannot themselves obey further

logical laws.

(There is not, as Russell supposed, for every ‘type’ a special

law of contradiction; but one is sufficient, since it is not applied

to itself.)

6.1231 The mark of logical propositions is not their general validity.

To be general is only to be accidentally valid for all things.

An ungeneralized proposition can be tautologous just as well as

a generalized one.

6.1232 Logical general validity, we could call essential as opposed to

accidental general validity, e.g. of the proposition ‘all men are

mortal’. Propositions like Russell’s ‘axiom of reducibility’ are

not logical propositions, and this explains our feeling that, if

true, they can only be true by a happy chance.

6.1233 We can imagine a world in which the axiom of reducibility is

not valid. But it is clear that logic has nothing to do with the

question whether our world is really of this kind or not.

6.124 The logical propositions describe the scaffolding of the world,

or rather they present it. They ‘treat’ of nothing. They pre-

suppose that names have meaning, and that elementary propo-

sitions have sense. And this is their connexion with the world.

It is clear that it must show something about the world that

certain combinations of symbols--which essentially have a def-

inite character--are tautologies. Herein lies the decisive point.

We said that in the symbols which we use much is arbitrary,

much not. In logic only this expresses: but this means that in

logic it is not we who express, by means of signs, what we want,

but in logic the nature of the essentially necessary signs itself

asserts. That is to say, if we know the logical syntax of any sign

language, then all the propositions of logic are already given.

6.125 It is possible, even in the old logic, to give at the outset a de-

scription of all ‘true’ logical propositions.

6.1251 Hence there can never be surprises in logic.

6.126 Whether a proposition belongs to logic can be determined by

determining the logical properties of the symbol.

And this we do when we prove a logical proposition. For

without troubling ourselves about a sense and a meaning, we

form the logical propositions out of others by mere symbolic

rules.

We prove a logical proposition by creating it out of other

logicalpropositionsbyapplyinginsuccessioncertainoperations,

which again generate tautologies out of the first. (And from a

tautology only tautologies follow.)

Naturally this way of showing that its propositions are tau-

tologies is quite unessential to logic. Because the propositions,

from which the proof starts, must show without proof that they

are tautologies.

6.1261 In logic process and result are equivalent. (Therefore no sur-

prises.)

6.1262 Proof in logic is only a mechanical expedient to facilitate the

recognition of tautology, where it is complicated.

6.1263 Itwouldbetooremarkable, ifonecouldproveasignificantprop-

osition logically from another, and a logical proposition also. It

is clear from the beginning that the logical proof of a significant

proposition and the proof in logic must be two quite different

things.

6.1264 The significant proposition asserts something, and its proof

shows that it is so; in logic every proposition is the form of a

proof.

Every proposition of logic is a modus ponens presented in

signs. (And the modus ponens can not be expressed by a prop-

osition.)

6.1265 Logic can always be conceived to be such that every proposition

is its own proof.

6.127 All propositions of logic are of equal rank; there are not some

which are essentially primitive and others deduced from these.

Every tautology itself shows that it is a tautology.

6.1271 It is clear that the number of ‘primitive propositions of logic’

is arbitrary, for we could deduce logic from one primitive prop-

osition by simply forming, for example, the logical product of

Frege’s primitive propositions. (Frege would perhaps say that

this would no longer be immediately self-evident. But it is re-

markable that so exact a thinker as Frege should have appealed

to the degree of self-evidence as the criterion of a logical propo-

sition.)

6.13 Logic is not a theory but a reflexion of the world.

Logic is transcendental.

6.2 Mathematics is a logical method.

The propositions of mathematics are equations, and there-

fore pseudo-propositions.

6.21 Mathematical propositions express no thoughts.

6.211 In life it is never a mathematical proposition which we need,

but we use mathematical propositions only in order to infer

from propositions which do not belong to mathematics to others

which equally do not belong to mathematics.

(Inphilosophythequestion‘Whydowereallyusethatword,

that proposition?’ constantly leads to valuable results.)

6.22 The logic of the world which the propositions of logic show in

tautologies, mathematics shows in equations.

