Tractatus Logico-Philosophicus
Logisch-philosophische Abhandlung
By Ludwig WittgensteinFirst published by Kegan Paul (London), 1922.
S IDE - BY- SIDE - BY- SIDE EDITION, VERSION 0.63 (J ULY 14, 2022),
containing the original German, alongside both the Ogden/Ramsey, and Pears/McGuinness English translations.
Available at: http://people.umass.edu/klement/tlp/

Introduction
By Bertrand Russell, F. R. S.
M R . W ITTGENSTEIN ’ S Tractatus Logico-Philosophicus, whether or what we assert is never quite precise. Thus, logic has two problems
not it prove to give the ultimate truth on the matters with which it to deal with in regard to Symbolism: (1) the conditions for sense
deals, certainly deserves, by its breadth and scope and profundity, to rather than nonsense in combinations of symbols; (2) the conditions
be considered an important event in the philosophical world. Starting for uniqueness of meaning or reference in symbols or combinations offrom the principles of Symbolism and the relations which are necessary symbols. A logically perfect language has rules of syntax which prevent
between words and things in any language, it applies the result of this nonsense, and has single symbols which always have a definite andinquiry to various departments of traditional philosophy, showing in unique meaning. Mr. Wittgenstein is concerned with the conditions foreach case how traditional philosophy and traditional solutions arise a logically perfect language—not that any language is logically perfect,
out of ignorance of the principles of Symbolism and out of misuse of or that we believe ourselves capable, here and now, of constructing alanguage.
logically perfect language, but that the whole function of language is
The logical structure of propositions and the nature of logical in- to have meaning, and it only fulfills this function in proportion as itference are first dealt with. Thence we pass successively to Theory of approaches to the ideal language which we postulate.
Knowledge, Principles of Physics, Ethics, and finally the Mystical (dasThe essential business of language is to assert or deny facts. Given
Mystische).
the syntax of language, the meaning of a sentence is determined as
In order to understand Mr. Wittgenstein’s book, it is necessary soon as the meaning of the component words is known. In order that ato realize what is the problem with which he is concerned. In the certain sentence should assert a certain fact there must, however thepart of his theory which deals with Symbolism he is concerned with language may be constructed, be something in common between thethe conditions which would have to be fulfilled by a logically perfect structure of the sentence and the structure of the fact. This is perhaps
language. There are various problems as regards language. First, the most fundamental thesis of Mr. Wittgenstein’s theory. That which
there is the problem what actually occurs in our minds when we use has to be in common between the sentence and the fact cannot, helanguage with the intention of meaning something by it; this problem contends, be itself in turn said in language. It can, in his phraseology,
belongs to psychology. Secondly, there is the problem as to what is only be shown, not said, for whatever we may say will still need to
the relation subsisting between thoughts, words, or sentences, and have the same structure.
that which they refer to or mean; this problem belongs to epistemology.
The first requisite of an ideal language would be that there should
Thirdly, there is the problem of using sentences so as to convey truth be one name for every simple, and never the same name for two difrather that falsehood; this belongs to the special sciences dealing with ferent simples. A name is a simple symbol in the sense that it has no
the subject-matter of the sentences in question. Fourthly, there is the parts which are themselves symbols. In a logically perfect language
question: what relation must one fact (such as a sentence) have to nothing that is not simple will have a simple symbol. The symbol for
another in order to be capable of being a symbol for that other? This the whole will be a “complex”, containing the symbols for the parts. (Inlast is a logical question, and is the one with which Mr. Wittgenstein speaking of a “complex” we are, as will appear later, sinning against
is concerned. He is concerned with the conditions for accurate Symbol- the rules of philosophical grammar, but this is unavoidable at theism, i.e. for Symbolism in which a sentence “means” something quite outset. “Most propositions and questions that have been written about
definite. In practice, language is always more or less vague, so that philosophical matters are not false but senseless. We cannot, therefore,answer questions of this kind at all, but only state their senselessness. that the one can be a picture of the other at all. What the picture must
Most questions and propositions of the philosophers result from the have in common with reality in order to be able to represent it afterfact that we do not understand the logic of our language. They are its manner—rightly or falsely—is its form of representation” (2.161,of the same kind as the question whether the Good is more or less 2.17).
identical than the Beautiful” (4.003).) What is complex in the world
We speak of a logical picture of a reality when we wish to imply
is a fact. Facts which are not compounded of other facts are what Mr. only so much resemblance as is essential to its being a picture in any
Wittgenstein calls Sachverhalte, whereas a fact which may consist of sense, that is to say, when we wish to imply no more than identity
two or more facts is a Tatsache: thus, for example “Socrates is wise” of logical form. The logical picture of a fact, he says, is a Gedanke.
is a Sachverhalt, as well as a Tatsache, whereas “Socrates is wise and A picture can correspond or not correspond with the fact and be acPlato is his pupil” is a Tatsache but not a Sachverhalt.
cordingly true or false, but in both cases it shares the logical form
He compares linguistic expression to projection in geometry. A with the fact. The sense in which he speaks of pictures is illustrated
geometrical figure may be projected in many ways: each of these ways by his statement: “The gramophone record, the musical thought, thecorresponds to a different language, but the projective properties of score, the waves of sound, all stand to one another in that pictorial
the original figure remain unchanged whichever of these ways may internal relation which holds between language and the world. To
be adopted. These projective properties correspond to that which in all of them the logical structure is common. (Like the two youths,
his theory the proposition and the fact must have in common, if the their two horses and their lilies in the story. They are all in a certain
proposition is to assert the fact.
sense one)” (4.014). The possibility of a proposition representing a
In certain elementary ways this is, of course, obvious. It is impossi- fact rests upon the fact that in it objects are represented by signs.
ble, for example, to make a statement about two men (assuming for the The so-called logical “constants” are not represented by signs, but are
moment that the men may be treated as simples), without employing themselves present in the proposition as in the fact. The proposition
two names, and if you are going to assert a relation between the two and the fact must exhibit the same logical “manifold”, and this cannot
men it will be necessary that the sentence in which you make the be itself represented since it has to be in common between the fact
assertion shall establish a relation between the two names. If we say and the picture. Mr. Wittgenstein maintains that everything properly“Plato loves Socrates”, the word “loves” which occurs between the word philosophical belongs to what can only be shown, or to what is in“Plato” and the word “Socrates” establishes a certain relation between common between a fact and its logical picture. It results from this view
these two words, and it is owing to this fact that our sentence is able that nothing correct can be said in philosophy. Every philosophicalto assert a relation between the persons named by the words “Plato” proposition is bad grammar, and the best that we can hope to achieveand “Socrates”. “We must not say, the complex sign ‘aRb’ says that ‘a by philosophical discussion is to lead people to see that philosophicalstands in a certain relation R to b’; but we must say, that ‘a’ stands in discussion is a mistake. “Philosophy is not one of the natural sciences.
a certain relation to ‘ b’ says that aRb” (3.1432).
(The word ‘philosophy’ must mean something which stands above or
Mr. Wittgenstein begins his theory of Symbolism with the state- below, but not beside the natural sciences.) The object of philosophy
ment (2.1): “We make to ourselves pictures of facts.” A picture, he says, is the logical clarification of thoughts. Philosophy is not a theory but
is a model of the reality, and to the objects in the reality correspond the an activity. A philosophical work consists essentially of elucidations.
elements of the picture: the picture itself is a fact. The fact that things The result of philosophy is not a number of ‘philosophical propositions’,
have a certain relation to each other is represented by the fact that in but to make propositions clear. Philosophy should make clear andthe picture its elements have a certain relation to one another. “In the delimit sharply the thoughts which otherwise are, as it were, opaque
picture and the pictured there must be something identical in order and blurred” (4.111 and 4.112). In accordance with this principle thethings that have to be said in leading the reader to understand Mr. atomic fact is called an atomic proposition. All atomic propositions are
Wittgenstein’s theory are all of them things which that theory itself logically independent of each other. No atomic proposition implies any
condemns as meaningless. With this proviso we will endeavour to other or is inconsistent with any other. Thus the whole business of
convey the picture of the world which seems to underlie his system.
logical inference is concerned with propositions which are not atomic.
The world consists of facts: facts cannot strictly speaking be de- Such propositions may be called molecular.
fined, but we can explain what we mean by saying that facts are what
Wittgenstein’s theory of molecular propositions turns upon his
makes propositions true, or false. Facts may contain parts which are theory of the construction of truth-functions.
facts or may contain no such parts; for example: “Socrates was a wise
A truth-function of a proposition p is a proposition containing p
Athenian”, consists of the two facts, “Socrates was wise”, and “Socrates and such that its truth or falsehood depends only upon the truth or
was an Athenian.” A fact which has no parts that are facts is called by falsehood of p, and similarly a truth-function of several propositions
Mr. Wittgenstein a Sachverhalt. This is the same thing that he calls p, q, r, . . . is one containing p, q, r, . . . and such that its truth or falsean atomic fact. An atomic fact, although it contains no parts that are hood depends only upon the truth or falsehood of p, q, r, . . . It mightfacts, nevertheless does contain parts. If we may regard “Socrates is seem at first sight as though there were other functions of propositions
wise” as an atomic fact we perceive that it contains the constituents besides truth-functions; such, for example, would be “A believes p”,“Socrates” and “wise”. If an atomic fact is analyzed as fully as possible for in general A will believe some true propositions and some false(theoretical, not practical possibility is meant) the constituents finally ones: unless he is an exceptionally gifted individual, we cannot infer
reached may be called “simples” or “objects”. It is a logical necessity that p is true from the fact that he believes it or that p is false fromdemanded by theory, like an electron. His ground for maintaining the fact that he does not believe it. Other apparent exceptions wouldthat there must be simples is that every complex presupposes a fact. be such as “ p is a very complex proposition” or “ p is a proposition
It is not necessarily assumed that the complexity of facts is finite; about Socrates”. Mr. Wittgenstein maintains, however, for reasons
even if every fact consisted of an infinite number of atomic facts and if which will appear presently, that such exceptions are only apparent,
every atomic fact consisted of an infinite number of objects there would and that every function of a proposition is really a truth-function. It
still be objects and atomic facts (4.2211). The assertion that there is follows that if we can define truth-functions generally, we can obtaina certain complex reduces to the assertion that its constituents are a general definition of all propositions in terms of the original set of
related in a certain way, which is the assertion of a fact: thus if we atomic propositions. This Wittgenstein proceeds to do.
give a name to the complex the name only has meaning in virtue of
It has been shown by Dr. Sheffer (Trans. Am. Math. Soc., Vol. XIV.
the truth of a certain proposition, namely the proposition asserting pp. 481–488) that all truth-functions of a given set of propositions can
the relatedness of the constituents of the complex. Thus the naming be constructed out of either of the two functions “not- p or not- q” orof complexes presupposes propositions, while propositions presuppose “not- p and not- q”. Wittgenstein makes use of the latter, assuming a
the naming of simples. In this way the naming of simples is shown to knowledge of Dr. Sheffer’s work. The manner in which other truthbe what is logically first in logic.
functions are constructed out of “not- p and not- q” is easy to see. “Not- p
The world is fully described if all atomic facts are known, together and not- p” is equivalent to “not- p”, hence we obtain a definition ofwith the fact that these are all of them. The world is not described negation in terms of our primitive function: hence we can define “ p orby merely naming all the objects in it; it is necessary also to know q”, since this is the negation of “not- p and not- q”, i.e. of our primitive
the atomic facts of which these objects are constituents. Given this function. The development of other truth-functions out of “not- p” andtotality of atomic facts, every true proposition, however complex, can “ p or q” is given in detail at the beginning of Principia Mathematica.
theoretically be inferred. A proposition (true or false) asserting an This gives all that is wanted when the propositions which are argu-

ments to our truth-function are given by enumeration. Wittgenstein,(a). Sheffer’s proof that all truth-functions can be obtained out of
however, by a very interesting analysis succeeds in extending the pro- simultaneous negation, i.e. out of “not- p and not- q”;
cess to general propositions, i.e. to cases where the propositions which(b). Mr. Wittgenstein’s theory of the derivation of general proposiare arguments to our truth-function are not given by enumeration but tions from conjunctions and disjunctions;
are given as all those satisfying some condition. For example, let f x be(c). The assertion that a proposition can only occur in anothera propositional function (i.e. a function whose values are propositions), proposition as argument to a truth-function. Given these three founsuch as “ x is human”—then the various values of f x form a set of dations, it follows that all propositions which are not atomic can bepropositions. We may extend the idea “not- p and not- q” so as to apply derived from such as are, by a uniform process, and it is this process
to the simultaneous denial of all the propositions which are values which is indicated by Mr. Wittgenstein’s symbol.
of f x. In this way we arrive at the proposition which is ordinarily
From this uniform method of construction we arrive at an amazing
represented in mathematical logic by the words “ f x is false for all simplification of the theory of inference, as well as a definition of thevalues of x”. The negation of this would be the proposition “there is sort of propositions that belong to logic. The method of generation
at least one x for which f x is true” which is represented by “(∃ x). f x”. which has just been described, enables Wittgenstein to say that all
If we had started with not- f x instead of f x we should have arrived at propositions can be constructed in the above manner from atomic
the proposition “ f x is true for all values of x” which is represented by propositions, and in this way the totality of propositions is defined.
“( x). f x”. Wittgenstein’s method of dealing with general propositions (The apparent exceptions which we mentioned above are dealt with in[i.e. “( x). f x” and “(∃ x). f x”] differs from previous methods by the fact a manner which we shall consider later.) Wittgenstein is enabled tothat the generality comes only in specifying the set of propositions assert that propositions are all that follows from the totality of atomic
concerned, and when this has been done the building up of truth- propositions (together with the fact that it is the totality of them); thatfunctions proceeds exactly as it would in the case of a finite number of a proposition is always a truth-function of atomic propositions; andenumerated arguments p, q, r, . . .
that if p follows from q the meaning of p is contained in the meaning
Mr. Wittgenstein’s explanation of his symbolism at this point is not of q, from which of course it results that nothing can be deduced fromquite fully given in the text. The symbol he uses is [ p, ξ, N(ξ)]. The an atomic proposition. All the propositions of logic, he maintains, arefollowing is the explanation of this symbol:
tautologies, such, for example, as “ p or not p”.
The fact that nothing can be deduced from an atomic propositionp stands for all atomic propositions.
has interesting applications, for example, to causality. There cannot, inξ stands for any set of propositions.
Wittgenstein’s logic, be any such thing as a causal nexus. “The events
N(ξ) stands for the negation of all the propositions making up ξ.
of the future”, he says, “cannot be inferred from those of the present.
Superstition is the belief in the causal nexus.” That the sun will rise
The whole symbol [ p, ξ, N(ξ)] means whatever can be obtained by
to-morrow is a hypothesis. We do not in fact know whether it will rise,taking any selection of atomic propositions, negating them all, thensince there is no compulsion according to which one thing must happen
taking any selection of the set of propositions now obtained, together
because another happens.
with any of the originals—and so on indefinitely. This is, he says, theLet us now take up another subject—that of names. In Wittgengeneral truth-function and also the general form of proposition. Whatstein’s theoretical logical language, names are only given to simples.
is meant is somewhat less complicated than it sounds. The symbol is
We do not give two names to one thing, or one name to two things.
intended to describe a process by the help of which, given the atomic
There is no way whatever, according to him, by which we can describe
propositions, all others can be manufactured. The process depends
the totality of things that can be named, in other words, the totality ofupon:what there is in the world. In order to be able to do this we should have that whatever can be said has to be about bounded portions of theto know of some property which must belong to every thing by a logical world. This view may have been originally suggested by notation, andnecessity. It has been sought to find such a property in self-identity, but if so, that is much in its favor, for a good notation has a subtlety andthe conception of identity is subjected by Wittgenstein to a destructive suggestiveness which at times make it seem almost like a live teacher.
criticism from which there seems no escape. The definition of identity Notational irregularities are often the first sign of philosophical errors,by means of the identity of indiscernibles is rejected, because the iden- and a perfect notation would be a substitute for thought. But althoughtity of indiscernibles appears to be not a logically necessary principle. notation may have first suggested to Mr. Wittgenstein the limitation
According to this principle x is identical with y if every property of x is of logic to things within the world as opposed to the world as a whole,a property of y, but it would, after all be logically possible for two things yet the view, once suggested, is seen to have much else to recommend
to have exactly the same properties. If this does not in fact happen that it. Whether it is ultimately true I do not, for my part, profess to know.
is an accidental characteristic of the world, not a logically necessary In this Introduction I am concerned to expound it, not to pronounce
characteristic, and accidental characteristics of the world must, of upon it. According to this view we could only say things about thecourse, not be admitted into the structure of logic. Mr. Wittgenstein world as a whole if we could get outside the world, if, that is to say, it
accordingly banishes identity and adopts the convention that different ceased to be for us the whole world. Our world may be bounded forletters are to mean different things. In practice, identity is needed as some superior being who can survey it from above, but for us, however
between a name and a description or between two descriptions. It is finite it may be, it cannot have a boundary, since it has nothing outsideneeded for such propositions as “Socrates is the philosopher who drank it. Wittgenstein uses, as an analogy, the field of vision. Our field of
the hemlock”, or “The even prime is the next number after 1.” For such vision does not, for us, have a visual boundary, just because there
uses of identity it is easy to provide on Wittgenstein’s system.
is nothing outside it, and in like manner our logical world has no
The rejection of identity removes one method of speaking of the logical boundary because our logic knows of nothing outside it. These
totality of things, and it will be found that any other method that may considerations lead him to a somewhat curious discussion of Solipsism.
be suggested is equally fallacious: so, at least, Wittgenstein contends Logic, he says, fills the world. The boundaries of the world are also its
and, I think, rightly. This amounts to saying that “object” is a pseudo- boundaries. In logic, therefore, we cannot say, there is this and this inconcept. To say “ x is an object” is to say nothing. It follows from this the world, but not that, for to say so would apparently presuppose thatthat we cannot make such statements as “there are more than three we exclude certain possibilities, and this cannot be the case, since it
objects in the world”, or “there are an infinite number of objects in the would require that logic should go beyond the boundaries of the world
world”. Objects can only be mentioned in connexion with some definite as if it could contemplate these boundaries from the other side also.
property. We can say “there are more than three objects which are What we cannot think we cannot think, therefore we also cannot say
human”, or “there are more than three objects which are red”, for in what we cannot think.
these statements the word object can be replaced by a variable in theThis, he says, gives the key to solipsism. What Solipsism intends islanguage of logic, the variable being one which satisfies in the first quite correct, but this cannot be said, it can only be shown. That thecase the function “ x is human”; in the second the function “ x is red”. world is my world appears in the fact that the boundaries of language
But when we attempt to say “there are more than three objects”, this (the only language I understand) indicate the boundaries of my world.
substitution of the variable for the word “object” becomes impossible, The metaphysical subject does not belong to the world but is a boundary
and the proposition is therefore seen to be meaningless.
of the world.
We here touch one instance of Wittgenstein’s fundamental thesis,
We must take up next the question of molecular propositions whichthat it is impossible to say anything about the world as a whole, and are at first sight not truth-functions, of the propositions that theycontain, such, for example, as “A believes p.”
relation of intention, or association, or what-not. The psychologicalWittgenstein introduces this subject in the statement of his po- part of meaning, however, does not concern the logician. What does
sition, namely, that all molecular functions are truth-functions. He concern him in this problem of belief is the logical schema. It is clear
says (5.54): “In the general propositional form, propositions occur in a that, when a person believes a proposition, the person, considered as aproposition only as bases of truth-operations.” At first sight, he goes on metaphysical subject, does not have to be assumed in order to explain
to explain, it seems as if a propositions could also occur in other ways, what is happening. What has to be explained is the relation between
e.g. “A believes p.” Here it seems superficially as if the proposition p the set of words which is the proposition considered as a fact on its
stood in a sort of relation to the object A. “But it is clear that ‘A believes own account, and the “objective” fact which makes the proposition truethat p,’ ‘A thinks p,’ ‘A says p’ are of the form “‘ p’ says p”; and here or false. This reduces ultimately to the question of the meaning of
we have no co-ordination of a fact and an object, but a co-ordination of propositions, that is to say, the meaning of propositions is the only
facts by means of a co-ordination of their objects” (5.542).
non-psychological portion of the problem involved in the analysis of
What Mr. Wittgenstein says here is said so shortly that its point is belief. This problem is simply one of a relation of two facts, namely, thenot likely to be clear to those who have not in mind the controversies relation between the series of words used by the believer and the factwith which he is concerned. The theory which which he is disagreeing which makes these words true or false. The series of words is a fact just
will be found in my articles on the nature of truth and falsehood as much as what makes it true or false is a fact. The relation between
in Philosophical Essays and Proceedings of the Aristotelian Society, these two facts is not unanalyzable, since the meaning of a proposition
1906–7. The problem at issue is the problem of the logical form of results from the meaning of its constituent words. The meaning of thebelief, i.e. what is the schema representing what occurs when a man series of words which is a proposition is a function of the meaning ofbelieves. Of course, the problem applies not only to belief, but also to the separate words. Accordingly, the proposition as a whole does nota host of other mental phenomena which may be called propositional really enter into what has to be explained in explaining the meaning
attitudes: doubting, considering, desiring, etc. In all these cases it of a propositions. It would perhaps help to suggest the point of view
seems natural to express the phenomenon in the form “A doubts p”, which I am trying to indicate, to say that in the cases which have been
“A considers p”, “A desires p”, etc., which makes it appear as though considering the proposition occurs as a fact, not as a proposition. Suchwe were dealing with a relation between a person and a proposition. a statement, however, must not be taken too literally. The real point isThis cannot, of course, be the ultimate analysis, since persons are that in believing, desiring, etc., what is logically fundamental is thefictions and so are propositions, except in the sense in which they relation of a proposition considered as a fact, to the fact which makes
are facts on their own account. A proposition, considered as a fact it true or false, and that this relation of two facts is reducible to aon its own account, may be a set of words which a man says over to relation of their constituents. Thus the proposition does not occur at
himself, or a complex image, or train of images passing through his all in the same sense in which it occurs in a truth-function.
mind, or a set of incipient bodily movements. It may be any one of
There are some respects, in which, as it seems to me, Mr. Wittgeninnumerable different things. The proposition as a fact on its own stein’s theory stands in need of greater technical development. Thisaccount, for example, the actual set of words the man pronounces applies in particular to his theory of number (6.02ff.) which, as it
to himself, is not relevant to logic. What is relevant to logic is that stands, is only capable of dealing with finite numbers. No logic can becommon element among all these facts, which enables him, as we say, considered adequate until it has been shown to be capable of dealing
to mean the fact which the proposition asserts. To psychology, of course, with transfinite numbers. I do not think there is anything in Mr.
more is relevant; for a symbol does not mean what it symbolizes in Wittgenstein’s system to make it impossible for him to fill this lacuna.
virtue of a logical relation alone, but in virtue also of a psychologicalMore interesting than such questions of comparative detail is Mr.

Wittgenstein’s attitude towards the mystical. His attitude upon this by Mr. Wittgenstein among the things that can be spoken of, for this is
grows naturally out of his doctrine in pure logic, according to which no other than the totality of things in the world, and thus involves thethe logical proposition is a picture (true or false) of the fact, and has in attempt to conceive the world as a whole; “the feeling of the world ascommon with the fact a certain structure. It is this common structure a bounded whole is the mystical”; hence the totality of the values ofwhich makes it capable of being a picture of the fact, but the structure x is mystical (6.45). This is expressly argued when Mr. Wittgensteincannot itself be put into words, since it is a structure of words, as denies that we can make propositions as to how many things there are
well as of the fact to which they refer. Everything, therefore, which in the world, as for example, that there are more than three.
is involved in the very idea of the expressiveness of language must
These difficulties suggest to my mind some such possibility as this:remain incapable of being expressed in language, and is, therefore, that every language has, as Mr. Wittgenstein says, a structure concerninexpressible in a perfectly precise sense. This inexpressible contains, ing which in the language, nothing can be said, but that there may be
according to Mr. Wittgenstein, the whole of logic and philosophy. The another language dealing with the structure of the first language, andright method of teaching philosophy, he says, would be to confine one- having itself a new structure, and that to this hierarchy of languages
self to propositions of the sciences, stated with all possible clearness there may be no limit. Mr. Wittgenstein would of course reply that his
and exactness, leaving philosophical assertions to the learner, and whole theory is applicable unchanged to the totality of such languages.
proving to him, whenever he made them, that they are meaningless. The only retort would be to deny that there is any such totality. The
It is true that the fate of Socrates might befall a man who attempted totalities concerning which Mr. Wittgenstein holds that it is impossible
this method of teaching, but we are not to be deterred by that fear, if to speak logically are nevertheless thought by him to exist, and are theit is the only right method. It is not this that causes some hesitation subject-matter of his mysticism. The totality resulting from our hierarin accepting Mr. Wittgenstein’s position, in spite of the very powerful chy would be not merely logically inexpressible, but a fiction, a merearguments which he brings to its support. What causes hesitation is delusion, and in this way the supposed sphere of the mystical would be
the fact that, after all, Mr. Wittgenstein manages to say a good deal abolished. Such a hypothesis is very difficult, and I can see objections
about what cannot be said, thus suggesting to the sceptical reader that to it which at the moment I do not know how to answer. Yet I do
possibly there may be some loophole through a hierarchy of languages, not see how any easier hypothesis can escape from Mr. Wittgenstein’sor by some other exit. The whole subject of ethics, for example, is placed conclusions. Even if this very difficult hypothesis should prove tenable,
by Mr. Wittgenstein in the mystical, inexpressible region. Nevertheless it would leave untouched a very large part of Mr. Wittgenstein’s theory,
he is capable of conveying his ethical opinions. His defence would be though possibly not the part upon which he himself would wish tothat what he calls the mystical can be shown, although it cannot be lay most stress. As one with a long experience of the difficulties ofsaid. It may be that this defence is adequate, but, for my part, I confess logic and of the deceptiveness of theories which seem irrefutable, I findthat it leaves me with a certain sense of intellectual discomfort.
myself unable to be sure of the rightness of a theory, merely on the
There is one purely logical problem in regard to which these diffi- ground that I cannot see any point on which it is wrong. But to haveculties are peculiarly acute. I mean the problem of generality. In the constructed a theory of logic which is not at any point obviously wrong
theory of generality it is necessary to consider all propositions of the is to have achieved a work of extraordinary difficulty and importance.
form f x where f x is a given propositional function. This belongs to the This merit, in my opinion, belongs to Mr. Wittgenstein’s book, andpart of logic which can be expressed, according to Mr. Wittgenstein’s makes it one which no serious philosopher can afford to neglect.
system. But the totality of possible values of x which might seem to be
B ERTRAND R USSELL .
involved in the totality of propositions of the form f x is not admitted
May 1922.

Tractatus Logico-Philosophicus

D EDICATEDTO THE M EMORY OF M Y F RIEND

D AVID H. P INSENT

M o t t o: . . . und alles, was man weiss, nicht bloss rauschen und brausen gehört hat, lässt sich in drei Worten sagen.

–K ÜRNBERGER .

Vorwort (Preface)
GermanOgden

Pears/McGuinness

Dieses Buch wird vielleicht nur der verstehen, der die Gedanken, die darin ausgedrückt
sind – oder doch ähnliche Gedanken – schon
selbst einmal gedacht hat. – Es ist also kein
Lehrbuch. – Sein Zweck wäre erreicht, wenn
es Einem, der es mit Verständnis liest Vergnügen bereitete.
Das Buch behandelt die philosophischen
Probleme und zeigt – wie ich glaube – dass die
Fragestellung dieser Probleme auf dem Mißverständnis der Logik unserer Sprache beruht.
Man könnte den ganzen Sinn des Buches etwa
in die Worte fassen: Was sich überhaupt sagen
lässt, lässt sich klar sagen; und wovon man
nicht reden kann, darüber muss man schweigen.
Das Buch will also dem Denken eine Grenze ziehen, oder vielmehr – nicht dem Denken,
sondern dem Ausdruck der Gedanken: Dennum dem Denken eine Grenze zu ziehen, müssten wir beide Seiten dieser Grenze denken
können (wir müssten also denken können, wassich nicht denken lässt).
Die Grenze wird also nur in der Sprachegezogen werden können und was jenseits derGrenze liegt, wird einfach Unsinn sein.
Wieweit meine Bestrebungen mit denen
anderer Philosophen zusammenfallen, will ich
nicht beurteilen. Ja, was ich hier geschrieben
habe macht im Einzelnen überhaupt nicht den
Anspruch auf Neuheit; und darum gebe ich
auch keine Quellen an, weil es mir gleichgültig
ist, ob das was ich gedacht habe, vor mir schon

This book will perhaps only be understoodby those who have themselves already thought
the thoughts which are expressed in it—or
similar thoughts. It is therefore not a textbook. Its object would be attained if there were
one person who read it with understanding
and to whom it afforded pleasure.
The book deals with the problems of philosophy and shows, as I believe, that the methodof formulating these problems rests on the misunderstanding of the logic of our language. Its
whole meaning could be summed up somewhat
as follows: What can be said at all can be said
clearly; and whereof one cannot speak thereof
one must be silent.

