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"TRACTATUS LOGICO-PHILOSOPHICUS"

Perhaps this book will be understood only by someone who has himself
already had the thoughts that are expressed in it—or at least similar
thoughts.—So it is not a textbook.—Its purpose would be achieved if it gave
pleasure to one person who read and understood it.

The book deals with the problems of philosophy, and shows, I believe, that
the reason why these problems are posed is that the logic of our language is
misunderstood. The whole sense of the book might be summed up the following
words: what can be said at all can be said clearly, and what we cannot talk
about we must pass over in silence.

Thus the aim of the book is to draw a limit to thought, or rather—not to
thought, but to the expression of thoughts: for in order to be able to draw a
limit to thought, we should have to find both sides of the limit thinkable (i.e.
we should have to be able to think what cannot be thought).

It will therefore only be in language that the limit can be drawn, and what
lies on the other side of the limit will simply be nonsense.

I do not wish to judge how far my efforts coincide with those of other
philosophers. Indeed, what I have written here makes no claim to novelty in
detail, and the reason why I give no sources is that it is a matter of
indifference to me whether the thoughts that I have had have been anticipated by
someone else.

I will only mention that I am indebted to Frege's great works and to the
writings of my friend Mr Bertrand Russell for much of the stimulation of my
thoughts.

If this work has any value, it consists in two things: the first is that
thoughts are expressed in it, and on this score the better the thoughts are
expressed—the more the nail has been hit on the head—the greater will be its
value.—Here I am conscious of having fallen a long way short of what is
possible. Simply because my powers are too slight for the accomplishment of the
task.—May others come and do it better.

On the other hand the truth of the thoughts that are here communicated seems
to me unassailable and definitive. I therefore believe myself to have found, on
all essential points, the final solution of the problems. And if I am not
mistaken in this belief, then the second thing in which the of this work
consists is that it shows how little is achieved when these problems are solved.

L.W. Vienna, 1918

## Contents

**1. The world is all that is the case. **

2. What is the case—a fact—is the existence of states of affairs.

3. A logical picture of facts is a thought.

4. A thought is a proposition with a sense.

5. A proposition is a truth-function of elementary propositions.

6. The general form of a truth-function is [p, E, N(E)].

7. What we cannot speak about we must pass over in silence.

## 1. The world is all 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 their being all the facts.

1.12 For the totality of facts determines what is the case, and also whatever
is not the case.

1.13 The facts in logical space are the world.

1.2 The world divides into facts.

1.21 Each item can be the case or not the case while everything else remains
the same.

## 2. What is the case—a fact—is the existence of states of affairs.

2.01 A state of affairs (a state of things) is a combination of objects
(things).

2.011 It is essential to things that they should be possible constituents of
states of affairs.

2.012 In logic nothing is accidental: if a thing can occur in a state of
affairs, the possibility of the state of affairs must be written into the thing
itself.

2.0121 It would seem to be a sort of accident, if it turned out that a
situation would fit a thing that could already exist entirely on its own. If
things can occur in states of affairs, this possibility must be in them from the
beginning. (Nothing in the province of logic can be merely possible. Logic deals
with every possibility and all possibilities are its facts.) Just as we are
quite unable to imagine spatial objects outside space or temporal objects
outside time, so too there is no object that we can imagine excluded from the
possibility of combining with others. If I can imagine objects combined in
states of affairs, I cannot imagine them excluded from the possibility of such
combinations.

2.0122 Things are independent in so far as they can occur in all possible
situations, but this form of independence is a form of connexion with states of
affairs, a form of dependence. (It is impossible for words to appear in two
different roles: by themselves, and in propositions.)

2.0123 If I know an object I also know all its possible occurrences in states
of affairs. (Every one of these possibilities must be part of the nature of the
object.) A new possibility cannot be discovered later.

2.01231 If I am to know an object, thought I need not know its external
properties, I must know all its internal properties.

2.0124 If all objects are given, then at the same time all possible states of
affairs are also given.

2.013 Each thing is, as it were, in a space of possible states of affairs.
This space I can imagine empty, but I cannot imagine the thing without the
space.

2.0131 A spatial object must be situated in infinite space. (A spatial point
is an argument-place.) A speck in the visual field, thought it need not be red,
must have some colour: it is, so to speak, surrounded by colour-space. Notes
must have some pitch, objects of the sense of touch some degree of hardness, and
so on.

2.014 Objects contain the possibility of all situations.

2.0141 The possibility of its occurring in states of affairs is the form of
an object.

2.02 Objects are simple.

2.0201 Every statement about complexes can be resolved into a statement about
their constituents and into the propositions that describe the complexes
completely.

2.021 Objects make up the substance of the world. That is why they cannot be
composite.

2.0211 If they world had no substance, then whether a proposition had sense
would depend on whether another proposition was true.

2.0212 In that case we could not sketch any picture of the world (true or
false).

2.022 It is obvious that an imagined world, however difference it may be from
the real one, must have something—a form—in common with it.

2.023 Objects are just what constitute this unalterable form.

2.0231 The substance of the world can only determine a form, and not any
material properties. For it is only by means of propositions that material
properties are represented—only by the configuration of objects that they are
produced.

2.0232 In a manner of speaking, objects are colourless.

2.0233 If two objects have the same logical form, the only distinction
between them, apart from their external properties, is that they are different.

2.02331 Either a thing has properties that nothing else has, in which case we
can immediately use a description to distinguish it from the others and refer to
it; or, on the other hand, there are several things that have the whole set of
their properties in common, in which case it is quite impossible to indicate one
of them. For it there is nothing to distinguish a thing, I cannot distinguish
it, since otherwise it would be distinguished after all.

2.024 The substance is what subsists independently of what is the case.

2.025 It is form and content.

2.0251 Space, time, colour (being coloured) are forms of objects.

2.026 There must be objects, if the world is to have unalterable form.

2.027 Objects, the unalterable, and the subsistent are one and the same.

2.0271 Objects are what is unalterable and subsistent; their configuration is
what is changing and unstable.

2.0272 The configuration of objects produces states of affairs.

2.03 In a state of affairs objects fit into one another like the links of a
chain.

2.031 In a state of affairs objects stand in a determinate relation to one
another.

2.032 The determinate way in which objects are connected in a state of
affairs is the structure of the state of affairs.

2.033 Form is the possibility of structure.

2.034 The structure of a fact consists of the structures of states of
affairs.

2.04 The totality of existing states of affairs is the world.

2.05 The totality of existing states of affairs also determines which states
of affairs do not exist.

2.06 The existence and non-existence of states of affairs is reality. (We
call the existence of states of affairs a positive fact, and their non-existence
a negative fact.)

2.061 States of affairs are independent of one another.

2.062 From the existence or non-existence of one state of affairs it is
impossible to infer the existence or non-existence of another.

2.063 The sum-total of reality is the world.

2.1 We picture facts to ourselves.

2.11 A picture presents a situation in logical space, the existence and
non-existence of states of affairs.

2.12 A picture is a model of reality.

2.13 In a picture objects have the elements of the picture corresponding to
them.

2.131 In a picture the elements of the picture are the representatives of
objects.

2.14 What constitutes a picture is that its elements are related to one
another in a determinate way.

2.141 A picture is a fact.

2.15 The fact that the elements of a picture are related to one another in a
determinate way represents that things are related to one another in the same
way. Let us call this connexion of its elements the structure of the picture,
and let us call the possibility of this structure the pictorial form of the
picture.

2.151 Pictorial form is the possibility that things are related to one
another in the same way as the elements of the picture.

2.1511 That is how a picture is attached to reality; it reaches right out to
it.

2.1512 It is laid against reality like a measure.

2.15121 Only the end-points of the graduating lines actually touch the object
that is to be measured.

2.1514 So a picture, conceived in this way, also includes the pictorial
relationship, which makes it into a picture.

2.1515 These correlations are, as it were, the feelers of the picture's
elements, with which the picture touches reality.

2.16 If a fact is to be a picture, it must have something in common with what
it depicts.

2.161 There must be something identical in a picture and what it depicts, to
enable the one to be a picture of the other at all.

2.17 What a picture must have in common with reality, in order to be able to
depict it—correctly or incorrectly—in the way that it does, is its pictorial
form.

2.171 A picture can depict any reality whose form it has. A spatial picture
can depict anything spatial, a coloured one anything coloured, etc.

2.172 A picture cannot, however, depict its pictorial form: it displays it.

2.173 A picture represents its subject from a position outside it. (Its
standpoint is its representational form.) That is why a picture represents its
subject correctly or incorrectly.

2.174 A picture cannot, however, place itself outside its representational
form.

2.18 What any picture, of whatever form, must have in common with reality, in
order to be able to depict it—correctly or incorrectly—in any way at all, is
logical form, i.e. the form of reality.

2.181 A picture whose pictorial form is logical form is called a logical
picture.

2.182 Every picture is at the same time a logical one. (On the other hand,
not every picture is, for example, a spatial one.)

2.19 Logical pictures can depict the world.

2.2 A picture has logico-pictorial form in common with what it depicts.

2.201 A picture depicts reality by representing a possibility of existence
and non-existence of states of affairs.

2.202 A picture contains the possibility of the situation that it represents.

2.203 A picture agrees with reality or fails to agree; it is correct or
incorrect, true or false.

2.22 What a picture represents it represents independently of its truth or
falsity, by means of its pictorial form.

2.221 What a picture represents is its sense.

2.222 The agreement or disagreement or its sense with reality constitutes its
truth or falsity.

2.223 In order to tell whether a picture is true or false we must compare it
with reality.

2.224 It is impossible to tell from the picture alone whether it is true or
false.

2.225 There are no pictures that are true a priori.

## 3. A logical picture of facts is a thought.

3.001 'A state of affairs is thinkable': what this means is that we can
picture it to ourselves.

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

3.02 A thought contains the possibility of the situation of which it is the
thought. What is thinkable is possible too.

3.03 Thought can never be of anything illogical, since, if it were, we should
have to think illogically.

3.031 It used to be said that God could create anything except what would be
contrary to the laws of logic. The truth is that we could not say what an
'illogical' world would look like.

3.032 It is as impossible to represent in language anything that 'contradicts
logic' as it is in geometry to represent by its coordinates a figure that
contradicts the laws of space, or to give the coordinates of a point that does
not exist.

3.0321 Though a state of affairs that would contravene the laws of physics
can be represented by us spatially, one that would contravene the laws of
geometry cannot.

3.04 It a thought were correct a priori, it would be a thought whose
possibility ensured its truth.

3.05 A priori knowledge that a thought was true would be possible only it its
truth were recognizable from the thought itself (without anything a to compare
it with).

3.1 In a proposition a thought finds an expression that can be perceived by
the senses.

3.11 We use the perceptible sign of a proposition (spoken or written, etc.)
as a projection of a possible situation. The method of projection is to think of
the sense of the proposition.

3.12 I call the sign with which we express a thought a propositional sign.
And a proposition is a propositional sign in its projective relation to the
world.

3.13 A proposition, therefore, does not actually contain its sense, but does
contain the possibility of expressing it. ('The content of a proposition' means
the content of a proposition that has sense.) A proposition contains the form,
but not the content, of its sense.

3.14 What constitutes a propositional sign is that in its elements (the
words) stand in a determinate relation to one another. A propositional sign is a
fact.

3.141 A proposition is not a blend of words.(Just as a theme in music is not
a blend of notes.) A proposition is articulate.

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

3.143 Although a propositional sign is a fact, this is obscured by the usual
form of expression in writing or print. For in a printed proposition, for
example, no essential difference is apparent between a propositional sign and a
word. (That is what made it possible for Frege to call a proposition a composite
name.)

3.1431 The essence of a propositional sign is very clearly seen if we imagine
one composed of spatial objects (such as tables, chairs, and books) instead of
written signs.

3.1432 Instead of, 'The complex sign "aRb" says that a stands to b in the
relation R' we ought to put, 'That "a" stands to "b" in a certain relation says
that aRb.'

3.144 Situations can be described but not given names.

3.2 In a proposition a thought can be expressed in such a way that elements
of the propositional sign correspond to the objects of the thought.

3.201 I call such elements 'simple signs', and such a proposition 'complete
analysed'.

3.202 The simple signs employed in propositions are called names.

3.203 A name means an object. The object is its meaning. ('A' is the same
sign as 'A'.)

3.21 The configuration of objects in a situation corresponds to the
configuration of simple signs in the propositional sign.

