by Ludwig Wittgenstein
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
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
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 of 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 problem are
Propositions under "1, 2,
- The world is all that is the case.
- The world is the totality of facts, not of things.
- The world is determined by the facts, and by their being
all the facts.
- For the totality of facts determines what is the case,
and also whatever is not the case.
- The facts in logical space are the world.
- The world divides into facts.
- Each item can be the case or not the case while
everything else remains the same.
- What is the case--a fact--is the existence of states of
- A state of affairs (a state of things) is a combination
of objects (things).
- It is essential to things that they should be possible
constituents of states of affairs.
- 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.
- 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.
- 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.)
- 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.
- If I am to know an object, though I need not know its
external properties, I must know all its internal
- If all objects are given, then at the same time all
possible states of affairs are also given.
- 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.
- 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.
- Objects contain the possibility of all situations.
- The possibility of its occurring in states of affairs is
the form of an object.
- Objects are simple.
- Every statement about complexes can be resolved into a
statement about their constituents and into the
propositions that describe the complexes completely.
- Objects make up the substance of the world. That is why
they cannot be composite.
- If they world had no substance, then whether a
proposition had sense would depend on whether another
proposition was true.
- In that case we could not sketch any picture of the world
(true or false).
- 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.
- Objects are just what constitute this unalterable form.
- 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.
- In a manner of speaking, objects are colourless.
- If two objects have the same logical form, the only
distinction between them, apart from their external
properties, is that they are different.
- 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.
- The substance is what subsists independently of what is
- It is form and content.
- Space, time, colour (being coloured) are forms of
- There must be objects, if the world is to have
- Objects, the unalterable, and the subsistent are one and
- Objects are what is unalterable and subsistent; their
configuration is what is changing and unstable.
- The configuration of objects produces states of affairs.
- In a state of affairs objects fit into one another like
the links of a chain.
- In a state of affairs objects stand in a determinate
relation to one another.
- The determinate way in which objects are connected in a
state of affairs is the structure of the state of
- Form is the possibility of structure.
- The structure of a fact consists of the structures of
states of affairs.
- The totality of existing states of affairs is the world.
- The totality of existing states of affairs also
determines which states of affairs do not exist.
- 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.)
- States of affairs are independent of one another.
- From the existence or non-existence of one state of
affairs it is impossible to infer the existence or
non-existence of another.
- The sum-total of reality is the world.
- We picture facts to ourselves.
- A picture presents a situation in logical space, the
existence and non-existence of states of affairs.
- A picture is a model of reality.
- In a picture objects have the elements of the picture
corresponding to them.
- In a picture the elements of the picture are the
representatives of objects.
- What constitutes a picture is that its elements are
related to one another in a determinate way.
- A picture is a fact.
- 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.
- Pictorial form is the possibility that things are related
to one another in the same way as the elements of the
- That is how a picture is attached to reality; it reaches
right out to it.
- It is laid against reality like a measure.
- Only the end-points of the graduating lines actually
touch the object that is to be measured.
- So a picture, conceived in this way, also includes the
pictorial relationship, which makes it into a picture.
- These correlations are, as it were, the feelers of the
picture's elements, with which the picture touches
- If a fact is to be a picture, it must have something in
common with what it depicts.
- 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.
- 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.
- A picture can depict any reality whose form it has. A
spatial picture can depict anything spatial, a coloured
one anything coloured, etc.
- A picture cannot, however, depict its pictorial form: it
- 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
- A picture cannot, however, place itself outside its
- 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.
- A picture whose pictorial form is logical form is called
a logical picture.
- Every picture is at the same time a logical one. (On the
other hand, not every picture is, for example, a spatial
- Logical pictures can depict the world.
- A picture has logico-pictorial form in common with what
- A picture depicts reality by representing a possibility
of existence and non-existence of states of affairs.
- A picture contains the possibility of the situation that
- A picture agrees with reality or fails to agree; it is
correct or incorrect, true or false.
- What a picture represents it represents independently of
its truth or falsity, by means of its pictorial form.
- What a picture represents is its sense.