Perhaps this book will be understood only
by someone who has himself already had thethoughts that are expressed in it—or at least
similar thoughts.—So it is not a textbook.—Its
purpose would be achieved if it gave pleasure
to one person who read and understood it.
The book deals with the problems of philosophy, and shows, I believe, that the reason
why these problems are posed is that the logicof our language is misunderstood. The wholesense of the book might be summed up in thefollowing words: what can be said at all canbe said clearly, and what we cannot talk about
we must pass over in silence.

The book will, therefore, draw a limit tothinking, or rather—not to thinking, but tothe expression of thoughts; for, in order todraw a limit to thinking we should have to beable to think both sides of this limit (we should
therefore have to be able to think what cannot
be thought).
The limit can, therefore, only be drawn inlanguage and what lies on the other side ofthe limit will be simply nonsense.
How far my efforts agree with those ofother philosophers I will not decide. Indeed
what I have here written makes no claim tonovelty in points of detail; and therefore I
give no sources, because it is indifferent tome whether what I have thought has alreadybeen thought before me by another.

Thus the aim of the book is to draw a limit
to thought, or rather—not to thought, but tothe expression of thoughts: for in order to beable to draw a limit to thought, we should haveto find both sides of the limit thinkable (i.e. weshould have to be able to think what cannot
be thought).
It will therefore only be in language thatthe limit can be drawn, and what lies on theother side of the limit will simply be nonsense.
I do not wish to judge how far my efforts
coincide with those of other philosophers. Indeed, what I have written here makes no claim
to novelty in detail, and the reason why I give
no sources is that it is a matter of indifference
to me whether the thoughts that I have had
have been anticipated by someone else.

ein anderer gedacht hat.
Nur das will ich erwähnen, dass ich den
großartigen Werken Freges und den Arbeiten meines Freundes Herrn Bertrand Russell
einen großen Teil der Anregung zu meinen
Gedanken schulde.
Wenn diese Arbeit einen Wert hat, so besteht er in Zweierlei. Erstens darin, dass in ihr
Gedanken ausgedrückt sind, und dieser Wert
wird umso größer sein, je besser die Gedanken
ausgedrückt sind. Je mehr der Nagel auf den
Kopf getroffen ist. – Hier bin ich mir bewusst,
weit hinter dem Möglichen zurückgeblieben zu
sein. Einfach darum, weil meine Kraft zur Bewältigung der Aufgabe zu gering ist. – Mögen
andere kommen und es besser machen.
Dagegen scheint mir die W a h r h e i t derhier mitgeteilten Gedanken unantastbar und
definitiv. Ich bin also der Meinung, die Probleme im Wesentlichen endgültig gelöst zu haben. Und wenn ich mich hierin nicht irre, so
besteht nun der Wert dieser Arbeit zweitens
darin, dass sie zeigt, wie wenig damit getan
ist, dass diese Probleme gelöst sind.

I will only mention that to the great
I will only mention that I am indebted toworks of Frege and the writings of my friend Frege’s great works and of the writings of myBertrand Russell I owe in large measure the friend Mr. Bertrand Russell for much of thestimulation of my thoughts.
stimulation of my thoughts.
If this work has a value it consists in two
things. First that in it thoughts are expressed,
and this value will be the greater the betterthe thoughts are expressed. The more thenail has been hit on the head.—Here I am
conscious that I have fallen far short of thepossible. Simply because my powers are insufficient to cope with the task.—May others
come and do it better.
On the other hand the truth of the thoughts
communicated here seems to me unassailable
and definitive. I am, therefore, of the opinion
that the problems have in essentials been finally solved. And if I am not mistaken in this,then the value of this work secondly consistsin the fact that it shows how little has been
done when these problems have been solved.

L. W.
Wien, 1918

L. W.
Vienna, 1918

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 thethoughts 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 comeand do it better.
On the other hand the truth of the thoughtsthat are here communicated seems to me unassailable and definitive. I therefore believe myself to have found, on all essential points, thefinal solution of the problems. And if I am
not mistaken in this belief, then the second
thing in which the value of this work consists
is that it shows how little is achieved whenthese problems are solved.
L. W.
Vienna, 1918

Tractatus Logico-Philosophicus

1*
1.1
1.11

1.12

1.13
1.2
1.21

2
2.01
2.011

2.012

2.0121

GermanOgden

Die Welt ist alles, was der Fall ist.
Die Welt ist die Gesamtheit der Tatsachen, nicht der Dinge.
Die Welt ist durch die Tatsachen bestimmt und dadurch, dass es a l l e Tatsachen sind.
Denn, die Gesamtheit der Tatsachen
bestimmt, was der Fall ist und auch, wasalles nicht der Fall ist.
Die Tatsachen im logischen Raum sinddie Welt.
Die Welt zerfällt in Tatsachen.
Eines kann der Fall sein oder nicht derFall sein und alles übrige gleich bleiben.

The world is everything that is the case.
The world is all that is the case.
The world is the totality of facts, not of
The world is the totality of facts, not ofthings.
things.
The world is determined by the facts,
The world is determined by the facts,
and by these being all the facts.
and by their being all the facts.

Was der Fall ist, die Tatsache, ist das
Bestehen von Sachverhalten.
Der Sachverhalt ist eine Verbindungvon Gegenständen. (Sachen, Dingen.)Es ist dem Ding wesentlich, der Bestandteil eines Sachverhaltes sein zu können.
In der Logik ist nichts zufällig: Wenndas Ding im Sachverhalt vorkommen
k a n n, so muss die Möglichkeit des Sachverhaltes im Ding bereits präjudiziert sein.
Es erschiene gleichsam als Zufall, wenn
dem Ding, das allein für sich bestehen

Pears/McGuinness

For the totality of facts determines both
For the totality of facts determines
what is the case, and also all that is not what is the case, and also whatever is not
the case.
the case.
The facts in logical space are the world.
The facts in logical space are the world.
The world divides into facts.
Any one can either be the case or not be
the case, and everything else remain thesame.
What is the case, the fact, is the existence of atomic facts.
An atomic fact is a combination of objects (entities, things).
It is essential to a thing that it can bea constituent part of an atomic fact.

The world divides into facts.
Each item can be the case or not thecase while everything else remains thesame.
What is the case—a fact—is the existence of states of affairs.
A state of affairs (a state of things) is acombination 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 an atomic fact the possibilityof that atomic fact must already be prejudged in the thing.
It would, so to speak, appear as an accident, when to a thing that could exist alone

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* [German] Die Decimalzahlen als Nummern der einzelnen Sätze deuten das logische Gewicht der Sätze an, den Nachdruck, der auf ihnen in meiner Darstellung liegt.
Die Sätze n.1, n.2, n.3, etc., sind Bemerkungen zum Satze No. n; die Sätze n.m1, n.m2, etc. Bemerkungen zum Satze No. n.m; und so weiter. / [Ogden] The decimal figures
as numbers of the separate propositions indicate the logical importance of the propositions, the emphasis laid upon them in my exposition. The propositions n.1, n.2, n.3,
etc., are comments on proposition No. n; the propositions n.m1, n.m2, etc., are comments on the proposition No. n.m; and so on. / [Pears & McGuinness] The decimal
numbers assigned to the individual propositions indicate the logical importance of the propositions, the stress laid on them in my exposition. The propositions n.1, n.2, n.3,
etc. are comments on proposition no. n; the propositions n.m1, n.m2, etc. are comments on proposition no. n.m; and so on.

könnte, nachträglich eine Sachlage passen
würde.
Wenn die Dinge in Sachverhalten vorkommen können, so muss dies schon in
ihnen liegen.
(Etwas Logisches kann nicht nurmöglich sein. Die Logik handelt von jeder
Möglichkeit und alle Möglichkeiten sindihre Tatsachen.)
Wie wir uns räumliche Gegenständeüberhaupt nicht außerhalb des Raumes,
zeitliche nicht außerhalb der Zeit denken
können, so können wir uns k e i n e n Gegenstand außerhalb der Möglichkeit seiner
Verbindung mit anderen denken.
Wenn ich mir den Gegenstand im Verbande des Sachverhalts denken kann, sokann ich ihn nicht außerhalb der M ö g l i c h k e i t dieses Verbandes denken.
Das Ding ist selbständig, insofern es in
2.0122
allen m ö g l i c h e n Sachlagen vorkommen kann, aber diese Form der Selbständigkeit ist eine Form des Zusammenhangs
mit dem Sachverhalt, eine Form der Unselbständigkeit. (Es ist unmöglich, dassWorte in zwei verschiedenen Weisen auftreten, allein und im Satz.)
2.0123
Wenn ich den Gegenstand kenne, so
kenne ich auch sämtliche Möglichkeiten
seines Vorkommens in Sachverhalten.
(Jede solche Möglichkeit muss in derNatur des Gegenstandes liegen.)Es kann nicht nachträglich eine neue
Möglichkeit gefunden werden.
2.01231
Um einen Gegenstand zu kennen, mussich zwar nicht seine externen – aber ichmuss alle seine internen Eigenschaften

on its own account, subsequently a state of thing that could already exist entirely on
affairs could be made to fit.
its own.
If things can occur in atomic facts, thisIf things can occur in states of affairs,
possibility must already lie in them.
this possibility must be in them from the
beginning.
(A logical entity cannot be merely pos(Nothing in the province of logic cansible. Logic treats of every possibility, and be merely possible. Logic deals with evall possibilities are its facts.)
ery possibility and all possibilities are its
facts.)
Just as we cannot think of spatial obJust as we are quite unable to imagine
jects at all apart from space, or temporal spatial objects outside space or temporal
objects apart from time, so we cannot think objects outside time, so too there is no obof any object apart from the possibility of ject that we can imagine excluded from theits connexion with other things.
possibility of combining with others.
If I can think of an object in the context
If I can imagine objects combined inof an atomic fact, I cannot think of it apart states of affairs, I cannot imagine themfrom the possibility of this context.
excluded from the possibility of such combinations.
The thing is independent, in so far as
Things are independent in so far as
it can occur in all possible circumstances, they can occur in all possible situations,but this form of independence is a form of but this form of independence is a form
connexion with the atomic fact, a form of of connexion with states of affairs, a form
dependence. (It is impossible for words to of dependence. (It is impossible for words
occur in two different ways, alone and in to appear in two different roles: by themthe proposition.)selves, and in propositions.)
If I know an object, then I also know all
the possibilities of its occurrence in atomic
facts.
(Every such possibility must lie in thenature of the object.)A new possibility cannot subsequently
be found.
In order to know an object, I must know
not its external but all its internal qualities.

If I know an object I also know all its
possible occurrences in states of affairs.
(Every one of these possibilities mustbe 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 properties.

2.01242.013

2.01312.014
2.01412.02
2.02012.021

2.02112.0212kennen.
Sind alle Gegenstände gegeben, so sind
damit auch alle m ö g l i c h e n Sachverhalte gegeben.
Jedes Ding ist, gleichsam, in einem
Raume möglicher Sachverhalte. Diesen
Raum kann ich mir leer denken, nicht aberdas Ding ohne den Raum.
Der räumliche Gegenstand muss im unendlichen Raume liegen. (Der Raumpunkt
ist eine Argumentstelle.)
Der Fleck im Gesichtsfeld muss zwar
nicht rot sein, aber eine Farbe muss er haben: er hat sozusagen den Farbenraum umsich. Der Ton muss e i n e Höhe haben, derGegenstand des Tastsinnes e i n e Härte,usw.
Die Gegenstände enthalten die Möglichkeit aller Sachlagen.
Die Möglichkeit seines Vorkommens in
Sachverhalten, ist die Form des Gegenstandes.
Der Gegenstand ist einfach.
Jede Aussage über Komplexe lässt sichin eine Aussage über deren Bestandteile
und in diejenigen Sätze zerlegen, welche
die Komplexe vollständig beschreiben.

If all objects are given, then thereby are
If all objects are given, then at the same
all possible atomic facts also given.
time all possible states of affairs are also
given.
Every thing is, as it were, in a space ofEach thing is, as it were, in a space ofpossible atomic facts. I can think of this possible states of affairs. This space I canspace as empty, but not of the thing with- imagine empty, but I cannot imagine theout the space.
thing without the space.
A spatial object must lie in infinite
A spatial object must be situated in infispace. (A point in space is an argument nite space. (A spatial point is an argumentplace.)
place.)
A speck in a visual field need not be
A speck in the visual field, thought it
red, but it must have a colour; it has, so need not be red, must have some colour: itto speak, a colour space round it. A tone is, so to speak, surrounded by colour-space.
must have a pitch, the object of the sense Notes must have some pitch, objects of theof touch a hardness, etc.
sense of touch some degree of hardness,and so on.
Objects contain the possibility of all
Objects contain the possibility of all sitstates of affairs.
uations.
The possibility of its occurrence in
The possibility of its occurring in states
atomic facts is the form of the object.
of affairs is the form of an object.

The object is simple.
Every statement about complexes canbe analysed into a statement about theirconstituent parts, and into those propositions which completely describe the complexes.
Die Gegenstände bilden die Substanz
Objects form the substance of the world.
der Welt. Darum können sie nicht zusam- Therefore they cannot be compound.
mengesetzt sein.
Hätte die Welt keine Substanz, so würIf the world had no substance, thende, ob ein Satz Sinn hat, davon abhängen, whether a proposition had sense would deob ein anderer Satz wahr ist.
pend on whether another proposition wastrue.
Es wäre dann unmöglich, ein Bild derIt would then be impossible to form a
Welt (wahr oder falsch) zu entwerfen.
picture of the world (true or false).

Objects are simple.
Every statement about complexes canbe resolved into a statement about theirconstituents and into the propositions that
describe the complexes completely.
Objects make up the substance of theworld. That is why they cannot be composite.
If the world had no substance, thenwhether a proposition had sense would depend on whether another proposition wastrue.
In that case we could not sketch any
picture of the world (true or false).

2.0222.023
2.0231

Es ist offenbar, dass auch eine von derwirklichen noch so verschieden gedachte
Welt Etwas – eine Form – mit der wirklichen gemein haben muss.
Diese feste Form besteht eben aus den
Gegenständen.
Die Substanz der Welt k a n n nur eine
Form und keine materiellen Eigenschaften
bestimmen. Denn diese werden erst durch
die Sätze dargestellt – erst durch die Konfiguration der Gegenstände gebildet.

Beiläufig gesprochen: Die Gegenstände
sind farblos.
2.0233Zwei Gegenstände von der gleichen logischen Form sind – abgesehen von ihren
externen Eigenschaften – von einander nur
dadurch unterschieden, dass sie verschieden sind.
2.02331
Entweder ein Ding hat Eigenschaften,
die kein anderes hat, dann kann man es
ohneweiteres durch eine Beschreibung aus
den anderen herausheben, und darauf hinweisen; oder aber, es gibt mehrere Dinge,
die ihre sämtlichen Eigenschaften gemeinsam haben, dann ist es überhaupt unmöglich auf eines von ihnen zu zeigen.
Denn, ist das Ding durch nichts hervorgehoben, so kann ich es nicht hervorheben,
denn sonst ist es eben hervorgehoben.
2.024
Die Substanz ist das, was unabhängig
von dem was der Fall ist, besteht.
2.025
Sie ist Form und Inhalt.
2.0251
Raum, Zeit und Farbe (Färbigkeit) sind
Formen der Gegenstände.
2.026
Nur wenn es Gegenstände gibt, kannes eine feste Form der Welt geben.
2.0232

It is clear that however different from
It is obvious that an imagined world,
the real one an imagined world may be, it however different it may be from the realmust have something—a form—in common one, must have something—a form—inwith the real world.
common with it.
This fixed form consists of the objects.
Objects are just what constitute thisunalterable form.
The substance of the world can only
The substance of the world can only
determine a form and not any material determine a form, and not any material
properties. For these are first presented properties. For it is only by means of propoby the propositions—first formed by the sitions that material properties are repconfiguration of the objects.
resented—only by the configuration of objects that they are produced.
Roughly speaking: objects are colourIn a manner of speaking, objects areless.
colourless.
Two objects of the same logical form
If two objects have the same logicalare—apart from their external proper- form, the only distinction between them,ties—only differentiated from one another apart from their external properties, isin that they are different.
that they are different.
Either a thing has properties which noother has, and then one can distinguish it
straight away from the others by a description and refer to it; or, on the other hand,
there are several things which have thetotality of their properties in common, andthen it is quite impossible to point to any
one of them.
For if a thing is not distinguished byanything, I cannot distinguish it—for otherwise it would be distinguished.
Substance is what exists independentlyof what is the case.
It is form and content.
Space, time and colour (colouredness)
are forms of objects.
Only if there are objects can there be a
fixed form of the world.

Either a thing has properties that nothing else has, in which case we can immediately use a description to distinguish itfrom the others and refer to it; or, on theother hand, there are several things thathave the whole set of their properties incommon, in which case it is quite impossible to indicate one of them.
For if there is nothing to distinguish athing, I cannot distinguish it, since otherwise it would be distinguished after all.
The substance is what subsists independently of what is the case.
It is form and content.
Space, time, colour (being coloured) areforms of objects.
There must be objects, if the world is tohave unalterable form.

2.0272.02712.0272
2.03
2.031

2.032

2.033
2.034
2.04
2.05

2.06

2.061
2.062

2.063
2.1

Das Feste, das Bestehende und der Gegenstand sind Eins.
Der Gegenstand ist das Feste, Bestehende; die Konfiguration ist das Wechselnde, Unbeständige.
Die Konfiguration der Gegenstände bildet den Sachverhalt.
Im Sachverhalt hängen die Gegenstände ineinander, wie die Glieder einer Kette.
Im Sachverhalt verhalten sich die Gegenstände in bestimmter Art und Weise
zueinander.
Die Art und Weise, wie die Gegenstände im Sachverhalt zusammenhängen, ist
die Struktur des Sachverhaltes.
Die Form ist die Möglichkeit der Struktur.
Die Struktur der Tatsache besteht aus
den Strukturen der Sachverhalte.
Die Gesamtheit der bestehenden Sachverhalte ist die Welt.
Die Gesamtheit der bestehenden Sachverhalte bestimmt auch, welche Sachverhalte nicht bestehen.
Das Bestehen und Nichtbestehen von
Sachverhalten ist die Wirklichkeit.
(Das Bestehen von Sachverhalten nennen wir auch eine positive, das Nichtbestehen eine negative Tatsache.)
Die Sachverhalte sind von einander unabhängig.
Aus dem Bestehen oder Nichtbesteheneines Sachverhaltes kann nicht auf das Bestehen oder Nichtbestehen eines anderengeschlossen werden.
Die gesamte Wirklichkeit ist die Welt.
Wir machen uns Bilder der Tatsachen.

The fixed, the existent and the object
Objects, the unalterable, and the subare one.
sistent are one and the same.
The object is the fixed, the existent; theObjects are what is unalterable andconfiguration is the changing, the variable. subsistent; their configuration is what ischanging and unstable.
The configuration of the objects forms
The configuration of objects produces
the atomic fact.
states of affairs.
In the atomic fact objects hang one in
In a state of affairs objects fit into one
another, like the links of a chain.
another like the links of a chain.
In the atomic fact the objects are comIn a state of affairs objects stand in abined in a definite way.
determinate relation to one another.
The way in which objects hang together
in the atomic fact is the structure of theatomic fact.
The form is the possibility of the structure.
The structure of the fact consists of thestructures of the atomic facts.
The totality of existent atomic facts is
the world.
The totality of existent atomic facts also
determines which atomic facts do not exist.
The existence and non-existence of
atomic facts is the reality.
(The existence of atomic facts we also
call a positive fact, their non-existence a
negative fact.)
Atomic facts are independent of one another.
From the existence or non-existence ofan atomic fact we cannot infer the existence or non-existence of another.
The total reality is the world.
We make to ourselves pictures of facts.

The determinate way in which objects
are connected in a state of affairs is the
structure of the state of affairs.
Form is the possibility of structure.
The structure of a fact consists of thestructures 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 ofstates 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.

2.11

2.12
2.13
2.131
2.14

2.141
2.15

2.151

Das Bild stellt die Sachlage im logischen Raume, das Bestehen und Nichtbestehen von Sachverhalten vor.
Das Bild ist ein Modell der Wirklichkeit.
Den Gegenständen entsprechen im Bilde die Elemente des Bildes.
Die Elemente des Bildes vertreten im
Bild die Gegenstände.
Das Bild besteht darin, dass sich seine
Elemente in bestimmter Art und Weise zu
einander verhalten.
Das Bild ist eine Tatsache.
Dass sich die Elemente des Bildes inbestimmter Art und Weise zu einander verhalten, stellt vor, dass sich die Sachen sozu einander verhalten.
Dieser Zusammenhang der Elemente
des Bildes heiße seine Struktur und ihre
Möglichkeit seine Form der Abbildung.
Die Form der Abbildung ist die Möglichkeit, dass sich die Dinge so zu einanderverhalten, wie die Elemente des Bildes.

Das Bild ist s o mit der Wirklichkeit
verknüpft – es reicht bis zu ihr.
2.1512
Es ist wie ein Maßstab an die Wirklichkeit angelegt.
2.15121
Nur die äußersten Punkte der Teilstriche b e r ü h r e n den zu messenden Gegenstand.
2.1513
Nach dieser Auffassung gehört also
zum Bilde auch noch die abbildende Beziehung, die es zum Bild macht.
2.1514
Die abbildende Beziehung besteht aus
den Zuordnungen der Elemente des Bildes
2.1511The picture presents the facts in logical
A picture presents a situation in logicalspace, the existence and non-existence of space, the existence and non-existence of
atomic facts.
states of affairs.
The picture is a model of reality.
A picture is a model of reality.
To the objects correspond in the picture
the elements of the picture.
The elements of the picture stand, inthe picture, for the objects.
The picture consists in the fact that its
elements are combined with one anotherin a definite way.
The picture is a fact.
That the elements of the picture are
combined with one another in a definite
way, represents that the things are so combined with one another.
This connexion of the elements of thepicture is called its structure, and the possibility of this structure is called the formof representation of the picture.
The form of representation is the possibility that the things are combined with
one another as are the elements of the picture.
Thus the picture is linked with reality;
it reaches up to it.
It is like a scale applied to reality.

In a picture objects have the elementsof 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 thepictorial form of the picture.
Pictorial form is the possibility thatthings are related to one another in thesame way as the elements of the picture.
That is how a picture is attached toreality; it reaches right out to it.
It is laid against reality like a measure.

Only the outermost points of the dividOnly the end-points of the graduating
ing lines touch the object to be measured. lines actually touch the object that is to bemeasured.
According to this view the representSo a picture, conceived in this way, also
ing relation which makes it a picture, also includes the pictorial relationship, which
belongs to the picture.
makes it into a picture.
The representing relation consists of
The pictorial relationship consists ofthe co-ordinations of the elements of the the correlations of the picture’s elements

2.1515

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2.172
2.173

2.174
2.18

2.181

2.182

und der Sachen.
Diese Zuordnungen sind gleichsam die
Fühler der Bildelemente, mit denen das
Bild die Wirklichkeit berührt.
Die Tatsache muss, um Bild zu sein,
etwas mit dem Abgebildeten gemeinsam
haben.
In Bild und Abgebildetem muss etwas
identisch sein, damit das eine überhaupt
ein Bild des anderen sein kann.
Was das Bild mit der Wirklichkeit gemein haben muss, um sie auf seine Art und
Weise – richtig oder falsch – abbilden zu
können, ist seine Form der Abbildung.
Das Bild kann jede Wirklichkeit abbilden, deren Form es hat.
Das räumliche Bild alles Räumliche,
das farbige alles Farbige, etc.

picture and the things.
These co-ordinations are as it were thefeelers of its elements with which the picture touches reality.
In order to be a picture a fact musthave something in common with what it
pictures.
In the picture and the pictured there
must be something identical in order thatthe one can be a picture of the other at all.
What the picture must have in common
with reality in order to be able to represent
it after its manner—rightly or falsely—is
its form of representation.
The picture can represent every reality
whose form it has.
The spatial picture, everything spatial,
the coloured, everything coloured, etc.

Seine Form der Abbildung aber, kanndas Bild nicht abbilden; es weist sie auf.
Das Bild stellt sein Objekt von außerhalb dar (sein Standpunkt ist seine Form
der Darstellung), darum stellt das Bildsein Objekt richtig oder falsch dar.

The picture, however, cannot represent
its form of representation; it shows it forth.
The picture represents its object fromwithout (its standpoint is its form of representation), therefore the picture represents
its object rightly or falsely.

Das Bild kann sich aber nicht außerhalb seiner Form der Darstellung stellen.
Was jedes Bild, welcher Form immer,
mit der Wirklichkeit gemein haben muss,um sie überhaupt – richtig oder falsch –abbilden zu können, ist die logische Form,das ist, die Form der Wirklichkeit.
Ist die Form der Abbildung die logischeForm, so heißt das Bild das logische Bild.

But the picture cannot place itself outside of its form of representation.
What every picture, of whatever form,
must have in common with reality in order
to be able to represent it at all—rightly
or falsely—is the logical form, that is, theform of reality.
If the form of representation is the logical form, then the picture is called a logicalpicture.
Jedes Bild ist a u c h ein logisches. (DaEvery picture is also a logical picture.

with things.
These correlations are, as it were, thefeelers of the picture’s elements, withwhich the picture touches reality.
If a fact is to be a picture, it must havesomething in common with what it depicts.
There must be something identical in a
picture and what it depicts, to enable theone 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 waythat 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 displays it.
A picture represents its subject froma position outside it. (Its standpoint is
its representational form.) That is why
a picture represents its subject correctly or
incorrectly.
A picture cannot, however, place itself
outside its representational form.
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 logicalform is called a logical picture.
Every picture is at the same time a log-

2.19
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2.202
2.203
2.21

2.22

2.221
2.222

2.223

2.224
2.225
3
3.001

gegen ist z. B. nicht jedes Bild ein räumliches.)
Das logische Bild kann die Welt abbilden.
Das Bild hat mit dem Abgebildeten die
logische Form der Abbildung gemein.