3.221 Objects can only be named. Signs are their representatives. I can only
speak about them: I cannot put them into words. Propositions can only say how
things are, not what they are.

3.23 The requirement that simple signs be possible is the requirement that
sense be determinate.

3.24 A proposition about a complex stands in an internal relation to a
proposition about a constituent of the complex. A complex can be given only by
its description, which will be right or wrong. A proposition that mentions a
complex will not be nonsensical, if the complex does not exits, but simply
false. When a propositional element signifies a complex, this can be seen from
an indeterminateness in the propositions in which it occurs. In such cases we
know that the proposition leaves something undetermined. (In fact the notation
for generality contains a prototype.) The contraction of a symbol for a complex
into a simple symbol can be expressed in a definition.

3.25 A proposition cannot be dissected any further by means of a definition:
it is a primitive sign.

3.261 Every sign that has a definition signifies via the signs that serve to
define it; and the definitions point the way. Two signs cannot signify in the
same manner if one is primitive and the other is defined by means of primitive
signs. Names cannot be anatomized by means of definitions. (Nor can any sign
that has a meaning independently and on its own.)

3.262 What signs fail to express, their application shows. What signs slur
over, their application says clearly.

3.263 The meanings of primitive signs can be explained by means of
elucidations. Elucidations are propositions that stood if the meanings of those
signs are already known.

3.3 Only propositions have sense; only in the nexus of a proposition does a
name have meaning.

3.31 I call any part of a proposition that characterizes its sense an
expression (or a symbol). (A proposition is itself an expression.) Everything
essential to their sense that propositions can have in common with one another
is an expression. An expression is the mark of a form and a content.

3.311 An expression presupposes the forms of all the propositions in which it
can occur. It is the common characteristic mark of a class of propositions.

3.312 It is therefore presented by means of the general form of the
propositions that it characterizes. In fact, in this form the expression will be
constant and everything else variable.

3.313 Thus an expression is presented by means of a variable whose values are
the propositions that contain the expression. (In the limiting case the variable
becomes a constant, the expression becomes a proposition.) I call such a
variable a 'propositional variable'.

3.314 An expression has meaning only in a proposition. All variables can be
construed as propositional variables. (Even variable names.)

3.315 If we turn a constituent of a proposition into a variable, there is a
class of propositions all of which are values of the resulting variable
proposition. In general, this class too will be dependent on the meaning that
our arbitrary conventions have given to parts of the original proposition. But
if all the signs in it that have arbitrarily determined meanings are turned into
variables, we shall still get a class of this kind. This one, however, is not
dependent on any convention, but solely on the nature of the pro position. It
corresponds to a logical form—a logical prototype.

3.316 What values a propositional variable may take is something that is
stipulated. The stipulation of values is the variable.

3.317 To stipulate values for a propositional variable is to give the
propositions whose common characteristic the variable is. The stipulation is a
description of those propositions. The stipulation will therefore be concerned
only with symbols, not with their meaning. And the only thing essential to the
stipulation is that it is merely a description of symbols and states nothing
about what is signified. How the description of the propositions is produced is
not essential.

3.318 Like Frege and Russell I construe a proposition as a function of the
expressions contained in it.

3.32 A sign is what can be perceived of a symbol.

3.321 So one and the same sign (written or spoken, etc.) can be common to two
different symbols—in which case they will signify in different ways.

3.322 Our use of the same sign to signify two different objects can never
indicate a common characteristic of the two, if we use it with two different
modes of signification. For the sign, of course, is arbitrary. So we could
choose two different signs instead, and then what would be left in common on the
signifying side?

3.323 In everyday language it very frequently happens that the same word has
different modes of signification—and so belongs to different symbols—or that two
words that have different modes of signification are employed in propositions in
what is superficially the same way. Thus the word 'is' figures as the copula, as
a sign for identity, and as an expression for existence; 'exist' figures as an
intransitive verb like 'go', and 'identical' as an adjective; we speak of
something, but also of something's happening. (In the proposition, 'Green is
green'—where the first word is the proper name of a person and the last an
adjective—these words do not merely have different meanings: they are different
symbols.)

3.324 In this way the most fundamental confusions are easily produced (the
whole of philosophy is full of them).

3.325 In order to avoid such errors we must make use of a sign-language that
excludes them by not using the same sign for different symbols and by not using
in a superficially similar way signs that have different modes of signification:
that is to say, a sign-language that is governed by logical grammar—by logical
syntax. (The conceptual notation of Frege and Russell is such a language,
though, it is true, it fails to exclude all mistakes.)

3.326 In order to recognize a symbol by its sign we must observe how it is
used with a sense.

3.327 A sign does not determine a logical form unless it is taken together
with its logico-syntactical employment.

3.328 If a sign is useless, it is meaningless. That is the point of Occam's
maxim. (If everything behaves as if a sign had meaning, then it does have
meaning.)

3.33 In logical syntax the meaning of a sign should never play a role. It
must be possible to establish logical syntax without mentioning the meaning of a
sign: only the description of expressions may be presupposed.

3.331 From this observation we turn to Russell's 'theory of types'. It can be
seen that Russell must be wrong, because he had to mention the meaning of signs
when establishing the rules for them.

3.332 No proposition can make a statement about itself, because a
propositional sign cannot be contained in itself (that is the whole of the
'theory of types').

3.333 The reason why a function cannot be its own argument is that the sign
for a function already contains the prototype of its argument, and it cannot
contain itself. For let us suppose that the function F(fx) could be its own
argument: in that case there would be a proposition 'F(F(fx))', in which the
outer function F and the inner function F must have different meanings, since
the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)). Only
the letter 'F' is common to the two functions, but the letter by itself
signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write
'(do): F(Ou). Ou = Fu'. That disposes of Russell's paradox.

3.334 The rules of logical syntax must go without saying, once we know how
each individual sign signifies.

3.34 A proposition possesses essential and accidental features. Accidental
features are those that result from the particular way in which the
propositional sign is produced. Essential features are those without which the
proposition could not express its sense.

3.341 So what is essential in a proposition is what all propositions that can
express the same sense have in common. And similarly, in general, what is
essential in a symbol is what all symbols that can serve the same purpose have
in common.

3.3411 So one could say that the real name of an object was what all symbols
that signified it had in common. Thus, one by one, all kinds of composition
would prove to be unessential to a name.

3.342 Although there is something arbitrary in our notations, this much is
not arbitrary—that when we have determined one thing arbitrarily, something else
is necessarily the case. (This derives from the essence of notation.)

3.3421 A particular mode of signifying may be unimportant but it is always
important that it is a possible mode of signifying. And that is generally so in
philosophy: again and again the individual case turns out to be unimportant, but
the possibility of each individual case discloses something about the essence of
the world.

3.343 Definitions are rules for translating from one language into another.
Any correct sign-language must be translatable into any other in accordance with
such rules: it is this that they all have in common.

3.344 What signifies in a symbol is what is common to all the symbols that
the rules of logical syntax allow us to substitute for it.

3.3441 For instance, we can express what is common to all notations for
truth-functions in the following way: they have in common that, for example, the
notation that uses 'Pp' ('not p') and 'p C g' ('p or g') can be substituted for
any of them. (This serves to characterize the way in which something general can
be disclosed by the possibility of a specific notation.)

3.3442 Nor does analysis resolve the sign for a complex in an arbitrary way,
so that it would have a different resolution every time that it was incorporated
in a different proposition.

3.4 A proposition determines a place in logical space. The existence of this
logical place is guaranteed by the mere existence of the constituents—by the
existence of the proposition with a sense.

3.41 The propositional sign with logical coordinates—that is the logical
place.

3.411 In geometry and logic alike a place is a possibility: something can
exist in it.

3.42 A proposition can determine only one place in logical space:
nevertheless the whole of logical space must already be given by it. (Otherwise
negation, logical sum, logical product, etc., would introduce more and more new
elements in co-ordination.) (The logical scaffolding surrounding a picture
determines logical space. The force of a proposition reaches through the whole
of logical space.)

3.5 A propositional sign, applied and thought out, is a thought.

## 4. A thought is a proposition with a sense.

4.001 The totality of propositions is language.

4.022 Man possesses the ability to construct languages capable of expressing
every sense, without having any idea how each word has meaning or what its
meaning is—just as people speak without knowing how the individual sounds are
produced. Everyday language is a part of the human organism and is no less
complicated than it. It is not humanly possible to gather immediately from it
what the logic of language is. Language disguises thought. So much so, that from
the outward form of the clothing it is impossible to infer the form of the
thought beneath it, because the outward form of the clothing is not designed to
reveal the form of the body, but for entirely different purposes. The tacit
conventions on which the understanding of everyday language depends are
enormously complicated.

4.003 Most of the propositions and questions to be found in philosophical
works are not false but nonsensical. Consequently we cannot give any answer to
questions of this kind, but can only point out that they are nonsensical. Most
of the propositions and questions of philosophers arise from our failure to
understand the logic of our language. (They belong to the same class as the
question whether the good is more or less identical than the beautiful.) And it
is not surprising that the deepest problems are in fact not problems at all.

4.0031 All philosophy is a 'critique of language' (though not in Mauthner's
sense). It was Russell who performed the service of showing that the apparent
logical form of a proposition need not be its real one.

4.01 A proposition is a picture of reality. A proposition is a model of
reality as we imagine it.

4.011 At first sight a proposition—one set out on the printed page, for
example—does not seem to be a picture of the reality with which it is concerned.
But neither do written notes seem at first sight to be a picture of a piece of
music, nor our phonetic notation (the alphabet) to be a picture of our speech.
And yet these sign-languages prove to be pictures, even in the ordinary sense,
of what they represent.

4.012 It is obvious that a proposition of the form 'aRb' strikes us as a
picture. In this case the sign is obviously a likeness of what is signified.

4.013 And if we penetrate to the essence of this pictorial character, we see
that it is not impaired by apparent irregularities (such as the use [sharp] of
and [flat] in musical notation). For even these irregularities depict what they
are intended to express; only they do it in a different way.

4.014 A gramophone record, the musical idea, the written notes, and the
sound-waves, all stand to one another in the same internal relation of depicting
that holds between language and the world. They are all constructed according to
a common logical pattern. (Like the two youths in the fairy-tale, their two
horses, and their lilies. They are all in a certain sense one.)

4.0141 There is a general rule by means of which the musician can obtain the
symphony from the score, and which makes it possible to derive the symphony from
the groove on the gramophone record, and, using the first rule, to derive the
score again. That is what constitutes the inner similarity between these things
which seem to be constructed in such entirely different ways. And that rule is
the law of projection which projects the symphony into the language of musical
notation. It is the rule for translating this language into the language of
gramophone records.

4.015 The possibility of all imagery, of all our pictorial modes of
expression, is contained in the logic of depiction.

4.016 In order to understand the essential nature of a proposition, we should
consider hieroglyphic script, which depicts the facts that it describes. And
alphabetic script developed out of it without losing what was essential to
depiction.

4.02 We can see this from the fact that we understand the sense of a
propositional sign without its having been explained to us.

4.021 A proposition is a picture of reality: for if I understand a
proposition, I know the situation that it represents. And I understand the
proposition without having had its sense explained to me.

4.022 A proposition shows its sense. A proposition shows how things stand if
it is true. And it says that they do so stand.

4.023 A proposition must restrict reality to two alternatives: yes or no. In
order to do that, it must describe reality completely. A proposition is a
description of a state of affairs. Just as a description of an object describes
it by giving its external properties, so a proposition describes reality by its
internal properties. A proposition constructs a world with the help of a logical
scaffolding, so that one can actually see from the proposition how everything
stands logically if it is true. One can draw inferences from a false
proposition.

4.024 To understand a proposition means to know what is the case if it is
true. (One can understand it, therefore, without knowing whether it is true.) It
is understood by anyone who understands its constituents.

4.025 When translating one language into another, we do not proceed by
translating each proposition of the one into a proposition of the other, but
merely by translating the constituents of propositions. (And the dictionary
translates not only substantives, but also verbs, adjectives, and conjunctions,
etc.; and it treats them all in the same way.)

4.026 The meanings of simple signs (words) must be explained to us if we are
to understand them. With propositions, however, we make ourselves understood.