- The agreement or disagreement or its sense with reality
constitutes its truth or falsity.
- In order to tell whether a picture is true or false we
must compare it with reality.
- It is impossible to tell from the picture alone whether
it is true or false.
- There are no pictures that are true a priori.
- A logical picture of facts is a thought.
- 'A state of affairs is thinkable': what this means is
that we can picture it to ourselves.
- The totality of true thoughts is a picture of the world.
- A thought contains the possibility of the situation of
which it is the thought. What is thinkable is possible
- Thought can never be of anything illogical, since, if it
were, we should have to think illogically.
- 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
- 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.
- 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.
- It a thought were correct a priori, it would be a thought
whose possibility ensured its truth.
- 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).
- In a proposition a thought finds an expression that can
be perceived by the senses.
- 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
- 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.
- 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.
- 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.
- A proposition is not a blend of words.(Just as a theme in
music is not a blend of notes.) A proposition is
- Only facts can express a sense, a set of names cannot.
- 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.)
- 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.
- 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.'
- Situations can be described but not given names.
- 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.
- I call such elements 'simple signs', and such a
proposition 'complete analysed'.
- The simple signs employed in propositions are called
- A name means an object. The object is its meaning. ('A'
is the same sign as 'A'.)
- The configuration of objects in a situation corresponds
to the configuration of simple signs in the propositional
- 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.
- The requirement that simple signs be possible is the
requirement that sense be determinate.
- 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.
- A proposition cannot be dissected any further by means of
a definition: it is a primitive sign.
- 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.)
- What signs fail to express, their application shows. What
signs slur over, their application says clearly.
- 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.
- Only propositions have sense; only in the nexus of a
proposition does a name have meaning.
- 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.
- 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.
- 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
- 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'.
- An expression has meaning only in a proposition. All
variables can be construed as propositional variables.
(Even variable names.)
- 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.
- What values a propositional variable may take is
something that is stipulated. The stipulation of values
is the variable.
- 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.
- Like Frege and Russell I construe a proposition as a
function of the expressions contained in it.
- A sign is what can be perceived of a symbol.
- 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.
- 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?
- 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
- In this way the most fundamental confusions are easily
produced (the whole of philosophy is full of them).
- 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
- In order to recognize a symbol by its sign we must
observe how it is used with a sense.
- A sign does not determine a logical form unless it is
taken together with its logico-syntactical employment.
- 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.)
- 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.
- 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.
- 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').
- 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
- The rules of logical syntax must go without saying, once
we know how each individual sign signifies.
- 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.
- 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.
- 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.
- 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.)
- 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.
- 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.
- 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.
- 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.)
- 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
- 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.
- The propositional sign with logical co-ordinates--that is
the logical place.
- In geometry and logic alike a place is a possibility:
something can exist in it.
- 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
- A propositional sign, applied and thought out, is a
Propositions under "4"
- A thought is a proposition with a sense.
- The totality of propositions is language.
- 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
- 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.
- 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.
- A proposition is a picture of reality. A proposition is a
model of reality as we imagine it.
- 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.
- 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.
- 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
- 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.)
- 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
- The possibility of all imagery, of all our pictorial
modes of expression, is contained in the logic of
- 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.
- We can see this from the fact that we understand the
sense of a propositional sign without its having been
explained to us.
- 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.
- A proposition shows its sense. A proposition shows how
things stand if it is true. And it says that they do so
- 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.
- 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.
- 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.)
- The meanings of simple signs (words) must be explained to
us if we are to understand them. With propositions,
however, we make ourselves understood.
- It belongs to the essence of a proposition that it should
be able to communicate a new sense to us.
- 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.
- 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'.
- 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
- 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.
- 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.)
- 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.)
- This mathematical multiplicity, of course, cannot itself
be the subject of depiction. One cannot get away from it
- 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.
- 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.
- Reality is compared with propositions.
- A proposition can be true or false only in virtue of
being a picture of reality.
- 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.
- 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
- 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
- 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.
- 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.
- 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 prelimary
to a proposition.
- Propositions represent the existence and non-existence of
states of affairs.