(On the other hand, for example, not every
picture is spatial.)
The logical picture can depict theworld.
The picture has the logical form of representation in common with what it pictures.
Das Bild bildet die Wirklichkeit ab, inThe picture depicts reality by repredem es eine Möglichkeit des Bestehens senting a possibility of the existence andund Nichtbestehens von Sachverhalten non-existence of atomic facts.
darstellt.
Das Bild stellt eine mögliche Sachlage
The picture represents a possible state
im logischen Raume dar.
of affairs in logical space.
Das Bild enthält die Möglichkeit der
The picture contains the possibility of
Sachlage, die es darstellt.
the state of affairs which it represents.
Das Bild stimmt mit der Wirklichkeit
The picture agrees with reality or not;überein oder nicht; es ist richtig oder un- it is right or wrong, true or false.
richtig, wahr oder falsch.
Das Bild stellt dar, was es darstellt, unThe picture represents what it repreabhängig von seiner Wahr- oder Falschheit, sents, independently of its truth or falsedurch die Form der Abbildung.
hood, through the form of representation.
Was das Bild darstellt, ist sein Sinn.
What the picture represents is its
sense.
In der Übereinstimmung oder NichtIn the agreement or disagreement ofübereinstimmung seines Sinnes mit der its sense with reality, its truth or falsity
Wirklichkeit, besteht seine Wahrheit oder consists.
Falschheit.
Um zu erkennen, ob das Bild wahr oder
In order to discover whether the picture
falsch ist, müssen wir es mit der Wirklich- is true or false we must compare it withkeit vergleichen.
reality.
Aus dem Bild allein ist nicht zu erkenIt cannot be discovered from the picturenen, ob es wahr oder falsch ist.
alone whether it is true or false.
Ein a priori wahres Bild gibt es nicht.
There is no picture which is a priori
true.
Das logische Bild der Tatsachen ist der
The logical picture of the facts is theGedanke.
thought.
„Ein Sachverhalt ist denkbar“ heißt:“An atomic fact is thinkable”—means:

ical one. (On the other hand, not every
picture is, for example, a spatial one.)
Logical pictures can depict the world.
A picture has logico-pictorial form incommon with what it depicts.
A picture depicts reality by representing a possibility of existence and nonexistence of states of affairs.
A picture represents a possible situation in logical space.
A picture contains the possibility of thesituation that it represents.
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, bymeans of its pictorial form.
What a picture represents is its sense.
The agreement or disagreement of its
sense with reality constitutes its truth or
falsity.
In order to tell whether a picture is trueor false we must compare it with reality.
It is impossible to tell from the picturealone whether it is true or false.
There are no pictures that are true apriori.
A logical picture of facts is a thought.
‘A state of affairs is thinkable’: what

Wir können uns ein Bild von ihm machen. we can imagine it.
3.01
3.02

3.03

3.031

3.032

3.0321

3.04

3.05

3.1

3.11

Die Gesamtheit der wahren Gedanken
sind ein Bild der Welt.
Der Gedanke enthält die Möglichkeit
der Sachlage, die er denkt. Was denkbar
ist, ist auch möglich.
Wir können nichts Unlogisches denken,
weil wir sonst unlogisch denken müssten.
Man sagte einmal, dass Gott alles
schaffen könne, nur nichts, was den logischen Gesetzen zuwider wäre. – Wir können nämlich von einer „unlogischen“ Welt
nicht s a g e n, wie sie aussähe.
Etwas „der Logik widersprechendes“ inder Sprache darstellen, kann man ebensowenig, wie in der Geometrie eine den Gesetzen des Raumes widersprechende Figur
durch ihre Koordinaten darstellen; oder
die Koordinaten eines Punktes angeben,
welcher nicht existiert.
Wohl können wir einen Sachverhalt
räumlich darstellen, welcher den Gesetzen
der Physik, aber keinen, der den Gesetzen
der Geometrie zuwiderliefe.
Ein a priori richtiger Gedanke wäre einsolcher, dessen Möglichkeit seine Wahrheit
bedingte.
Nur so könnten wir a priori wissen,dass ein Gedanke wahr ist, wenn aus dem
Gedanken selbst (ohne Vergleichsobjekt)seine Wahrheit zu erkennen wäre.
Im Satz drückt sich der Gedanke sinnlich wahrnehmbar aus.
Wir benützen das sinnlich wahrnehm-

The totality of true thoughts is a picture of the world.
The thought contains the possibility ofthe state of affairs which it thinks. Whatis thinkable is also possible.
We cannot think anything unlogical, forotherwise we should have to think unlogically.
It used to be said that God could create
everything, except what was contrary to
the laws of logic. The truth is, we could not
say of an “unlogical” world how it wouldlook.
To present in language anything which“contradicts logic” is as impossible as ingeometry to present by its co-ordinates afigure which contradicts the laws of space;or to give the co-ordinates of a point which
does not exist.
We could present spatially an atomic
fact which contradicted the laws of physics,
but not one which contradicted the laws ofgeometry.
An a priori true thought would be one
whose possibility guaranteed its truth.
Only if we could know a priori that athought is true if its truth was to be recognized from the thought itself (without anobject of comparison).
In the proposition the thought is expressed perceptibly through the senses.
We use the sensibly perceptible sign

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 ofthe situation of which it is the thought.
What is thinkable is possible too.
Thought can never be of anything illogical, since, if it were, we should have tothink 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 look
like.
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 lawsof space, or to give the co-ordinates of apoint 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.
If a thought were correct a priori, it
would be a thought whose possibility ensured its truth.
A priori knowledge that a thought wastrue would be possible only if its truth wererecognizable from the thought itself (without anything to compare it with).
In a proposition a thought finds an
expression that can be perceived by thesenses.
We use the perceptible sign of a proposi-

3.12

3.13

3.14

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3.142

3.143

bare Zeichen (Laut- oder Schriftzeichenetc.) des Satzes als Projektion der möglichen Sachlage.
Die Projektionsmethode ist das Denken
des Satz-Sinnes.
Das Zeichen, durch welches wir den Gedanken ausdrücken, nenne ich das Satzzeichen. Und der Satz ist das Satzzeichen inseiner projektiven Beziehung zur Welt.
Zum Satz gehört alles, was zur Projektion gehört; aber nicht das Projizierte.
Also die Möglichkeit des Projizierten,
aber nicht dieses selbst.
Im Satz ist also sein Sinn noch nicht
enthalten, wohl aber die Möglichkeit, ihn
auszudrücken.
(„Der Inhalt des Satzes“ heißt der Inhalt des sinnvollen Satzes.)
Im Satz ist die Form seines Sinnes enthalten, aber nicht dessen Inhalt.
Das Satzzeichen besteht darin, dasssich seine Elemente, die Wörter, in ihm
auf bestimmte Art und Weise zu einanderverhalten.
Das Satzzeichen ist eine Tatsache.
Der Satz ist kein Wörtergemisch. –(Wie das musikalische Thema kein Gemisch von Tönen.)
Der Satz ist artikuliert.
Nur Tatsachen können einen Sinn ausdrücken, eine Klasse von Namen kann es
nicht.
Dass das Satzzeichen eine Tatsache ist,
wird durch die gewöhnliche Ausdrucksform der Schrift oder des Druckes verschleiert.

(sound or written sign, etc.) of the proposition as a projection of the possible state ofaffairs.
The method of projection is the thinking of the sense of the proposition.
The sign through which we express thethought I call the propositional sign. And
the proposition is the propositional sign in
its projective relation to the world.
To the proposition belongs everythingwhich belongs to the projection; but notwhat is projected.
Therefore the possibility of what is projected but not this itself.
In the proposition, therefore, its sense
is not yet contained, but the possibility of
expressing it.
(“The content of the proposition” means
the content of the significant proposition.)In the proposition the form of its sense
is contained, but not its content.
The propositional sign consists in thefact that its elements, the words, are combined in it in a definite way.

tion (spoken or written, etc.) as a projectionof a possible situation.
The method of projection is to think of
the sense of the proposition.
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 includes all that the projection includes, but not what is projected.
Therefore, though what is projected is
not itself included, its possibility is.
A proposition, therefore, does not actually contain its sense, but does contain thepossibility 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 signis that in it its elements (the words) standin a determinate relation to one another.

The propositional sign is a fact.
A propositional sign is a fact.
The proposition is not a mixture of
A proposition is not a blend ofwords (just as the musical theme is not words.—(Just as a theme in music is not aa mixture of tones).
blend of notes.)
The proposition is articulate.
A proposition is articulate.
Only facts can express a sense, a class
Only facts can express a sense, a set ofof names cannot.
names cannot.
That the propositional sign is a fact is
Although a propositional sign is a fact,
concealed by the ordinary form of expres- this is obscured by the usual form of exsion, written or printed.
pression in writing or print.

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3.202
3.203

3.21

Denn im gedruckten Satz z. B. sieht das
For in the printed proposition, for exSatzzeichen nicht wesentlich verschieden ample, the sign of a proposition does not
aus vom Wort.
appear essentially different from a word.
(So war es möglich, dass Frege den Satz
(Thus it was possible for Frege to calleinen zusammengesetzten Namen nann- the proposition a compounded name.)te.)Sehr klar wird das Wesen des SatzzeiThe essential nature of the proposichens, wenn wir es uns, statt aus Schrift- tional sign becomes very clear when wezeichen, aus räumlichen Gegenständen (et- imagine it made up of spatial objects (suchwa Tischen, Stühlen, Büchern) zusammen- as tables, chairs, books) instead of written
gesetzt denken.
signs.
Die gegenseitige räumliche Lage dieser
The mutual spatial position of these
Dinge drückt dann den Sinn des Satzes things then expresses the sense of the
aus.
proposition.
Nicht: „Das komplexe Zeichen ‚aRb‘
We must not say, “The complex signsagt, dass a in der Beziehung R zu b steht“, ‘aRb’ says ‘a stands in relation R to b’”; but
sondern: D a s s „a“ in einer gewissen Be- we must say, “That ‘a’ stands in a certain
ziehung zu „ b“ steht, sagt, d a s s aRb.
relation to ‘ b’ says that aRb”.
Sachlagen kann man beschreiben,
States of affairs can be described but
nicht b e n e n n e n.
not named.
(Namen gleichen Punkten, Sätze Pfei(Names resemble points; propositionslen, sie haben Sinn.)
resemble arrows, they have sense.)
Im Satze kann der Gedanke so ausgeIn propositions thoughts can be so exdrückt sein, dass den Gegenständen des pressed that to the objects of the thoughts
Gedankens Elemente des Satzzeichens ent- correspond the elements of the proposisprechen.
tional sign.
Diese Elemente nenne ich „einfaThese elements I call “simple signs”
che Zeichen“ und den Satz „vollständig and the proposition “completely analysed”.
analysiert“.
Die im Satze angewandten einfachen
The simple signs employed in proposiZeichen heißen Namen.
tions are called names.
Der Name bedeutet den Gegenstand.
The name means the object. The object
Der Gegenstand ist seine Bedeutung. („ A “ is its meaning. (“ A ” is the same sign as
ist dasselbe Zeichen wie „ A “.)“ A ”.)
Der Konfiguration der einfachen ZeiTo the configuration of the simple signs
chen im Satzzeichen entspricht die Konfi- in the propositional sign corresponds theguration der Gegenstände in der Sachlage. configuration of the objects in the state of

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 isvery clearly seen if we imagine one composed of spatial objects (such as tables,
chairs, and books) instead of written signs.
Then the spatial arrangement of thesethings will express the sense of the proposition.
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” ina certain relation says that aRb.’
Situations can be described but notgiven names.
(Names are like points; propositions
like arrows—they have sense.)
In a proposition a thought can be expressed in such a way that elements of the
propositional sign correspond to the objectsof the thought.
I call such elements ‘simple signs’, andsuch a proposition ‘complete analysed’.
The simple signs employed in propositions are called names.
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 sign.

3.22
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3.23

3.24

3.25
3.251

Der Name vertritt im Satz den Gegenstand.
Die Gegenstände kann ich nur n e n n e n. Zeichen vertreten sie. Ich kann nur
v o n ihnen sprechen, s i e a u s s p r e c h e n kann ich nicht. Ein Satz kann nur
sagen, w i e ein Ding ist, nicht w a s es
ist.
Die Forderung der Möglichkeit der einfachen Zeichen ist die Forderung der Bestimmtheit des Sinnes.
Der Satz, welcher vom Komplex handelt, steht in interner Beziehung zum Satze, der von dessen Bestandteil handelt.
Der Komplex kann nur durch seine Beschreibung gegeben sein, und diese wird
stimmen oder nicht stimmen. Der Satz, inwelchem von einem Komplex die Rede ist,
wird, wenn dieser nicht existiert, nicht unsinnig, sondern einfach falsch sein.
Dass ein Satzelement einen Komplex
bezeichnet, kann man aus einer Unbestimmtheit in den Sätzen sehen, worin es
vorkommt. Wir w i s s e n, durch diesen
Satz ist noch nicht alles bestimmt. (DieAllgemeinheitsbezeichnung e n t h ä l t ja
ein Urbild.)
Die Zusammenfassung des Symbols eines Komplexes in ein einfaches Symbol
kann durch eine Definition ausgedrückt
werden.
Es gibt eine und nur eine vollständige
Analyse des Satzes.
Der Satz drückt auf bestimmte, klar
angebbare Weise aus, was er ausdrückt:Der Satz ist artikuliert.

affairs.
In the proposition the name represents
the object.
Objects I can only name. Signs represent them. I can only speak of them. I
cannot assert them. A proposition can only
say how a thing is, not what it is.

The postulate of the possibility of thesimple signs is the postulate of the determinateness of the sense.
A proposition about a complex stands ininternal relation to the proposition about
its constituent part.
A complex can only be given by its description, and this will either be right orwrong. The proposition in which there ismention of a complex, if this does not exist,
becomes not nonsense but simply false.

In a proposition a name is the representative of an object.
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 bepossible 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 notexist, but simply false.

That a propositional element signifies
a complex can be seen from an indeterminateness in the propositions in which it
occurs. We know that everything is not yet
determined by this proposition. (The notation for generality contains a prototype.)

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 thatthe proposition leaves something undetermined. (In fact the notation for generality
contains a prototype.)
The combination of the symbols of a
The contraction of a symbol for a comcomplex in a simple symbol can be ex- plex into a simple symbol can be expressedpressed by a definition.
in a definition.
There is one and only one complete
A proposition has one and only one comanalysis of the proposition.
plete analysis.
The proposition expresses what it exWhat a proposition expresses it expresses in a definite and clearly specifiable presses in a determinate manner, which
way: the proposition is articulate.
can be set out clearly: a proposition is ar-

3.26

Der Name ist durch keine Definition
weiter zu zergliedern: er ist ein Urzeichen.

3.261

Jedes definierte Zeichen bezeichnet
ü b e r jene Zeichen, durch welche es definiert wurde; und die Definitionen weisenden Weg.
Zwei Zeichen, ein Urzeichen, und ein
durch Urzeichen definiertes, können nicht
auf dieselbe Art und Weise bezeichnen. Namen k a n n man nicht durch Definitionen
auseinanderlegen. (Kein Zeichen, welches
allein, selbständig eine Bedeutung hat.)
Was in den Zeichen nicht zum Ausdruck kommt, das zeigt ihre Anwendung.
Was die Zeichen verschlucken, das spricht
ihre Anwendung aus.
Die Bedeutungen von Urzeichen können durch Erläuterungen erklärt werden.
Erläuterungen sind Sätze, welche die Urzeichen enthalten. Sie können also nur verstanden werden, wenn die Bedeutungendieser Zeichen bereits bekannt sind.
Nur der Satz hat Sinn; nur im Zusammenhang des Satzes hat ein Name Bedeutung.
Jeden Teil des Satzes, der seinen Sinncharakterisiert, nenne ich einen Ausdruck(ein Symbol).
(Der Satz selbst ist ein Ausdruck.)

3.262

3.263

3.3

3.31

ticulate.
The name cannot be analysed further
A name cannot be dissected any further
by any definition. It is a primitive sign.
by means of a definition: it is a primitive
sign.
Every defined sign signifies via those
Every sign that has a definition signisigns by which it is defined, and the defini- fies via the signs that serve to define it;tions show the way.
and the definitions point the way.
Two signs, one a primitive sign, andone defined by primitive signs, cannot
signify in the same way. Names cannot
be taken to pieces by definition (nor any
sign which alone and independently has ameaning).
What does not get expressed in the sign
is shown by its application. What the signs
conceal, their application declares.

Two signs cannot signify in the samemanner if one is primitive and the otheris defined by means of primitive signs.
Names cannot be anatomized by meansof definitions. (Nor can any sign that has ameaning independently and on its own.)
What signs fail to express, their application shows. What signs slur over, theirapplication says clearly.

The meanings of primitive signs can be
explained by elucidations. Elucidations arepropositions which contain the primitive
signs. They can, therefore, only be understood when the meanings of these signs are
already known.
Only the proposition has sense; only inthe context of a proposition has a namemeaning.
Every part of a proposition which characterizes its sense I call an expression (asymbol).
(The proposition itself is an expression.)
Ausdruck ist alles, für den Sinn desExpressions are everything—essential
Satzes wesentliche, was Sätze miteinander for the sense of the proposition—that
gemein haben können.
propositions can have in common with one
another.
Der Ausdruck kennzeichnet eine Form
An expression characterizes a form and

The meanings of primitive signs can be
explained by means of elucidations. Elucidations are propositions that contain theprimitive signs. So they can only be understood if the meanings of those signs are
already known.
Only propositions have sense; only inthe nexus of a proposition does a namehave meaning.
I call any part of a proposition thatcharacterizes its sense an expression (ora 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3.311

3.312

3.313

3.314

3.315

und einen Inhalt.
Der Ausdruck setzt die Formen aller
Sätze voraus, in welchen er vorkommen
kann. Er ist das gemeinsame charakteristische Merkmal einer Klasse von Sätzen.
Er wird also dargestellt durch die allgemeine Form der Sätze, die er charakterisiert.
Und zwar wird in dieser Form der Ausdruck k o n s t a n t und alles übrige v a r i a b e l sein.
Der Ausdruck wird also durch eine Variable dargestellt, deren Werte die Sätze
sind, die den Ausdruck enthalten.
(Im Grenzfall wird die Variable zur
Konstanten, der Ausdruck zum Satz.)
Ich nenne eine solche Variable„Satzvariable“.
Der Ausdruck hat nur im Satz Bedeutung. Jede Variable lässt sich als Satzvariable auffassen.
(Auch der variable Name.)
Verwandeln wir einen Bestandteil eines Satzes in eine Variable, so gibt es eine
Klasse von Sätzen, welche sämtlich Werte des so entstandenen variablen Satzessind. Diese Klasse hängt im allgemeinennoch davon ab, was wir, nach willkürlicher Übereinkunft, mit Teilen jenes Satzes meinen. Verwandeln wir aber alle jene
Zeichen, deren Bedeutung willkürlich bestimmt wurde, in Variable, so gibt es nun
noch immer eine solche Klasse. Diese aber
ist nun von keiner Übereinkunft abhängig,
sondern nur noch von der Natur des Satzes. Sie entspricht einer logischen Form –a content.
An expression presupposes the forms ofall propositions in which it can occur. It isthe common characteristic mark of a classof propositions.
It is therefore represented by the general form of the propositions which it characterizes.
And in this form the expression is constant and everything else variable.

a content.
An expression presupposes the formsof all the propositions in which it can occur.
It is the common characteristic mark of aclass 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 variable.

An expression is thus presented by avariable, whose values are the propositions
which contain the expression.
(In the limiting case the variable becomes constant, the expression a proposition.)
I call such a variable a “propositionalvariable”.
An expression has meaning only in a
proposition. Every variable can be conceived as a propositional variable.
(Including the variable name.)
If we change a constituent part of a
proposition into a variable, there is a classof propositions which are all the values of
the resulting variable proposition. This
class in general still depends on what, byarbitrary agreement, we mean by partsof that proposition. But if we change all
those signs, whose meaning was arbitrarily
determined, into variables, there always
remains such a class. But this is now nolonger dependent on any agreement; it depends only on the nature of the proposition.
It corresponds to a logical form, to a logicalprototype.

Thus an expression is presented bymeans of a variable whose values are thepropositions that contain the expression.
(In the limiting case the variable becomes a constant, the expression becomesa proposition.)
I call such a variable a ‘propositionalvariable’.
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, thisclass too will be dependent on the meaning that our arbitrary conventions havegiven 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 thiskind. This one, however, is not dependent
on any convention, but solely on the nature of the proposition. It corresponds to alogical form—a logical prototype.

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einem logischen Urbild.
Welche Werte die Satzvariable annehmen darf, wird festgesetzt.
Die Festsetzung der Werte i s t die Variable.
Die Festsetzung der Werte der Satzvariablen ist die A n g a b e d e r S ä t z e,deren gemeinsames Merkmal die Variable
ist.
Die Festsetzung ist eine Beschreibung
dieser Sätze.
Die Festsetzung wird also nur von Symbolen, nicht von deren Bedeutung handeln.
Und n u r dies ist der Festsetzung
wesentlich, d a s s s i e n u r e i n e B e schreibung von Symbolen ist
und nichts über das Bezeichn e t e a u s s a g t.
Wie die Beschreibung der Sätze geschieht, ist unwesentlich.
Den Satz fasse ich – wie Frege und Russell – als Funktion der in ihm enthaltenen
Ausdrücke auf.
Das Zeichen ist das sinnlich Wahrnehmbare am Symbol.
Zwei verschiedene Symbole können also das Zeichen (Schriftzeichen oder Lautzeichen etc.) miteinander gemein haben –sie bezeichnen dann auf verschiedene Art
und Weise.
Es kann nie das gemeinsame Merkmal zweier Gegenstände anzeigen, dasswir sie mit demselben Zeichen, aber durch
zwei verschiedene B e z e i c h n u n g s w e i s e n bezeichnen. Denn das Zeichen
ist ja willkürlich. Man könnte also auchWhat values the propositional variable
can assume is determined.
The determination of the values is thevariable.
The determination of the values of the
propositional variable is done by indicating
the propositions whose common mark thevariable is.
The determination is a description ofthese propositions.
The determination will therefore deal
only with symbols not with their meaning.

What values a propositional variable
may take is something that is stipulated.
The stipulation of values is the variable.
To stipulate values for a propositionalvariable is to give the propositions whose
common characteristic the variable is.

The way in which we describe thepropositions is not essential.
I conceive the proposition—like Frege
and Russell—as a function of the expressions contained in it.
The sign is the part of the symbol perceptible by the senses.
Two different symbols can therefore
have the sign (the written sign or the soundsign) in common—they then signify in different ways.

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 asymbol.
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.

It can never indicate the common characteristic of two objects that we symbolize
them with the same signs but by different
methods of symbolizing. For the sign isarbitrary. We could therefore equally wellchoose two different signs and where thenOur use of the same sign to signify two
different objects can never indicate a common characteristic of the two, if we use itwith two different modes of signification.
For the sign, of course, is arbitrary. So we
could choose two different signs instead,

The stipulation is a description of those
propositions.
The stipulation will therefore be concerned only with symbols, not with their
meaning.
And only this is essential to the deterAnd the only thing essential to the stipmination, that it is only a description of ulation is that it is merely a description ofsymbols and asserts nothing about what is symbols and states nothing about what issymbolized.
signified.

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zwei verschiedene Zeichen wählen, und wo
bliebe dann das Gemeinsame in der Bezeichnung?
In der Umgangssprache kommt es ungemein häufig vor, dass dasselbe Wort auf
verschiedene Art und Weise bezeichnet –also verschiedenen Symbolen angehört –,oder, dass zwei Wörter, die auf verschiedene Art und Weise bezeichnen, äußerlichin der gleichen Weise im Satz angewandt
werden.
So erscheint das Wort „ist“ als Kopula,als Gleichheitszeichen und als Ausdruck
der Existenz; „existieren“ als intransitives
Zeitwort wie „gehen“; „identisch“ als Eigenschaftswort; wir reden von E t w a s, aberauch davon, dass e t w a s geschieht.
(Im Satze „Grün ist grün“ – wo das erste Wort ein Personenname, das letzte ein
Eigenschaftswort ist – haben diese Worte
nicht einfach verschiedene Bedeutung, sondern es sind v e r s c h i e d e n e S y m b o l e.)
So entstehen leicht die fundamentalsten Verwechslungen (deren die ganze Philosophie voll ist).
Um diesen Irrtümern zu entgehen,müssen wir eine Zeichensprache verwenden, welche sie ausschließt, indem sie nicht
das gleiche Zeichen in verschiedenen Symbolen, und Zeichen, welche auf verschiedene Art bezeichnen, nicht äußerlich auf die
gleiche Art verwendet. Eine Zeichensprache also, die der l o g i s c h e n Grammatik– der logischen Syntax – gehorcht.
(Die Begriffsschrift Frege’s und Russell’s ist eine solche Sprache, die allerdings

would be what was common in the symbol- and then what would be left in common on
ization?
the signifying side?
In the language of everyday life it very
often happens that the same word signifies in two different ways—and therefore
belongs to two different symbols—or that
two words, which signify in different ways,
are apparently applied in the same way inthe proposition.

In everyday language it very frequently
happens that the same word has different
modes of signification—and so belongs todifferent symbols—or that two words thathave different modes of signification are
employed in propositions in what is superficially the same way.

Thus the word “is” appears as the copula, as the sign of equality, and as the expression of existence; “to exist” as an intransitive verb like “to go”; “identical” asan adjective; we speak of something but
also of the fact of something happening.
(Inthe proposition “Green is green”—where
the first word is a proper name as the lastan adjective—these words have not merely
different meanings but they are different
symbols.)
Thus there easily arise the most fundamental confusions (of which the whole ofphilosophy is full).
In order to avoid these errors, we must
employ a symbolism which excludes them,by not applying the same sign in different
symbols and by not applying signs in thesame way which signify in different ways.
A symbolism, that is to say, which obeys
the rules of logical grammar—of logicalsyntax.

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 alsoof something’s happening.
(Intheproposition, ‘Green is green’—where thefirst word is the proper name of a person
and the last an adjective—these words do
not merely have different meanings: they
are different symbols.)
In this way the most fundamental confusions are easily produced (the whole ofphilosophy 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, asign-language that is governed by logicalgrammar—by logical syntax.

(The logical symbolism of Frege and(The conceptual notation of Frege andRussell is such a language, which, however, Russell is such a language, though, it is3.326

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3.333

noch nicht alle Fehler ausschließt.)
Um das Symbol am Zeichen zu erkennen, muss man auf den sinnvollen Gebrauch achten.
Das Zeichen bestimmt erst mit seiner
logisch-syntaktischen Verwendung zusammen eine logische Form.
Wirdein
Zeichen n i c h t g e b r a u c h t, so ist es
bedeutungslos. Das ist der Sinn der Devise
Occams.
(Wenn sich alles so verhält als hätte
ein Zeichen Bedeutung, dann hat es auch
Bedeutung.)
In der logischen Syntax darf nie die Bedeutung eines Zeichens eine Rolle spielen;sie muss sich aufstellen lassen, ohne dassdabei von der B e d e u t u n g eines Zeichens die Rede wäre, sie darf n u r die Beschreibung der Ausdrücke voraussetzen.
Von dieser Bemerkung sehen wir in
Russells „Theory of types“ hinüber: Der
Irrtum Russells zeigt sich darin, dass er
bei der Aufstellung der Zeichenregeln vonder Bedeutung der Zeichen reden musste.
Kein Satz kann etwas über sich selbst
aussagen, weil das Satzzeichen nicht insich selbst enthalten sein kann (das ist dieganze „Theory of types“).
Eine Funktion kann darum nicht ihr
eigenes Argument sein, weil das Funktionszeichen bereits das Urbild seines Arguments enthält und es sich nicht selbst
enthalten kann.
Nehmen wir nämlich an, die Funktion
F ( f x) könnte ihr eigenes Argument sein;dann gäbe es also einen Satz: „F (F ( f x))“does still not exclude all errors.)
true, it fails to exclude all mistakes.)
In order to recognize the symbol in the
In order to recognize a symbol by its
sign we must consider the significant use. sign we must observe how it is used with asense.
The sign determines a logical form only
A sign does not determine a logical form
together with its logical syntactic applica- unless it is taken together with its logicotion.
syntactical employment.
If a sign is not necessary then it is meanIf a sign is useless, it is meaningless.
ingless. That is the meaning of Occam’s That is the point of Occam’s maxim.
razor.
(If everything in the symbolism works
as though a sign had meaning, then it hasmeaning.)
In logical syntax the meaning of a sign
ought never to play a rôle; it must admit ofbeing established without mention being
thereby made of the meaning of a sign; itought to presuppose only the description of
the expressions.
From this observation we get a further
view—into Russell’s Theory of Types. Russell’s error is shown by the fact that indrawing up his symbolic rules he has tospeak about the things his signs mean.
No proposition can say anything about
itself, because the propositional sign cannot be contained in itself (that is the “whole
theory of types”).
A function cannot be its own argument,
because the functional sign already contains the prototype of its own argument
and it cannot contain itself.

(If 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 thedescription 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 tomention 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 thewhole of the ‘theory of types’).
The reason why a function cannot be
its own argument is that the sign for afunction already contains the prototype of
its argument, and it cannot contain itself.

If, for example, we suppose that the
For let us suppose that the function
function F ( f x) could be its own argu- F ( f x) could be its own argument: inment, then there would be a proposition that case there would be a proposition

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3.3411

3.342

und in diesem müssen die äußere Funktion F und die innere Funtion F verschiedene Bedeutungen haben, denn die innere
hat die Form φ( f x), die äußere die Formψ(φ( f x)). Gemeinsam ist den beiden Funktionen nur der Buchstabe „F “, der aber allein nichts bezeichnet.
Dies wird sofort klar, wenn wir statt„F (F u)“ schreiben „(∃φ): F (φ u) . φ u = F u“.

“F (F ( f x))”, and in this the outer function F
and the inner function F must have different meanings; for the inner has the formφ( f x), the outer the form ψ(φ( f x)). Common to both functions is only the letter “F ”,
which by itself signifies nothing.

Hiermit erledigt sich Russells Paradox.
Die Regeln der logischen Syntax müssen sich von selbst verstehen, wenn man
nur weiß, wie ein jedes Zeichen bezeichnet.
Der Satz besitzt wesentliche und zufällige Züge.
Zufällig sind die Züge, die von der besonderen Art der Hervorbringung des Satzzeichens herrühren. Wesentlich diejenigen,welche allein den Satz befähigen, seinen
Sinn auszudrücken.
Das Wesentliche am Satz ist also das,
was allen Sätzen, welche den gleichen Sinnausdrücken können, gemeinsam ist.
Und ebenso ist allgemein das Wesentliche am Symbol das, was alle Symbole, diedenselben Zweck erfüllen können, gemeinsam haben.
Man könnte also sagen: Der eigentliche Name ist das, was alle Symbole, dieden Gegenstand bezeichnen, gemeinsam
haben. Es würde sich so successive ergeben, dass keinerlei Zusammensetzung fürden Namen wesentlich ist.
An unseren Notationen ist zwar etwas
willkürlich, aber d a s ist nicht willkürlich: Dass, w e n n wir etwas willkürlich

Herewith Russell’s paradox vanishes.
The rules of logical syntax must followof themselves, if we only know how every
single sign signifies.
A proposition possesses essential andaccidental features.
Accidental are the features which are
due to a particular way of producing the
propositional sign. Essential are thosewhich alone enable the proposition to express its sense.
The essential in a proposition is therefore that which is common to all propositions which can express the same sense.
And in the same way in general theessential in a symbol is that which all symbols which can fulfill the same purpose
have in common.
One could therefore say the real nameis that which all symbols, which signify
an object, have in common. It would thenfollow, step by step, that no sort of composition was essential for a name.

This is at once clear, if instead of“F (Fu)” we write “(∃φ): F (φ u) . φ u = Fu”.