4.027 It belongs to the essence of a proposition that it should be able to
communicate a new sense to us.

4.03 A proposition must use old expressions to communicate a new sense. A
proposition communicates a situation to us, and so it must be essentially
connected with the situation. And the connexion is precisely that it is its
logical picture. A proposition states something only in so far as it is a
picture.

4.031 In a proposition a situation is, as it were, constructed by way of
experiment. Instead of, 'This proposition has such and such a sense, we can
simply say, 'This proposition represents such and such a situation'.

4.0311 One name stands for one thing, another for another thing, and they are
combined with one another. In this way the whole group—like a tableau
vivant—presents a state of affairs.

4.0312 The possibility of propositions is based on the principle that objects
have signs as their representatives. My fundamental idea is that the 'logical
constants' are not representatives; that there can be no representatives of the
logic of facts.

4.032 It is only in so far as a proposition is logically articulated that it
is a picture of a situation. (Even the proposition, 'Ambulo', is composite: for
its stem with a different ending yields a different sense, and so does its
ending with a different stem.)

4.04 In a proposition there must be exactly as many distinguishable parts as
in the situation that it represents. The two must possess the same logical
(mathematical) multiplicity. (Compare Hertz's Mechanics on dynamical models.)

4.041 This mathematical multiplicity, of course, cannot itself be the subject
of depiction. One cannot get away from it when depicting.

4.0411. If, for example, we wanted to express what we now write as '(x). fx'
by putting an affix in front of 'fx'—for instance by writing 'Gen. fx'—it would
not be adequate: we should not know what was being generalized. If we wanted to
signalize it with an affix 'g'—for instance by writing 'f(xg)'—that would not be
adequate either: we should not know the scope of the generality-sign. If we were
to try to do it by introducing a mark into the argument-places—for instance by
writing '(G,G). F(G,G)' —it would not be adequate: we should not be able to
establish the identity of the variables. And so on. All these modes of
signifying are inadequate because they lack the necessary mathematical
multiplicity.

4.0412 For the same reason the idealist's appeal to 'spatial spectacles' is
inadequate to explain the seeing of spatial relations, because it cannot explain
the multiplicity of these relations.

4.05 Reality is compared with propositions.

4.06 A proposition can be true or false only in virtue of being a picture of
reality.

4.061 It must not be overlooked that a proposition has a sense that is
independent of the facts: otherwise one can easily suppose that true and false
are relations of equal status between signs and what they signify. In that case
one could say, for example, that 'p' signified in the true way what 'Pp'
signified in the false way, etc.

4.062 Can we not make ourselves understood with false propositions just as we
have done up till now with true ones?—So long as it is known that they are meant
to be false.—No! For a proposition is true if we use it to say that things stand
in a certain way, and they do; and if by 'p' we mean Pp and things stand as we
mean that they do, then, construed in the new way, 'p' is true and not false.

4.0621 But it is important that the signs 'p' and 'Pp' can say the same
thing. For it shows that nothing in reality corresponds to the sign 'P'. The
occurrence of negation in a proposition is not enough to characterize its sense
(PPp = p). The propositions 'p' and 'Pp' have opposite sense, but there
corresponds to them one and the same reality.

4.063 An analogy to illustrate the concept of truth: imagine a black spot on
white paper: you can describe the shape of the spot by saying, for each point on
the sheet, whether it is black or white. To the fact that a point is black there
corresponds a positive fact, and to the fact that a point is white (not black),
a negative fact. If I designate a point on the sheet (a truth-value according to
Frege), then this corresponds to the supposition that is put forward for
judgement, etc. etc. But in order to be able to say that a point is black or
white, I must first know when a point is called black, and when white: in order
to be able to say,'"p" is true (or false)', I must have determined in what
circumstances I call 'p' true, and in so doing I determine the sense of the
proposition. Now the point where the simile breaks down is this: we can indicate
a point on the paper even if we do not know what black and white are, but if a
proposition has no sense, nothing corresponds to it, since it does not designate
a thing (a truth-value) which might have properties called 'false' or 'true'.
The verb of a proposition is not 'is true' or 'is false', as Frege thought:
rather, that which 'is true' must already contain the verb.

4.064 Every proposition must already have a sense: it cannot be given a sense
by affirmation. Indeed its sense is just what is affirmed. And the same applies
to negation, etc.

4.0641 One could say that negation must be related to the logical place
determined by the negated proposition. The negating proposition determines a
logical place different from that of the negated proposition. The negating
proposition determines a logical place with the help of the logical place of the
negated proposition. For it describes it as lying outside the latter's logical
place. The negated proposition can be negated again, and this in itself shows
that what is negated is already a proposition, and not merely something that is
preliminary to a proposition.

4.1 Propositions represent the existence and non-existence of states of
affairs.

4.11 The totality of true propositions is the whole of natural science (or
the whole corpus of the natural sciences).

4.111 Philosophy is not one of the natural sciences. (The word 'philosophy'
must mean something whose place is above or below the natural sciences, not
beside them.)

4.112 Philosophy aims at the logical clarification of thoughts. Philosophy is
not a body of doctrine but an activity. A philosophical work consists
essentially of elucidations. Philosophy does not result in 'philosophical
propositions', but rather in the clarification of propositions. Without
philosophy thoughts are, as it were, cloudy and indistinct: its task is to make
them clear and to give them sharp boundaries.

4.1121 Psychology is no more closely related to philosophy than any other
natural science. Theory of knowledge is the philosophy of psychology. Does not
my study of sign-language correspond to the study of thought-processes, which
philosophers used to consider so essential to the philosophy of logic? Only in
most cases they got entangled in unessential psychological investigations, and
with my method too there is an analogous risk.

4.1122 Darwin's theory has no more to do with philosophy than any other
hypothesis in natural science.

4.113 Philosophy sets limits to the much disputed sphere of natural science.

4.114 It must set limits to what can be thought; and, in doing so, to what
cannot be thought. It must set limits to what cannot be thought by working
outwards through what can be thought.

4.115 It will signify what cannot be said, by presenting clearly what can be
said.

4.116 Everything that can be thought at all can be thought clearly.
Everything that can be put into words can be put clearly. 4.12 Propositions can
represent the whole of reality, but they cannot represent what they must have in
common with reality in order to be able to represent it—logical form. In order
to be able to represent logical form, we should have to be able to station
ourselves with propositions somewhere outside logic, that is to say outside the
world.

4.121 Propositions cannot represent logical form: it is mirrored in them.
What finds its reflection in language, language cannot represent. What expresses
itself in language, we cannot express by means of language. Propositions show
the logical form of reality. They display it.

4.1211 Thus one proposition 'fa' shows that the object a occurs in its sense,
two propositions 'fa' and 'ga' show that the same object is mentioned in both of
them. If two propositions contradict one another, then their structure shows it;
the same is true if one of them follows from the other. And so on.

4.1212 What can be shown, cannot be said.

4.1213 Now, too, we understand our feeling that once we have a sign-language
in which everything is all right, we already have a correct logical point of
view.

4.122 In a certain sense we can talk about formal properties of objects and
states of affairs, or, in the case of facts, about structural properties: and in
the same sense about formal relations and structural relations. (Instead of
'structural property' I also say 'internal property'; instead of 'structural
relation', 'internal relation'. I introduce these expressions in order to
indicate the source of the confusion between internal relations and relations
proper (external relations), which is very widespread among philosophers.) It is
impossible, however, to assert by means of propositions that such internal
properties and relations obtain: rather, this makes itself manifest in the
propositions that represent the relevant states of affairs and are concerned
with the relevant objects.

4.1221 An internal property of a fact can also be bed a feature of that fact
(in the sense in which we speak of facial features, for example).

4.123 A property is internal if it is unthinkable that its object should not
possess it. (This shade of blue and that one stand, eo ipso, in the internal
relation of lighter to darker. It is unthinkable that these two objects should
not stand in this relation.) (Here the shifting use of the word 'object'
corresponds to the shifting use of the words 'property' and 'relation'.)

4.124 The existence of an internal property of a possible situation is not
expressed by means of a proposition: rather, it expresses itself in the
proposition representing the situation, by means of an internal property of that
proposition. It would be just as nonsensical to assert that a proposition had a
formal property as to deny it.

4.1241 It is impossible to distinguish forms from one another by saying that
one has this property and another that property: for this presupposes that it
makes sense to ascribe either property to either form.

4.125 The existence of an internal relation between possible situations
expresses itself in language by means of an internal relation between the
propositions representing them.

4.1251 Here we have the answer to the vexed question 'whether all relations
are internal or external'.

4.1252 I call a series that is ordered by an internal relation a series of
forms. The order of the number-series is not governed by an external relation
but by an internal relation. The same is true of the series of propositions
'aRb', '(d: c): aRx. xRb', '(d x,y): aRx. xRy. yRb', and so forth. (If b stands
in one of these relations to a, I call b a successor of a.)

4.126 We can now talk about formal concepts, in the same sense that we speak
of formal properties. (I introduce this expression in order to exhibit the
source of the confusion between formal concepts and concepts proper, which
pervades the whole of traditional logic.) When something falls under a formal
concept as one of its objects, this cannot be expressed by means of a
proposition. Instead it is shown in the very sign for this object. (A name shows
that it signifies an object, a sign for a number that it signifies a number,
etc.) Formal concepts cannot, in fact, be represented by means of a function, as
concepts proper can. For their characteristics, formal properties, are not
expressed by means of functions. The expression for a formal property is a
feature of certain symbols. So the sign for the characteristics of a formal
concept is a distinctive feature of all symbols whose meanings fall under the
concept. So the expression for a formal concept is a propositional variable in
which this distinctive feature alone is constant.

4.127 The propositional variable signifies the formal concept, and its values
signify the objects that fall under the concept.

4.1271 Every variable is the sign for a formal concept. For every variable
represents a constant form that all its values possess, and this can be regarded
as a formal property of those values.

4.1272 Thus the variable name 'x' is the proper sign for the pseudo-concept
object. Wherever the word 'object' ('thing', etc.) is correctly used, it is
expressed in conceptual notation by a variable name. For example, in the
proposition, 'There are 2 objects which.. .', it is expressed by ' (dx,y)... '.
Wherever it is used in a different way, that is as a proper concept-word,
nonsensical pseudo-propositions are the result. So one cannot say, for example,
'There are objects', as one might say, 'There are books'. And it is just as
impossible to say, 'There are 100 objects', or, 'There are!0 objects'. And it is
nonsensical to speak of the total number of objects. The same applies to the
words 'complex', 'fact', 'function', 'number', etc. They all signify formal
concepts, and are represented in conceptual notation by variables, not by
functions or classes (as Frege and Russell believed). '1 is a number', 'There is
only one zero', and all similar expressions are nonsensical. (It is just as
nonsensical to say, 'There is only one 1', as it would be to say, '2 + 2 at 3
o'clock equals 4'.)

4.12721 A formal concept is given immediately any object falling under it is
given. It is not possible, therefore, to introduce as primitive ideas objects
belonging to a formal concept and the formal concept itself. So it is
impossible, for example, to introduce as primitive ideas both the concept of a
function and specific functions, as Russell does; or the concept of a number and
particular numbers.

4.1273 If we want to express in conceptual notation the general proposition,
'b is a successor of a', then we require an expression for the general term of
the series of forms 'aRb', '(d: c): aRx. xRb', '(d x,y) : aRx. xRy. yRb',..., In
order to express the general term of a series of forms, we must use a variable,
because the concept 'term of that series of forms' is a formal concept. (This is
what Frege and Russell overlooked: consequently the way in which they want to
express general propositions like the one above is incorrect; it contains a
vicious circle.) We can determine the general term of a series of forms by
giving its first term and the general form of the operation that produces the
next term out of the proposition that precedes it.

4.1274 To ask whether a formal concept exists is nonsensical. For no
proposition can be the answer to such a question. (So, for example, the
question, 'Are there unanalysable subject-predicate propositions?' cannot be
asked.)

4.128 Logical forms are without number. Hence there are no pre-eminent
numbers in logic, and hence there is no possibility of philosophical monism or
dualism, etc.

4.2 The sense of a proposition is its agreement and disagreement with
possibilities of existence and non-existence of states of affairs. 4.21 The
simplest kind of proposition, an elementary proposition, asserts the existence
of a state of affairs.