- The totality of true propositions is the whole of natural
science (or the whole corpus of the natural sciences).
- 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.)
- 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.
- 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
- Darwin's theory has no more to do with philosophy than
any other hypothesis in natural science.
- Philosophy sets limits to the much disputed sphere of
- 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
- It will signify what cannot be said, by presenting
clearly what can be said.
- Everything that can be thought at all can be thought
clearly. Everything that can be put into words can be put
- 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.
- 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
- 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.
- What can be shown, cannot be said.
- 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.
- 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
- 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).
- 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'.)
- 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.
- 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.
- The existence of an internal relation between possible
situations expresses itself in language by means of an
internal relation between the propositions representing
- Here we have the answer to the vexed question 'whether
all relations are internal or external'.
- 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
'(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.)
- 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.
- The propositional variable signifies the formal concept,
and its values signify the objects that fall under the
- 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.
- 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
- 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
- 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
'(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
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.
- 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.)
- Logical forms are without number. Hence there are no
preeminent numbers in logic, and hence there is no
possibility of philosophical monism or dualism, etc.
- The sense of a proposition is its agreement and
disagreement with possibilities of existence and
non-existence of states of affairs.
- The simplest kind of proposition, an elementary
proposition, asserts the existence of a state of affairs.
- It is a sign of a proposition's being elementary that
there can be no elementary proposition contradicting it.
- An elementary proposition consists of names. It is a
nexus, a concatenation, of names.
- 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.
- 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
- It is only in the nexus of an elementary proposition that
a name occurs in a proposition.
- 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',
- 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.)
- Expressions of the form 'a = b' are, therefore, mere
representational devices. They state nothing about the
meaning of the signs 'a' and 'b'.
- 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
- If an elementary proposition is true, the state of
affairs exists: if an elementary proposition is false,
the state of affairs does not exist.
- 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
- There correspond to these combinations the same number of
possibilities of truth--and falsity--for n elementary
- Truth-possibilities of elementary propositions mean
Possibilities of existence and non-existence of states of
- 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):
- A proposition is an expression of agreement and
disagreement with truth-possibilities of elementary
- Truth-possibilities of elementary propositions are the
conditions of the truth and falsity of propositions.
- 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.
- For n elementary propositions there are ways in which a
proposition can agree and disagree with their truth
- 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.
- 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
- The sign that results from correlating the mark 'I"
with truth-possibilities is a propositional sign.
- 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.
- 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.)
- 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.
- 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.
- 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
- Tautologies and contradictions are not, however,
nonsensical. They are part of the symbolism, much as '0'
is part of the symbolism of arithmetic.
- 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.
- 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.
- 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.)
- 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.
- 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.
- 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 .
- 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.
- 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
- 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.)
- The general propositional form is a variable.
Propositions under "5"
- A proposition is a truth-function of elementary
propositions. (An elementary proposition is a
truth-function of itself.)
- Elementary propositions are the truth-arguments of
- The arguments of functions are readily confused with the
aff ixes 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.
- Truth-functions can be arranged in series. That is the
foundation of the theory of probability.
- The truth-functions of a given number of elementary
proposi tions can always be set out in a schema of the
(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
(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.
- 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.
- 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
- The truth-grounds of the one are contained in those of
the other: p follows from q.
- If p follows from q, the sense of 'p' is contained in the
sense of 'q'.
- 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
- A proposition affirms every proposition that follows from
- '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.
- When the truth of one proposition follows from the truth
of others, we can see this from the structure of the
- 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.
- 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.)
- 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.
- All deductions are made a priori.
- One elementary proposition cannot be deduced form
- There is no possible way of making an inference form the
existence of one situation to the existence of another,
entirely different situation.
- There is no causal nexus to justify such an inference.
- We cannot infer the events of the future from those of
the present. Belief in the causal nexus is superstition.
- 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
- 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
- If one proposition follows from another, then the latter
says more than the former, and the former less than the
- If p follows from q and q from p, then they are one and
- A tautology follows from all propositions: it says
- 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
- 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
- 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.
- There is no special object peculiar to probability
- 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.)