‘F (F ( f x))’, in which the outer function F
and the inner function F must have different meanings, since the inner one has theform φ( f x) and the outer one has the formψ(φ( f x)). 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 ‘(∃φ): F (φ u) . φ u =Fu’.
That disposes of Russell’s paradox.
The rules of logical syntax must go
without saying, once we know how eachindividual sign signifies.
A proposition possesses essential andaccidental features.
Accidental features are those that result from the particular way in which thepropositional sign is produced. Essential
features are those without which the proposition could not express its sense.
So what is essential in a proposition iswhat all propositions that can express thesame sense have in common.
And similarly, in general, what is essential in a symbol is what all symbols thatcan serve the same purpose have in common.
So one could say that the real name ofan object was what all symbols that signified it had in common. Thus, one by one,all kinds of composition would prove to beunessential to a name.

In our notations there is indeed someAlthough there is something arbitrary
thing arbitrary, but this is not arbitrary, in our notations, this much is not arbinamely that if we have determined any- trary—that when we have determined one

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3.34413.3442

bestimmt haben, dann etwas anderes derFall sein muss. (Dies hängt von dem W e s e n der Notation ab.)Eine besondere Bezeichnungsweise
mag unwichtig sein, aber wichtig ist es immer, dass diese eine m ö g l i c h e Bezeichnungsweise ist. Und so verhält es sich inder Philosophie überhaupt: Das Einzelne
erweist sich immer wieder als unwichtig,
aber die Möglichkeit jedes Einzelnen gibt
uns einen Aufschluss über das Wesen der
Welt.
Definitionen sind Regeln der Übersetzung von einer Sprache in eine andere. Jede richtige Zeichensprache muss sich in jede andere nach solchen Regeln übersetzen
lassen: D i e s ist, was sie alle gemeinsam
haben.
Das, was am Symbol bezeichnet, ist das
Gemeinsame aller jener Symbole, durch
die das erste den Regeln der logischen Syntax zufolge ersetzt werden kann.
Man kann z. B. das Gemeinsame aller
Notationen für die Wahrheitsfunktionen
so ausdrücken: Es ist ihnen gemeinsam,
dass sich alle – z. B. – durch die Notationvon „∼ p“ („nicht p“) und „ p ∨ q“ („ p oderq“) e r s e t z e n l a s s e n.
(Hiermit ist die Art und Weise gekennzeichnet, wie eine spezielle mögliche Notation uns allgemeine Aufschlüsse gebenkann.)
Das Zeichen des Komplexes löst sichauch bei der Analyse nicht willkürlich auf,
so dass etwa seine Auflösung in jedem Satzgefüge eine andere wäre.

thing arbitrarily, then something else must
be the case. (This results from the essenceof the notation.)
A particular method of symbolizing
may be unimportant, but it is always important that this is a possible method ofsymbolizing. And this happens as a rule inphilosophy: The single thing proves over
and over again to be unimportant, but thepossibility of every single thing reveals
something about the nature of the world.

thing arbitrarily, something else is necessarily the case. (This derives from theessence of notation.)
A particular mode of signifying may be
unimportant but it is always importantthat it is a possible mode of signifying. And
that is generally so in philosophy: again
and again the individual case turns out tobe unimportant, but the possibility of eachindividual case discloses something about
the essence of the world.

Definitions are rules for the translationof one language into another. Every correct
symbolism must be translatable into every
other according to such rules. It is thiswhich all have in common.

Definitions are rules for translatingfrom one language into another. Any correct sign-language must be translatable
into any other in accordance with suchrules: it is this that they all have in common.
What signifies in a symbol is what iscommon to all the symbols that the rulesof logical syntax allow us to substitute forit.
For instance, we can express what iscommon to all notations for truth-functionsin the following way: they have in common
that, for example, the notation that uses‘∼ p’ (‘not p’) and ‘ p ∨ q’ (‘ p or q’) can besubstituted for any of them.
(This serves to characterize the way inwhich something general can be disclosedby the possibility of a specific notation.)

What signifies in the symbol is what iscommon to all those symbols by which it
can be replaced according to the rules oflogical syntax.
We can, for example, express what
is common to all notations for the truthfunctions as follows: It is common to themthat they all, for example, can be replacedby the notations of “∼ p” (“not p”) and“ p ∨ q” (“ p or q”).
(Herewith is indicated the way in whicha special possible notation can give us general information.)

The sign of the complex is not arbitrarNor does analysis resolve the sign for
ily resolved in the analysis, in such a way a complex in an arbitrary way, so that itthat its resolution would be different in would have a different resolution every
every propositional structure.
time that it was incorporated in a different
proposition.

3.4

3.41
3.411

3.42

3.5
4
4.001
4.002

Der Satz bestimmt einen Ort im logischen Raum. Die Existenz dieses logischen
Ortes ist durch die Existenz der Bestandteile allein verbürgt, durch die Existenz
des sinnvollen Satzes.
Das Satzzeichen und die logischen Koordinaten: Das ist der logische Ort.
Der geometrische und der logische Ort
stimmen darin überein, dass beide die Möglichkeit einer Existenz sind.
Obwohl der Satz nur einen Ort des logischen Raumes bestimmen darf, so mussdoch durch ihn schon der ganze logische
Raum gegeben sein.
(Sonst würden durch die Verneinung,die logische Summe, das logische Produkt,etc. immer neue Elemente – in Koordination – eingeführt.)(Das logische Gerüst um das Bild herum bestimmt den logischen Raum. Der
Satz durchgreift den ganzen logischen
Raum.)
Das angewandte, gedachte, Satzzeichen ist der Gedanke.
Der Gedanke ist der sinnvolle Satz.

The proposition determines a place inlogical space: the existence of this logicalplace is guaranteed by the existence of theconstituent parts alone, by the existence ofthe significant proposition.
The propositional sign and the logicalco-ordinates: that is the logical place.
The geometrical and the logical placeagree in that each is the possibility of an
existence.
Although a proposition may only determine one place in logical space, the whole
logical space must already be given by it.

A proposition determines a place in logical space. The existence of this logicalplace is guaranteed by the mere existenceof the constituents—by the existence of the
proposition with a sense.
The propositional sign with logical coordinates—that is the logical place.
In geometry and logic alike a place is apossibility: something can exist in it.

(The logical scaffolding round the picture determines the logical space. The
proposition reaches through the whole logical space.)
The applied, thought, propositionalsign, is the thought.
The thought is the significant proposition.
The totality of propositions is the language.
Man possesses the capacity of constructing languages, in which every sense
can be expressed, without having an idea
how and what each word means—just as
one speaks without knowing how the single
sounds are produced.

(The logical scaffolding surrounding a
picture determines logical space. The force
of a proposition reaches through the wholeof logical space.)
A propositional sign, applied andthought out, is a thought.
A thought is a proposition with a sense.

A proposition can determine only one
place in logical space: nevertheless thewhole of logical space must already begiven by it.
(Otherwise denial, the logical sum, the(Otherwise negation, logical sum, logilogical product, etc., would always intro- cal product, etc., would introduce more andduce new elements—in co-ordination.)more new elements—in co-ordination.)

Die Gesamtheit der Sätze ist die Sprache.
Der Mensch besitzt die Fähigkeit Sprachen zu bauen, womit sich jeder Sinn ausdrücken lässt, ohne eine Ahnung davon zu
haben, wie und was jedes Wort bedeutet. –Wie man auch spricht, ohne zu wissen, wie
die einzelnen Laute hervorgebracht werden.
Die Umgangssprache ist ein Teil desColloquial language is a part of the humenschlichen Organismus und nicht weni- man organism and is not less complicated

The totality of propositions is language.
Man possesses the ability to constructlanguages capable of expressing every
sense, without having any idea how eachword 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4.0034.00314.01

ger kompliziert als dieser.
than it.
than it.
Es ist menschenunmöglich, die SprachFrom it it is humanly impossible to
It is not humanly possible to gather
logik aus ihr unmittelbar zu entnehmen.
gather immediately the logic of language. immediately from it what the logic of language is.
Die Sprache verkleidet den Gedanken.
Language disguises the thought; soLanguage disguises thought. So much
Und zwar so, dass man nach der äußeren that from the external form of the clothes so, that from the outward form of the clothForm des Kleides, nicht auf die Form des one cannot infer the form of the thought ing it is impossible to infer the form of thebekleideten Gedankens schließen kann; they clothe, because the external form of thought beneath it, because the outward
weil die äußere Form des Kleides nach the clothes is constructed with quite an- form of the clothing is not designed to reganz anderen Zwecken gebildet ist als da- other object than to let the form of the body veal the form of the body, but for entirely
nach, die Form des Körpers erkennen zu be recognized.
different purposes.
lassen.
Die stillschweigenden Abmachungen
The silent adjustments to understandThe tacit conventions on which the unzum Verständnis der Umgangssprache colloquial language are enormously compli- derstanding of everyday language depends
sind enorm kompliziert.
cated.
are enormously complicated.
Die meisten Sätze und Fragen, welMost propositions and questions, that
Most of the propositions and questionsche über philosophische Dinge geschrieben have been written about philosophical mat- to be found in philosophical works are notworden sind, sind nicht falsch, sondern un- ters, are not false, but senseless. We can- false but nonsensical. Consequently wesinnig. Wir können daher Fragen dieser not, therefore, answer questions of this cannot give any answer to questions of thisArt überhaupt nicht beantworten, sondern kind at all, but only state their senseless- kind, but can only point out that they are
nur ihre Unsinnigkeit feststellen. Die mei- ness. Most questions and propositions of nonsensical. Most of the propositions andsten Fragen und Sätze der Philosophen be- the philosophers result from the fact that questions of philosophers arise from ourruhen darauf, dass wir unsere Sprachlogik we do not understand the logic of our lan- failure to understand the logic of our lannicht verstehen.
guage.
guage.
(Sie sind von der Art der Frage, ob das(They are of the same kind as the ques(They belong to the same class as the
Gute mehr oder weniger identisch sei als tion whether the Good is more or less iden- question whether the good is more or lessdas Schöne.)tical than the Beautiful.)
identical than the beautiful.)
Und es ist nicht verwunderlich, dassAnd so it is not to be wondered at that
And it is not surprising that the deepest
die tiefsten Probleme eigentlich k e i n e the deepest problems are really no prob- problems are in fact not problems at all.
Probleme sind.
lems.
Alle Philosophie ist „Sprachkritik“. (AlAll philosophy is “Critique of language”
All philosophy is a ‘critique of language’lerdings nicht im Sinne Mauthners.) Rus- (but not at all in Mauthner’s sense). Rus- (though not in Mauthner’s sense). It wassells Verdienst ist es, gezeigt zu haben, sell’s merit is to have shown that the ap- Russell who performed the service of showdass die scheinbare logische Form des Sat- parent logical form of the proposition need ing that the apparent logical form of azes nicht seine wirkliche sein muss.
not be its real form.
proposition need not be its real one.
Der Satz ist ein Bild der Wirklichkeit.
The proposition is a picture of reality.
A proposition is a picture of reality.
Der Satz ist ein Modell der WirklichThe proposition is a model of the reality
A proposition is a model of reality as4.011

4.012

4.013

4.014

4.0141keit, so wie wir sie uns denken.
Auf den ersten Blick scheint der Satz– wie er etwa auf dem Papier gedruckt
steht – kein Bild der Wirklichkeit zu sein,von der er handelt. Aber auch die Notenschrift scheint auf den ersten Blickkein Bild der Musik zu sein, und unsere Lautzeichen-(Buchstaben-)Schrift kein
Bild unserer Lautsprache.
Und doch erweisen sich diese Zeichensprachen auch im gewöhnlichen Sinne als
Bilder dessen, was sie darstellen.
Offenbar ist, dass wir einen Satz vonder Form „aRb“ als Bild empfinden. Hier
ist das Zeichen offenbar ein Gleichnis desBezeichneten.
Und wenn wir in das Wesentliche dieser Bildhaftigkeit eindringen, so sehenwir, dass dieselbe durch s c h e i n b a r e
U n r e g e l m ä ß i g k e i t e n (wie die Verwendung der ] und [ in der Notenschrift)n i c h t gestört wird.
Denn auch diese Unregelmäßigkeiten
bilden das ab, was sie ausdrücken sollen;
nur auf eine andere Art und Weise.
Die Grammophonplatte, der musikalische Gedanke, die Notenschrift, die Schallwellen, stehen alle in jener abbildenden
internen Beziehung zu einander, die zwischen Sprache und Welt besteht.
Ihnen allen ist der logische Bau gemeinsam.
(Wie im Märchen die zwei Jünglinge,
ihre zwei Pferde und ihre Lilien. Sie sindalle in gewissem Sinne Eins.)
Dass es eine allgemeine Regel gibt,
durch die der Musiker aus der Partitur die

as we think it is.
At the first glance the proposition—say
as it stands printed on paper—does notseem to be a picture of the reality of which
it treats. But nor does the musical scoreappear at first sight to be a picture of a musical piece; nor does our phonetic spelling(letters) seem to be a picture of our spoken
language.
And yet these symbolisms prove to be
pictures—even in the ordinary sense of theword—of what they represent.
It is obvious that we perceive a proposition of the form aRb as a picture. Here thesign is obviously a likeness of the signified.

we imagine it.
At first sight a proposition—one set out
on the printed page, for example—does notseem to be a picture of the reality withwhich it is concerned. But neither do written notes seem at first sight to be a pictureof a piece of music, nor our phonetic notation (the alphabet) to be a picture of ourspeech.
And yet these sign-languages prove tobe pictures, even in the ordinary sense, ofwhat they represent.
It is obvious that a proposition of theform ‘aRb’ strikes us as a picture. In thiscase the sign is obviously a likeness of what
is signified.
And if we penetrate to the essence of
And if we penetrate to the essence of
this pictorial nature we see that this is not this pictorial character, we see that it is not
disturbed by apparent irregularities (like impaired by apparent irregularities (suchthe use of ] and [ in the score).
as the use of ] and [ in musical notation).

For these irregularities also picture
what they are to express; only in anotherway.
The gramophone record, the musicalthought, the score, the waves of sound, allstand to one another in that pictorial internal relation, which holds between language and the world.
To all of them the logical structure iscommon.
(Like the two youths, their two horses
and their lilies in the story. They are all ina certain sense one.)
In the fact that there is a general ruleby which the musician is able to read the

For even these irregularities depict
what they are intended to express; only
they do it in a different way.
A gramophone record, the musical idea,
the written notes, and the sound-waves, allstand to one another in the same internal
relation of depicting that holds between
language and the world.
They are all constructed according to acommon 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 ofwhich the musician can obtain the sym-Symphonie entnehmen kann, durch welche
man aus der Linie auf der Grammophonplatte die Symphonie und nach der ersten
Regel wieder die Partitur ableiten kann,darin besteht eben die innere Ähnlichkeit
dieser scheinbar so ganz verschiedenen Gebilde. Und jene Regel ist das Gesetz derProjektion, welches die Symphonie in die
Notensprache projiziert. Sie ist die Regelder Übersetzung der Notensprache in die
Sprache der Grammophonplatte.
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4.023

symphony out of the score, and that there
is a rule by which one could reconstruct thesymphony from the line on a gramophonerecord and from this again—by means of
the first rule—construct the score, hereinlies the internal similarity between thesethings which at first sight seem to be entirely different. And the rule is the lawof projection which projects the symphony
into the language of the musical score. It is
the rule of translation of this language into
the language of the gramophone record.
The possibility of all similes, of all theimages of our language, rests on the logic
of representation.
In order to understand the essence ofthe proposition, consider hieroglyphic writing, which pictures the facts it describes.

Die Möglichkeit aller Gleichnisse, derganzen Bildhaftigkeit unserer Ausdrucksweise, ruht in der Logik der Abbildung.
Um das Wesen des Satzes zu verstehen,denken wir an die Hieroglyphenschrift,
welche die Tatsachen die sie beschreibt abbildet.
Und aus ihr wurde die Buchstabenschrift, ohne das Wesentliche der Abbildung zu verlieren.
Dies sehen wir daraus, dass wir den
Sinn des Satzzeichens verstehen, ohne
dass er uns erklärt wurde.

And from it came the alphabet without the essence of the representation being
lost.
This we see from the fact that we understand the sense of the propositional sign,
without having had it explained to us.

Der Satz ist ein Bild der Wirklichkeit:
Denn ich kenne die von ihm dargestellte
Sachlage, wenn ich den Satz verstehe. Und
den Satz verstehe ich, ohne dass mir sein
Sinn erklärt wurde.
Der Satz z e i g t seinen Sinn.
Der Satz z e i g t, wie es sich verhält,
w e n n er wahr ist. Und er s a g t, d a s s
es sich so verhält.
Die Wirklichkeit muss durch den Satzauf ja oder nein fixiert sein.

The proposition is a picture of reality,
for I know the state of affairs presented
by it, if I understand the proposition. And
I understand the proposition, without its
sense having been explained to me.
The proposition shows its sense.
The proposition shows how things
stand, if it is true. And it says, that they
do so stand.
The proposition determines reality tothis extent, that one only needs to say “Yes”

phony from the score, and which makes it
possible to derive the symphony from thegroove 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 beconstructed in such entirely different ways.
And that rule is the law of projection which
projects the symphony into the languageof musical notation. It is the rule for translating this language into the language ofgramophone records.
The possibility of all imagery, of all our
pictorial modes of expression, is containedin the logic of depiction.
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 weunderstand the sense of a propositionalsign without its having been explained tous.
A proposition is a picture of reality: for
if I understand a proposition, I know thesituation 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 sostand.
A proposition must restrict reality to
two alternatives: yes or no.

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4.027

Dazu muss sie durch ihn vollständig
beschrieben werden.
Der Satz ist die Beschreibung eines
Sachverhaltes.
Wie die Beschreibung einen Gegenstand nach seinen externen Eigenschaften, so beschreibt der Satz die Wirklichkeit
nach ihren internen Eigenschaften.
Der Satz konstruiert eine Welt mit Hilfe eines logischen Gerüstes und darum
kann man am Satz auch sehen, wie sichalles Logische verhält, w e n n er wahr
ist. Man kann aus einem falschen SatzS c h l ü s s e z i e h e n.
Einen Satz verstehen, heißt, wissen
was der Fall ist, wenn er wahr ist.
(Man kann ihn also verstehen, ohne zu
wissen, ob er wahr ist.)
Man versteht ihn, wenn man seine Bestandteile versteht.
Die Übersetzung einer Sprache in eine
andere geht nicht so vor sich, dass man jeden S a t z der einen in einen S a t z deranderen übersetzt, sondern nur die Satzbestandteile werden übersetzt.
(Und das Wörterbuch übersetzt nicht
nur Substantiva, sondern auch Zeit-,
Eigenschafts- und Bindewörter etc.; undes behandelt sie alle gleich.)
Die Bedeutung der einfachen Zeichen(der Wörter) müssen uns erklärt werden,
dass wir sie verstehen.
Mit den Sätzen aber verständigen wir
uns.
Es liegt im Wesen des Satzes, dass er
uns einen n e u e n Sinn mitteilen kann.

or “No” to it to make it agree with reality.
Reality must therefore be completely
described by the proposition.
A proposition is the description of afact.
As the description of an object describes
it by its external properties so propositions
describe reality by its internal properties.
The proposition constructs a world withthe help of a logical scaffolding, and therefore one can actually see in the proposition
all the logical features possessed by reality
if it is true. One can draw conclusions froma false proposition.
To understand a proposition means toknow what is the case, if it is true.
(One can therefore understand it without knowing whether it is true or not.)One understands it if one understands
it constituent parts.
The translation of one language intoanother is not a process of translating each
proposition of the one into a proposition of
the other, but only the constituent parts ofpropositions are translated.
(And the dictionary does not only translate substantives but also adverbs and conjunctions, etc., and it treats them all alike.)
The meanings of the simple signs (thewords) must be explained to us, if we areto understand them.
By means of propositions we explain
ourselves.
It is essential to propositions, that they
can communicate a new sense to us.

In order to do that, it must describe
reality completely.
A proposition is a description of a stateof 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 withthe 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 toknow 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 translatingeach 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 makeourselves understood.
It belongs to the essence of a proposition that it should be able to communicate

4.03

4.031

Ein Satz muss mit alten Ausdrücken
einen neuen Sinn mitteilen.
Der Satz teilt uns eine Sachlage mit,
also muss er w e s e n t l i c h mit der Sachlage zusammenhängen.
Und der Zusammenhang ist eben, dasser ihr logisches Bild ist.
Der Satz sagt nur insoweit etwas aus,
als er ein Bild ist.
Im Satz wird gleichsam eine Sachlage
probeweise zusammengestellt.
Man kann geradezu sagen: statt, dieser
Satz hat diesen und diesen Sinn; dieser
Satz stellt diese und diese Sachlage dar.

4.03114.03124.032

Ein Name steht für ein Ding, ein anderer für ein anderes Ding und untereinander sind sie verbunden, so stellt das Ganze– wie ein lebendes Bild – den Sachverhalt
vor.
Die Möglichkeit des Satzes beruht auf
dem Prinzip der Vertretung von Gegenständen durch Zeichen.
Mein Grundgedanke ist, dass die „logischen Konstanten“ nicht vertreten. Dasssich die L o g i k der Tatsachen nicht vertreten lässt.
Nur insoweit ist der Satz ein Bild derSachlage, als er logisch gegliedert ist.
(Auch der Satz: „ambulo“, ist zusammengesetzt, denn sein Stamm ergibt mit
einer anderen Endung, und seine Endung
mit einem anderen Stamm, einen anderen
Sinn.)A proposition must communicate a newsense with old words.
The proposition communicates to us astate of affairs, therefore it must be essentially connected with the state of affairs.
And the connexion is, in fact, that it is
its logical picture.
The proposition only asserts something,in so far as it is a picture.
In the proposition a state of affairs is,
as it were, put together for the sake of experiment.
One can say, instead of, This proposition has such and such a sense, This proposition represents such and such a state ofaffairs.
One name stands for one thing, andanother for another thing, and they areconnected together. And so the whole, likea living picture, presents the atomic fact.
The possibility of propositions is based
upon the principle of the representation of
objects by signs.
My fundamental thought is that the“logical constants” do not represent. Thatthe logic of the facts cannot be represented.
The proposition is a picture of its stateof affairs, only in so far as it is logically
articulated.
(Even the proposition “ambulo” is composite, for its stem gives a different sensewith another termination, or its termination with another stem.)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 inso 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 suchand such a sense’, we can simply say, ‘This
proposition represents such and such a situation’.
One name stands for one thing, anotherfor another thing, and they are combinedwith one another. In this way the wholegroup—like a tableau vivant—presents astate of affairs.
The possibility of propositions is basedon the principle that objects have signs as
their representatives.
My fundamental idea is that the ‘logicalconstants’ 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 ofa 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.)

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4.05

Am Satz muss gerade soviel zu unterIn the proposition there must be exactly
scheiden sein, als an der Sachlage, die er as many things distinguishable as there
darstellt.
are in the state of affairs, which it represents.
Die beiden müssen die gleiche logische
They must both possess the same logi(mathematische) Mannigfaltigkeit besit- cal (mathematical) multiplicity (cf. Hertz’s
zen. (Vergleiche Hertz’s „Mechanik,“ über Mechanics, on Dynamic Models).
Dynamische Modelle.)
Diese mathematische Mannigfaltigkeit
This mathematical multiplicity natukann man natürlich nicht selbst wieder rally cannot in its turn be represented.
abbilden. Aus ihr kann man beim Abbilden One cannot get outside it in the represennicht heraus.
tation.
Wollten wir z. B. das, was wir durch
If we tried, for example, to express„( x). f x“ ausdrücken, durch Vorsetzen eines what is expressed by “( x). f x” by putting
Indexes vor „ f x“ ausdrücken – etwa so: an index before f x, like: “Gen. f x”, it would„Alg. f x“ – es würde nicht genügen – wir not do, we should not know what was genwüssten nicht, was verallgemeinert wurde. eralized. If we tried to show it by an index
Wollten wir es durch einen Index „a “ an- g, like: “ f ( x g )” it would not do—we should
zeigen – etwa so: „ f ( xa )“ – es würde auch not know the scope of the generalization.
nicht genügen – wir wüssten nicht den Bereich der Allgemeinheitsbezeichnung.
Wollten wir es durch Einführung einer
If we were to try it by introducing
Marke in die Argumentstellen versuchen – a mark in the argument places, likeetwa so: „( A, A ). F ( A, A )“ – es würde nicht “(G,G ). F (G,G )”, it would not do—we could
genügen – wir könnten die Identität der not determine the identity of the variables,
Variablen nicht feststellen. U.s.w.
etc.
Alle diese Bezeichnungsweisen genügen nicht, weil sie nicht die notwendige
mathematische Mannigfaltigkeit haben.
Aus demselben Grunde genügt die idealistische Erklärung des Sehens der räumlichen Beziehungen durch die „Raumbrille“
nicht, weil sie nicht die Mannigfaltigkeit
dieser Beziehungen erklären kann.
Die Wirklichkeit wird mit dem Satz verglichen.

All these ways of symbolizing are inadequate because they have not the necessary
mathematical multiplicity.
For the same reason the idealist explanation of the seeing of spatial relations
through “spatial spectacles” does not do,
because it cannot explain the multiplicityof these relations.
Reality is compared with the proposition.

In a proposition there must be exactly
as many distinguishable parts as in thesituation that it represents.
The two must possess the same logical (mathematical) multiplicity. (Compare
Hertz’s Mechanics on dynamical models.)
This mathematical multiplicity, ofcourse, cannot itself be the subject of depiction. One cannot get away from it when
depicting.
If, for example, we wanted to express
what we now write as ‘( x). f x’ by putting an
affix in front of ‘ f x’—for instance by writing ‘Gen. f x’—it would not be adequate: weshould not know what was being generalized. If we wanted to signalize it with an affix ‘ g’—for instance by writing ‘ f ( x g )’—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 beable 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 multiplicityof these relations.
Reality is compared with propositions.

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Nur dadurch kann der Satz wahr oderfalsch sein, indem er ein Bild der Wirklichkeit ist.
Beachtet man nicht, dass der Satzeinen von den Tatsachen unabhängigen
Sinn hat, so kann man leicht glauben, dasswahr und falsch gleichberechtigte Beziehungen von Zeichen und Bezeichnetemsind.
Man könnte dann z. B. sagen, dass „ p“
auf die wahre Art bezeichnet, was „∼ p“ aufdie falsche Art, etc.
Kann man sich nicht mit falschen Sätzen, wie bisher mit wahren, verständigen?
Solange man nur weiß, dass sie falsch gemeint sind. Nein! Denn, wahr ist ein Satz,wenn es sich so verhält, wie wir es durch
ihn sagen; und wenn wir mit „ p“ ∼ p meinen, und es sich so verhält wie wir es meinen, so ist „ p“ in der neuen Auffassung
wahr und nicht falsch.
Dass aber die Zeichen „ p“ und „∼ p“ dasgleiche sagen k ö n n e n, ist wichtig. Dennes zeigt, dass dem Zeichen „∼“ in der Wirklichkeit nichts entspricht.
Dass in einem Satz die Verneinung vorkommt, ist noch kein Merkmal seines Sinnes (∼∼ p = p).
Die Sätze „ p“ und „∼ p“ haben entgegengesetzten Sinn, aber es entspricht ihnen
eine und dieselbe Wirklichkeit.
Ein Bild zur Erklärung des Wahrheitsbegriffes: Schwarzer Fleck auf weißem Papier; die Form des Fleckes kann man beschreiben, indem man für jeden Punkt der
Fläche angibt, ob er weiß oder schwarz

Propositions can be true or false only
A proposition can be true or false only
by being pictures of the reality.
in virtue of being a picture of reality.
If one does not observe that propositions have a sense independent of the facts,
one can easily believe that true and false
are two relations between signs and things
signified with equal rights.
One could, then, for example, say that“ p” signifies in the true way what “∼ p” signifies in the false way, etc.
Can we not make ourselves understoodby means of false propositions as hitherto
with true ones, so long as we know thatthey are meant to be false? No! For aproposition is true, if what we assert bymeans of it is the case; and if by “ p” wemean ∼ p, and what we mean is the case,then “ p” in the new conception is true andnot false.
That, however, the signs “ p” and “∼ p”
can say the same thing is important, forit shows that the sign “∼” corresponds tonothing in reality.
That negation occurs in a proposition,
is no characteristic of its sense (∼∼ p = p).
The propositions “ p” and “∼ p” have opposite senses, but to them corresponds one
and the same reality.
An illustration to explain the conceptof truth. A black spot on white paper; theform of the spot can be described by saying of each point of the plane whether itis white or black. To the fact that a point

It must not be overlooked that a proposition has a sense that is independent of
the facts: otherwise one can easily supposethat 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 ‘∼ p’signified in the false way, etc.
Can we not make ourselves understoodwith false propositions just as we havedone up till now with true ones?—So longas it is known that they are meant to befalse.—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 ∼ p
and things stand as we mean that they do,then, construed in the new way, ‘ p’ is true
and not false.
But it is important that the signs ‘ p’and ‘∼ p’ can say the same thing. For itshows that nothing in reality corresponds
to the sign ‘∼’.
The occurrence of negation in a proposition is not enough to characterize its sense(∼∼ p = p).
The propositions ‘ p’ and ‘∼ p’ have opposite sense, but there corresponds to them
one and the same reality.
An analogy to illustrate the concept of
truth: imagine a black spot on white paper:
you can describe the shape of the spot bysaying, for each point on the sheet, whether
it is black or white. To the fact that a point4.0644.0641

ist. Der Tatsache, dass ein Punkt schwarz is black corresponds a positive fact; to theist, entspricht eine positive – der, dass ein fact that a point is white (not black), a negPunkt weiß (nicht schwarz) ist, eine negati- ative fact. If I indicate a point of the planeve Tatsache. Bezeichne ich einen Punkt der (a truth-value in Frege’s terminology), thisFläche (einen Fregeschen Wahrheitswert), corresponds to the assumption proposedso entspricht dies der Annahme, die zur for judgment, etc. etc.
Beurteilung aufgestellt wird, etc. etc.
Um aber sagen zu können, ein Punkt
But to be able to say that a point issei schwarz oder weiß, muss ich vorerst black or white, I must first know underwissen, wann man einen Punkt schwarz what conditions a point is called white or
und wann man ihn weiß nennt; um sagen black; in order to be able to say “ p” is true
zu können: „ p“ ist wahr (oder falsch), muss (or false) I must have determined underich bestimmt haben, unter welchen Um- what conditions I call “ p” true, and thereby
ständen ich „ p“ wahr nenne, und damit I determine the sense of the proposition.
bestimme ich den Sinn des Satzes.
Der Punkt, an dem das Gleichnis hinkt
The point at which the simile breaks
ist nun der: Wir können auf einen Punkt down is this: we can indicate a point on
des Papiers zeigen, auch ohne zu wissen, the paper, without knowing what white
was weiß und schwarz ist; einem Satz oh- and black are; but to a proposition withne Sinn aber entspricht gar nichts, denn out a sense corresponds nothing at all, forer bezeichnet kein Ding (Wahrheitswert) it signifies no thing (truth-value) whose
dessen Eigenschaften etwa „falsch“ oder properties are called “false” or “true”; the„wahr“ hießen; das Verbum eines Satzes verb of the proposition is not “is true” or “isist nicht „ist wahr“ oder „ist falsch“ – wie false”—as Frege thought—but that which
Frege glaubte –, sondern das, was „wahr “is true” must already contain the verb.
ist“, muss das Verbum schon enthalten.
Jeder Satz muss s c h o n einen Sinn
Every proposition must already have ahaben; die Bejahung kann ihn ihm nicht sense; assertion cannot give it a sense, forgeben, denn sie bejaht ja gerade den Sinn. what it asserts is the sense itself. And theUnd dasselbe gilt von der Verneinung, etc. same holds of denial, etc.
Man könnte sagen: Die Verneinung bezieht sich schon auf den logischen Ort, dender verneinte Satz bestimmt.
Der verneinende Satz bestimmt einena n d e r e n logischen Ort als der verneinte.