4.211 It is a sign of a proposition's being elementary that there can be no
elementary proposition contradicting it.

4.22 An elementary proposition consists of names. It is a nexus, a
concatenation, of names.

4.221 It is obvious that the analysis of propositions must bring us to
elementary propositions which consist of names in immediate combination. This
raises the question how such combination into propositions comes about.

4.2211 Even if the world is infinitely complex, so that every fact consists
of infinitely many states of affairs and every state of affairs is composed of
infinitely many objects, there would still have to be objects and states of
affairs.

4.23 It is only in the nexus of an elementary proposition that a name occurs
in a proposition.

4.24 Names are the simple symbols: I indicate them by single letters ('x',
'y', 'z'). I write elementary propositions as functions of names, so that they
have the form 'fx', 'O (x,y)', etc. Or I indicate them by the letters 'p', 'q',
'r'.

4.241 When I use two signs with one and the same meaning, I express this by
putting the sign '=' between them. So 'a = b' means that the sign 'b' can be
substituted for the sign 'a'. (If I use an equation to introduce a new sign 'b',
laying down that it shall serve as a substitute for a sign a that is already
known, then, like Russell, I write the equation—definition—in the form 'a = b
Def.' A definition is a rule dealing with signs.)

4.242 Expressions of the form 'a = b' are, therefore, mere representational
devices. They state 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 understand a proposition in which two
names occur without knowing whether their meaning is the same or different?
Suppose I know the meaning of an English word and of a German word that means
the same: then it is impossible for me to be unaware that they do mean the same;
I must be capable of translating each into the other. Expressions like 'a = a',
and those derived from them, are neither elementary propositions nor is there
any other way in which they have sense. (This will become evident later.)

4.25 If an elementary proposition is true, the state of affairs exists: if an
elementary proposition is false, the state of affairs does not exist.

4.26 If all true elementary propositions are given, the result is a complete
description of the world. The world is completely described by giving all
elementary propositions, and adding which of them are true and which false. For
n states of affairs, there are possibilities of existence and non-existence. Of
these states of affairs any combination can exist and the remainder not exist.

4.28 There correspond to these combinations the same number of possibilities
of truth—and falsity—for n elementary propositions.

4.3 Truth-possibilities of elementary propositions mean Possibilities of
existence and non-existence of states of affairs.

4.31 We can represent truth-possibilities by schemata of the following kind
('T' means 'true', 'F' means 'false'; the rows of 'T's' and 'F's' under the row
of elementary propositions symbolize their truth-possibilities in a way that can
easily be understood):

4.4 A proposition is an expression of agreement and disagreement with
truth-possibilities of elementary propositions.

4.41 Truth-possibilities of elementary propositions are the conditions of the
truth and falsity of propositions.

4.411 It immediately strikes one as probable that the introduction of
elementary propositions provides the basis for understanding all other kinds of
proposition. Indeed the understanding of general propositions palpably depends
on the understanding of elementary propositions.

4.42 For n elementary propositions there are ways in which a proposition can
agree and disagree with their truth possibilities.

4.43 We can express agreement with truth-possibilities by correlating the
mark 'T' (true) with them in the schema. The absence of this mark means
disagreement.

4.431 The expression of agreement and disagreement with the truth
possibilities of elementary propositions expresses the truth-conditions of a
proposition. A proposition is the expression of its truth-conditions. (Thus
Frege was quite right to use them as a starting point when he explained the
signs of his conceptual notation. But the explanation of the concept of truth
that Frege gives is mistaken: if 'the true' and 'the false' were really objects,
and were the arguments in Pp etc., then Frege's method of determining the sense
of 'Pp' would leave it absolutely undetermined.)

4.44 The sign that results from correlating the mark 'I' with
truth-possibilities is a propositional sign.

4.441 It is clear that a complex of the signs 'F' and 'T' has no object (or
complex of objects) corresponding to it, just as there is none corresponding to
the horizontal and vertical lines or to the brackets.—There are no 'logical
objects'. Of course the same applies to all signs that express what the schemata
of 'T's' and 'F's' express.

4.442 For example, the following is a propositional sign: (Frege's 'judgement
stroke' '|-' is logically quite meaningless: in the works of Frege (and Russell)
it simply indicates that these authors hold the propositions marked with this
sign to be true. Thus '|-' is no more a component part of a proposition than is,
for instance, the proposition's number. It is quite impossible for a proposition
to state that it itself is true.) If the order or the truth-possibilities in a
scheme is fixed once and for all by a combinatory rule, then the last column by
itself will be an expression of the truth-conditions. If we now write this
column as a row, the propositional sign will become '(TT-T) (p,q)' or more
explicitly '(TTFT) (p,q)' (The number of places in the left-hand pair of
brackets is determined by the number of terms in the right-hand pair.)

4.45 For n elementary propositions there are Ln possible groups of
truth-conditions. The groups of truth-conditions that are obtainable from the
truth-possibilities of a given number of elementary propositions can be arranged
in a series.

4.46 Among the possible groups of truth-conditions there are two extreme
cases. In one of these cases the proposition is true for all the
truth-possibilities 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 contradictory. In the
first case we call the proposition a tautology; in the second, a contradiction.

4.461 Propositions show what they say; tautologies and contradictions show
that they say nothing. A tautology has no truth-conditions, since it is
unconditionally true: and a contradiction is true on no condition. Tautologies
and contradictions lack sense. (Like a point from which two arrows go out in
opposite directions to one another.) (For example, I know nothing about the
weather when I know that it is either raining or not raining.)

4.46211 Tautologies and contradictions are not, however, nonsensical. They
are part of the symbolism, much as '0' is part of the symbolism of arithmetic.

4.462 Tautologies and contradictions are not pictures of reality. They do not
represent any possible situations. For the former admit all possible situations,
and latter none. In a tautology the conditions of agreement with the world—the
representational relations—cancel one another, so that it does not stand in any
representational relation to reality.

4.463 The truth-conditions of a proposition determine the range that it
leaves open to the facts. (A proposition, a picture, or a model is, in the
negative sense, like a solid body that restricts the freedom of movement of
others, and in the positive sense, like a space bounded by solid substance in
which there is room for a body.) A tautology leaves open to reality the
whole—the infinite whole—of logical space: a contradiction fills the whole of
logical space leaving no point of it for reality. Thus neither of them can
determine reality in any way.

4.464 A tautology's truth is certain, a proposition's possible, a
contradiction's impossible. (Certain, possible, impossible: here we have the
first indication of the scale that we need in the theory of probability.)

4.465 The logical product of a tautology and a proposition says the same
thing as the proposition. This product, therefore, is identical with the
proposition. For it is impossible to alter what is essential to a symbol without
altering its sense.

4.466 What corresponds to a determinate logical combination of signs is a
determinate logical combination of their meanings. It is only to the uncombined
signs that absolutely any combination corresponds. In other words, propositions
that are true for every situation cannot be combinations of signs at all, since,
if they were, only determinate combinations of objects could correspond to them.
(And what is not a logical combination has no combination of objects
corresponding to it.) Tautology and contradiction are the limiting cases—indeed
the disintegration—of the combination of signs.

4.4661 Admittedly the signs are still combined with one another even in
tautologies and contradictions—i.e. they stand in certain relations to one
another: but these relations have no meaning, they are not essential to the
symbol.

4.5 It now seems possible to give the most general propositional form: that
is, to give a description of the propositions of any sign-language whatsoever in
such a way that every possible sense can be expressed by a symbol satisfying the
description, and every symbol satisfying the description can express a sense,
provided that the meanings of the names are suitably chosen. It is clear that
only what is essential to the most general propositional form may be included in
its description—for otherwise it would not be the most general form. The
existence of a general propositional form is proved by the fact that there
cannot be a proposition whose form could not have been foreseen (i.e.
constructed). The general form of a proposition is: This is how things stand.

4.51 Suppose that I am given all elementary propositions: then I can simply
ask what propositions I can construct out of them. And there I have all
propositions, and that fixes their limits.

4.52 Propositions comprise all that follows from the totality of all
elementary propositions (and, of course, from its being the totality of them all
). (Thus, in a certain sense, it could be said that all propositions were
generalizations of elementary propositions.)

4.53 The general propositional form is a variable.

## 5. A proposition is a truth-function of elementary propositions.

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

5.01 Elementary propositions are the truth-arguments of propositions.

5.02 The arguments of functions are readily confused with the affixes of
names. For both arguments and affixes enable me to recognize the meaning of the
signs containing them. For example, when Russell writes '+c', the 'c' is an
affix which indicates that the sign as a whole is the addition-sign for cardinal
numbers. But the use of this sign is the result of arbitrary convention and it
would be quite possible to choose a simple sign instead of '+c'; in 'Pp'
however, 'p' is not an affix but an argument: the sense of 'Pp' cannot be
understood unless the sense of 'p' has been understood already. (In the name
Julius Caesar 'Julius' is an affix. An affix is always part of a description of
the object to whose name we attach it: e.g. the Caesar of the Julian gens.) If I
am not mistaken, Frege's theory about the meaning of propositions and functions
is based on the confusion between an argument and an affix. Frege regarded the
propositions of logic as names, and their arguments as the affixes of those
names.

5.1 Truth-functions can be arranged in series. That is the foundation of the
theory of probability.

5.101 The truth-functions of a given number of elementary propositions can
always be set out in a schema of the following kind: (TTTT) (p, q) Tautology (If
p then p, and if q then q.) (p z p. q z q) (FTTT) (p, q) In words: Not both p
and q. (P(p. q)) (TFTT) (p, q) ": If q then p. (q z p) (TTFT) (p, q) ": If p
then q. (p z q) (TTTF) (p, q) ": p or q. (p C q) (FFTT) (p, q) ": Not g. (Pq)
(FTFT) (p, q) ": Not p. (Pp) (FTTF) (p, q) " : p or q, but not both. (p. Pq: C:
q. Pp) (TFFT) (p, q) ": If p then p, and if q then p. (p + q) (TFTF) (p, q) ": p
(TTFF) (p, q) ": q (FFFT) (p, q) ": Neither p nor q. (Pp. Pq or p | q) (FFTF)
(p, q) ": p and not q. (p. Pq) (FTFF) (p, q) ": q and not p. (q. Pp) (TFFF)
(p,q) ": q and p. (q. p) (FFFF) (p, q) Contradiction (p and not p, and q and not
q.) (p. Pp. q. Pq) I will give the name truth-grounds of a proposition to those
truth-possibilities of its truth-arguments that make it true.

5.11 If all the truth-grounds that are common to a number of propositions are
at the same time truth-grounds of a certain proposition, then we say that the
truth of that proposition follows from the truth of the others.

5.12 In particular, the truth of a proposition 'p' follows from the truth of
another proposition 'q' is all the truth-grounds of the latter are truth-grounds
of the former.

5.121 The truth-grounds of the one are contained in those of the other: p
follows from q.

5.122 If p follows from q, the sense of 'p' is contained in the sense of 'q'.

5.123 If a god creates a world in which certain propositions are true, then
by that very act he also creates a world in which all the propositions that
follow from them come true. And similarly he could not create a world in which
the proposition 'p' was true without creating all its objects.

5.124 A proposition affirms every proposition that follows from it.

5.1241 'p. q' is one of the propositions that affirm 'p' and at the same time
one of the propositions that affirm 'q'. Two propositions are opposed to one
another if there is no proposition with a sense, that affirms them both. Every
proposition that contradicts another negate it.

5.13 When the truth of one proposition follows from the truth of others, we
can see this from the structure of the proposition.

5.131 If the truth of one proposition follows from the truth of others, this
finds expression in relations in which the forms of the propositions stand to
one another: nor is it necessary for us to set up these relations between them,
by combining them with one another in a single proposition; on the contrary, the
relations are internal, and their existence is an immediate result of the
existence of the propositions.

5.1311 When we infer q from p C q and Pp, the relation between the
propositional forms of 'p C q' and 'Pp' is masked, in this case, by our mode of
signifying. But if instead of 'p C q' we write, for example, 'p|q. |. p|q', and
instead of 'Pp', 'p|p' (p|q = neither p nor q), then the inner connexion becomes
obvious. (The possibility of inference from (x). fx to fa shows that the symbol
(x). fx itself has generality in it.)