- In itself, a proposition is neither probable nor
improbable. Either an event occurs or it does not: there
is no middle way.
- 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.
- 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.
- 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.
- The structures of propositions stand in internal
relations to one another.
- 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).
- An operation is the expression of a relation between the
structures of its result and of its bases.
- The operation is what has to be done to the one
proposition in order to make the other out of it.
- And that will, of course, depend on their formal
properties, on the internal similarity of their forms.
- The internal relation by which a series is ordered is
equivalent to the operation that produces one term from
- 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.
- Truth-functions of elementary propositions are results of
operations with elementary propositions as bases. (These
operations I call truth-operations.)
- 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.)
- 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.)
- An operation is not the mark of a form, but only of a
difference between forms.
- 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
- 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.)
- A function cannot be its own argument, whereas an
operation can take one of its own results as its base.
- 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.)
- 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.
- 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.
- The concept of successive applications of an operation is
equivalent to the concept 'and so on'.
- One operation can counteract the effect of another.
Operations can cancel one another.
- An operation can vanish (e.g. negation in 'PPp' : PPp =
- 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
- 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
- All truth-functions are results of successive
applications to elementary propositions of a finite
number of truth-operations.
- At this point it becomes manifest that there are no
'logical objects' or 'logical constants' (in Frege's and
- 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
- 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.
- 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.
- 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.
- 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'.
- If we are given a proposition, then with it we are also
given the results of all truth-operations that have it as
- 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.
- 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
- 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.
- 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.
- 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.
- 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.
- 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
- 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
- Signs for logical operations are punctuation-marks,
- 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
- The general propositional form is the essence of a
- To give the essence of a proposition means to give the
essence of all description, and thus the essence of the
- The description of the most general propositional form is
the description of the one and only general primitive
sign in logic.
- 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.
- 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.
- We cannot give a sign the wrong sense.
- 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
- 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.
- The number of fundamental operations that are necessary
depends solely on our notation.
- All that is required is that we should construct a system
of signs with a particular number of dimensions--with a
particular mathematical multiplicity
- 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.
- Every truth-function is a result of successive
applications to elementary propositions of the operation
This operation negates all the propositions in the
right-hand pair of brackets, and I call it the negation
of those propositions.
- 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 ali 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.
- So instead of '(-----T)(E, ....)', I write 'N(E)'. N(E)
is the negation of all the values of the propositional
- 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.
- 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).
- 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.
- '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
- 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.
- 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.
- 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.
- 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.
- If E has as its values all the values of a function fx
for all values of x, then N(E) = P(dx) . fx.
- 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.
- What is peculiar to the generality-sign is first, that it
indicates a logical prototype, and secondly, that it
gives prominence to constants.
- The generality-sign occurs as an argument.
- 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.
- 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.
- We can describe the world completely by means of fully
generalized propositions, i.e. without first correlating
any name with a particular object.
- 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
- 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.)
- 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.
- 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.
- 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 .)
- 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.
- 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)'.
- 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)'.
- 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'.
- The identity-sign, therefore, is not an essential
constituent of conceptual notation.
- 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
- 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
- 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.)
- 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
- In the general propositional form propositions occur in
other propositions only as bases of truth-operations.
- 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.)
- 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
- 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.
- 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.)
- 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).
- 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
- 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
- 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?'
- 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?
- 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.)
- It would be completely arbitrary to give any specific
- 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
- 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
- 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.
- There cannot be a hierarchy of the forms of elementary
propositions. We can foresee only what we ourselves
- 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.
- 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.
- 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.)
- 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.
- If I cannot say a priori what elementary propositions
there are, then the attempt to do so must lead to obvious
- The limits of my language mean the limits of my world.
- 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
- 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.
- The world and life are one.
- I am my world. (The microcosm.)
- 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.--
- The subject does not belong to the world: rather, it is a
limit of the world.
- 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.
- For the form of the visual field is surely not like this
- 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
- 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
- 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.
Propositions under "6"
- The general form of a truth-function is [p, E, N(E)].
This is the general form of a proposition.