One could say, the denial is already related to the logical place determined by the
proposition that is denied.
The denying proposition determines alogical place other than does the proposition denied.

is black there corresponds a positive fact,
and to the fact that a point is white (notblack), a negative fact. If I designate apoint 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 apoint 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” istrue (or false)’, I must have determinedin what circumstances I call ‘ p’ true, andin 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 thepaper even if we do not know what black
and white are, but if a proposition has nosense, nothing corresponds to it, since itdoes not designate a thing (a truth-value)
which might have properties called ‘false’or ‘true’. The verb of a proposition is not ‘istrue’ or ‘is false’, as Frege thought: rather,that which ‘is true’ must already containthe verb.
Every proposition must already havea sense: it cannot be given a sense by affirmation. Indeed its sense is just what isaffirmed. 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 thenegated proposition.

Der verneinende Satz bestimmt einen
The denying proposition determines alogischen Ort mit Hilfe des logischen Ortes logical place, with the help of the logicaldes verneinten Satzes, indem er jenen als place of the proposition denied, by saying
außerhalb diesem liegend beschreibt.
that it lies outside the latter place.

Psychology is no nearer related to philosophy, than is any other natural science.

The negating proposition determines alogical place with the help of the logicalplace of the negated proposition. For it
describes it as lying outside the latter’s
logical place.
The negated proposition can be negatedagain, and this in itself shows that what isnegated is already a proposition, and not
merely something that is preliminary to a
proposition.
Propositions represent the existence
and non-existence of states of affairs.
The totality of true propositions is thewhole of natural science (or the whole corpus of the natural sciences).
Philosophy is not one of the natural sciences.
(The word ‘philosophy’ must meansomething 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 inthe clarification of propositions.
Without philosophy thoughts are, as it
were, cloudy and indistinct: its task is tomake them clear and to give them sharpboundaries.
Psychology is no more closely related tophilosophy than any other natural science.

The theory of knowledge is the philoso-

Theory of knowledge is the philosophy

Dass man den verneinten Satz wieder
That one can deny again the deniedverneinen kann, zeigt schon, dass das, was proposition, shows that what is denied isverneint wird, schon ein Satz und nicht already a proposition and not merely theerst die Vorbereitung zu einem Satze ist.
preliminary to a proposition.
4.1
4.11

4.111

4.112

4.1121

Der Satz stellt das Bestehen und Nichtbestehen der Sachverhalte dar.
Die Gesamtheit der wahren Sätze ist
die gesamte Naturwissenschaft (oder die
Gesamtheit der Naturwissenschaften).
Die Philosophie ist keine der Naturwissenschaften.
(Das Wort „Philosophie“ muss etwas bedeuten, was über oder unter, aber nicht
neben den Naturwissenschaften steht.)
Der Zweck der Philosophie ist die logische Klärung der Gedanken.
Die Philosophie ist keine Lehre, sondern eine Tätigkeit.
Ein philosophisches Werk besteht wesentlich aus Erläuterungen.
Das Resultat der Philosophie sind nicht„philosophische Sätze“, sondern das Klarwerden von Sätzen.
Die Philosophie soll die Gedanken, diesonst, gleichsam, trübe und verschwommen sind, klar machen und scharf abgrenzen.
Die Psychologie ist der Philosophie
nicht verwandter als irgend eine andere
Naturwissenschaft.
Erkenntnistheorie ist die Philosophie

A proposition presents the existence
and non-existence of atomic facts.
The totality of true propositions is thetotal natural science (or the totality of thenatural sciences).
Philosophy is not one of the natural sciences.
(The word “philosophy” must meansomething which stands above or below,
but not beside the natural sciences.)
The object of philosophy is the logicalclarification of thoughts.
Philosophy is not a theory but an activity.
A philosophical work consists essentially of elucidations.
The result of philosophy is not a number of “philosophical propositions”, but tomake propositions clear.
Philosophy should make clear and delimit sharply the thoughts which otherwise
are, as it were, opaque and blurred.

4.1122

4.113
4.114

der Psychologie.
Entspricht nicht mein Studium der Zeichensprache dem Studium der Denkprozesse, welches die Philosophen für die Philosophie der Logik für so wesentlich hielten? Nur verwickelten sie sich meistens
in unwesentliche psychologische Untersuchungen und eine analoge Gefahr gibt es
auch bei meiner Methode.
Die Darwinsche Theorie hat mit der
Philosophie nicht mehr zu schaffen als irgendeine andere Hypothese der Naturwissenschaft.
Die Philosophie begrenzt das bestreitbare Gebiet der Naturwissenschaft.
Sie soll das Denkbare abgrenzen unddamit das Undenkbare.
Sie soll das Undenkbare von innen
durch das Denkbare begrenzen.

4.115
4.116

4.12

Sie wird das Unsagbare bedeuten, indem sie das Sagbare klar darstellt.
Alles was überhaupt gedacht werdenkann, kann klar gedacht werden. Alles,
was sich aussprechen lässt, lässt sich klar
aussprechen.
Der Satz kann die gesamte Wirklichkeit darstellen, aber er kann nicht das darstellen, was er mit der Wirklichkeit gemeinhaben muss, um sie darstellen zu können
– die logische Form.
Um die logische Form darstellen zu
können, müssten wir uns mit dem Satze außerhalb der Logik aufstellen können,
das heißt außerhalb der Welt.

phy of psychology.
Does not my study of sign-language correspond to the study of thought processes
which philosophers held to be so essential
to the philosophy of logic? Only they got
entangled for the most part in unessential
psychological investigations, and there isan analogous danger for my method.

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 inmost cases they got entangled in unessential psychological investigations, and withmy method too there is an analogous risk.

The Darwinian theory has no more to
Darwin’s theory has no more to do withdo with philosophy than has any other hy- philosophy than any other hypothesis inpothesis of natural science.
natural science.
Philosophy limits the disputable sphere
Philosophy sets limits to the much disof natural science.
puted sphere of natural science.
It should limit the thinkable andIt must set limits to what can be
thereby the unthinkable.
thought; and, in doing so, to what cannot
be thought.
It should limit the unthinkable fromIt must set limits to what cannot bewithin through the thinkable.
thought by working outwards throughwhat can be thought.
It will mean the unspeakable by clearly
It will signify what cannot be said, by
displaying the speakable.
presenting clearly what can be said.
Everything that can be thought at all
Everything that can be thought at all
can be thought clearly. Everything that can be thought clearly. Everything thatcan be said can be said clearly.
can be put into words can be put clearly.
Propositions can represent the whole
reality, but they cannot represent what
they must have in common with reality inorder to be able to represent it—the logicalform.
To be able to represent the logical form,
we should have to be able to put ourselveswith the propositions outside logic, that is
outside the world.

Propositions can represent the wholeof reality, but they cannot represent what
they must have in common with reality
in order to be able to represent it—logicalform.
In order to be able to represent logical form, we should have to be able tostation ourselves with propositions somewhere outside logic, that is to say outside
the world.

4.1214.12114.12124.1213

4.122

Der Satz kann die logische Form nicht
Propositions cannot represent the logidarstellen, sie spiegelt sich in ihm.
cal form: this mirrors itself in the propositions.
Was sich in der Sprache spiegelt, kannThat which mirrors itself in language,
sie nicht darstellen.
language cannot represent.
Was s i c h in der Sprache ausdrückt,That which expresses itself in lankönnen w i r nicht durch sie ausdrücken. guage, we cannot express by language.
Der Satz z e i g t die logische Form der
The propositions show the logical form
Wirklichkeit.
of reality.
Er weist sie auf.
They exhibit it.
So zeigt ein Satz „ f a“, dass in seinem
Thus a proposition “ f a” shows that in
Sinn der Gegenstand a vorkommt, zwei its sense the object a occurs, two propoSätze „ f a“ und „ ga“, dass in ihnen beiden sitions “ f a” and “ ga” that they are bothvon demselben Gegenstand die Rede ist.
about the same object.
Wenn zwei Sätze einander widerspreIf two propositions contradict one anchen, so zeigt dies ihre Struktur; ebenso, other, this is shown by their structure; simwenn einer aus dem anderen folgt. U.s.w. ilarly if one follows from another, etc.
Was gezeigt werden k a n n, k a n nnicht gesagt werden.
Jetzt verstehen wir auch unser Gefühl:
dass wir im Besitze einer richtigen logischen Auffassung seien, wenn nur einmal
alles in unserer Zeichensprache stimmt.
Wir können in gewissem Sinne von formalen Eigenschaften der Gegenstände undSachverhalte bezw. von Eigenschaften derStruktur der Tatsachen reden, und in demselben Sinne von formalen Relationen undRelationen von Strukturen.
(Statt Eigenschaft der Struktur sageich auch „interne Eigenschaft“; statt Relation der Strukturen „interne Relation“.
Ich führe diese Ausdrücke ein, um den
Grund der bei den Philosophen sehr verbreiteten Verwechslung zwischen den internen Relationen und den eigentlichen

What can be shown cannot be said.

Propositions cannot represent logicalform: it is mirrored in them.
What finds its reflection in language,
language cannot represent.
What expresses itself in language, wecannot express by means of language.
Propositions show the logical form of
reality.
They display it.
Thus one proposition ‘ f a’ shows thatthe object a occurs in its sense, two propositions ‘ f a’ and ‘ ga’ show that the same
object is mentioned in both of them.
If two propositions contradict one another, then their structure shows it; thesame is true if one of them follows from theother. And so on.
What can be shown, cannot be said.

Now we understand our feeling that we
Now, too, we understand our feelingare in possession of the right logical concep- that once we have a sign-language in whichtion, if only all is right in our symbolism. everything is all right, we already have acorrect logical point of view.
We can speak in a certain sense of forIn a certain sense we can talk about
mal properties of objects and atomic facts, formal properties of objects and states of
or of properties of the structure of facts, affairs, or, in the case of facts, about strucand in the same sense of formal relations tural properties: and in the same sense
and relations of structures.
about formal relations and structural relations.
(Instead of property of the structure(Instead of ‘structural property’ I
I also say “internal property”; instead of also say ‘internal property’; instead ofrelation of structures “internal relation”.
‘structural relation’, ‘internal relation’.
I introduce these expressions in order
I introduce these expressions in orderto show the reason for the confusion, very to indicate the source of the confusion
widespread among philosophers, between between internal relations and relations
internal relations and proper (external) re- proper (external relations), which is very(externen) Relationen zu zeigen.)
Das Bestehen solcher interner Eigenschaften und Relationen kann aber nicht
durch Sätze behauptet werden, sondern
es zeigt sich in den Sätzen, welche jene
Sachverhalte darstellen und von jenen Gegenständen handeln.
4.1221

4.123

4.124

4.1241Eine interne Eigenschaft einer Tatsache können wir auch einen Zug dieser Tatsache nennen. (In dem Sinn, in welchem
wir etwa von Gesichtszügen sprechen.)Eine Eigenschaft ist intern, wenn es undenkbar ist, dass ihr Gegenstand sie nicht
besitzt.
(Diese blaue Farbe und jene stehen in
der internen Relation von heller und dunkler eo ipso. Es ist undenkbar, dass d i e s ebeiden Gegenstände nicht in dieser Relation stünden.)(Hier entspricht dem schwankenden
Gebrauch der Worte „Eigenschaft“ und„Relation“ der schwankende Gebrauch desWortes „Gegenstand“.)
Das Bestehen einer internen Eigenschaft einer möglichen Sachlage wird nicht
durch einen Satz ausgedrückt, sondern es
drückt sich in dem sie darstellenden Satz
durch eine interne Eigenschaft dieses Satzes aus.
Es wäre ebenso unsinnig, dem Satzeeine formale Eigenschaft zuzusprechen, alssie ihm abzusprechen.
Formen kann man nicht dadurch voneinander unterscheiden, dass man sagt,
die eine habe diese, die andere aber jene
Eigenschaft; denn dies setzt voraus, dasslations.)
widespread among philosophers.)
The holding of such internal properties
It is impossible, however, to assert byand relations cannot, however, be asserted means of propositions that such internal
by propositions, but it shows itself in the properties and relations obtain: rather,
propositions, which present the facts and this makes itself manifest in the propositreat of the objects in question.
tions that represent the relevant states ofaffairs and are concerned with the relevant
objects.
An internal property of a fact we also
An internal property of a fact can also
call a feature of this fact. (In the sense in be called a feature of that fact (in the sense
which we speak of facial features.)in which we speak of facial features, forexample).
A property is internal if it is unthinkA property is internal if it is unthinkable that its object does not possess it.
able that its object should not possess it.
(This bright blue colour and that standin the internal relation of bright anddarker eo ipso. It is unthinkable that these
two objects should not stand in this relation.)(Here to the shifting use of the words“property” and “relation” there corresponds
the shifting use of the word “object”.)(This shade of blue and that one stand,eo ipso, in the internal relation of lighterto 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 ofa possible state of affairs is not expressedby a proposition, but it expresses itself inthe proposition which presents that stateof affairs, by an internal property of this
proposition.
It would be as senseless to ascribe aformal property to a proposition as to denyit the formal property.
One cannot distinguish forms from one
another by saying that one has this property, the other that: for this assumes that
there is a sense in asserting either property

The existence of an internal property
of a possible situation is not expressed bymeans of a proposition: rather, it expresses
itself in the proposition representing thesituation, by means of an internal propertyof that proposition.
It would be just as nonsensical to assertthat 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4.1254.12514.12524.126

es einen Sinn habe, beide Eigenschaften
von beiden Formen auszusagen.
Das Bestehen einer internen Relation
zwischen möglichen Sachlagen drückt sichsprachlich durch eine interne Relation zwischen den sie darstellenden Sätzen aus.

of either form.

ascribe either property to either form.

The existence of an internal relation
The existence of an internal relation bebetween possible states of affairs expresses tween possible situations expresses itself
itself in language by an internal relation in language by means of an internal relabetween the propositions presenting them. tion between the propositions representingthem.
Now this settles the disputed question
Here we have the answer to the vexed“whether all relations are internal or exter- question ‘whether all relations are internalnal”.
or external’.
Series which are ordered by internal
I call a series that is ordered by an inrelations I call formal series.
ternal relation a series of forms.

Hier erledigt sich nun die Streitfrage, „ob alle Relationen intern oder externseien“.
Reihen, welche durch i n t e r n e Relationen geordnet sind, nenne ich Formenreihen.
Die Zahlenreihe ist nicht nach einer
The series of numbers is ordered not byexternen, sondern nach einer internen Re- an external, but by an internal relation.
lation geordnet.
Ebenso die Reihe der Sätze „aRb“,
Similarly the series of propositions“aRb”,„(∃ x): aRx . xRb“,“(∃ x): aRx . xRb”,„(∃ x, y): aRx . xR y . yRb“, u. s. f.
“(∃ x, y): aRx . xR y . yRb”, etc.
(Steht b in einer dieser Beziehungen zu(If b stands in one of these relations toa, so nenne ich b einen Nachfolder von a.) a, I call b a successor of a.)
In dem Sinne, in welchem wir von forIn the sense in which we speak of formalen Eigenschaften sprechen, können wir mal properties we can now speak also ofnun auch von formalen Begriffen reden.
formal concepts.
(Ich führe diesen Ausdruck ein, um den(I introduce this expression in order toGrund der Verwechslung der formalen Be- make clear the confusion of formal congriffe mit den eigentlichen Begriffen, wel- cepts with proper concepts which runsche die ganze alte Logik durchzieht, klar through the whole of the old logic.)zu machen.)
Dass etwas unter einen formalen BeThat anything falls under a formal congriff als dessen Gegenstand fällt, kann cept as an object belonging to it, cannot benicht durch einen Satz ausgedrückt wer- expressed by a proposition. But it is shown
den. Sondern es zeigt sich an dem Zeichen in the symbol for the object itself. (Thedieses Gegenstandes selbst. (Der Name name shows that it signifies an object, thezeigt, dass er einen Gegenstand bezeich- numerical sign that it signifies a number,net, das Zahlenzeichen, dass es eine Zahl etc.)

The order of the number-series is notgoverned by an external relation but by aninternal relation.
The same is true of the series of propositions ‘aRb’,‘(∃ x): aRx . xRb’,‘(∃ x, y): aRx . xR y . yRb’, and so forth.
(If b stands in one of these relations toa, 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 toexhibit the source of the confusion between
formal concepts and concepts proper, whichpervades 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 thisobject. (A name shows that it signifies anobject, a sign for a number that it signifies
a number, etc.)

4.127

4.1271

4.1272

bezeichnet etc.)
Die formalen Begriffe können ja nicht,
wie die eigentlichen Begriffe, durch eine
Funktion dargestellt werden.
Denn ihre Merkmale, die formalen Eigenschaften, werden nicht durch Funktionen ausgedrückt.
Der Ausdruck der formalen Eigenschaft ist ein Zug gewisser Symbole.
Das Zeichen der Merkmale eines formalen Begriffes ist also ein charakteristischer
Zug aller Symbole, deren Bedeutungen unter den Begriff fallen.
Der Ausdruck des formalen Begriffes
also, eine Satzvariable, in welcher nur dieser charakteristische Zug konstant ist.
Die Satzvariable bezeichnet den formalen Begriff und ihre Werte die Gegenstände, welche unter diesen Begriff fallen.
Jede Variable ist das Zeichen eines formalen Begriffes.
Denn jede Variable stellt eine konstante Form dar, welche alle ihre Werte besitzen, und die als formale Eigenschaft dieser
Werte aufgefasst werden kann.
So ist der variable Name „ x“ das eigentliche Zeichen des Scheinbegriffes G e g e n s t a n d.
Wo immer das Wort „Gegenstand“(„Ding“, „Sache“, etc.) richtig gebraucht
wird, wird es in der Begriffsschrift durch
den variablen Namen ausgedrückt.
Zum Beispiel in dem Satz „es gibt 2 Gegenstände, welche . . . “ durch „(∃ x, y) . . .“.

Formal concepts, cannot, like proper
Formal concepts cannot, in fact, be repconcepts, be presented by a function.
resented by means of a function, as concepts proper can.
For their characteristics, the formal
For their characteristics, formal propproperties, are not expressed by the func- erties, are not expressed by means of functions.
tions.
The expression of a formal property is
The expression for a formal property isa feature of certain symbols.
a feature of certain symbols.
The sign that signifies the characterSo the sign for the characteristics of a
istics of a formal concept is, therefore, a formal concept is a distinctive feature ofcharacteristic feature of all symbols, whose all symbols whose meanings fall under themeanings fall under the concept.
concept.
The expression of the formal concept is
So the expression for a formal concept
therefore a propositional variable in which is a propositional variable in which this
only this characteristic feature is constant. distinctive feature alone is constant.
The propositional variable signifies the
The propositional variable signifies theformal concept, and its values signify the formal concept, and its values signify theobjects which fall under this concept.
objects that fall under the concept.
Every variable is the sign of a formal
Every variable is the sign for a formal
concept.
concept.
For every variable presents a constantFor every variable represents a conform, which all its values possess, and stant form that all its values possess, andwhich can be conceived as a formal prop- this can be regarded as a formal property
erty of these values.
of those values.
So the variable name “ x” is the proper
Thus the variable name ‘ x’ is the proper
sign of the pseudo-concept object.
sign for the pseudo-concept object.
Wherever the word “object” (“thing”,
Wherever the word ‘object’ (‘thing’, etc.)“entity”, etc.) is rightly used, it is expressed is correctly used, it is expressed in concepin logical symbolism by the variable name. tual notation by a variable name.

For example in the proposition “there
For example, in the proposition, ‘There
are two objects which . . . ”, by “(∃ x, y) . . .”.
are 2 objects which . . . ’, it is expressed by‘(∃ x, y) . . .’.
Wo immer es anders, also als eigentliWherever it is used otherwise, i.e. as a
Wherever it is used in a different way,
ches Begriffswort gebraucht wird, entste- proper concept word, there arise senseless that is as a proper concept-word, nonsensi-

hen unsinnige Scheinsätze.
pseudo-propositions.
So kann man z. B. nicht sagen „Es
So one cannot, e.g. say “There are obgibt Gegenstände“, wie man etwa sagt: jects” as one says “There are books”. Nor„Es gibt Bücher“. Und ebenso wenig: „Es “There are 100 objects” or “There are ℵ0
gibt 100 Gegenstände“, oder „Es gibt ℵ0 objects”.
Gegenstände“.
Und es ist unsinnig, von der A n z a h l
And it is senseless to speak of the numa l l e r G e g e n s t ä n d e zu sprechen.
ber of all objects.
Dasselbe gilt von den Worten
The same holds of the words “Complex”,„Komplex“, „Tatsache“, „Funktion“, „Zahl“, “Fact”, “Function”, “Number”, etc.
etc.
Sie alle bezeichnen formale Begriffe
They all signify formal concepts andund werden in der Begriffsschrift durch Va- are presented in logical symbolism by variriable, nicht durch Funktionen oder Klas- ables, not by functions or classes (as Fregesen dargestellt. (Wie Frege und Russell and Russell thought).
glaubten.)
Ausdrücke wie „1 ist eine Zahl“, „es gibt
Expressions like “1 is a number”, “there
nur Eine Null“ und alle ähnlichen sind un- is only one number nought”, and all likesinnig.
them are senseless.
(Es ist ebenso unsinnig zu sagen: „es(It is as senseless to say, “there is only
gibt nur eine 1“, als es unsinnig wäre, zu one 1” as it would be to say: 2 + 2 is at 3
sagen: 2 + 2 ist um 3 Uhr gleich 4.)
o’clock equal to 4.)4.12721
Der formale Begriff ist mit einem GeThe formal concept is already given
genstand, der unter ihn fällt, bereits gege- with an object, which falls under it. One
ben. Man kann also nicht Gegenstände ei- cannot, therefore, introduce both, the obnes formalen Begriffes u n d den formalen jects which fall under a formal concept and
Begriff selbst als Grundbegriffe einführen. the formal concept itself, as primitive ideas.
Man kann also z. B. nicht den Begriff der One cannot, therefore, e.g. introduce (asFunktion, und auch spezielle Funktionen Russell does) the concept of function and(wie Russell) als Grundbegriffe einführen; also special functions as primitive ideas; oroder den Begriff der Zahl und bestimmte the concept of number and definite numZahlen.
bers.
4.1273
Wollen wir den allgemeinen Satz: „ b ist
If we want to express in logical symbolein Nachfolger von a“ in der Begriffsschrift ism the general proposition “ b is a succesausdrücken, so brauchen wir hierzu einen sor of a” we need for this an expression for
Ausdruck für das allgemeine Glied der For- the general term of the formal series:
menreihe:

cal pseudo-propositions are the result.
So one cannot say, for example, ‘There
are objects’, as one might say, ‘There arebooks’. And it is just as impossible to say,
‘There are 100 objects’, or, ‘There are ℵ0
objects’.
And it is nonsensical to speak of thetotal number of objects.
The same applies to the words‘complex’, ‘fact’, ‘function’, ‘number’, etc.
They all signify formal concepts, andare represented in conceptual notation byvariables, not by functions or classes (asFrege 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, ‘Thereis only one 1’, as it would be to say, ‘2 + 2 at
3 o’clock equals 4’.)
A formal concept is given immediately
any object falling under it is given. It is notpossible, 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 asprimitive ideas both the concept of a function and specific functions, as Russell does;or the concept of a number and particularnumbers.
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

4.1274

4.128

4.2

4.21

aRb,
aRb,
(∃ x): aRx . xRb,(∃ x): aRx . xRb,(∃ x, y): aRx . xR y . yRb,(∃ x, y): aRx . xR y . yRb,
... .
... .
Das allgemeine Glied einer Formenreihe The general term of a formal series cankann man nur durch eine Variable aus- only be expressed by a variable, for the
drücken, denn der Begriff: Glied dieser concept symbolized by “term of this formal
Formenreihe, ist ein f o r m a l e r Begriff. series” is a formal concept. (This Frege and(Dies haben Frege und Russell übersehen; Russell overlooked; the way in which theydie Art und Weise, wie sie allgemeine Sät- express general propositions like the above
ze wie den obigen ausdrücken wollen, ist is, therefore, false; it contains a vicious
daher falsch; sie enthält einen circulus vi- circle.)tiosus.)
Wir können das allgemeine Glied derWe can determine the general term ofFormenreihe bestimmen, indem wir ihr the formal series by giving its first term
erstes Glied angeben und die allgemeine and the general form of the operation,
Form der Operation, welche das folgende which generates the following term out of
Glied aus dem vorhergehenden Satz er- the preceding proposition.
zeugt.
Die Frage nach der Existenz eines forThe question about the existence of amalen Begriffes ist unsinnig. Denn kein formal concept is senseless. For no proposiSatz kann eine solche Frage beantworten. tion can answer such a question.
(Man kann also z. B. nicht fragen: „Gibt(For example, one cannot ask:es
unanalysierbare “Are there unanalysable subject-predicate
Subjekt-Prädikatsätze?“)
propositions?”)
Die logischen Formen sind zahl l o s.
The logical forms are anumerical.
Darum gibt es in der Logik keine ausgeTherefore there are in logic no prezeichneten Zahlen und darum gibt es kei- eminent numbers, and therefore there isnen philosophischen Monismus oder Dua- no philosophical monism or dualism, etc.
lismus, etc.
Der Sinn des Satzes ist seine ÜberThe sense of a proposition is its agreeeinstimmung und Nichtübereinstimmung ment and disagreement with the possibilmit den Möglichkeiten des Bestehens und ities of the existence and non-existence ofNichtbestehens der Sachverhalte.
the atomic facts.
Der einfachste Satz, der Elementarsatz,
The simplest proposition, the elemenbehauptet das Bestehen eines Sachverhal- tary proposition, asserts the existence ofaRb,(∃ x): aRx . xRb,(∃ x, y): aRx . xR y . yRb,... .
In order to express the general term of aseries of forms, we must use a variable,
because the concept ‘term of that seriesof forms’ is a formal concept. (This iswhat Frege and Russell overlooked: consequently the way in which they want toexpress general propositions like the one
above is incorrect; it contains a vicious circle.)
We can determine the general term of aseries of forms by giving its first term andthe 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 pre-eminent 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 ofaffairs.
The simplest kind of proposition, an elementary proposition, asserts the existence4.211

4.22

4.221

4.2211

4.23

4.24

4.241

tes.
Ein Zeichen des Elementarsatzes ist
es, dass kein Elementarsatz mit ihm inWiderspruch stehen kann.
Der Elementarsatz besteht aus Namen.
Er ist ein Zusammenhang, eine Verkettung, von Namen.
Es ist offenbar, dass wir bei der Analyse der Sätze auf Elementarsätze kommen
müssen, die aus Namen in unmittelbarer
Verbindung bestehen.
Es frägt sich hier, wie kommt der Satzverband zustande.
Auch wenn die Welt unendlich komplex ist, so dass jede Tatsache aus unendlich vielen Sachverhalten besteht und jeder
Sachverhalt aus unendlich vielen Gegenständen zusammengesetzt ist, auch dann
müsste es Gegenstände und Sachverhalte
geben.
Der Name kommt im Satz nur im Zusammenhange des Elementarsatzes vor.

an atomic fact.
It is a sign of an elementary proposition, that no elementary proposition cancontradict it.
The elementary proposition consists ofnames. It is a connexion, a concatenation,of names.
It is obvious that in the analysis of
propositions we must come to elementary
propositions, which consist of names in immediate combination.
The question arises here, how the
propositional connexion comes to be.
Even if the world is infinitely complex,
so that every fact consists of an infinite
number of atomic facts and every atomic
fact is composed of an infinite number ofobjects, even then there must be objects
and atomic facts.