5.132 If p follows from q, I can make an inference from q to p, deduce p from
q. The nature of the inference can be gathered only from the two propositions.
They themselves are the only possible justification of the inference. 'Laws of
inference', which are supposed to justify inferences, as in the works of Frege
and Russell, have no sense, and would be superfluous.

5.133 All deductions are made a priori.

5.134 One elementary proposition cannot be deduced form another.

5.135 There is no possible way of making an inference form the existence of
one situation to the existence of another, entirely different situation.

5.136 There is no causal nexus to justify such an inference.

5.1361 We cannot infer the events of the future from those of the present.
Belief in the causal nexus is superstition.

5.1362 The freedom of the will consists in the impossibility of knowing
actions that still lie in the future. We could know them only if causality were
an inner necessity like that of logical inference.—The connexion between
knowledge and what is known is that of logical necessity. ('A knows that p is
the case', has no sense if p is a tautology.)

5.1363 If the truth of a proposition does not follow from the fact that it is
self-evident to us, then its self-evidence in no way justifies our belief in its
truth.

5.14 If one proposition follows from another, then the latter says more than
the former, and the former less than the latter.

5.141 If p follows from q and q from p, then they are one and same
proposition.

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

5.143 Contradiction is that common factor of propositions which no
proposition has in common with another. Tautology is the common factor of all
propositions that have nothing in common with one another. Contradiction, one
might say, vanishes outside all propositions: tautology vanishes inside them.
Contradiction is the outer limit of propositions: tautology is the unsubstantial
point at their centre.

5.15 If Tr is the number of the truth-grounds of a proposition 'r', and if
Trs is the number of the truth-grounds of a proposition 's' that are at the same
time truth-grounds of 'r', then we call the ratio Trs: Tr the degree of
probability that the proposition 'r' gives to the proposition 's'. 5.151 In a
schema like the one above in

5.101, let Tr be the number of 'T's' in the proposition r, and let Trs, be
the number of 'T's' in the proposition s that stand in columns in which the
proposition r has 'T's'. Then the proposition r gives to the proposition s the
probability Trs: Tr.

5.1511 There is no special object peculiar to probability propositions.

5.152 When propositions have no truth-arguments in common with one another,
we call them independent of one another. Two elementary propositions give one
another the probability 1/2. If p follows from q, then the proposition 'q' gives
to the proposition 'p' the probability 1. The certainty of logical inference is
a limiting case of probability. (Application of this to tautology and
contradiction.)

5.153 In itself, a proposition is neither probable nor improbable. Either an
event occurs or it does not: there is no middle way.

5.154 Suppose that an urn contains black and white balls in equal numbers
(and none of any other kind). I draw one ball after another, putting them back
into the urn. By this experiment I can establish that the number of black balls
drawn and the number of white balls drawn approximate to one another as the draw
continues. So this is not a mathematical truth. Now, if I say, 'The probability
of my drawing a white ball is equal to the probability of my drawing a black
one', this means that all the circumstances that I know of (including the laws
of nature assumed as hypotheses) give no more probability to the occurrence of
the one event than to that of the other. That is to say, they give each the
probability 1/2 as can easily be gathered from the above definitions. What I
confirm by the experiment is that the occurrence of the two events is
independent of the circumstances of which I have no more detailed knowledge.

5.155 The minimal unit for a probability proposition is this: The
circumstances—of which I have no further knowledge—give such and such a degree
of probability to the occurrence of a particular event.

5.156 It is in this way that probability is a generalization. It involves a
general description of a propositional form. We use probability only in default
of certainty—if our knowledge of a fact is not indeed complete, but we do know
something about its form. (A proposition may well be an incomplete picture of a
certain situation, but it is always a complete picture of something.) A
probability proposition is a sort of excerpt from other propositions.

5.2 The structures of propositions stand in internal relations to one
another.

5.21 In order to give prominence to these internal relations we can adopt the
following mode of expression: we can represent a proposition as the result of an
operation that produces it out of other propositions (which are the bases of the
operation).

5.22 An operation is the expression of a relation between the structures of
its result and of its bases.

5.23 The operation is what has to be done to the one proposition in order to
make the other out of it.

5.231 And that will, of course, depend on their formal properties, on the
internal similarity of their forms.

5.232 The internal relation by which a series is ordered is equivalent to the
operation that produces one term from another.

5.233 Operations cannot make their appearance before the point at which one
proposition is generated out of another in a logically meaningful way; i.e. the
point at which the logical construction of propositions begins.

5.234 Truth-functions of elementary propositions are results of operations
with elementary propositions as bases. (These operations I call
truth-operations.)

5.2341 The sense of a truth-function of p is a function of the sense of p.
Negation, logical addition, logical multiplication, etc. etc. are operations.
(Negation reverses the sense of a proposition.)

5.24 An operation manifests itself in a variable; it shows how we can get
from one form of proposition to another. It gives expression to the difference
between the forms. (And what the bases of an operation and its result have in
common is just the bases themselves.)

5.241 An operation is not the mark of a form, but only of a difference
between forms.

5.242 The operation that produces 'q' from 'p' also produces 'r' from 'q',
and so on. There is only one way of expressing this: 'p', 'q', 'r', etc. have to
be variables that give expression in a general way to certain formal relations.

5.25 The occurrence of an operation does not characterize the sense of a
proposition. Indeed, no statement is made by an operation, but only by its
result, and this depends on the bases of the operation. (Operations and
functions must not be confused with each other.)

5.251 A function cannot be its own argument, whereas an operation can take
one of its own results as its base.

5.252 It is only in this way that the step from one term of a series of forms
to another is possible (from one type to another in the hierarchies of Russell
and Whitehead). (Russell and Whitehead did not admit the possibility of such
steps, but repeatedly availed themselves of it.)

5.2521 If an operation is applied repeatedly to its own results, I speak of
successive applications of it. ('O'O'O'a' is the result of three successive
applications of the operation 'O'E' to 'a'.) In a similar sense I speak of
successive applications of more than one operation to a number of propositions.

5.2522 Accordingly I use the sign '[a, x, O'x]' for the general term of the
series of forms a, O'a, O'O'a,.... This bracketed expression is a variable: the
first term of the bracketed expression is the beginning of the series of forms,
the second is the form of a term x arbitrarily selected from the series, and the
third is the form of the term that immediately follows x in the series.

5.2523 The concept of successive applications of an operation is equivalent
to the concept 'and so on'.

5.253 One operation can counteract the effect of another. Operations can
cancel one another.

5.254 An operation can vanish (e.g. negation in 'PPp': PPp = p).

5.3 All propositions are results of truth-operations on elementary
propositions. A truth-operation is the way in which a truth-function is produced
out of elementary propositions. It is of the essence of truth-operations that,
just as elementary propositions yield a truth-function of themselves, so too in
the same way truth-functions yield a further truth-function. When a
truth-operation is applied to truth-functions of elementary propositions, it
always generates another truth-function of elementary propositions, another
proposition. When a truth-operation is applied to the results of
truth-operations on elementary propositions, there is always a single operation
on elementary propositions that has the same result. Every proposition is the
result of truth-operations on elementary propositions.

5.31 The schemata in 4.31 have a meaning even when 'p', 'q', 'r', etc. are
not elementary propositions. And it is easy to see that the propositional sign
in 4.442 expresses a single truth-function of elementary propositions even when
'p' and 'q' are truth-functions of elementary propositions.

5.32 All truth-functions are results of successive applications to elementary
propositions of a finite number of truth-operations.

5.4 At this point it becomes manifest that there are no 'logical objects' or
'logical constants' (in Frege's and Russell's sense).

5.41 The reason is that the results of truth-operations on truth-functions
are always identical whenever they are one and the same truth-function of
elementary propositions.

5.42 It is self-evident that C, z, etc. are not relations in the sense in
which right and left etc. are relations. The interdefinability of Frege's and
Russell's 'primitive signs' of logic is enough to show that they are not
primitive signs, still less signs for relations. And it is obvious that the 'z'
defined by means of 'P' and 'C' is identical with the one that figures with 'P'
in the definition of 'C'; and that the second 'C' is identical with the first
one; and so on.

5.43 Even at first sight it seems scarcely credible that there should follow
from one fact p infinitely many others, namely PPp, PPPPp, etc. And it is no
less remarkable that the infinite number of propositions of logic (mathematics)
follow from half a dozen 'primitive propositions'. But in fact all the
propositions of logic say the same thing, to wit nothing.

5.44 Truth-functions are not material functions. For example, an affirmation
can be produced by double negation: in such a case does it follow that in some
sense negation is contained in affirmation? Does 'PPp' negate Pp, or does it
affirm p—or both? The proposition 'PPp' is not about negation, as if negation
were an object: on the other hand, the possibility of negation is already
written into affirmation. And if there were an object called 'P', it would
follow that 'PPp' said something different from what 'p' said, just because the
one proposition would then be about P and the other would not.

5.441 This vanishing of the apparent logical constants also occurs in the
case of 'P(dx). Pfx', which says the same as '(x). fx', and in the case of
'(dx). fx. x = a', which says the same as 'fa'.

5.442 If we are given a proposition, then with it we are also given the
results of all truth-operations that have it as their base.

5.45 If there are primitive logical signs, then any logic that fails to show
clearly how they are placed relatively to one another and to justify their
existence will be incorrect. The construction of logic out of its primitive
signs must be made clear.

5.451 If logic has primitive ideas, they must be independent of one another.
If a primitive idea has been introduced, it must have been introduced in all the
combinations in which it ever occurs. It cannot, therefore, be introduced first
for one combination and later reintroduced for another. For example, once
negation has been introduced, we must understand it both in propositions of the
form 'Pp' and in propositions like 'P(p C q)', '(dx). Pfx', etc. We must not
introduce it first for the one class of cases and then for the other, since it
would then be left in doubt whether its meaning were the same in both cases, and
no reason would have been given for combining the signs in the same way in both
cases. (In short, Frege's remarks about introducing signs by means of
definitions (in The Fundamental Laws of Arithmetic ) also apply, mutatis
mutandis, to the introduction of primitive signs.)

5.452 The introduction of any new device into the symbolism of logic is
necessarily a momentous event. In logic a new device should not be introduced in
brackets or in a footnote with what one might call a completely innocent air.
(Thus in Russell and Whitehead's Principia Mathematica there occur definitions
and primitive propositions expressed in words. Why this sudden appearance of
words? It would require a justification, but none is given, or could be given,
since the procedure is in fact illicit.) But if the introduction of a new device
has proved necessary at a certain point, we must immediately ask ourselves, 'At
what points is the employment of this device now unavoidable?' and its place in
logic must be made clear.

5.453 All numbers in logic stand in need of justification. Or rather, it must
become evident that there are no numbers in logic. There are no pre-eminent
numbers.

5.454 In logic there is no co-ordinate status, and there can be no
classification. In logic there can be no distinction between the general and the
specific.

5.4541 The solutions of the problems of logic must be simple, since they set
the standard of simplicity. Men have always had a presentiment that there must
be a realm in which the answers to questions are symmetrically combined—a
priori—to form a self-contained system. A realm subject to the law: Simplex
sigillum veri.

5.46 If we introduced logical signs properly, then we should also have
introduced at the same time the sense of all combinations of them; i.e. not only
'p C q' but 'P(p C q)' as well, etc. etc. We should also have introduced at the
same time the effect of all possible combinations of brackets. And thus it would
have been made clear that the real general primitive signs are not 'p C q',
'(dx). fx', etc. but the most general form of their combinations.

5.461 Though it seems unimportant, it is in fact significant that the
pseudo-relations of logic, such as C and z, need brackets—unlike real relations.
Indeed, the use of brackets with these apparently primitive signs is itself an
indication that they are not primitive signs. And surely no one is going to
believe brackets have an independent meaning. 5.4611 Signs for logical
operations are punctuation-marks.

5.47 It is clear that whatever we can say in advance about the form of all
propositions, we must be able to say all at once. An elementary proposition
really contains all logical operations in itself. For 'fa' says the same thing
as '(dx). fx. x = a' Wherever there is compositeness, argument and function are
present, and where these are present, we already have all the logical constants.
One could say that the sole logical constant was what all propositions, by their
very nature, had in common with one another. But that is the general
propositional form.