6.23 If two expressions are connected by the sign of equality, this

means that they can be substituted for one another. But

whether this is the case must show itself in the two expressions

themselves.

It characterizes the logical form of two expressions, that they

can be substituted for one another.

6.231 It is a property of affirmation that it can be conceived as double

denial.

It is a property of ‘1+1+1+1’ that it can be conceived as

‘(1+1)+(1+1)’.

6.232 Frege says that these expressions have the same meaning but

different senses.

But what is essential about equation is that it is not neces-

saryinordertoshowthatbothexpressions, whichareconnected

by the sign of equality, have the same meaning: for this can be

perceived from the two expressions themselves.

6.2321 And, that the propositions of mathematics can be proved means

nothing else than that their correctness can be seen without our

having to compare what they express with the facts as regards

correctness.

6.2322 The identity of the meaning of two expressions cannot be as-

serted. For in order to be able to assert anything about their

meaning, I must know their meaning, and if I know their mean-

ing, I know whether they mean the same or something different.

6.2323 The equation characterizes only the standpoint from which I

consider the two expressions, that is to say the standpoint of

their equality of meaning.

6.233 To the question whether we need intuition for the solution of

mathematical problems it must be answered that language itself

here supplies the necessary intuition.

6.2331 The process of calculation brings about just this intuition.

Calculation is not an experiment.

6.234 Mathematics is a method of logic.

6.2341 The essential of mathematical method is working with equa-

tions. On this method depends the fact that every proposition

of mathematics must be self-intelligible.

6.24 The method by which mathematics arrives at its equations is

the method of substitution.

For equations express the substitutability of two expressions,

and we proceed from a number of equations to new equations,

replacingexpressionsbyothersinaccordancewiththeequations.

6.241 Thus the proof of the proposition 2×2 = 4 runs:

(Ων)µ′x = Ων×µ′x Def.

Ω2×2′x = (Ω2)2′x = (Ω2)1+1′x = Ω2′Ω2′x = Ω1+1′Ω1+1′x

= (Ω′Ω)′(Ω′Ω)′x = Ω′Ω′Ω′Ω′x = Ω1+1+1+1′x = Ω4′x.

6.3 Logical research means the investigation of all regularity. And

outside logic all is accident.

6.31 The so-called law of induction cannot in any case be a logical

law, for it is obviously a significant proposition.--And therefore

it cannot be a law a priori either.

*

6.32 The law of causality is not a law but the form of a law.

6.321 ‘Law of Causality’ is a class name. And as in mechanics there

are, for instance, minimum-laws, such as that of least action, so

in physics there are causal laws, laws of the causality form.

6.3211 Men had indeed an idea that there must be a ‘law of least ac-

tion’, before they knew exactly how it ran. (Here, as always,

the a priori certain proves to be something purely logical.)

6.33 We do not believe a priori in a law of conservation, but we know

a priori the possibility of a logical form.

6.34 Allpropositions, suchasthelawofcausation, thelawofcontinu-

ity in nature, the law of least expenditure in nature, etc. etc., all

these are a priori intuitions of possible forms of the propositions

of science.

6.341 Newtonian mechanics, for example, brings the description of the

universe to a unified form. Let us imagine a white surface with

irregular black spots. We now say: Whatever kind of picture

these make I can always get as near as I like to its description,

if I cover the surface with a sufficiently fine square network and

now say of every square that it is white or black. In this way

I shall have brought the description of the surface to a unified

form. This form is arbitrary, because I could have applied with

equal success a net with a triangular or hexagonal mesh. It can

happen that the description would have been simpler with the

aid of a triangular mesh; that is to say we might have described

the surface more accurately with a triangular, and coarser, than

with the finer square mesh, or vice versa, and so on. To the

different networks correspond different systems of describing the

world. Mechanics determine a form of description by saying: All

propositionsinthedescriptionoftheworldmustbeobtainedina

given way from a number of given propositions--the mechanical

axioms. It thus provides the bricks for building the edifice of

science, and says: Whatever building thou wouldst erect, thou

shalt construct it in some manner with these bricks and these

alone.

(As with the system of numbers one must be able to write

downanyarbitrarynumber, sowiththesystemofmechanicsone

must be able to write down any arbitrary physical proposition.)