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 ofnames. It is a nexus, a concatenation, ofnames.
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 ofaffairs is composed of infinitely many objects, there would still have to be objects
and states of affairs.

The name occurs in the proposition only
It is only in the nexus of an elemenin the context of the elementary proposi- tary proposition that a name occurs in ation.
proposition.
Die Namen sind die einfachen Symbole,
The names are the simple symbols, I
Names are the simple symbols: I indiich deute sie durch einzelne Buchstaben indicate them by single letters ( x, y, z).
cate them by single letters (‘ x’, ‘ y’, ‘ z’).
(„ x“, „ y“, „ z“) an.
Den Elementarsatz schreibe ich als
The elementary proposition I write as
I write elementary propositions as funcFunktion der Namen in der Form: „ f x“, function of the names, in the form “ f x”, tions of names, so that they have the form„φ( x, y)“, etc.
“φ( x, y)”, etc.
‘ f x’, ‘φ( x, y)’, etc.
Oder ich deute ihn durch die BuchstaOr I indicate it by the letters p, q, r .
Or I indicate them by the letters ‘ p’, ‘ q’,
ben p, q, r an.
‘ r ’.
Gebrauche ich zwei Zeichen in ein und
If I use two signs with one and the same
When I use two signs with one and thederselben Bedeutung, so drücke ich dies meaning, I express this by putting between same meaning, I express this by putting
aus, indem ich zwischen beide das Zeichen them the sign “=”.
the sign ‘=’ between them.
„=“ setze.
„a = b“ heißt also: das Zeichen „a“ ist“a = b” means then, that the sign “a” isSo ‘a = b’ means that the sign ‘ b’ can be
durch das Zeichen „ b“ ersetzbar.
replaceable by the sign “ b”.
substituted for the sign ‘a’.

4.242

4.243(Führe ich durch eine Gleichung ein
neues Zeichen „ b“ ein, indem ich bestimme, es solle ein bereits bekanntes Zeichen„a“ ersetzen, so schreibe ich die Gleichung– Definition – (wie Russell) in der Form„a = b Def.“. Die Definition ist eine Zeichenregel.)
Ausdrücke von der Form „a = b“ sindalso nur Behelfe der Darstellung; sie sagen
nichts über die Bedeutung der Zeichen „a“,
„ b“ aus.
Können wir zwei Namen verstehen, ohne zu wissen, ob sie dasselbe Ding oderzwei verschiedene Dinge bezeichnen? –Können wir einen Satz, worin zwei Namenvorkommen, verstehen, ohne zu wissen, ob
sie Dasselbe oder Verschiedenes bedeuten?
Kenne ich etwa die Bedeutung eines
englischen und eines gleichbedeutenden
deutschen Wortes, so ist es unmöglich, dassich nicht weiß, dass die beiden gleichbedeutend sind; es ist unmöglich, dass ich sie
nicht ineinander übersetzen kann.
Ausdrücke wie „a = a“, oder von diesen abgeleitete, sind weder Elementarsätze, noch sonst sinnvolle Zeichen. (Dies wird
sich später zeigen.)(If I introduce by an equation a newsign “ b”, by determining that it shall replace a previously known sign “a”, I write
the equation—definition—(like Russell) in
the form “a = b Def.”. A definition is a symbolic rule.)(If I use an equation to introduce a newsign ‘ b’, laying down that it shall serve asa substitute for a sign ‘a’ that is alreadyknown, then, like Russell, I write the equation—definition—in the form ‘a = b Def.’ Adefinition is a rule dealing with signs.)

Expressions of the form “a = b” aretherefore only expedients in presentation:
They assert nothing about the meaning of
the signs “a” and “ b”.
Can we understand two names without
knowing whether they signify the samething or two different things? Can we understand a proposition in which two names
occur, without knowing if they mean thesame or different things?
If I know the meaning of an English
and a synonymous German word, it is impossible for me not to know that they aresynonymous, it is impossible for me not tobe able to translate them into one another.

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 samething or two different things?—Can weunderstand 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 thatmeans the same: then it is impossible forme to be unaware that they do mean thesame; 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 inwhich they have sense. (This will become
evident later.)
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 descriptionof the world. The world is completely described by giving all elementary propositions, and adding which of them are true

Expressions like “a = a”, or expressions
deduced from these are neither elementary propositions nor otherwise significant
signs. (This will be shown later.)

4.25

Ist der Elementarsatz wahr, so besteht
If the elementary proposition is true,
der Sachverhalt; ist der Elementarsatz the atomic fact exists; if it is false thefalsch, so besteht der Sachverhalt nicht.
atomic fact does not exist.

4.26

Die Angabe aller wahren Elementarsätze beschreibt die Welt vollständig. Die Welt
ist vollständig beschrieben durch die Angaben aller Elementarsätze plus der Angabe,welche von ihnen wahr und welche falsch

The specification of all true elementary propositions describes the world completely. The world is completely describedby the specification of all elementary propositions plus the specification, which of

4.27

4.28

4.3

4.31

4.4

sind.
them are true and which false.
Bezüglich des Bestehens und NichtbeWith regard to the existence of n atomic
Ã !
n n
X
stehens von n Sachverhalten gibt es Kn = facts there are Kn =possibilities.
ν
ν
=
0
Ã !
n n
X
Möglichkeiten.
ν=0 ν
Es können alle Kombinationen der
It is possible for all combinations of
Sachverhalte bestehen, die andern nicht atomic facts to exist, and the others notbestehen.
to exist.
Diesen Kombinationen entsprechen
To these combinations correspond theebenso viele Möglichkeiten der Wahrheit – same number of possibilities of the
und Falschheit – von n Elementarsätzen. truth—and falsehood—of n elementary
propositions.
Die Wahrheitsmöglichkeiten der EleThe truth-possibilities of the elemenmentarsätze bedeuten die Möglichkeiten tary propositions mean the possibilities
des Bestehens und Nichtbestehens der of the existence and non-existence of the
Sachverhalte.
atomic facts.
Die Wahrheitsmöglichkeiten können
The truth-possibilities can be presented
wir durch Schemata folgender Art darstel- by schemata of the following kind (“T”
len („W“ bedeutet „wahr“, „F“, „falsch“. Die means “true”, “F” “false”. The rows of T’s
Reihen der „W“ und „F“ unter der Reihe and F’s under the row of the elementary
der Elementarsätze bedeuten in leichtver- propositions mean their truth-possibilities
ständlicher Symbolik deren Wahrheitsmög- in an easily intelligible symbolism).
lichkeiten):
p q
r
p q r

and which false.
For n states of affairs, there are Kn =
Ã !
n
X n
possibilities of existence and nonν=0 ν
existence.
Of these states of affairs any combination can exist and the remainder not exist.
There correspond to these combinations the same number of possibilitiesof truth—and falsity—for n elementary
propositions.
Truth-possibilities of elementary propositions mean possibilities of existence andnon-existence of states of affairs.
We can represent truth-possibilities byschemata 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 ina way that can easily be understood):

p

q

r

W W W
T T T
T T T
p q
p q
p q
F W W
F T T
F T T
p
p
p
W F W
W W
T F T
T T
T F T
T T
W W F
F W
W
T T F
F T
T
T T F
F T
T
F F W
W F
F
F F T
T F
F
F F T
T F
F
F W F
F F
F T F
F F
F T F
F F
W F F
T F F
T F F
F F F
F F F
F F F
Der Satz ist der Ausdruck der ÜbereinA proposition is the expression of agreeA proposition is an expression ofstimmung und Nichtübereinstimmung mit ment and disagreement with the truth- agreement and disagreement with truthden Wahrheitsmöglichkeiten der Elemen- possibilities of the elementary proposi- possibilities of elementary propositions.

4.41

4.411

4.42

4.43

4.431

tarsätze.
Die Wahrheitsmöglichkeiten der Elementarsätze sind die Bedingungen derWahrheit und Falschheit der Sätze.
Es ist von vornherein wahrscheinlich,dass die Einführung der Elementarsätze
für das Verständnis aller anderen Satzarten grundlegend ist. Ja, das Verständnis
der allgemeinen Sätze hängt f ü h l b a r
von dem der Elementarsätze ab.

tions.
The truth-possibilities of the elementary propositions are the conditions of thetruth and falsehood of the propositions.
It seems probable even at first sightthat the introduction of the elementary
propositions is fundamental for the comprehension of the other kinds of propositions. Indeed the comprehension of thegeneral propositions depends palpably onthat of the elementary propositions.
Bezüglich der Übereinstimmung undWith regard to the agreement and dis-Truth-possibilities of elementary propositions are the conditions of the truth andfalsity of propositions.
It immediately strikes one as probablethat 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 ofelementary propositions.
For n elementary propositions there
Ã !
Kn
X
Kn
Nichtübereinstimmung eines Satzes mit agreement of a proposition with the truth- are= Ln ways in which a propoκ=0 κ
den Wahrheitsmöglichkeiten von n Ele- possibilities of n elementary propositions sition can agree and disagree with their
Ã !
Ã !
Kn
Kn
X
X
Kn
Kn
mentarsätzen gibt es
truth possibilities.
= Ln Mög- there are= Ln possibilities.
κ=0 κ
κ=0 κ
lichkeiten.
Die Übereinstimmung mit den WahrAgreement with the truth-possibilities
We can express agreement with truthheitsmöglichkeiten können wir dadurch can be expressed by co-ordinating with possibilities by correlating the mark ‘T’ausdrücken, indem wir ihnen im Schema them in the schema the mark “T” (true).
(true) with them in the schema.
etwa das Abzeichen „W“ (wahr) zuordnen.
Das Fehlen dieses Abzeichens bedeutet
Absence of this mark means disagreeThe absence of this mark means disdie Nichtübereinstimmung.
ment.
agreement.
Der Ausdruck der Übereinstimmung
The expression of the agreement and
The expression of agreement and disund Nichtübereinstimmung mit den Wahr- disagreement with the truth-possibilities agreement with the truth possibilities of elheitsmöglichkeiten der Elementarsätze of the elementary propositions expresses ementary propositions expresses the truthdrückt die Wahrheitsbedingungen des Sat- the truth-conditions of the proposition.
conditions of a proposition.
zes aus.
Der Satz ist der Ausdruck seiner WahrThe proposition is the expression of its
A proposition is the expression of its
heitsbedingungen.
truth-conditions.
truth-conditions.
(Frege hat sie daher ganz richtig als Er(Frege has therefore quite rightly put(Thus Frege was quite right to use
klärung der Zeichen seiner Begriffsschrift them at the beginning, as explaining the them as a starting point when he explained
vorausgeschickt. Nur ist die Erklärung des signs of his logical symbolism. Only Frege’s the signs of his conceptual notation. But
Wahrheitsbegriffes bei Frege falsch: Wären explanation of the truth-concept is false: if the explanation of the concept of truth that„das Wahre“ und „das Falsche“ wirklich Ge- “the true” and “the false” were real objects Frege gives is mistaken: if ‘the true’ and

4.44

4.441

4.442

genstände und die Argumente in ∼ p etc. and the arguments in ∼ p, etc., then the ‘the false’ were really objects, and weredann wäre nach Freges Bestimmung der sense of ∼ p would by no means be deter- the arguments in ∼ p etc., then Frege’s
Sinn von „∼ p“ keineswegs bestimmt.)
mined by Frege’s determination.)
method of determining the sense of ‘∼ p’would leave it absolutely undetermined.)
Das Zeichen, welches durch die ZuordThe sign which arises from the coThe sign that results from correlatingnung jener Abzeichen „W“ und der Wahr- ordination of that mark “T” with the truth- the mark ‘T’ with truth-possibilities is aheitsmöglichkeiten entsteht, ist ein Satz- possibilities is a propositional sign.
propositional sign.
zeichen.
Es ist klar, dass dem Komplex der ZeiIt is clear that to the complex of the
It is clear that a complex of the signs ‘F’chen „F“ und „W“ kein Gegenstand (oder signs “F” and “T” no object (or complex of and ‘T’ has no object (or complex of objects)
Komplex von Gegenständen) entspricht; so objects) corresponds; any more than to hor- corresponding to it, just as there is none
wenig, wie den horizontalen und vertikalen izontal and vertical lines or to brackets. corresponding to the horizontal and vertiStrichen oder den Klammern. – „Logische There are no “logical objects”.
cal lines or to the brackets.—There are no
Gegenstände“ gibt es nicht.
‘logical objects’.
Analoges gilt natürlich für alle Zeichen,
Something analogous holds of course
Of course the same applies to all signs
die dasselbe ausdrücken wie die Schemata for all signs, which express the same as that express what the schemata of ‘T’s’ andder „W“ und „F“.
the schemata of “T” and “F”.
‘F’s’ express.
Es ist z. B.:
Thus e.g.
For example, the following is a propositional sign:
p q
“
“ p q
‘ p q
’

„

W
F
W
F

W
W
F
F

W
W
W

ein Satzzeichen.
(Frege’s „Urteilsstrich“ „`“ ist logischganz bedeutunglos; er zeigt bei Frege (undRussell) nur an, dass diese Autoren die sobezeichneten Sätze für wahr halten. „`“ gehört daher ebenso wenig zum Satzgefüge,
wie etwa die Nummer des Satzes. Ein Satzkann unmöglich von sich selbst aussagen,dass er wahr ist.)

T
F
T
F

T
T
F
F

T
T
T

”

is a propositional sign.
(Frege’s assertion sign “`” is logicallyaltogether meaningless; in Frege (and Russell) it only shows that these authors holdas true the propositions marked in this way.
“`” belongs therefore to the propositions nomore than does the number of the proposition. A proposition cannot possibly assertof itself that it is true.)

T
F
T
F

T
T
F
F

T
T
T

(Frege’s ‘judgement stroke’ ‘`’ is logically quite meaningless: in the works ofFrege (and Russell) it simply indicatesthat 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.)Ist die Reihenfolge der WahrheitsmögIf the sequence of the truth-possibilities
If the order or the truth-possibilities
lichkeiten im Schema durch eine Kombina- in the schema is once for all determined by in a schema is fixed once and for all by a

4.45

4.46

4.461

tionsregel ein für allemal festgesetzt, dann
ist die letzte Kolonne allein schon ein Ausdruck der Wahrheitsbedingungen. Schreiben wir diese Kolonne als Reihe hin, so
wird das Satzzeichen zu

a rule of combination, then the last column is by itself an expression of the truthconditions. If we write this column as a
row the propositional sign becomes:

combinatory rule, then the last column byitself will be an expression of the truthconditions. If we now write this column asa row, the propositional sign will become„ (WW–W) ( p, q)“oder deutlicher“ (TT–T) ( p, q)”,or more plainly:“ (TT–T) ( p, q)”,or more explicitly„ (WWFW) ( p, q)“.
(Die Anzahl der Stellen in der linken
Klammer ist durch die Anzahl der Glieder
in der rechten bestimmt.)
Für n Elementarsätze gibt es Ln mögliche Gruppen von Wahrheitsbedingungen.
Die Gruppen von Wahrheitsbedingungen, welche zu den Wahrheitsmöglichkeiten einer Anzahl von Elementarsätzen gehören, lassen sich in eine Reihe ordnen.
Unter den möglichen Gruppen von
Wahrheitsbedingungen gibt es zwei extreme Fälle.
In dem einen Fall ist der Satz für sämtliche Wahrheitsmöglichkeiten der Elementarsätze wahr. Wir sagen, die Wahrheitsbedingungen sind t a u t o l o g i s c h.
Im zweiten Fall ist der Satz für sämtliche Wahrheitsmöglichkeiten falsch: Die
Wahrheitsbedingungen sind k o n t r a d i k t o r i s c h.
Im ersten Fall nennen wir den Satz eine Tautologie, im zweiten Fall eine Kontradiktion.
Der Satz zeigt was er sagt, die Tautologie und die Kontradiktion, dass sie nichts
sagen.
Die Tautologie hat keine Wahrheitsbedingungen, denn sie ist bedingungslos“ (TTFT) ( p, q)”.
(The number of places in the left-handbracket is determined by the number ofterms in the right-hand bracket.)
For n elementary propositions there
are Ln possible groups of truth-conditions.
The groups of truth-conditions which
belong to the truth-possibilities of a number of elementary propositions can be ordered in a series.
Among the possible groups of truthconditions there are two extreme cases.

“ (TTFT) ( p, q)”.
(The number of places in the left-handpair 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 agiven number of elementary propositions
can be arranged in a series.
Among the possible groups of truthconditions there are two extreme cases.

In the one case the proposition is true
for all the truth-possibilities of the elementary propositions. We say that the truthconditions are tautological.
In the second case the proposition isfalse for all the truth-possibilities. The
truth-conditions are self-contradictory.

In one of these cases the propositionis true for all the truth-possibilities of theelementary propositions. We say that thetruth-conditions are tautological.
In the second case the proposition isfalse for all the truth-possibilities: thetruth-conditions are contradictory.

In the first case we call the proposition
a tautology, in the second case a contradiction.
The proposition shows what it says, thetautology and the contradiction that they
say nothing.
The tautology has no truth-conditions,for it is unconditionally true; and the con-

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 con-

4.4611

4.462

4.463

wahr; und die Kontradiktion ist unter keiner Bedingung wahr.
Tautologie und Kontradiktion sindsinnlos.
(Wie der Punkt, von dem zwei Pfeile inentgegengesetzter Richtung auseinandergehen.)(Ich weiß z. B. nichts über das Wetter,wenn ich weiß, dass es regnet oder nicht
regnet.)Tautologie und Kontradiktion sind abernicht unsinnig; sie gehören zum Symbolismus, und zwar ähnlich wie die „0“ zumSymbolismus der Arithmetik.
Tautologie und Kontradiktion sindnicht Bilder der Wirklichkeit. Sie stellenkeine mögliche Sachlage dar. Denn jene
lässt j e d e mögliche Sachlage zu, diesek e i n e.
In der Tautologie heben die Bedingungen der Übereinstimmung mit der Welt –die darstellenden Beziehungen – einanderauf, so dass sie in keiner darstellenden Beziehung zur Wirklichkeit steht.
Die Wahrheitsbedingungen bestimmen
den Spielraum, der den Tatsachen durch
den Satz gelassen wird.
(Der Satz, das Bild, das Modell, sindim negativen Sinne wie ein fester Körper,
der die Bewegungsfreiheit der anderen beschränkt; im positiven Sinne, wie der vonfester Substanz begrenzte Raum, worin ein
Körper Platz hat.)
Die Tautologie lässt der Wirklichkeit
den ganzen – unendlichen – logischen
Raum; die Kontradiktion erfüllt den ganzen logischen Raum und lässt der Wirklich-tradiction is on no condition true.

tradiction is true on no condition.

Tautology and contradiction are withTautologies and contradictions lackout sense.
sense.
(Like the point from which two arrows(Like a point from which two arrows go
go out in opposite directions.)out in opposite directions to one another.)(I know, e.g. nothing about the weather,(For example, I know nothing about thewhen I know that it rains or does not rain.) weather when I know that it is either raining or not raining.)
Tautology and contradiction are, howTautologies and contradictions are not,ever, not nonsensical; they are part of the however, nonsensical. They are part of thesymbolism, in the same way that “0” is part symbolism, much as ‘0’ is part of the symof the symbolism of Arithmetic.
bolism of arithmetic.
Tautology and contradiction are not picTautologies and contradictions are nottures of the reality. They present no possi- pictures of reality. They do not represent
ble state of affairs. For the one allows every any possible situations. For the former
possible state of affairs, the other none.
admit all possible situations, and latternone.
In the tautology the conditions of agreeIn a tautology the conditions of agreement with the world—the presenting rela- ment with the world—the representationaltions—cancel one another, so that it stands relations—cancel one another, so that itin no presenting relation to reality.
does not stand in any representational relation to reality.
The truth-conditions determine the
The truth-conditions of a proposition
range, which is left to the facts by the determine the range that it leaves open to
proposition.
the facts.
(The proposition, the picture, the(A proposition, a picture, or a model
model, are in a negative sense like a solid is, in the negative sense, like a solid bodybody, which restricts the free movement of that restricts the freedom of movement ofanother: in a positive sense, like the space others, and, in the positive sense, like alimited by solid substance, in which a body space bounded by solid substance in which
may be placed.)there is room for a body.)
Tautology leaves to reality the whole
A tautology leaves open to reality theinfinite logical space; contradiction fills the whole—the infinite whole—of logical space:whole logical space and leaves no point to a contradiction fills the whole of logicalreality. Neither of them, therefore, can in space leaving no point of it for reality. Thus

4.464

4.465

4.466

4.4661keit keinen Punkt. Keine von beiden kanndaher die Wirklichkeit irgendwie bestimmen.
Die Wahrheit der Tautologie ist gewiss,
des Satzes möglich, der Kontradiktion unmöglich.
(Gewiss, möglich, unmöglich: Hier haben wir das Anzeichen jener Gradation, die
wir in der Wahrscheinlichkeitslehre brauchen.)
Das logische Produkt einer Tautologie
und eines Satzes sagt dasselbe, wie derSatz. Also ist jenes Produkt identisch mit
dem Satz. Denn man kann das Wesentliche des Symbols nicht ändern, ohne seinen
Sinn zu ändern.
Einer bestimmten logischen Verbindung von Zeichen entspricht eine bestimmte logische Verbindung ihrer Bedeutungen; j e d e b e l i e b i g e Verbindung entspricht nur den unverbundenen Zeichen.
Das heißt, Sätze, die für jede Sachlage
wahr sind, können überhaupt keine Zeichenverbindungen sein, denn sonst könnten ihnen nur bestimmte Verbindungenvon Gegenständen entsprechen.
(Und keiner logischen Verbindung entspricht k e i n e Verbindung der Gegenstände.)
Tautologie und Kontradiktion sind die
Grenzfälle der Zeichenverbindung, nämlich ihre Auflösung.
Freilich sind auch in der Tautologie
und Kontradiktion die Zeichen noch mit
einander verbunden, d. h. sie stehen in Beziehungen zu einander, aber diese Beziehungen sind bedeutungslos, dem S y m -any way determine reality.

neither of them can determine reality in
any way.

The truth of tautology is certain, ofA tautology’s truth is certain, a proposipropositions possible, of contradiction im- tion’s possible, a contradiction’s impossible.
possible.
(Certain, possible, impossible: here we(Certain, possible, impossible: here wehave an indication of that gradation which have the first indication of the scale that
we need in the theory of probability.)
we need in the theory of probability.)
The logical product of a tautology anda proposition says the same as the proposition. Therefore that product is identicalwith the proposition. For the essence of thesymbol cannot be altered without altering
its sense.
To a definite logical combination ofsigns corresponds a definite logical combination of their meanings; every arbitrary
combination only corresponds to the unconnected signs.
That is, propositions which are true forevery state of affairs cannot be combinations of signs at all, for otherwise there
could only correspond to them definite combinations of objects.
(And to no logical combination corresponds no combination of the objects.)
Tautology and contradiction are thelimiting cases of the combination of symbols, namely their dissolution.
Of course the signs are also combinedwith one another in the tautology and contradiction, i.e. they stand in relations to
one another, but these relations are meaningless, unessential to the symbol.

The logical product of a tautology anda proposition says the same thing as the
proposition. This product, therefore, isidentical with the proposition. For it is
impossible to alter what is essential to asymbol 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 aretrue 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 thelimiting cases—indeed the disintegration—of the combination of signs.
Admittedly the signs are still combinedwith one another even in tautologies andcontradictions—i.e. they stand in certain
relations to one another: but these relations have no meaning, they are not essen-

4.5

4.51

4.52

4.53

b o l unwesentlich.
Nun scheint es möglich zu sein, die allgemeinste Satzform anzugeben: das heißt,
eine Beschreibung der Sätze i r g e n d e i n e r Zeichensprache zu geben, so dass jeder mögliche Sinn durch ein Symbol, aufwelches die Beschreibung passt, ausgedrückt werden kann, und dass jedes Symbol, worauf die Beschreibung passt, einen
Sinn ausdrücken kann, wenn die Bedeutungen der Namen entsprechend gewählt
werden.
Es ist klar, dass bei der Beschreibung
der allgemeinsten Satzform n u r ihr Wesentliches beschrieben werden darf, – sonstwäre sie nämlich nicht die allgemeinste.

Now it appears to be possible to givethe most general form of proposition; i.e.
to give a description of the propositions ofsome one sign language, so that every possible sense can be expressed by a symbol,
which falls under the description, and sothat every symbol which falls under the description can express a sense, if the meanings of the names are chosen accordingly.

It is clear that in the descriptionof the most general form of proposition
only what is essential to it may be described—otherwise it would not be themost general form.
Dass es eine allgemeine Satzform gibt,
That there is a general form is proved
wird dadurch bewiesen, dass es keinen by the fact that there cannot be a proposiSatz geben darf, dessen Form man nicht tion whose form could not have been forehätte voraussehen (d. h. konstruieren) kön- seen (i.e. constructed). The general form ofnen. Die allgemeine Form des Satzes ist: proposition is: Such and such is the case.
Es verhält sich so und so.
Angenommen, mir wären a l l e EleSuppose all elementary propositions
mentarsätze gegeben: Dann lässt sich ein- were given me: then we can simply ask:
fach fragen: Welche Sätze kann ich aus what propositions I can build out of them.
ihnen bilden. Und das sind a l l e Sätze And these are all propositions and so are
und s o sind sie begrenzt.
they limited.
Die Sätze sind Alles, was aus der GeThe propositions are everything whichsamtheit aller Elementarsätze folgt (natür- follows from the totality of all elementarylich auch daraus, dass es die G e s a m t - propositions (of course also from the facth e i t a l l e r ist). (So könnte man in ge- that it is the totality of them all). (So, inwissem Sinne sagen, dass a l l e Sätze some sense, one could say, that all proposiVerallgemeinerungen der Elementarsätze tions are generalizations of the elementarysind.)
propositions.)
Die allgemeine Satzform ist eine VariaThe general proposition form is a varible.
able.

tial to the symbol.
It now seems possible to give the mostgeneral propositional form: that is, to givea description of the propositions of any
sign-language whatsoever in such a waythat every possible sense can be expressedby a symbol satisfying the description, andevery 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 tothe 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 nothave 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 thatfixes their limits.
Propositions comprise all that follows
from the totality of all elementary propositions (and, of course, from its being thetotality 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.

5

5.01
5.02

5.1

Der Satz ist eine Wahrheitsfunktion
Propositions are truth-functions of eleder Elementarsätze.
mentary propositions.
(Der Elementarsatz ist eine Wahrheits(An elementary proposition is a truthfunktion seiner selbst.)
function of itself.)
Die Elementarsätze sind die WahrheitsThe elementary propositions are theargumente des Satzes.
truth-arguments of propositions.
Es liegt nahe, die Argumente von FunkIt is natural to confuse the arguments
tionen mit den Indices von Namen zu ver- of functions with the indices of names. For
wechseln. Ich erkenne nämlich sowohl am I recognize the meaning of the sign conArgument wie am Index die Bedeutung des taining it from the argument just as muchsie enthaltenden Zeichens.
as from the index.
In Russells „+ c “ ist z. B. „ c“ ein Index,
In Russell’s “+ c ”, for example, “ c” is an
der darauf hinweist, dass das ganze Zei- index which indicates that the whole sign
chen das Additionszeichen für Kardinal- is the addition sign for cardinal numbers.
zahlen ist. Aber diese Bezeichnung beruht But this way of symbolizing depends on
auf willkürlicher Übereinkunft und man arbitrary agreement, and one could choose
könnte statt „+ c “ auch ein einfaches Zei- a simple sign instead of “+ c ”: but in “∼ p”
chen wählen; in „∼ p“ aber ist „ p“ kein In- “ p” is not an index but an argument; thedex, sondern ein Argument: der Sinn von sense of “∼ p” cannot be understood, unless„∼ p“ k a n n n i c h t verstanden werden, the sense of “ p” has previously been underohne dass vorher der Sinn von „ p“ verstan- stood. (In the name Julius Cæsar, Julius
den worden wäre. (Im Namen Julius Cäsar is an index. The index is always part ofist „Julius“ ein Index. Der Index ist immer a description of the object to whose nameein Teil einer Beschreibung des Gegenstan- we attach it, e.g. The Cæsar of the Juliandes, dessen Namen wir ihn anhängen. Z. B. gens.)
D e r Cäsar aus dem Geschlechte der Julier.)
Die Verwechslung von Argument und
The confusion of argument and index is,
Index liegt, wenn ich mich nicht irre, der if I am not mistaken, at the root of Frege’s
Theorie Freges von der Bedeutung der Sät- theory of the meaning of propositions andze und Funktionen zugrunde. Für Frege functions. For Frege the propositions ofwaren die Sätze der Logik Namen, und de- logic were names and their arguments theren Argumente die Indices dieser Namen. indices of these names.