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

5.4711 To give the essence of a proposition means to give the essence of all
description, and thus 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 look after itself. If a sign is possible, then it is also
capable of signifying. Whatever is possible in logic is also permitted. (The
reason why 'Socrates is identical' means nothing is that there is no property
called 'identical'. The proposition is nonsensical because we have failed to
make an arbitrary determination, and not because the symbol, in itself, would be
illegitimate.) In a certain sense, we cannot make mistakes in logic.

5.4731 Self-evidence, which Russell talked about so much, can become
dispensable in logic, only because language itself prevents every logical
mistake.—What makes logic a priori is the impossibility of illogical thought.

5.4732 We cannot give a sign the wrong sense.

5,47321 Occam's maxim is, of course, not an arbitrary rule, nor one that is
justified by its success in practice: its point is that unnecessary units in a
sign-language mean nothing. Signs that serve one purpose are logically
equivalent, and signs that serve none are logically meaningless.

5.4733 Frege says that any legitimately constructed proposition must have a
sense. And I say that any possible proposition is legitimately constructed, and,
if it has no sense, that can only be because we have failed to give a meaning to
some of its constituents. (Even if we think that we have done so.) Thus the
reason why 'Socrates is identical' says nothing is that we have not given any
adjectival meaning to the word 'identical'. For when it appears as a sign for
identity, it symbolizes in an entirely different way—the signifying relation is
a different one—therefore the symbols also are entirely different in the two
cases: the two symbols have only the sign in common, and that is an accident.

5.474 The number of fundamental operations that are necessary depends solely
on our notation.

5.475 All that is required is that we should construct a system of signs with
a particular number of dimensions—with a particular mathematical multiplicity.

5.476 It is clear that this is not a question of a number of primitive ideas
that have to be signified, but rather of the expression of a rule.

5.5 Every truth-function is a result of successive applications to elementary
propositions of the operation '(——-T)(E,....)'. This operation negates all the
propositions in the right-hand pair of brackets, and I call it the negation of
those propositions.

5.501 When a bracketed expression has propositions as its terms—and the order
of the terms inside the brackets is indifferent—then I indicate it by a sign of
the form '(E)'. '(E)' is a variable whose values are terms of the bracketed
expression and the bar over the variable indicates that it is the representative
of all its values in the brackets. (E.g. if E has the three values P,Q, R, then
(E) = (P, Q, R). ) What the values of the variable are is something that is
stipulated. The stipulation is a description of the propositions that have the
variable as their representative. How the description of the terms of the
bracketed expression is produced is not essential. We can distinguish three
kinds of description: 1. Direct enumeration, in which case we can simply
substitute for the variable the constants that are its values; 2. Giving a
function fx whose values for all values of x are the propositions to be
described; 3. Giving a formal law that governs the construction of the
propositions, in which case the bracketed expression has as its members all the
terms of a series of forms.

5.502 So instead of '(——-T)(E,....)', I write 'N(E)'. N(E) is the negation of
all the values of the propositional variable E.

5.503 It is obvious that we can easily express how propositions may be
constructed with this operation, and how they may not be constructed with it; so
it must be possible to find an exact expression for this.

5.51 If E has only one value, then N(E) = Pp (not p); if it has two values,
then N(E) = Pp. Pq. (neither p nor g).

5.511 How can logic—all-embracing logic, which mirrors the world—use such
peculiar crotchets and contrivances? Only because they are all connected with
one another in an infinitely fine network, the great mirror.

5.512 'Pp' is true if 'p' is false. Therefore, in the proposition 'Pp', when
it is true, 'p' is a false proposition. How then can the stroke 'P' make it
agree with reality? But in 'Pp' it is not 'P' that negates, it is rather what is
common to all the signs of this notation that negate p. That is to say the
common rule that governs the construction of 'Pp', 'PPPp', 'Pp C Pp', 'Pp. Pp',
etc. etc. (ad inf.). And this common factor mirrors negation.

5.513 We might say that what is common to all symbols that affirm both p and
q is the proposition 'p. q'; and that what is common to all symbols that affirm
either p or q is the proposition 'p C q'. And similarly we can say that two
propositions are opposed to one another if they have nothing in common with one
another, and that every proposition has only one negative, since there is only
one proposition that lies completely outside it. Thus in Russell's notation too
it is manifest that 'q: p C Pp' says the same thing as 'q', that 'p C Pq' says
nothing.

5.514 Once a notation has been established, there will be in it a rule
governing the construction of all propositions that negate p, a rule governing
the construction of all propositions that affirm p, and a rule governing the
construction of all propositions that affirm p or q; and so on. These rules are
equivalent to the symbols; and in them their sense is mirrored.

5.515 It must be manifest in our symbols that it can only be propositions
that are combined with one another by 'C', '.', etc. And this is indeed the
case, since the symbol in 'p' and 'q' itself presupposes 'C', 'P', etc. If the
sign 'p' in 'p C q' does not stand for a complex sign, then it cannot have sense
by itself: but in that case the signs 'p C p', 'p. p', etc., which have the same
sense as p, must also lack sense. But if 'p C p' has no sense, then 'p C q'
cannot have a sense either.

5.5151 Must the sign of a negative proposition be constructed with that of
the positive proposition? Why should it not be possible to express a negative
proposition by means of a negative fact? (E.g. suppose that "a' does not stand
in a certain relation to 'b'; then this might be used to say that aRb was not
the case.) But really even in this case the negative proposition is constructed
by an indirect use of the positive. The positive proposition necessarily
presupposes the existence of the negative proposition and vice versa.

5.52 If E has as its values all the values of a function fx for all values of
x, then N(E) = P(dx). fx.

5.521 I dissociate the concept all from truth-functions. Frege and Russell
introduced generality in association with logical productor logical sum. This
made it difficult to understand the propositions '(dx). fx' and '(x) . fx', in
which both ideas are embedded.

5.522 What is peculiar to the generality-sign is first, that it indicates a
logical prototype, and secondly, that it gives prominence to constants.

5.523 The generality-sign occurs as an argument.

5.524 If objects are given, then at the same time we are given all objects.
If elementary propositions are given, then at the same time all elementary
propositions are given.

5.525 It is incorrect to render the proposition '(dx). fx' in the words, 'fx
is possible' as Russell does. The certainty, possibility, or impossibility of a
situation is not expressed by a proposition, but by an expression's being a
tautology, a proposition with a sense, or a contradiction. The precedent to
which we are constantly inclined to appeal must reside in the symbol itself.

5.526 We can describe the world completely by means of fully generalized
propositions, i.e. without first correlating any name with a particular object.

5.5261 A fully generalized proposition, like every other proposition, is
composite. (This is shown by the fact that in '(dx, O). Ox' we have to mention
'O' and 's' separately. They both, independently, stand in signifying relations
to the world, just as is the case in ungeneralized propositions.) It is a mark
of a composite symbol that it has something in common with other symbols.

5.5262 The truth or falsity of every proposition does make some alteration in
the general construction of the world. And the range that the totality of
elementary propositions leaves open for its construction is exactly the same as
that which is delimited by entirely general propositions. (If an elementary
proposition is true, that means, at any rate, one more true elementary
proposition.)

5.53 Identity of object I express by identity of sign, and not by using a
sign for identity. Difference of objects I express by difference of signs.

5.5301 It is self-evident that identity is not a relation between objects.
This becomes very clear if one considers, for example, the proposition '(x) :
fx. z. x = a'. What this proposition says is simply that only a satisfies the
function f, and not that only things that have a certain relation to a satisfy
the function, Of course, it might then be said that only a did have this
relation to a; but in order to express that, we should need the identity-sign
itself.

5.5302 Russell's definition of '=' is inadequate, because according to it we
cannot say that two objects have all their properties in common. (Even if this
proposition is never correct, it still has sense.)

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 at all.

5.531 Thus I do not write 'f(a, b). a = b', but 'f(a, a)' (or 'f(b, b)); and
not 'f(a,b). Pa = b', but 'f(a, b)'.

5.532 And analogously I do not write '(dx, y). f(x, y). x = y', but '(dx) .
f(x, x)'; and not '(dx, y). f(x, y). Px = y', but '(dx, y). f(x, y)'. 5.5321
Thus, for example, instead of '(x): fx z x = a' we write '(dx). fx . z: (dx, y).
fx. fy'. And the proposition, 'Only one x satisfies f( )', will read '(dx). fx:
P(dx, y). fx. fy'.

5.533 The identity-sign, therefore, is not an essential constituent of
conceptual notation.

5.534 And now we see that in a correct conceptual notation
pseudo-propositions like 'a = a', 'a = b. b = c. z a = c', '(x). x = x', '(dx).
x = a', etc. cannot even be written down.

5.535 This also disposes of all the problems that were connected with such
pseudo-propositions. All the problems that Russell's 'axiom of infinity' brings
with it can be solved at this point. What the axiom of infinity is intended to
say would express itself in language through the existence of infinitely many
names with different meanings.

5.5351 There are certain cases in which one is tempted to use expressions of
the form 'a = a' or 'p z p' and the like. In fact, this happens when one wants
to talk about prototypes, e.g. about proposition, thing, etc. Thus in Russell's
Principles of Mathematics 'p is a proposition'—which is nonsense—was given the
symbolic rendering 'p z p' and placed as an hypothesis in front of certain
propositions in order to exclude from their argument-places everything but
propositions. (It is nonsense to place the hypothesis 'p z p' in front of a
proposition, in order to ensure that its arguments shall have the right form, if
only because with a non-proposition as argument the hypothesis becomes not false
but nonsensical, and because arguments of the wrong kind make the proposition
itself nonsensical, so that it preserves itself from wrong arguments just as
well, or as badly, as the hypothesis without sense that was appended for that
purpose.)

5.5352 In the same way people have wanted to express, 'There are no things ',
by writing 'P(dx). x = x'. But even if this were a proposition, would it not be
equally true if in fact 'there were things' but they were not identical with
themselves?

5.54 In the general propositional form propositions occur in other
propositions only as bases of truth-operations.

5.541 At first sight it looks as if it were also possible for one proposition
to occur in another in a different way. Particularly with certain forms of
proposition in psychology, such as 'A believes that p is the case' and A has the
thought p', etc. For if these are considered superficially, it looks as if the
proposition p stood in some kind of relation to an object A. (And in modern
theory of knowledge (Russell, Moore, etc.) these propositions have actually been
construed in this way.)

5.542 It is clear, however, that 'A believes that p', 'A has the thought p',
and 'A says p' are of the form '"p" says p': and this does not involve a
correlation of a fact with an object, but rather the correlation of facts by
means of the correlation of their objects.

5.5421 This shows too that there is no such thing as the soul—the subject,
etc.—as it is conceived in the superficial psychology of the present day. Indeed
a composite soul would no longer be a soul.

5.5422 The correct explanation of the form of the proposition, 'A makes the
judgement p', must show that it is impossible for a judgement to be a piece of
nonsense. (Russell's theory does not satisfy this requirement.)

5.5423 To perceive a complex means to perceive that its constituents are
related to one another in such and such a way. This no doubt also explains why
there are two possible ways of seeing the figure as a cube; and all similar
phenomena. For we really see two different facts. (If I look in the first place
at the corners marked a and only glance at the b's, then the a's appear to be in
front, and vice versa).

5.55 We now have to answer a priori the question about all the possible forms
of elementary propositions. Elementary propositions consist of names. Since,
however, we are unable to give the number of names with different meanings, we
are also unable to give the composition of elementary propositions.

5.551 Our fundamental principle is that whenever a question can be decided by
logic at all it must be possible to decide it without more ado. (And if we get
into a position where we have to look at the world for an answer to such a
problem, that shows that we are on a completely wrong track.)

5.552 The 'experience' that we need in order to understand logic is not that
something or other is the state of things, but that something is: that, however,
is not an experience. Logic is prior to every experience—that something is so.
It is prior to the question 'How?' not prior to the question 'What?'

5.5521 And if this were not so, how could we apply logic? We might put it in
this way: if there would be a logic even if there were no world, how then could
there be a logic given that 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 is this supposed to
be decided?—By experience? (There is no pre-eminent number.)

5.554 It would be completely arbitrary to give any specific form.

5.5541 It is supposed to be possible to answer a priori the question whether
I can get into a position in which I need the sign for a 27-termed relation in
order to signify something.