6.342 And now we see the relative position of logic and mechanics.

(We could construct the network out of figures of different kinds,

as out of triangles and hexagons together.) That a picture like

that instanced above can be described by a network of a given

form asserts nothing about the picture. (For this holds of every

picture of this kind.) But this does characterize the picture, the

fact, namely, that it can be completely described by a definite

net of definite fineness.

So too the fact that it can be described by Newtonian me-

chanics asserts nothing about the world; but this asserts some-

thing, namely, that it can be described in that particular way in

which it is described, as is indeed the case. The fact, too, that it

can be described more simply by one system of mechanics than

by another says something about the world.

6.343 Mechanics is an attempt to construct according to a single plan

all true propositions which we need for the description of the

world.

6.3431 Through the whole apparatus of logic the physical laws still

speak of the objects of the world.

6.3432 We must not forget that the description of the world by me-

chanics is always quite general. There is, for example, never any

mention of particular material points in it, but always only of

some points or other.

6.35 Although the spots in our picture are geometrical figures, ge-

ometry can obviously say nothing about their actual form and

position. Butthenetworkispurely geometrical, andallitsprop-

erties can be given a priori.

Laws, like the law of causation, etc., treat of the network and

not of what the network described.

6.36 If there were a law of causality, it might run: ‘There are natural

laws’.

But that can clearly not be said: it shows itself.

6.361 In the terminology of Hertz we might say: Only uniform con-

nexions are thinkable.

6.3611 We cannot compare any process with the ‘passage of time’--

there is no such thing--but only with another process (say, with

the movement of the chronometer).

Hence the description of the temporal sequence of events is

only possible if we support ourselves on another process.

It is exactly analogous for space. When, for example, we say

that neither of two events (which mutually exclude one another)

can occur, because there is no cause why the one should occur

rather than the other, it is really a matter of our being unable

to describe one of the two events unless there is some sort of

asymmetry. And if there is such an asymmetry, we can regard

this as the cause of the occurrence of the one and of the non-

occurrence of the other.

6.36111 The Kantian problem of the right and left hand which cannot

be made to cover one another already exists in the plane, and

even in one-dimensional space; where the two congruent figures

a and b cannot be made to cover one another without moving

them out of this space. The right and left hand are in fact

completely congruent. And the fact that they cannot be made

to cover one another has nothing to do with it.

A right-hand glove could be put on a left hand if it could be

turned round in four-dimensional space.

6.362 What can be described can happen too, and what is excluded

by the law of causality cannot be described.

6.363 The process of induction is the process of assuming the simplest

law that can be made to harmonize with our experience.

6.3631 This process, however, has no logical foundation but only a psy-

chological one.

It is clear that there are no grounds for believing that the

simplest course of events will really happen.

6.36311 That the sun will rise to-morrow, is an hypothesis; and that

means that we do not know whether it will rise.

6.37 A necessity for one thing to happen because another has hap-

pened does not exist. There is only logical necessity.

6.371 At the basis of the whole modern view of the world lies the

illusion that the so-called laws of nature are the explanations of

natural phenomena.

6.372 So people stop short at natural laws as at something unassail-

able, as did the ancients at God and Fate.

And they both are right and wrong. But the ancients were

clearer, insofarastheyrecognizedoneclearconclusion, whereas

inthemodernsystemitshouldappearasthougheverything were

explained.

6.373 The world is independent of my will.

6.374 Even if everything we wished were to happen, this would only

be, so to speak, a favour of fate, for there is no logical connexion

between will and world, which would guarantee this, and the

assumed physical connexion itself we could not again will.

6.375 As there is only a logical necessity, so there is only a logical

impossibility.

6.3751 For two colours, e.g. to be at one place in the visual field, is

impossible, logically impossible, for it is excluded by the logical

structure of colour.

Let us consider how this contradiction presents itself in phys-

ics. Somewhat as follows: That a particle cannot at the same

time have two velocities, i.e. that at the same time it cannot be

in two places, i.e. that particles in different places at the same

time cannot be identical.

(It is clear that the logical product of two elementary prop-

ositions can neither be a tautology nor a contradiction. The as-

sertion that a point in the visual field has two different colours

at the same time, is a contradiction.)