A proposition is a truth-function of elementary propositions.
(An elementary proposition is a truthfunction of itself.)
Elementary propositions are the trutharguments of propositions.
The arguments of functions are readily
confused with the affixes of names. For
both arguments and affixes enable me torecognize the meaning of the signs containing them.
For example, when Russell writes ‘+ c ’,the ‘ c’ is an affix which indicates that thesign as a whole is the addition-sign for cardinal numbers. But the use of this sign isthe result of arbitrary convention and it
would be quite possible to choose a simple
sign instead of ‘+ c ’; in ‘∼ p’, however, ‘ p’ is
not an affix but an argument: the sense of‘∼ p’ cannot be understood unless the senseof ‘ p’ has been understood already. (In thename Julius Caesar ‘Julius’ is an affix. An
affix is always part of a description of theobject to whose name we attach it: e.g. theCaesar of the Julian gens.)

If I am not mistaken, Frege’s theory
about the meaning of propositions andfunctions is based on the confusion between an argument and an affix. Fregeregarded the propositions of logic as names,
and their arguments as the affixes of thosenames.
Die Wahrheitsfunktionen lassen sich in
The truth-functions can be ordered in
Truth-functions can be arranged in seReihen ordnen.
series.
ries.
Das ist die Grundlage der WahrscheinThat is the foundation of the theory ofThat is the foundation of the theory of5.101

5.11

5.12

5.121

5.122

lichkeitslehre.
probability.
probability.
Die Wahrheitsfunktionen jeder Anzahl
The truth-functions of every number of
The truth-functions of a given number
von Elementarsätzen lassen sich in einem elementary propositions can be written in of elementary propositions can always be
Schema folgender Art hinschreiben:
a schema of the following kind:set out in a schema of the following kind:(WWWW) ( p, q) Tautologie (Wenn p, so p; und wenn q, so q.)( p ⊃ p . q ⊃ q)
(F WWW) ( p, q) in Worten: Nicht beides p und q. (∼( p . q))(W F WW) ( p, q) ””
Wenn q, so p. ( q ⊃ p)
(WW F W) ( p, q) ””
Wenn p, so q. ( p ⊃ q)
(WWW F) ( p, q) ””
p oder q. ( p ∨ q)
(F F WW) ( p, q) ””
Nicht q. (∼ q)(F W F W) ( p, q) ””
Nicht p. (∼ p)(F WW F) ( p, q) ””
p, oder q, aber nicht beide.
( p . ∼ q :∨: q . ∼ p)
(W F F W) ( p, q) ””
Wenn p, so q; und wenn q, so p.
( p ≡ q)(W F W F) ( p, q) ””
p
(WW F F) ( p, q) ””
q
(F F F W) ( p, q) ””
Weder p noch q.
(∼ p . ∼ q) oder ( p | q)
(F F W F) ( p, q) ””
p und nicht q. ( p . ∼ q)
(F W F F) ( p, q) ””
q und nicht p. ( q . ∼ p)
(W F F F) ( p, q) ””
q und p. ( q . p)
(F F F F) ( p, q) Kontradiktion ( p und nicht p; und
q und nicht q.) ( p . ∼ p . q . ∼ q)

(TTTT) ( p, q) Tautology (if p then p; and if q then q)[ p ⊃ p . q ⊃ q]
(FTTT) ( p, q) in words: Not both p and q. [∼( p . q)](TFTT) ( p, q) ””
If q then p. [ q ⊃ p]
(TTFT) ( p, q) ””
If p then q. [ p ⊃ q]
(TTTF) ( p, q) ””
p or q. [ p ∨ q]
(FFTT) ( p, q) ””
Not q. [∼ q](FTFT) ( p, q) ””
Not p. [∼ p](FTTF) ( p, q) ””
p or q, but not both.
[ p . ∼ q :∨: q . ∼ p]
(TFFT) ( p, q) ””
If p, then q; and if q, then p.
[ p ≡ q](TFTF) ( p, q) ””
p
(TTFF) ( p, q) ””
q
(FFFT) ( p, q) ””
Neither p nor q.
[∼ p . ∼ q or p | q]
(FFTF) ( p, q) ””
p and not q. [ p . ∼ q]
(FTFF) ( p, q) ””
q and not p. [ q . ∼ p]
(TFFF) ( p, q) ””
p and q. [ p . q]
(FFFF) ( p, q) Contradiction ( p and not p; andq and not q.) [ p . ∼ p . q . ∼ q]

Diejenigen Wahrheitsmöglichkeiten
seiner Wahrheitsargumente, welche denSatz bewahrheiten, will ich seine W a h r h e i t s g r ü n d e nennen.
Sind die Wahrheitsgründe, die einer
Anzahl von Sätzen gemeinsam sind, sämtlich auch Wahrheitsgründe eines bestimmten Satzes, so sagen wir, die Wahrheit dieses Satzes folge aus der Wahrheit jener
Sätze.
Insbesondere folgt die Wahrheit eines
Satzes „ p“ aus der Wahrheit eines anderen „ q“, wenn alle Wahrheitsgründe deszweiten Wahrheitsgründe des ersten sind.
Die Wahrheitsgründe des einen sind indenen des anderen enthalten; p folgt aus
q.
Folgt p aus q, so ist der Sinn von „ p“
im Sinne von „ q“ enthalten.

Those truth-possibilities of its truthI will give the name truth-grounds of aarguments, which verify the proposition, proposition to those truth-possibilities of
I shall call its truth-grounds.
its truth-arguments that make it true.
If the truth-grounds which are common
to a number of propositions are all also
truth-grounds of some one proposition, we
say that the truth of this proposition follows from the truth of those propositions.
In particular the truth of a proposition
p follows from that of a proposition q, if all
the truth-grounds of the second are truthgrounds of the first.
The truth-grounds of q are containedin those of p; p follows from q.
If p follows from q, the sense of “ p” iscontained in that of “ q”.

(TTTT) ( p, q) Tautology (If p then p; and if q then q.)( p ⊃ p . q ⊃ q)
(FTTT) ( p, q) In words: Not both p and q. (∼( p . q))(TFTT) ( p, q) ”
” : If q then p. ( q ⊃ p)
(TTFT) ( p, q) ”
” : If p then q. ( p ⊃ q)
(TTTF) ( p, q) ”
” : p or q. ( p ∨ q)
(FFTT) ( p, q) ”
” : Not q. (∼ q)(FTFT) ( p, q) ”
” : Not p. (∼ p)(FTTF) ( p, q) ”
” : p or q, but not both.
( p . ∼ q :∨: q . ∼ p)
(TFFT) ( p, q) ”
” : If p then q, and if q then p.
( p ≡ q)(TFTF) ( p, q) ”
” : p
(TTFF) ( p, q) ””
q
(FFFT) ( p, q) ”
” : Neither p nor q.
(∼ p . ∼ q or p | q)
(FFTF) ( p, q) ”
” : p and not q. ( p . ∼ q)
(FTFF) ( p, q) ”
” : q and not p. ( q . ∼ p)
(TFFF) ( p, q) ”
” : q and p. ( q . p)
(FFFF) ( p, q) Contradiction ( p and not p, andq and not q.) ( p . ∼ p . q . ∼ q)

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 ofthat proposition follows from the truth ofthe others.
In particular, the truth of a proposition‘ p’ follows from the truth of another proposition ‘ q’ if all the truth-grounds of the latter are truth-grounds of the former.
The truth-grounds of the one are contained in those of the other: p follows fromq.
If p follows from q, the sense of ‘ p’ iscontained in the sense of ‘ q’.

5.123

5.124
5.1241

5.13

5.131

5.1311

Wenn ein Gott eine Welt erschafft,
worin gewisse Sätze wahr sind, so schafft
er damit auch schon eine Welt, in welcher
alle ihre Folgesätze stimmen. Und ähnlichkönnte er keine Welt schaffen, worin derSatz „ p“ wahr ist, ohne seine sämtlichen
Gegenstände zu schaffen.
Der Satz bejaht jeden Satz, der aus ihm
folgt.
„ p . q“ ist einer der Sätze, welche „ p“ bejahen, und zugleich einer der Sätze, welche„ q“ bejahen.
Zwei Sätze sind einander entgegengesetzt, wenn es keinen sinnvollen Satz gibt,
der sie beide bejaht.
Jeder Satz der einem anderen widerspricht, verneint ihn.
Dass die Wahrheit eines Satzes aus derWahrheit anderer Sätze folgt, ersehen wir
aus der Struktur der Sätze.
Folgt die Wahrheit eines Satzes ausder Wahrheit anderer, so drückt sich dies
durch Beziehungen aus, in welchen die Formen jener Sätze zu einander stehen; undzwar brauchen wir sie nicht erst in jene
Beziehungen zu setzen, indem wir sie in einem Satz miteinander verbinden, sondern
diese Beziehungen sind intern und bestehen, sobald, und dadurch dass, jene Sätze
bestehen.
Wenn wir von p ∨ q und ∼ p auf q schließen, so ist hier durch die Bezeichnungsweise die Beziehung der Satzformen von„ p ∨ q“ und „∼ p“ verhüllt. Schreiben wir
aber z. B. statt „ p ∨ q“ „ p | q .|. p | q“ undstatt „∼ p“ „ p | p“ ( p | q = weder p, noch q),

If a god creates a world in which certain
propositions are true, he creates thereby
also a world in which all propositions consequent on them are true. And similarly
he could not create a world in which theproposition “ p” is true without creating all
its objects.
A proposition asserts every proposition
which follows from it.
“ p . q” is one of the propositions whichassert “ p” and at the same time one of the
propositions which assert “ q”.
Two propositions are opposed to one another if there is no significant proposition
which asserts them both.
Every proposition which contradicts another, denies it.
That the truth of one proposition follows from the truth of other propositions,
we perceive from the structure of the propositions.
If the truth of one proposition follows
from the truth of others, this expresses
itself in relations in which the forms ofthese propositions stand to one another,and we do not need to put them in these
relations first by connecting them with one
another in a proposition; for these relations
are internal, and exist as soon as, and by
the very fact that, the propositions exist.
When we conclude from p ∨ q and ∼ p
to q the relation between the forms of the
propositions “ p ∨ q” and “∼ p” is here concealed by the method of symbolizing. But if
we write, e.g. instead of “ p ∨ q” “ p | q .|. p | q”
and instead of “∼ p” “ p | p” ( p | q = neither

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 cometrue. And similarly he could not create aworld in which the proposition ‘ p’ was true
without creating all its objects.
A proposition affirms every proposition
that follows from it.
‘ p . q’ is one of the propositions thataffirm ‘ 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 asense, that affirms them both.
Every proposition that contradicts another negates it.
When the truth of one proposition follows from the truth of others, we can see
this from the structure of the propositions.
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 thecontrary, the relations are internal, andtheir existence is an immediate result of
the existence of the propositions.
When we infer q from p ∨ q and ∼ p, therelation between the propositional formsof ‘ p ∨ q’ and ‘∼ p’ is masked, in this case,by our mode of signifying. But if instead of‘ p ∨ q’ we write, for example, ‘ p | q .|. p | q’,
and instead of ‘∼ p’, ‘ p | p’ ( p | q = neither p5.132

5.133
5.134
5.135

5.136
5.13615.1362

so wird der innere Zusammenhang offenbar.
(Dass man aus ( x). f x auf f a schließen
kann, das zeigt, dass die Allgemeinheit
auch im Symbol „( x). f x“ vorhanden ist.)
Folgt p aus q, so kann ich von q auf p
schließen; p aus q folgern.
Die Art des Schlusses ist allein aus den
beiden Sätzen zu entnehmen.
Nur sie selbst können den Schluss
rechtfertigen.
„Schlussgesetze“, welche – wie bei Frege und Russell – die Schlüsse rechtfertigen
sollen, sind sinnlos, und wären überflüssig.

p nor q), then the inner connexion becomes
obvious.
(The fact that we can infer f a from
( x). f x shows that generality is present also
in the symbol “( x). f x”.
If p follows from q, I can conclude fromq to p; infer p from q.
The method of inference is to be understood from the two propositions alone.
Only they themselves can justify theinference.
Laws of inference, which—as in Frege
and Russell—are to justify the conclusions,
are senseless and would be superfluous.

Alles Folgern geschieht a priori.
Aus einem Elementarsatz lässt sichkein anderer folgern.
Auf keine Weise kann aus dem Bestehen irgend einer Sachlage auf das Bestehen einer von ihr gänzlich verschiedenen
Sachlage geschlossen werden.
Einen Kausalnexus, der einen solchen
Schluss rechtfertigte, gibt es nicht.
Die Ereignisse der Zukunft k ö n n e n
wir nicht aus den gegenwärtigen erschließen.
Der Glaube an den Kausalnexus ist derA b e r g l a u b e.
Die Willensfreiheit besteht darin, dasszukünftige Handlungen jetzt nicht gewusst werden können. Nur dann könnten
wir sie wissen, wenn die Kausalität eine
i n n e r e Notwendigkeit wäre, wie die deslogischen Schlusses. – Der Zusammenhang
von Wissen und Gewusstem ist der der logischen Notwendigkeit.

All inference takes place a priori.
From an elementary proposition noother can be inferred.
In no way can an inference be madefrom the existence of one state of affairs to
the existence of another entirely different
from it.
There is no causal nexus which justifies
such an inference.
The events of the future cannot be inferred from those of the present.

nor q), then the inner connexion becomes
obvious.
(The possibility of inference from ( x). f x
to f a shows that the symbol ( x). f x 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 ofFrege and Russell, have no sense, andwould be superfluous.
All deductions are made a priori.
One elementary proposition cannot be
deduced from another.
There is no possible way of making an
inference from the existence of one situation to the existence of another, entirely
different situation.
There is no causal nexus to justify suchan inference.
We cannot infer the events of the futurefrom those of the present.

Superstition is the belief in the causal
nexus.
The freedom of the will consists in thefact that future actions cannot be known
now. We could only know them if causality
were an inner necessity, like that of logicaldeduction.—The connexion of knowledge
and what is known is that of logical necessity.

Superstition is nothing but belief in thecausal nexus.
The freedom of the will consists in theimpossibility of knowing actions that stilllie in the future. We could know them only
if causality were an inner necessity likethat of logical inference.—The connexion
between knowledge and what is known isthat of logical necessity.

5.1363

5.14

5.141
5.142
5.143

5.15

5.151

(„A weiß, dass p der Fall ist“ ist sinnlos,
wenn p eine Tautologie ist.)
Wenn daraus, dass ein Satz uns einleuchtet, nicht f o l g t, dass er wahr ist,
so ist das Einleuchten auch keine Rechtfertigung für unseren Glauben an seine
Wahrheit.
Folgt ein Satz aus einem anderen, so
sagt dieser mehr als jener, jener weniger
als dieser.
Folgt p aus q und q aus p, so sind sie
ein und derselbe Satz.
Die Tautologie folgt aus allen Sätzen:
sie sagt nichts.
Die Kontradiktion ist das Gemeinsame
der Sätze, was k e i n Satz mit einem anderen gemein hat. Die Tautologie ist das Gemeinsame aller Sätze, welche nichts miteinander gemein haben.
Die Kontradiktion verschwindet sozusagen außerhalb, die Tautologie innerhalb
aller Sätze.
Die Kontradiktion ist die äußere Grenze der Sätze, die Tautologie ihr substanzloser Mittelpunkt.
Ist Wr die Anzahl der Wahrheitsgründe
des Satzes „ r “, Wrs die Anzahl derjenigen
Wahrheitsgründe des Satzes „ s“, die zugleich Wahrheitsgründe von „ r “ sind, dannnennen wir das Verhältnis: Wrs : Wr das
Maß der W a h r s c h e i n l i c h k e i t, welche der Satz „ r “ dem Satz „ s“ gibt.
Sei in einem Schema wie dem obigen in
No. 5.101 Wr die Anzahl der „W“ im Satzer ; Wrs die Anzahl derjenigen „W“ im Satzes, die in gleichen Kolonnen mit „W“ desSatzes r stehen. Der Satz r gibt dann dem(“A knows that p is the case” is senseless if p is a tautology.)
If from the fact that a proposition is
obvious to us it does not follow that it istrue, then obviousness is no justificationfor our belief in its truth.

(‘A knows that p is the case’, has nosense if p is a tautology.)
If the truth of a proposition does notfollow from the fact that it is self-evident tous, then its self-evidence in no way justifies
our belief in its truth.

If a proposition follows from another,
then the latter says more than the former,
the former less than the latter.
If p follows from q and q from p thenthey are one and the same proposition.
A tautology follows from all propositions: it says nothing.
Contradiction is something shared bypropositions, which no proposition has incommon with another. Tautology is thatwhich is shared by all propositions, whichhave nothing in common with one another.
Contradiction vanishes so to speak outside, tautology inside all propositions.

If one proposition follows from another,
then the latter says more than the former,
and the former less than the latter.
If p follows from q and q from p, thenthey are one and the same proposition.
A tautology follows from all propositions: it says nothing.
Contradiction is that common factor of
propositions which no proposition has incommon with another. Tautology is thecommon factor of all propositions that havenothing in common with one another.
Contradiction, one might say, vanishes
outside all propositions: tautology vanishes inside them.
Contradiction is the outer limit ofpropositions: tautology is the unsubstantial point at their centre.
If Tr is the number of the truth-groundsof 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 degreeof probability that the proposition ‘ r ’ gives
to the proposition ‘ s’.
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’ inthe proposition s that stand in columns inwhich the proposition r has ‘T’s’. Then theContradiction is the external limit of
the propositions, tautology their substanceless centre.
If Tr is the number of the truth-groundsof the proposition “ r ”, Trs the number of
those truth-grounds of the proposition “ s”
which are at the same time truth-groundsof “ r ”, then we call the ratio Trs : Tr themeasure of the probability which the proposition “ r ” gives to the proposition “ s”.
Suppose in a schema like that above in
No. 5.101 Tr is the number of the “T” ’s in
the proposition r , Trs the number of those“T” ’s in the proposition s, which stand in
the same columns as “T” ’s of the proposi-Satze s die Wahrscheinlichkeit: Wrs : Wr .

5.1511

5.152

5.153

5.154

tion r ; then the proposition r gives to the proposition r gives to the proposition s the
proposition s the probability Trs : Tr .
probability Trs : Tr .
Es gibt keinen besonderen Gegenstand,
There is no special object peculiar to
There is no special object peculiar to
der den Wahrscheinlichkeitssätzen eigen probability propositions.
probability propositions.
wäre.
Sätze, welche keine WahrheitsarguPropositions which have no truthWhen propositions have no truthmente mit einander gemein haben, nennen arguments in common with one another arguments in common with one another,wir von einander unabhängig.
we call independent.
we call them independent of one another.
Zwei Elementarsätze geben einander
Independent propositions (e.g. any two
Two elementary propositions give one
1
die Wahrscheinlichkeit 2 .
elementary propositions) give to one an- another the probability 12 .
other the probability 12 .
Folgt p aus q, so gibt der Satz „ q“ demIf p follows from q, the proposition q
If p follows from q, then the proposition
Satz „ p“ die Wahrscheinlichkeit 1. Die Ge- gives to the proposition p the probability ‘ q’ gives to the proposition ‘ p’ the probabilwissheit des logischen Schlusses ist ein 1. The certainty of logical conclusion is a ity 1. The certainty of logical inference is a
Grenzfall der Wahrscheinlichkeit.
limiting case of probability.
limiting case of probability.
(Anwendung auf Tautologie und Kon(Application to tautology and contradic(Application of this to tautology andtradiktion.)tion.)
contradiction.)
Ein Satz ist an sich weder wahrscheinA proposition is in itself neither probIn itself, a proposition is neither probalich noch unwahrscheinlich. Ein Ereignis able nor improbable. An event occurs or ble nor improbable. Either an event occurs
trifft ein, oder es trifft nicht ein, ein Mittel- does not occur, there is no middle course.
or it does not: there is no middle way.
ding gibt es nicht.
In einer Urne seien gleichviel weiße
In an urn there are equal numbers ofSuppose that an urn contains black andund schwarze Kugeln (und keine anderen). white and black balls (and no others). I white balls in equal numbers (and none
Ich ziehe eine Kugel nach der anderen und draw one ball after another and put them of any other kind). I draw one ball afterlege sie wieder in die Urne zurück. Dann back in the urn. Then I can determine by another, putting them back into the urn.
kann ich durch den Versuch feststellen, the experiment that the numbers of the By this experiment I can establish thatdass sich die Zahlen der gezogenen schwar- black and white balls which are drawn ap- the number of black balls drawn and thezen und weißen Kugeln bei fortgesetztem proximate as the drawing continues.
number of white balls drawn approximate
Ziehen einander nähern.
to one another as the draw continues.
D a s ist also kein mathematisches FakSo this is not a mathematical fact.
So this is not a mathematical truth.
tum.
Wenn ich nun sage: Es ist gleich wahrIf then, I say, It is equally probable that
Now, if I say, ‘The probability of myscheinlich, dass ich eine weiße Kugel wie ei- I should draw a white and a black ball, this drawing a white ball is equal to the probne schwarze ziehen werde, so heißt das: Al- means, All the circumstances known to me ability of my drawing a black one’, thisle mir bekannten Umstände (die hypothe- (including the natural laws hypothetically means that all the circumstances that I
tisch angenommenen Naturgesetze mitin- assumed) give to the occurrence of the one know of (including the laws of nature asbegriffen) geben dem Eintreffen des einen event no more probability than to the oc- sumed as hypotheses) give no more prob-

5.155

5.156

5.2
5.21

5.22

Ereignisses nicht m e h r Wahrscheinlichkeit als dem Eintreffen des anderen. Das
heißt, sie geben – wie aus den obigen Erklärungen leicht zu entnehmen ist – jedem
die Wahrscheinlichkeit 12 .
Was ich durch den Versuch bestätige
ist, dass das Eintreffen der beiden Ereignisse von den Umständen, die ich nicht
näher kenne, unabhängig ist.
Die Einheit des Wahrscheinlichkeitssatzes ist: Die Umstände – die ich sonstnicht weiter kenne – geben dem Eintreffen
eines bestimmten Ereignisses den und den
Grad der Wahrscheinlichkeit.
So ist die Wahrscheinlichkeit eine Verallgemeinerung.
Sie involviert eine allgemeine Beschreibung einer Satzform.
Nur in Ermanglung der Gewissheit gebrauchen wir die Wahrscheinlichkeit. –Wenn wir zwar eine Tatsache nicht vollkommen kennen, wohl aber e t w a s überihre Form wissen.
(Ein Satz kann zwar ein unvollständiges Bild einer gewissen Sachlage sein, aberer ist immer e i n vollständiges Bild.)
Der Wahrscheinlichkeitssatz ist gleichsam ein Auszug aus anderen Sätzen.
Die Strukturen der Sätze stehen in internen Beziehungen zu einander.
Wir können diese internen Beziehungen dadurch in unserer Ausdrucksweise
hervorheben, dass wir einen Satz als Resultat einer Operation darstellen, die ihn
aus anderen Sätzen (den Basen der Operation) hervorbringt.
Die Operation ist der Ausdruck einer

currence of the other. That is they give—as
can easily be understood from the above
explanations—to each the probability 12 .

What I can verify by the experimentis that the occurrence of the two eventsis independent of the circumstances withwhich I have no closer acquaintance.
The unit of the probability proposition
is: The circumstances—with which I am
not further acquainted—give to the occurrence of a definite event such and such a
degree of probability.
Probability is a generalization.
It involves a general description of a
propositional form.
Only in default of certainty do we needprobability. If we are not completely acquainted with a fact, but know something
about its form.
(A proposition can, indeed, be an incomplete picture of a certain state of affairs,
but it is always a complete picture.)
The probability proposition is, as it
were, an extract from other propositions.
The structures of propositions stand to
one another in internal relations.
We can bring out these internal relations in our manner of expression, by presenting a proposition as the result of an operation which produces it from other propositions (the bases of the operation).

ability to the occurrence of the one event
than to that of the other. That is to say,they give each the probability 12 , as caneasily be gathered from the above definitions.
What I confirm by the experiment isthat 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—givesuch and such a degree of probability tothe 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 ofcertainty—if our knowledge of a fact is notindeed 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 ofexcerpt from other propositions.
The structures of propositions stand ininternal relations to one another.
In order to give prominence to theseinternal relations we can adopt the following mode of expression: we can represent
a proposition as the result of an operationthat produces it out of other propositions(which are the bases of the operation).
The operation is the expression of a reAn operation is the expression of a re-

5.23

5.231

5.232

5.233

5.234

5.2341

5.24

Beziehung zwischen den Strukturen ihres
Resultats und ihrer Basen.
Die Operation ist das, was mit dem
einen Satz geschehen muss, um aus ihm
den anderen zu machen.
Und das wird natürlich von ihren formalen Eigenschaften, von der internenÄhnlichkeit ihrer Formen abhängen.
Die interne Relation, die eine Reihe
ordnet, ist äquivalent mit der Operation,
durch welche ein Glied aus dem anderen
entsteht.
Die Operation kann erst dort auftreten,
wo ein Satz auf logisch bedeutungsvolle
Weise aus einem anderen entsteht. Also
dort, wo die logische Konstruktion des Satzes anfängt.
Die Wahrheitsfunktionen der Elementarsätze sind Resultate von Operationen,die die Elementarsätze als Basen haben.
(Ich nenne diese Operationen Wahrheitsoperationen.)
Der Sinn einer Wahrheitsfunktion vonp ist eine Funktion des Sinnes von p.
Verneinung, logische Addition, logische
Multiplikation, etc., etc. sind Operationen.
(Die Verneinung verkehrt den Sinn desSatzes.)Die Operation zeigt sich in einer Variablen; sie zeigt, wie man von einer Form von
Sätzen zu einer anderen gelangen kann.
Sie bringt den Unterschied der Formen
zum Ausdruck.
(Und das Gemeinsame zwischen den
Basen und dem Resultat der Operationsind eben die Basen.)lation between the structures of its result
and its bases.
The operation is that which must happen to a proposition in order to make another out of it.
And that will naturally depend on theirformal properties, on the internal similarity of their forms.
The internal relation which orders a series is equivalent to the operation by which
one term arises from another.

lation between the structures of its resultand of its bases.
The operation is what has to be done tothe one proposition in order to make theother out of it.
And that will, of course, depend on
their formal properties, on the internalsimilarity of their forms.
The internal relation by which a series
is ordered is equivalent to the operationthat produces one term from another.

The first place in which an operation
can occur is where a proposition arises
from another in a logically significant way;i.e. where the logical construction of the
proposition begins.

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.
The truth-functions of elementary
Truth-functions of elementary propoproposition, are results of operations which sitions are results of operations with elhave the elementary propositions as bases. ementary propositions as bases. (These(I call these operations, truth-operations.) operations I call truth-operations.)
The sense of a truth-function of p is afunction of the sense of p.
Denial, logical addition, logical multiplication, etc., etc., are operations.
(Denial reverses the sense of a proposition.)
An operation shows itself in a variable;
it shows how we can proceed from one formof proposition to another.
It gives expression to the difference between the forms.
(And that which is common to the bases,
and the result of an operation, is the bases
themselves.)

The sense of a truth-function of p is afunction 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 operationand its result have in common is just thebases themselves.)

5.241

5.242

5.25

5.251

5.252

5.25215.2522

Die Operation kennzeichnet keine
Form, sondern nur den Unterschied derFormen.
Dieselbe Operation, die „ q“ aus „ p“
macht, macht aus „ q“ „ r “ u. s. f. Dies kannnur darin ausgedrückt sein, dass „ p“, „ q“,„ r “, etc. Variable sind, die gewisse formale
Relationen allgemein zum Ausdruck bringen.
Das Vorkommen der Operation charakterisiert den Sinn des Satzes nicht.
Die Operation sagt ja nichts aus, nur
ihr Resultat, und dies hängt von den Basender Operation ab.
(Operation und Funktion dürfen nicht
miteinander verwechselt werden.)
Eine Funktion kann nicht ihr eigenes
Argument sein, wohl aber kann das Resultat einer Operation ihre eigene Basiswerden.
Nur so ist das Fortschreiten von Glied
zu Glied in einer Formenreihe (von Typezu Type in den Hierarchien Russells und
Whiteheads) möglich. (Russell und Whitehead haben die Möglichkeit dieses Fortschreitens nicht zugegeben, aber immer
wieder von ihr Gebrauch gemacht.)
Die fortgesetzte Anwendung einer Operation auf ihr eigenes Resultat nenne ichihre successive Anwendung („O’ O’ O’ a“ istdas Resultat der dreimaligen successiven
Anwendung von „O’ ξ“ auf „a“).
In einem ähnlichen Sinne rede ich vonder successiven Anwendung m e h r e r e rOperationen auf eine Anzahl von Sätzen.
Das allgemeine Glied einer Formenreihe a, O’ a, O’ O’ a, . . . schreibe ich daher so:

The operation does not characterize
a form but only the difference between
forms.
The same operation which makes “ q”
from “ p”, makes “ r ” from “ q”, and so on.
This can only be expressed by the fact that“ p”, “ q”, “ r ”, etc., are variables which give
general expression to certain formal relations.
The occurrence of an operation does notcharacterize the sense of a proposition.
For an operation does not assert anything; only its result does, and this depends on the bases of the operation.
(Operation and function must not be
confused with one another.)A function cannot be its own argument,
but the result of an operation can be its
own basis.