- What this says is just that every proposition is a result
of successive applications to elementary propositions of
the operation N(E)
- 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.
- 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.
- And this is how we arrive at numbers. I give the
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).
- A number is the exponent of an operation.
- 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.
- The general form of an integer is [0, E, E +1].
- The theory of classes is completely superfluous in
mathematics. This is connected with the fact that the
generality required in mathematics is not accidental
- The propositions of logic are tautologies.
- Therefore the propositions of logic say nothing. (They
are the analytic propositions.)
- 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.
- The correct explanation of the propositions of logic must
assign to them a unique status among all propositions.
- 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.
- 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
- 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.
- It is clear that one could achieve the same purpose by
using contradictions instead of tautologies.
- 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
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
- 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.
- 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.
- 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.
- 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
- 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.
- It also becomes clear now why logic was called the theory
of forms and of inference.
- 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.)
- 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
- 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
- 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.
- 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.
- It is possible--indeed possible even according to the old
conception of logic--to give in advance a description of
all 'true' logical propositions.
- Hence there can never be surprises in logic.
- 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
- In logic process and result are equivalent. (Hence the
absence of surprise.)
- Proof in logic is merely a mechanical expedient to
facilitate the recognition of tautologies in complicated
- 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.
- 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.)
- It is always possible to construe logic in such a way
that every proposition is its own proof.
- 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
- 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.)
- Logic is not a body of doctrine, but a mirror-image of
the world. Logic is transcendental.
- Mathematics is a logical method. The propositions of
mathematics are equations, and therefore
- A proposition of mathematics does not express a thought.
- 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.)
- The logic of the world, which is shown in tautologies by
the propositions of logic, is shown in equations by
- 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.
- 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)'.
- 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.
- 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.
- 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.
- An equation merely marks the point of view from which I
consider the two expressions: it marks their equivalence
- 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
- The process of calculating serves to bring about that
intuition. Calculation is not an experiment.
- Mathematics is a method of logic.
- 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
- 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
- Thus the proof of the proposition 2 t 2 = 4 runs as
(/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.
- The exploration of logic means the exploration of
everything that is subject to law . And outside logic
everything is accidental.
- 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.
- The law of causality is not a law but the form of a law.
- '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
- 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.)
- 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.
- All such propositions, including the principle of
sufficient reason, tile laws of continuity in nature and
of least effort in ature, etc. etc.--all these are a
priori insights about the forms in which the propositions
of science can be cast.
- 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.)
- 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
- Mechanics is an attempt to construct according to a
single plan all the true propositions that we need for
the description of the world.
- The laws of physics, with all their logical apparatus,
still speak, however indirectly, about the objects of the
- 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
- 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
- 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.
- One might say, using Hertt:'s terminology, that only
connexions that are subject to law are thinkable.
- 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.
- 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
- What can be described can happen too: and what the law of
causality is meant to exclude cannot even be described.
- The procedure of induction consists in accepting as true
the simplest law that can be reconciled with our
- 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.
- It is an hypothesis that the sun will rise tomorrow: and
this means that we do not know whether it will rise.
- There is no compulsion making one thing happen because
another has happened. The only necessity that exists is
- 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.
- 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
- The world is independent of my will.
- 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
- Just as the only necessity that exists is logical
necessity, so too the only impossibility that exists is
- 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
- All propositions are of equal value.
- 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.
- So too it is impossible for there to be propositions of
ethics. Propositions can express nothing that is higher.
- It is clear that ethics cannot be put into words. Ethics
is transcendental. (Ethics and aesthetics are one and the
- 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.)
- 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.
- 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.
- So too at death the world does not alter, but comes to an
- 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
- 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.)
- How things are in the world is a matter of complete
indifference for what is higher. God does not reveal
himself in the world.
- The facts all contribute only to setting the problem, not
to its solution.
- It is not how things are in the world that is mystical,
but that it exists.
- 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.
- 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
- 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.
- 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.
- 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?)
- There are, indeed, things that cannot be put into words.
They make themselves manifest. They are what is mystical.
- 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.
- 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.)
- What we cannot speak about we must pass over in silence.