5.5542 But is it really legitimate even to ask such a question? Can we set up
a form of sign without knowing whether anything can correspond to it? Does it
make sense to ask what there must be in order that something can be the case?

5.555 Clearly we have some concept of elementary propositions quite apart
from their particular logical forms. But when there is a system by which we can
create symbols, the system is what is important for logic and not the individual
symbols. And anyway, is it really possible that in logic I should have to deal
with forms that I can invent? What I have to deal with must be that which makes
it possible for me to invent them.

5.556 There cannot be a hierarchy of the forms of elementary propositions. We
can foresee only what we ourselves construct.

5.5561 Empirical reality is limited by the totality of objects. The limit
also makes itself manifest in the totality of elementary propositions.
Hierarchies are and must be independent of reality.

5.5562 If we know on purely logical grounds that there must be elementary
propositions, then everyone who understands propositions in their C form must
know It.

5.5563 In fact, all the propositions of our everyday language, just as they
stand, are in perfect logical order.—That utterly simple thing, which we have to
formulate here, is not a likeness of the truth, but the truth itself in its
entirety. (Our problems are not abstract, but perhaps the most concrete that
there are.)

5.557 The application of logic decides what elementary propositions there
are. What belongs to its application, logic cannot anticipate. It is clear that
logic must not clash with its application. But logic has to be in contact with
its application. Therefore logic and its application must not overlap.

5.5571 If I cannot say a priori what elementary propositions there are, then
the attempt to do so must lead to obvious nonsense. 5.6 The limits of my
language mean the limits of my world.

5.61 Logic pervades the world: the limits of the world are also its limits.
So we cannot say in logic, 'The world has this in it, and this, but not that.'
For that would appear to presuppose that we were excluding certain
possibilities, and this cannot be the case, since it would require that logic
should go beyond the limits of the world; for only in that way could it view
those limits from the other side as well. We cannot think what we cannot think;
so what we cannot think we cannot say either.

5.62 This remark provides the key to the problem, how much truth there is in
solipsism. For what the solipsist means is quite correct; only it cannot be
said, but makes itself manifest. The world is my world: this is manifest in the
fact that the limits of language (of that language which alone 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 There is no such thing as the subject that thinks or entertains ideas.
If I wrote a book called The World as l found it, I should have to include a
report on my body, and should have to say which parts were subordinate to my
will, and which were not, etc., this being a method of isolating the subject, or
rather of showing that in an important sense there is no subject; for it alone
could not be mentioned in that book.—

5.632 The subject does not belong to the world: rather, it is a limit of the
world.

5.633 Where in the world is a metaphysical subject to be found? You will say
that this is exactly like the case of the eye and the visual field. But really
you do not see the eye. And nothing in the visual field allows you to infer that
it is seen by an eye.

5.6331 For the form of the visual field is surely not like this

5.634 This is connected with the fact that no part of our experience is at
the same time a priori. Whatever we see could be other than it is. Whatever we
can describe at all could be other than it is. There is no a priori order of
things.

5.64 Here it can be seen that solipsism, when its implications are followed
out strictly, coincides with pure realism. The self of solipsism shrinks to a
point without extension, and there remains the reality co-ordinated with it.

5.641 Thus there really is a sense in which philosophy can talk about the
self in a non-psychological way. What brings the self into philosophy is the
fact that 'the world is my world'. The philosophical self is not the human
being, not the human body, or the human soul, with which psychology deals, but
rather the metaphysical subject, the limit of the world—not a part of it.

## 6. The general form of a truth-function is [p, E, N(E)].

### This is the general form of a proposition.

6.001 What this says is just that every proposition is a result of successive
applications to elementary propositions of the operation N(E)

6.002 If we are given the general form according to which propositions are
constructed, then with it we are also given the general form according to which
one proposition can be generated out of another by means of an operation.

6.01 Therefore the general form of an operation /'(n) is [E, N(E)]' (n) ( =
[n, E, N(E)]). This is the most general form of transition from one proposition
to another.

6.02 And this is how we arrive at numbers. I give the following definitions x
= /0x Def., /'/v'x = /v+1'x Def. So, in accordance with these rules, which deal
with signs, we write the series x, /'x, /'/'x, /'/'/'x,..., in the following way
/0'x, /0+1'x, /0+1+1'x, /0+1+1+1'x,.... Therefore, instead of '[x, E, /'E]', I
write '[/0'x, /v'x, /v+1'x]'. And I give the following definitions 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 of number is simply what is common to all numbers, the
general form of a number. The concept of number is the variable number. And the
concept of numerical equality is the general form of all particular cases of
numerical equality.

6.03 The general form of an integer is [0, E, E +1].

6.031 The theory of classes is completely superfluous in mathematics. This is
connected with the fact that the generality required in mathematics is not
accidental generality.

6.1 The propositions of logic are tautologies.

6.11 Therefore the propositions of logic say nothing. (They are the analytic
propositions.)

6.111 All theories that make a proposition of logic appear to have content
are false. One might think, for example, that the words 'true' and 'false'
signified two properties among other properties, and then it would seem to be a
remarkable fact that every proposition possessed one of these properties. On
this theory it seems to be anything but obvious, just as, for instance, the
proposition, 'All roses are either yellow or red', would not sound obvious even
if it were true. Indeed, the logical proposition acquires all the
characteristics of a proposition of natural science and this is the sure sign
that it has been construed wrongly.

6.112 The correct explanation of the propositions of logic must assign to
them a unique status among all propositions.

6.113 It is the peculiar mark of logical propositions that one can recognize
that they are true from the symbol alone, and this fact contains in itself the
whole philosophy of logic. And so too it is a very important fact that the truth
or falsity of non-logical propositions cannot be recognized from the
propositions alone.

6.12 The fact that the propositions of logic are tautologies shows the
formal—logical—properties of language and the world. The fact that a tautology
is yielded by this particular way of connecting its constituents characterizes
the logic of its constituents. If propositions are to yield a tautology when
they are connected in a certain way, they must have certain structural
properties. So their yielding a tautology when combined in this shows that they
possess these structural properties.

6.1201 For example, the fact that the propositions 'p' and 'Pp' in the
combination '(p. Pp)' yield a tautology shows that they contradict one another.
The fact that the propositions 'p z q', 'p', and 'q', combined with one another
in the form '(p z q). (p):z: (q)', yield a tautology shows that q follows from p
and p z q. The fact that '(x). fxx:z: fa' is a tautology shows that fa follows
from (x). fx. Etc. etc.

6.1202 It is clear that one could achieve the same purpose by using
contradictions instead of tautologies.

6.1203 In order to recognize an expression as a tautology, in cases where no
generality-sign occurs in it, one can employ the following intuitive method:
instead of 'p', 'q', 'r', etc. I write 'TpF', 'TqF', 'TrF', etc.
Truth-combinations I express by means of brackets, e.g. and I use lines to
express the correlation of the truth or falsity of the whole proposition with
the truth-combinations of its truth-arguments, in the following way So this
sign, for instance, would represent the proposition p z q. Now, by way of
example, I wish to examine the proposition P(p.Pp) (the law of contradiction) in
order to determine whether it is a tautology. In our notation the form 'PE' is
written as and the form 'E. n' as Hence the proposition P(p. Pp). reads as
follows If we here substitute 'p' for 'q' and examine how the outermost T and F
are connected with the innermost ones, the result will be that the truth of the
whole proposition is correlated with all the truth-combinations of its argument,
and its falsity with none of the truth-combinations.

6.121 The propositions of logic demonstrate the logical properties of
propositions by combining them so as to form propositions that say nothing. This
method could also be called a zero-method. In a logical proposition,
propositions are brought into equilibrium with one another, and the state of
equilibrium then indicates what the logical constitution of these propositions
must be.

6.122 It follows from this that we can actually do without logical
propositions; for in a suitable notation we can in fact recognize the formal
properties of propositions by mere inspection of the propositions themselves.

6.1221 If, for example, two propositions 'p' and 'q' in the combination 'p z
q' yield a tautology, then it is clear that q follows from p. For example, we
see from the two propositions themselves that 'q' follows from 'p z q. p', but
it is also possible to show it in this way: we combine them to form 'p z q. p:z:
q', and then show that this is a tautology.

6.1222 This throws some light on the question why logical propositions cannot
be confirmed by experience any more than they can be refuted by it. Not only
must a proposition of logic be irrefutable by any possible experience, but it
must also be unconfirmable by any possible experience.

6.1223 Now it becomes clear why people have often felt as if it were for us
to 'postulate' the 'truths of logic'. The reason is that we can postulate them
in so far as we can postulate an adequate notation.

6.1224 It also becomes clear now why logic was called the theory of forms and
of inference.

6.123 Clearly the laws of logic cannot in their turn be subject to laws of
logic. (There is not, as Russell thought, a special law of contradiction for
each 'type'; one law is enough, since it is not applied to itself.)

6.1231 The mark of a logical proposition is not general validity. To be
general means no more than to be accidentally valid for all things. An
ungeneralized proposition can be tautological just as well as a generalized one.

6.1232 The general validity of logic might be called essential, in contrast
with the accidental general validity of such propositions as 'All men are
mortal'. Propositions like Russell's 'axiom of reducibility' are not logical
propositions, and this explains our feeling that, even if they were true, their
truth could only be the result of a fortunate accident.

6.1233 It is possible to imagine a world in which the axiom of reducibility
is not valid. It is clear, however, that logic has nothing to do with the
question whether our world really is like that or not.

6.124 The propositions of logic describe the scaffolding of the world, or
rather they represent it. They have no 'subject-matter'. They presuppose that
names have meaning and elementary propositions sense; and that is their
connexion with the world. It is clear that something about the world must be
indicated by the fact that certain combinations of symbols—whose essence
involves the possession of a determinate character—are tautologies. This
contains the decisive point. We have said that some things are arbitrary in the
symbols that we use and that some things are not. In logic it is only the latter
that express: but that means that logic is not a field in which we express what
we wish with the help of signs, but rather one in which the nature of the
absolutely necessary signs speaks for itself. If we know the logical syntax of
any sign-language, then we have already been given all the propositions of
logic.

6.125 It is possible—indeed possible even according to the old conception of
logic—to give in advance a description of all 'true' logical propositions.

6.1251 Hence there can never be surprises in logic.

6.126 One can calculate whether a proposition belongs to logic, by
calculating the logical properties of the symbol. And this is what we do when we
'prove' a logical proposition. For, without bothering about sense or meaning, we
construct the logical proposition out of others using only rules that deal with
signs. The proof of logical propositions consists in the following process: we
produce them out of other logical propositions by successively applying certain
operations that always generate further tautologies out of the initial ones.
(And in fact only tautologies follow from a tautology.) Of course this way of
showing that the propositions of logic are tautologies is not at all essential
to logic, if only because the propositions from which the proof starts must show
without any proof that they are tautologies.

6.1261 In logic process and result are equivalent. (Hence the absence of
surprise.)

6.1262 Proof in logic is merely a mechanical expedient to facilitate the
recognition of tautologies in complicated cases.

6.1263 Indeed, it would be altogether too remarkable if a proposition that
had sense could be proved logically from others, and so too could a logical
proposition. It is clear from the start that a logical proof of a proposition
that has sense and a proof in logic must be two entirely different things.

6.1264 A proposition that has sense states something, which is shown by its
proof to be so. In logic every proposition is the form of a proof. Every
proposition of logic is a modus ponens represented in signs. (And one cannot
express the modus ponens by means of a proposition.)

6.1265 It is always possible to construe logic in such a way that every
proposition is its own proof.

6.127 All the propositions of logic are of equal status: it is not the case
that some of them are essentially derived propositions. Every tautology itself
shows that it is a tautology.

6.1271 It is clear that the number of the 'primitive propositions of logic'
is arbitrary, since one could derive logic from a single primitive proposition,
e.g. by simply constructing the logical product of Frege's primitive
propositions. (Frege would perhaps say that we should then no longer have an
immediately self-evident primitive proposition. But it is remarkable that a
thinker as rigorous as Frege appealed to the degree of self-evidence as the
criterion of a logical proposition.)

6.13 Logic is not a body of doctrine, but a mirror-image of the world. Logic
is transcendental.

6.2 Mathematics is a logical method. The propositions of mathematics are
equations, and therefore pseudo-propositions.

6.21 A proposition of mathematics does not express a thought.

6.211 Indeed in real life a mathematical proposition is never what we want.
Rather, we make use of mathematical propositions only in inferences from
propositions that do not belong to mathematics to others that likewise do not
belong to mathematics. (In philosophy the question, 'What do we actually use
this word or this proposition for?' repeatedly leads to valuable insights.)