6.4 All propositions are of equal value.

6.41 The sense of the world must lie outside the world. In the world

everything is as it is and happens as it does happen. In it there

is no value--and if there were, it would be of no value.

If there is a value which is of value, it must lie outside all

happening and being-so. For all happening and being-so is ac-

cidental.

What makes it non-accidental cannot lie in the world, for

otherwise this would again be accidental.

It must lie outside the world.

6.42 Hence also there can be no ethical propositions.

Propositions cannot express anything higher.

6.421 It is clear that ethics cannot be expressed.

Ethics are transcendental.

(Ethics and aesthetics are one.)

6.422 The first thought in setting up an ethical law of the form ‘thou

shalt ...’ is: And what if I do not do it. But it is clear that

ethics has nothing to do with punishment and reward in the

ordinarysense. Thisquestionastotheconsequences ofanaction

mustthereforebeirrelevant. Atleasttheseconsequenceswillnot

beevents. Fortheremustbesomethingrightinthatformulation

of the question. There must be some sort of ethical reward and

ethical punishment, but this must lie in the action itself.

(And this is clear also that the reward must be something

acceptable, and the punishment something unacceptable.)

6.423 Of the will as the bearer of the ethical we cannot speak.

And the will as a phenomenon is only of interest to psychol-

ogy.

6.43 If good or bad willing changes the world, it can only change the

limits of the world, not the facts; not the things that can be

expressed in language.

In brief, the world must thereby become quite another. It

must so to speak wax or wane as a whole.

The world of the happy is quite another than that of the

unhappy.

6.431 As in death, too, the world does not change, but ceases.

6.4311 Death is not an event of life. Death is not lived through.

If by eternity is understood not endless temporal duration

but timelessness, then he lives eternally who lives in the present.

Our life is endless in the way that our visual field is without

limit.

6.4312 The temporal immortality of the soul of man, that is to say, its

eternal survival also after death, is not only in no way guaran-

teed, but this assumption in the first place will not do for us

what we always tried to make it do. Is a riddle solved by the

fact that I survive for ever? Is this eternal life not as enigmatic

as our present one? The solution of the riddle of life in space

and time lies outside space and time.

(It is not problems of natural science which have to be

solved.)

6.432 How the world is, is completely indifferent for what is higher.

God does not reveal himself in the world.

6.4321 The facts all belong only to the task and not to its performance.

6.44 Not how the world is, is the mystical, but that it is.

6.45 The contemplation of the world sub specie aeterni is its contem-

plation as a limited whole.

The feeling of the world as a limited whole is the mystical

feeling.

6.5 For an answer which cannot be expressed the question too can-

not be expressed.

The riddle does not exist.

If a question can be put at all, then it can also be answered.

6.51 Scepticism is not irrefutable, but palpably senseless, if it would

doubt where a question cannot be asked.

Fordoubtcanonlyexistwherethereisaquestion; aquestion

only where there is an answer, and this only where something

can be said.

6.52 We feel that even if all possible scientific questions be answered,

the problems of life have still not been touched at all. Of course

there is then no question left, and just this is the answer.

6.521 The solution of the problem of life is seen in the vanishing of

this problem.

(Is not this the reason why men to whom after long doubting

the sense of life became clear, could not then say wherein this

sense consisted?)

6.522 There is indeed the inexpressible. This shows itself; it is the

mystical.

6.53 The right method of philosophy would be this. To say nothing

except what can be said, i.e. the propositions of natural science,

i.e. something that has nothing to do with philosophy: and then

always, when someone else wished to say something metaphys-

ical, to demonstrate to him that he had given no meaning to

certain signs in his propositions. This method would be un-

satisfying to the other--he would not have the feeling that we

were teaching him philosophy--but it would be the only strictly

correct method.

6.54 Mypropositionsareelucidatoryinthisway: hewhounderstands

me finally recognizes them as senseless, when he has climbed out

through them, on them, over them. (He must so to speak throw

away the ladder, after he has climbed up on it.)

He must surmount these propositions; then he sees the world

rightly.

7 Whereof one cannot speak, thereof one must be silent.