6.22 The logic of the world, which is shown in tautologies by the
propositions of logic, is shown in equations by mathematics.

6.23 If two expressions are combined by means of the sign of equality, that
means that they can be substituted for one another. But it must be manifest in
the two expressions themselves whether this is the case or not. When two
expressions can be substituted for one another, that characterizes their logical
form.

6.231 It is a property of affirmation that it can be construed as double
negation. It is a property of '1 + 1 + 1 + 1' that it can be construed as '(1 +
1) + (1 + 1)'.

6.232 Frege says that the two expressions have the same meaning but different
senses. But the essential point about an equation is that it is not necessary in
order to show that the two expressions connected by the sign of equality have
the same meaning, since this can be seen from the two expressions themselves.

6.2321 And the possibility of proving the propositions of mathematics means
simply that their correctness can be perceived without its being necessary that
what they express should itself be compared with the facts in order to determine
its correctness.

6.2322 It is impossible to assert the identity of meaning of two expressions.
For in order to be able to assert anything about their meaning, I must know
their meaning, and I cannot know their meaning without knowing whether what they
mean is the same or different.

6.2323 An equation merely marks the point of view from which I consider the
two expressions: it marks their equivalence in meaning.

6.233 The question whether intuition is needed for the solution of
mathematical problems must be given the answer that in this case language itself
provides the necessary intuition.

6.2331 The process of calculating serves to bring about that intuition.
Calculation is not an experiment.

6.234 Mathematics is a method of logic.

6.2341 It is the essential characteristic of mathematical method that it
employs equations. For it is because of this method that every proposition of
mathematics must go without saying.

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, starting from a number of equations, we advance to new equations by
substituting different expressions in accordance with the equations.

6.241 Thus the proof of the proposition 2 t 2 = 4 runs as follows: (/v)n'x =
/v x u'x Def., /2 x 2'x = (/2)2'x = (/2)1 + 1'x = /2' /2'x = /1 + 1'/1 + 1'x =
(/'/)'(/'/)'x =/'/'/'/'x = /1 + 1 + 1 + 1'x = /4'x. 6.3 The exploration of logic
means the exploration of everything that is subject to law. And outside logic
everything is accidental.

6.31 The so-called law of induction cannot possibly be a law of logic, since
it is obviously a proposition with sense.—-Nor, therefore, can it be an a priori
law.

6.32 The law of causality is not a law but the form of a law.

6.321 'Law of causality'—that is a general name. And just as in mechanics,
for example, there are 'minimum-principles', such as the law of least action, so
too in physics there are causal laws, laws of the causal form.

6.3211 Indeed people even surmised that there must be a 'law of least action'
before they knew exactly how it went. (Here, as always, what is certain a priori
proves to be something purely logical.)

6.33 We do not have an a priori belief in a law of conservation, but rather a
priori knowledge of the possibility of a logical form.

6.34 All such propositions, including the principle of sufficient reason,
tile laws of continuity in nature and of least effort in nature, etc. etc.—all
these are a priori insights about the forms in which the propositions of science
can be cast.

6.341 Newtonian mechanics, for example, imposes a unified form on the
description of the world. Let us imagine a white surface with irregular black
spots on it. We then say that whatever kind of picture these make, I can always
approximate as closely as I wish to the description of it by covering the
surface with a sufficiently fine square mesh, and then saying of every square
whether it is black or white. In this way I shall have imposed a unified form on
the description of the surface. The form is optional, since I could have
achieved the same result by using a net with a triangular or hexagonal mesh.
Possibly the use of a triangular mesh would have made the description simpler:
that is to say, it might be that we could describe the surface more accurately
with a coarse triangular mesh than with a fine square mesh (or conversely), and
so on. The different nets correspond to different systems for describing the
world. Mechanics determines one form of description of the world by saying that
all propositions used in the description of the world must be obtained in a
given way from a given set of propositions—the axioms of mechanics. It thus
supplies the bricks for building the edifice of science, and it says, 'Any
building that you want to erect, whatever it may be, must somehow be constructed
with these bricks, and with these alone.' (Just as with the number-system we
must be able to write down any number we wish, so with the system of mechanics
we must be able to write down any proposition of physics that we wish.)

6.342 And now we can see the relative position of logic and mechanics. (The
net might also consist of more than one kind of mesh: e.g. we could use both
triangles and hexagons.) The possibility of describing a picture like the one
mentioned above with a net of a given form tells us nothing about the picture.
(For that is true of all such pictures.) But what does characterize the picture
is that it can be described completely by a particular net with a particular
size of mesh. Similarly the possibility of describing the world by means of
Newtonian mechanics tells us nothing about the world: but what does tell us
something about it is the precise way in which it is possible to describe it by
these means. We are also told something about the world by the fact that it can
be described more simply with one system of mechanics than with another.

6.343 Mechanics is an attempt to construct according to a single plan all the
true propositions that we need for the description of the world.

6.3431 The laws of physics, with all their logical apparatus, still speak,
however indirectly, about the objects of the world.

6.3432 We ought not to forget that any description of the world by means of
mechanics will be of the completely general kind. For example, it will never
mention particular point-masses: it will only talk about any point-masses
whatsoever.

6.35 Although the spots in our picture are geometrical figures, nevertheless
geometry can obviously say nothing at all about their actual form and position.
The network, however, is purely geometrical; all its properties can be given a
priori. Laws like the principle of sufficient reason, etc. are about the net and
not about what the net describes.

6.36 If there were a law of causality, it might be put in the following way:
There are laws of nature. But of course that cannot be said: it makes itself
manifest.

6.361 One might say, using Hertt:'s terminology, that only connexions that
are subject to law are thinkable.

6.3611 We cannot compare a process with 'the passage of time'—there is no
such thing—but only with another process (such as the working of a chronometer).
Hence we can describe the lapse of time only by relying on some other process.
Something exactly analogous applies to space: e.g. when people say that neither
of two events (which exclude one another) can occur, because there is nothing to
cause the one to 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
to be found. And if such an asymmetry is to be found, we can regard it as the
cause of the occurrence of the one and the non-occurrence of the other.

6.36111 Kant's problem about the right hand and the left hand, which cannot
be made to coincide, exists even in two dimensions. Indeed, it exists in
one-dimensional space in which the two congruent figures, a and b, cannot be
made to coincide unless they are moved out of this space. The right hand and the
left hand are in fact completely congruent. It is quite irrelevant that they
cannot be made to coincide. A right-hand glove could be put on the left hand, if
it could be turned round in four-dimensional space.

6.362 What can be described can happen too: and what the law of causality is
meant to exclude cannot even be described.

6.363 The procedure of induction consists in accepting as true the simplest
law that can be reconciled with our experiences.

6.3631 This procedure, however, has no logical justification but only a
psychological one. It is clear that there are no grounds for believing that the
simplest eventuality will in fact be realized.

6.36311 It is an hypothesis that the sun will rise tomorrow: and this means
that we do not know whether it will rise.

6.37 There is no compulsion making one thing happen because another has
happened. The only necessity that exists is logical necessity.

6.371 The whole modern conception of the world is founded on the illusion
that the so-called laws of nature are the explanations of natural phenomena.

6.372 Thus people today stop at the laws of nature, treating them as
something inviolable, just as God and Fate were treated in past ages. And in
fact both are right and both wrong: though the view of the ancients is clearer
in so far as they have a clear and acknowledged terminus, while the modern
system tries to make it look as if everything were explained.

6.373 The world is independent of my will.

6.374 Even if all that we wish for were to happen, still this would only be a
favour granted by fate, so to speak: for there is no logical connexion between
the will and the world, which would guarantee it, and the supposed physical
connexion itself is surely not something that we could will.

6.375 Just as the only necessity that exists is logical necessity, so too the
only impossibility that exists is logical impossibility.

6.3751 For example, the simultaneous presence of two colours at the same
place in the visual field is impossible, in fact logically impossible, since it
is ruled out by the logical structure of colour. Let us think how this
contradiction appears in physics: more or less as follows—a particle cannot have
two velocities at the same time; that is to say, it cannot be in two places at
the same time; that is to say, particles that are in different places at the
same time cannot be identical. (It is clear that the logical product of two
elementary propositions can neither be a tautology nor a contradiction. The
statement 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 everything happens as it does happen: in it no value
exists—and if it did exist, it would have no value. If there is any value that
does have value, it must lie outside the whole sphere of what happens and is the
case. For all that happens and is the case is accidental. What makes it
non-accidental cannot lie within the world, since if it did it would itself be
accidental. It must lie outside the world.

6.42 So too it is impossible for there to be propositions of ethics.
Propositions can express nothing that is higher.

6.421 It is clear that ethics cannot be put into words. Ethics is
transcendental. (Ethics and aesthetics are one and the same.)

6.422 When an ethical law of the form, 'Thou shalt...' is laid down, one's
first thought is, 'And what if I do, not do it?' It is clear, however, that
ethics has nothing to do with punishment and reward in the usual sense of the
terms. So our question about the consequences of an action must be
unimportant.—At least those consequences should not be events. For there must be
something right about the question we posed. There must indeed be some kind of
ethical reward and ethical punishment, but they must reside in the action
itself. (And it is also clear that the reward must be something pleasant and the
punishment something unpleasant.)

6.423 It is impossible to speak about the will in so far as it is the subject
of ethical attributes. And the will as a phenomenon is of interest only to
psychology.

6.43 If the good or bad exercise of the will does alter the world, it can
alter only the limits of the world, not the facts—not what can be expressed by
means of language. In short the effect must be that it becomes an altogether
different world. It must, so to speak, wax and wane as a whole. The world of the
happy man is a different one from that of the unhappy man.

6.431 So too at death the world does not alter, but comes to an end.

6.4311 Death is not an event in life: we do not live to experience death. If
we take eternity to mean not infinite temporal duration but timelessness, then
eternal life belongs to those who live in the present. Our life has no end in
just the way in which our visual field has no limits.

6.4312 Not only is there no guarantee of the temporal immortality of the
human soul, that is to say of its eternal survival after death; but, in any
case, this assumption completely fails to accomplish the purpose for which it
has always been intended. Or is some riddle solved by my surviving for ever? Is
not this eternal life itself as much of a riddle as our present life? The
solution of the riddle of life in space and time lies outside space and time.
(It is certainly not the solution of any problems of natural science that is
required.)

6.432 How things are in the world is a matter of complete indifference for
what is higher. God does not reveal himself in the world.

6.4321 The facts all contribute only to setting the problem, not to its
solution.

6.44 It is not how things are in the world that is mystical, but that it
exists.

6.45 To view the world sub specie aeterni is to view it as a whole—a limited
whole. Feeling the world as a limited whole—it is this that is mystical.

6.5 When the answer cannot be put into words, neither can the question be put
into words. The riddle does not exist. If a question can be framed at all, it is
also possible to answer it.

6.51 Scepticism is not irrefutable, but obviously nonsensical, when it tries
to raise doubts where no questions can be asked. For doubt can exist only where
a question exists, a question only where an answer exists, and an answer only
where something can be said.

6.52 We feel that even when all possible scientific questions have been
answered, the problems of life remain completely untouched. Of course there are
then no questions left, and this itself is the answer.

6.521 The solution of the problem of life is seen in the vanishing of the
problem. (Is not this the reason why those who have found after a long period of
doubt that the sense of life became clear to them have then been unable to say
what constituted that sense?)

6.522 There are, indeed, things that cannot be put into words. They make
themselves manifest. They are what is mystical.

6.53 The correct method in philosophy would really be the following: to say
nothing except what can be said, i.e. propositions of natural science—i.e.
something that has nothing to do with philosophy—and then, whenever someone else
wanted to say something metaphysical, to demonstrate to him that he had failed
to give a meaning to certain signs in his propositions. Although it would not be
satisfying to the other person—he would not have the feeling that we were
teaching him philosophy—this method would be the only strictly correct one.

6.54 My propositions are elucidatory in this way: he who understands 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 transcend these propositions, and then he will see
the world aright.

## 7. What we cannot speak about we must pass over in silence.