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. Thereis only one way of expressing this: ‘ p’, ‘ q’,‘ r ’, etc. have to be variables that give expression in a general way to certain formal
relations.
The occurrence of an operation does notcharacterize the sense of a proposition.
Indeed, no statement is made by anoperation, but only by its result, and thisdepends 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.

Only in this way is the progress fromterm to term in a formal series possible(from type to type in the hierarchy of Russell and Whitehead). (Russell and Whitehead have not admitted the possibility ofthis progress but have made use of it all
the same.)
The repeated application of an operation to its own result I call its successive
application (“O’ O’ O’ a” is the result of thethreefold successive application of “O’ ξ” to“a”).
In a similar sense I speak of the successive application of several operations to anumber of propositions.
The general term of the formal series a, O’ a, O’ O’ a, . . . . I write thus:

It is only in this way that the step from
one term of a series of forms to anotheris possible (from one type to another inthe hierarchies of Russell and Whitehead).
(Russell and Whitehead did not admit thepossibility of such steps, but repeatedly
availed themselves of it.)
If an operation is applied repeatedly toits own results, I speak of successive applications of it. (‘O’ O’ O’ a’ is the result of
three successive applications of the operation ‘O’ ξ’ to ‘a’.)
In a similar sense I speak of successive
applications of more than one operation toa number of propositions.
Accordingly I use the sign ‘[a, x, O’ x]’
for the general term of the series of forms

5.2523

5.253

5.254
5.3

„[a, x, O’ x]“. Dieser Klammerausdruck ist “[a, x, O’ x]”. This expression in brackets
eine Variable. Das erste Glied des Klam- is a variable. The first term of the expresmerausdruckes ist der Anfang der Formen- sion is the beginning of the formal series,
reihe, das zweite die Form eines beliebi- the second the form of an arbitrary termgen Gliedes x der Reihe und das dritte die x of the series, and the third the form ofForm desjenigen Gliedes der Reihe, wel- that term of the series which immediately
ches auf x unmittelbar folgt.
follows x.
Der Begriff der successiven AnwenThe concept of the successive applicadung der Operation ist äquivalent mit dem tion of an operation is equivalent to the
Begriff „und so weiter“.
concept “and so on”.
Eine Operation kann die Wirkung einer
One operation can reverse the effectanderen rückgängig machen. Operationen of another. Operations can cancel one ankönnen einander aufheben.
other.
Die Operation kann verschwinden (z. B.
Operations can vanish (e.g. denial indie Verneinung in „∼∼ p“: ∼∼ p = p).
“∼∼ p”. ∼∼ p = p).
Alle Sätze sind Resultate von WahrAll propositions are results of truthheitsoperationen mit den Elementarsät- operations on the elementary propositions.
zen.
Die Wahrheitsoperation ist die Art und
The truth-operation is the way in which
Weise, wie aus den Elementarsätzen die a truth-function arises from elementary
Wahrheitsfunktion entsteht.
propositions.
Nach dem Wesen der WahrheitsoperatiAccording to the nature of truthon wird auf die gleiche Weise, wie aus den operations, in the same way as out of elElementarsätzen ihre Wahrheitsfunktion, ementary propositions arise their truthaus Wahrheitsfunktionen eine Neue. Je- functions, from truth-functions arises ade Wahrheitsoperation erzeugt aus Wahr- new one. Every truth-operation creates
heitsfunktionen von Elementarsätzen wie- from truth-functions of elementary propoder eine Wahrheitsfunktion von Elemen- sitions, another truth-function of elementarsätzen, einen Satz. Das Resultat jeder tary propositions i.e. a proposition. The
Wahrheitsoperation mit den Resultaten result of every truth-operation on the revon Wahrheitsoperationen mit Elementar- sults of truth-operations on elementary
sätzen ist wieder das Resultat E i n e r propositions is also the result of one truthWahrheitsoperation mit Elementarsätzen. operation on elementary propositions.

a, O’ a, O’ O’ a, . . . . This bracketed expression is a variable: the first term of thebracketed expression is the beginning of
the series of forms, the second is the formof a term x arbitrarily selected from the series, and the third is the form of the termthat immediately follows x in the series.
The concept of successive applicationsof an operation is equivalent to the concept‘and so on’.
One operation can counteract the effectof another. Operations can cancel one another.
An operation can vanish (e.g. negationin ‘∼∼ p’: ∼∼ p = p).
All propositions are results of truthoperations 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 yielda truth-function of themselves, so too in thesame 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 atruth-operation is applied to the resultsof truth-operations on elementary propositions, there is always a single operation onelementary propositions that has the same
result.
Jeder Satz ist das Resultat von WahrEvery proposition is the result of truthEvery proposition is the result of truthheitsoperationen mit Elementarsätzen.
operations on elementary propositions.
operations on elementary propositions.

5.31

5.32

5.4

5.41

5.42

5.43

Die Schemata No. 4.31 haben auchdann eine Bedeutung, wenn „ p“, „ q“, „ r “,
etc. nicht Elementarsätze sind.
Und es ist leicht zu sehen, dass dasSatzzeichen in No. 4.442, auch wenn „ p“
und „ q“ Wahrheitsfunktionen von Elementarsätzen sind, Eine Wahrheitsfunktion
von Elementarsätzen ausdrückt.
Alle Wahrheitsfunktionen sind Resultate der successiven Anwendung einer endlichen Anzahl von Wahrheitsoperationen
auf die Elementarsätze.
Hier zeigt es sich, dass es „logische
Gegenstände“, „logische Konstante“ (imSinne Freges und Russells) nicht gibt.

The Schemata No. 4.31 are also significant, if “ p”, “ q”, “ r ”, etc. are not elementary
propositions.
And it is easy to see that the propositional sign in No. 4.442 expresses one
truth-function of elementary propositions
even when “ p” and “ q” are truth-functionsof elementary propositions.
All truth-functions are results of thesuccessive application of a finite numberof truth-operations to elementary propositions.
Here it becomes clear that there are nosuch things as “logical objects” or “logicalconstants” (in the sense of Frege and Russell).
Denn: Alle Resultate von WahrheitsFor all those results of truth-operations
operationen mit Wahrheitsfunktionen sind on truth-functions are identical, which are
identisch, welche eine und dieselbe Wahr- one and the same truth-function of elemenheitsfunktion von Elementarsätzen sind. tary propositions.

Dass ∨, ⊃, etc. nicht Beziehungen im
Sinne von rechts und links etc. sind, leuchtet ein.
Die Möglichkeit des kreuzweisen Definierens der logischen „Urzeichen“ Freges
und Russells zeigt schon, dass diese keine
Urzeichen sind, und schon erst recht, dasssie keine Relationen bezeichnen.
Und es ist offenbar, dass das „⊃“, welches wir durch „∼“ und „∨“ definieren, identisch ist mit dem, durch welches wir „∨“mit „∼“ definieren, und dass dieses „∨“ mitdem ersten identisch ist. U. s. w.
Dass aus einer Tatsache p unendlich
viele a n d e r e folgen sollten, nämlich∼∼ p, ∼∼∼∼ p, etc., ist doch von vornherein

The schemata in 4.31 have a meaningeven 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-functionsof elementary propositions.
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 Russell’s sense).

The reason is that the results of truthoperations on truth-functions are always
identical whenever they are one and thesame truth-function of elementary propositions.
That ∨, ⊃, etc., are not relations in the
It is self-evident that ∨, ⊃, etc. are notsense of right and left, etc., is obvious.
relations in the sense in which right andleft etc. are relations.
The possibility of crosswise definition
The interdefinability of Frege’s andof the logical “primitive signs” of Frege and Russell’s ‘primitive signs’ of logic is enough
Russell shows by itself that these are not to show that they are not primitive signs,
primitive signs and that they signify no still less signs for relations.
relations.
And it is obvious that the “⊃” which we
And it is obvious that the ‘⊃’ defined bydefine by means of “∼” and “∨” is identical means of ‘∼’ and ‘∨’ is identical with thewith that by which we define “∨” with the one that figures with ‘∼’ in the definition
help of “∼”, and that this “∨” is the same of ‘∨’; and that the second ‘∨’ is identicalas the first, and so on.
with the first one; and so on.
That from a fact p an infinite number ofEven at first sight it seems scarcely
others should follow, namely, ∼∼ p, ∼∼∼∼ p, credible that there should follow from one
etc., is indeed hardly to be believed, and it fact p infinitely many others, namely ∼∼ p,5.44

kaum zu glauben. Und nicht weniger merk- is no less wonderful that the infinite numwürdig ist, dass die unendliche Anzahl der ber of propositions of logic (of mathematics)
Sätze der Logik (der Mathematik) aus ei- should follow from half a dozen “primitive
nem halben Dutzend „Grundgesetzen“ fol- propositions”.
gen.
Alle Sätze der Logik sagen aber dasselBut the propositions of logic say thebe. Nämlich nichts.
same thing. That is, nothing.
Die Wahrheitsfunktionen sind keine
Truth-functions are not material funcmateriellen Funktionen.
tions.
Wenn man z. B. eine Bejahung durch
If e.g. an affirmation can be produced
doppelte Verneinung erzeugen kann, ist by repeated denial, is the denial—in any
dann die Verneinung – in irgend einem sense—contained in the affirmation? Does
Sinn – in der Bejahung enthalten? Ver- “∼∼ p” deny ∼ p, or does it affirm p; or both?
neint „∼∼ p“ ∼ p, oder bejaht es p; oder beides?
Der Satz „∼∼ p“ handelt nicht von der
The proposition “∼∼ p” does not treat ofVerneinung wie von einem Gegenstand; denial as an object, but the possibility ofwohl aber ist die Möglichkeit der Vernei- denial is already prejudged in affirmation.
nung in der Bejahung bereits präjudiziert.
Und gäbe es einen Gegenstand, der „∼“And if there was an object called “∼”,
hieße, so müsste „∼∼ p“ etwas anderes sa- then “∼∼ p” would have to say something
gen als „ p“. Denn der eine Satz würde dann other than “ p”. For the one proposition
eben von ∼ handeln, der andere nicht.
would then treat of ∼, the other would not.

5.441

Dieses Verschwinden der scheinbarenThis disappearance of the apparent loglogischen Konstanten tritt auch ein, wenn ical constants also occurs if “∼ (∃ x). ∼ f x”„∼ (∃ x). ∼ f x“ dasselbe sagt wie „( x). f x“, says the same as “( x). f x”, or “(∃ x). f x . x =oder „(∃ x). f x . x = a“ dasselbe wie „ f a“.
a” the same as “ f a”.

5.442

Wenn uns ein Satz gegeben ist, so sindm i t i h m auch schon die Resultate aller
Wahrheitsoperationen, die ihn zur Basishaben, gegeben.
Gibt es logische Urzeichen, so muss eine richtige Logik ihre Stellung zueinanderklar machen und ihr Dasein rechtfertigen.
Der Bau der Logik a u s ihren Urzeichen

5.45

If a proposition is given to us then the
results of all truth-operations which haveit as their basis are given with it.

∼∼∼∼ p, etc. And it is no less remarkablethat the infinite number of propositionsof logic (mathematics) follow from half adozen ‘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 casedoes it follow that in some sense negationis contained in affirmation? Does ‘∼∼ p’
negate ∼ p, or does it affirm p—or both?
The proposition ‘∼∼ p’ is not about negation, as if negation were an object: on theother hand, the possibility of negation isalready written into affirmation.
And if there were an object called ‘∼’,it would follow that ‘∼∼ p’ said something
different from what ‘ p’ said, just because
the one proposition would then be about ∼and the other would not.
This vanishing of the apparent logical constants also occurs in the caseof ‘∼ (∃ x). ∼ f x’, which says the same as‘( x). f x’, and in the case of ‘(∃ x). f x . x = a’,which says the same as ‘ f a’.
If we are given a proposition, then with
it we are also given the results of all truthoperations that have it as their base.

If there are logical primitive signs a
If there are primitive logical signs, thencorrect logic must make clear their position any logic that fails to show clearly how
relative to one another and justify their they are placed relatively to one anotherexistence. The construction of logic out of and to justify their existence will be incor-

muss klar werden.
5.451

5.452

its primitive signs must become clear.

Hat die Logik Grundbegriffe, so müsIf logic has primitive ideas these mustsen sie von einander unabhängig sein. Ist be independent of one another. If a primein Grundbegriff eingeführt, so muss er in itive idea is introduced it must be introallen Verbindungen eingeführt sein, worin duced in all contexts in which it occurs at
er überhaupt vorkommt. Man kann ihn all. One cannot therefore introduce it foralso nicht zuerst für e i n e Verbindung, one context and then again for another. For
dann noch einmal für eine andere einfüh- example, if denial is introduced, we mustren. Z. B.: Ist die Verneinung eingeführt, understand it in propositions of the form
so müssen wir sie jetzt in Sätzen von der “∼ p”, just as in propositions like “∼( p ∨ q)”,Form „∼ p“ ebenso verstehen, wie in Sätzen “(∃ x). ∼ f x” and others. We may not first inwie „∼( p ∨ q)“, „(∃ x). ∼ f x“ u. a. Wir dürfen troduce it for one class of cases and then forsie nicht erst für die eine Klasse von Fäl- another, for it would then remain doubtfullen, dann für die andere einführen, denn whether its meaning in the two cases was
es bliebe dann zweifelhaft, ob ihre Bedeu- the same, and there would be no reason totung in beiden Fällen die gleiche wäre und use the same way of symbolizing in the two
es wäre kein Grund vorhanden, in beiden cases.
Fällen dieselbe Art der Zeichenverbindungzu benützen.
(Kurz, für die Einführung der Urzei(In short, what Frege (“Grundgesetzechen gilt, mutatis mutandis, dasselbe, was der Arithmetik”) has said about the inFrege („Grundgesetze der Arithmetik“) für troduction of signs by definitions holds,
die Einführung von Zeichen durch Defini- mutatis mutandis, for the introduction oftionen gesagt hat.)primitive signs also.)
Die Einführung eines neuen Behelfes
The introduction of a new expedient inin den Symbolismus der Logik muss immer the symbolism of logic must always be an
ein folgenschweres Ereignis sein. Kein neu- event full of consequences. No new symbol
er Behelf darf in die Logik – sozusagen, mit may be introduced in logic in brackets or inganz unschuldiger Miene – in Klammern the margin—with, so to speak, an entirely
oder unter dem Striche eingeführt werden. innocent face.
(So(Thus in the “Principia Mathematica”
kommen in den „Principia Mathematica“ of Russell and Whitehead there occur defivon Russell und Whitehead Definitionen nitions and primitive propositions in words.
und Grundgesetze in Worten vor. Warum Why suddenly words here? This would
hier plötzlich Worte? Dies bedürfte einer need a justification. There was none, and
Rechtfertigung. Sie fehlt und muss fehlen, can be none for the process is actually not

rect. 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 havebeen introduced in all the combinations inwhich it ever occurs. It cannot, therefore,
be introduced first for one combination andlater reintroduced for another. For example, once negation has been introduced, wemust understand it both in propositionsof the form ‘∼ p’ and in propositions like‘∼( p ∨ q)’, ‘(∃ x). ∼ f x’, etc. We must not introduce it first for the one class of cases andthen for the other, since it would then beleft in doubt whether its meaning were thesame in both cases, and no reason would
have been given for combining the signs inthe same way in both cases.
(In short, Frege’s remarks about introducing signs by means of definitions (inThe Fundamental Laws of Arithmetic) also
apply, mutatis mutandis, to the introduction of primitive signs.)
The introduction of any new device into
the symbolism of logic is necessarily amomentous event. In logic a new device
should not be introduced in brackets orin a footnote with what one might call acompletely innocent air.
(Thus in Russell and Whitehead’s Principia Mathematica there occur definitions
and primitive propositions expressed inwords. Why this sudden appearance ofwords? It would require a justification, but
none is given, or could be given, since the5.4535.454

5.45415.46

da das Vorgehen tatsächlich unerlaubt ist.)
Hat sich aber die Einführung eines neuen Behelfes an einer Stelle als nötig erwiesen, so muss man sich nun sofort fragen:Wo muss dieser Behelf nun i m m e r angewandt werden? Seine Stellung in der Logik
muss nun erklärt werden.
Alle Zahlen der Logik müssen sichrechtfertigen lassen.
Oder vielmehr: Es muss sich herausstellen, dass es in der Logik keine Zahlen
gibt.
Es gibt keine ausgezeichneten Zahlen.
In der Logik gibt es kein Nebeneinander, kann es keine Klassifikation geben.
In der Logik kann es nicht Allgemeineres und Spezielleres geben.
Die Lösungen der logischen Probleme
müssen einfach sein, denn sie setzen den
Standard der Einfachheit.
Die Menschen haben immer geahnt,
dass es ein Gebiet von Fragen geben müsse,deren Antworten – a priori – symmetrisch,und zu einem abgeschlossenen, regelmäßigen Gebilde vereint liegen.
Ein Gebiet, in dem der Satz gilt: simplex sigillum veri.
Wenn man die logischen Zeichen richtig
einführte, so hätte man damit auch schon
den Sinn aller ihrer Kombinationen eingeführt; also nicht nur „ p ∨ q“ sondern auchschon „∼( p ∨∼ q)“ etc. etc. Man hätte damit
auch schon die Wirkung aller nur möglichen Kombinationen von Klammern eingeführt. Und damit wäre es klar geworden,
dass die eigentlichen allgemeinen Urzeichen nicht die „ p ∨ q“, „(∃ x). f x“, etc. sind,allowed.)But if the introduction of a new expedient has proved necessary in one place,
we must immediately ask: Where is thisexpedient always to be used? Its position
in logic must be made clear.

procedure is in fact illicit.)
But if the introduction of a new device
has proved necessary at a certain point, wemust 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 must be capable of
All numbers in logic stand in need ofjustification.
justification.
Or rather it must become plain that
Or rather, it must become evident that
there are no numbers in logic.
there are no numbers in logic.
There are no pre-eminent numbers.
In logic there is no side by side, there
can be no classification.
In logic there cannot be a more generaland a more special.
The solution of logical problems must
be neat for they set the standard of neatness.
Men have always thought that there
must be a sphere of questions whose
answers—a priori—are symmetrical andunited into a closed regular structure.

A sphere in which the proposition, simplex sigillum veri, is valid.
When we have rightly introduced thelogical signs, the sense of all their combinations has been already introduced with
them: therefore not only “ p ∨ q” but also“∼( p ∨ ∼ q)”, etc. etc. We should then already have introduced the effect of all
possible combinations of brackets; and it
would then have become clear that theproper general primitive signs are not“ p ∨ q”, “(∃ x). f x”, etc., but the most general

There are no privileged 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 standardof simplicity.
Men have always had a presentimentthat 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 thesame time the sense of all combinations ofthem; i.e. not only ‘ p ∨ q’ but ‘∼( p ∨ ∼ 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 thereal general primitive signs are not ‘ p ∨ q’,‘(∃ x). f x’, etc. but the most general form of5.4615.4611
5.47

5.471
5.4711sondern die allgemeinste Form ihrer Kombinationen.
Bedeutungsvoll ist die scheinbar unwichtige Tatsache, dass die logischen
Scheinbeziehungen, wie ∨ und ⊃, derKlammern bedürfen – im Gegensatz zu
den wirklichen Beziehungen.
Die Benützung der Klammern mit jenen scheinbaren Urzeichen deutet ja schon
darauf hin, dass diese nicht die wirklichen
Urzeichen sind. Und es wird doch wohl
niemand glauben, dass die Klammern eineselbständige Bedeutung haben.
Die logischen Operationszeichen sindInterpunktionen.
Es ist klar, dass alles, was sich überhaupt v o n v o r n h e r e i n über die
Form aller Sätze sagen lässt, sich a u f
e i n m a l sagen lassen muss.
Sind ja schon im Elementarsatze alle logischen Operationen enthalten. Denn „ f a“sagt dasselbe wie

form of their combinations.

their combinations.

The apparently unimportant fact that
Though it seems unimportant, it is in
the apparent relations like ∨ and ⊃ need fact significant that the pseudo-relations
brackets—unlike real relations—is of great of logic, such as ∨ and ⊃, need brackimportance.
ets—unlike real relations.
The use of brackets with these apparent primitive signs shows that these are
not the real primitive signs; and nobody
of course would believe that the brackets
have meaning by themselves.

Indeed, the use of brackets with theseapparently primitive signs is itself an indication that they are not primitive signs.
And surely no one is going to believe brackets have an independent meaning.

Logical operation signs are punctuations.
It is clear that everything which canbe said beforehand about the form of all
propositions at all can be said on one occasion.
For all logical operations are alreadycontained in the elementary proposition.
For “ f a” says the same as

Signs for logical operations are
punctuation-marks.
It is clear that whatever we can say inadvance 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 ‘ f a’says the same thing as„ ( ∃ x ) . f x . x = a “.
“ ( ∃ x ) . f x . x = a ”.
‘ (∃ x). f x . x = a’.
Wo Zusammengesetztheit ist, da ist ArWhere there is composition, there is
Wherever there is compositeness, argugument und Funktion, und wo diese sind, argument and function, and where these ment and function are present, and wheresind bereits alle logischen Konstanten.
are, all logical constants already are.
these are present, we already have all the
logical constants.
Man könnte sagen: Die Eine logische
One could say: the one logical constant
One could say that the sole logical conKonstante ist das, was a l l e Sätze, ihrer is that which all propositions, according stant was what all propositions, by their
Natur nach, mit einander gemein haben. to their nature, have in common with one very nature, had in common with one ananother.
other.
Das aber ist die allgemeine Satzform.
That however is the general form of
But that is the general propositional
proposition.
form.
Die allgemeine Satzform ist das Wesen
The general form of proposition is the
The general propositional form is thedes Satzes.
essence of proposition.
essence of a proposition.
Das Wesen des Satzes angeben, heißt,
To give the essence of proposition
To give the essence of a proposition
das Wesen aller Beschreibung angeben, al- means to give the essence of all description, means to give the essence of all description,

so das Wesen der Welt.
Die Beschreibung der allgemeinsten
5.472
Satzform ist die Beschreibung des einen
und einzigen allgemeinen Urzeichens derLogik.
5.473
Die Logik muss für sich selber sorgen.
Ein m ö g l i c h e s Zeichen muss auchbezeichnen können. Alles was in der Logik
möglich ist, ist auch erlaubt. („Sokrates ist
identisch“ heißt darum nichts, weil es keine Eigenschaft gibt, die „identisch“ heißt.
Der Satz ist unsinnig, weil wir eine willkürliche Bestimmung nicht getroffen haben,aber nicht darum, weil das Symbol an undfür sich unerlaubt wäre.)Wir können uns, in gewissem Sinne,
nicht in der Logik irren.
5.4731
Das Einleuchten, von dem Russell soviel sprach, kann nur dadurch in der Logik entbehrlich werden, dass die Spracheselbst jeden logischen Fehler verhindert.
– Dass die Logik a priori ist, besteht darin, dass nicht unlogisch gedacht werden
k a n n.
5.4732
Wir können einem Zeichen nicht den
unrechten Sinn geben.
5.47321
Occams Devise ist natürlich keine willkürliche, oder durch ihren praktischen Erfolg gerechtfertigte Regel: Sie besagt, dassu n n ö t i g e Zeicheneinheiten nichts bedeuten.
Zeichen, die E i n e n Zweck erfüllen,sind logisch äquivalent, Zeichen, die k e i n e n Zweck erfüllen, logisch bedeutungslos.
5.4733
Frege sagt: Jeder rechtmäßig gebildete
Satz muss einen Sinn haben; und ich sage:

therefore the essence of the world.
The description of the most general
propositional form is the description of
the one and only general primitive signin logic.
Logic must take care of itself.
A possible sign must also be able tosignify. Everything which is possible inlogic is also permitted. (“Socrates is identical” means nothing because there is noproperty which is called “identical”. The
proposition is senseless because we havenot made some arbitrary determination,
not because the symbol is in itself unpermissible.)
In a certain sense we cannot make mistakes in logic.
Self-evidence, of which Russell has said
so much, can only be discarded in logic
by language itself preventing every logicalmistake. That logic is a priori consists inthe fact that we cannot think illogically.

and thus the essence of the world.
The description of the most general
propositional form is the description of
the one and only general primitive signin 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 havefailed 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 illogicalthought.

We cannot give a sign the wrong sense.

We cannot give a sign the wrong sense.

Occam’s razor is, of course, not an arbiOccam’s maxim is, of course, not an artrary rule nor one justified by its practical bitrary rule, nor one that is justified by its
success. It simply says that unnecessary success in practice: its point is that unelements in a symbolism mean nothing.
necessary units in a sign-language mean
nothing.
Signs which serve one purpose are logiSigns that serve one purpose are logically equivalent, signs which serve no pur- cally equivalent, and signs that serve nonepose are logically meaningless.
are logically meaningless.
Frege says: Every legitimately conFrege says that any legitimately constructed proposition must have a sense; structed proposition must have a sense.

5.474

5.475

5.476

5.5

Jeder mögliche Satz ist rechtmäßig gebildet, und wenn er keinen Sinn hat, so kanndas nur daran liegen, dass wir einigen seiner Bestandteile keine B e d e u t u n g gegeben haben.
(Wenn wir auch glauben, es getan zu
haben.)
So sagt „Sokrates ist identisch“ darum
nichts, weil wir dem Wort „identisch“ alsE i g e n s c h a f t s w o r t k e i n e Bedeutung gegeben haben. Denn, wenn es als
Gleichheitszeichen auftritt, so symbolisiert
es auf ganz andere Art und Weise – die bezeichnende Beziehung ist eine andere, –also ist auch das Symbol in beiden Fällen
ganz verschieden; die beiden Symbole haben nur das Zeichen zufällig miteinander
gemein.
Die Anzahl der nötigen Grundoperationen hängt n u r von unserer Notation ab.
Es kommt nur darauf an, ein Zeichensystem von einer bestimmten Anzahl von Dimensionen – von einer bestimmten mathematischen Mannigfaltigkeit – zu bilden.
Es ist klar, dass es sich hier nicht um
eine A n z a h l v o n G r u n d b e g r i f f e n handelt, die bezeichnet werden müssen, sondern um den Ausdruck einer Regel.
Jede Wahrheitsfunktion ist ein Resultat der successiven Anwendung der Operation (− − − − −W) (ξ, . . . . .) auf Elementarsätze.
Diese Operation verneint sämtliche
Sätze in der rechten Klammer, und ich nenne sie die Negation dieser Sätze.

and I say: Every possible proposition islegitimately constructed, and if it has nosense this can only be because we havegiven no meaning to some of its constituent
parts.
(Even if we believe that we have doneso.)Thus “Socrates is identical” says nothing, because we have given no meaning to
the word “identical” as adjective. For when
it occurs as the sign of equality it symbolizes in an entirely different way—thesymbolizing relation is another—therefore
the symbol is in the two cases entirely different; the two symbols have the sign incommon with one another only by accident.

And I say that any possible proposition islegitimately constructed, and, if it has nosense, that can only be because we havefailed to give a meaning to some of its constituents.
(Even if we think that we have doneso.)Thus the reason why ‘Socrates is identical’ says nothing is that we have notgiven any adjectival meaning to the word
‘identical’. For when it appears as a signfor identity, it symbolizes in an entirely
different way—the signifying relation is adifferent one—therefore the symbols also
are entirely different in the two cases: thetwo symbols have only the sign in common,and that is an accident.

The number of necessary fundamental
The number of fundamental operations
operations depends only on our notation.
that are necessary depends solely on ournotation.
It is only a question of constructing a
All that is required is that we shouldsystem of signs of a definite number of di- construct a system of signs with a particumensions—of a definite mathematical mul- lar number of dimensions—with a particutiplicity.
lar mathematical multiplicity.
It is clear that we are not concerned
It is clear that this is not a question ofhere with a number of primitive ideas a number of primitive ideas that have to bewhich must be signified but with the ex- signified, but rather of the expression of apression of a rule.
rule.
Every truth-function is a result of theEvery truth-function is a result of
successive application of the operation successive applications to elementary(− − − − −T) (ξ, . . . . .) to elementary propo- propositions of the operation ‘(− − − − −T)
sitions.
(ξ, . . . . .)’.
This operation denies all the proposiThis operation negates all the propotions in the right-hand bracket and I call sitions in the right-hand pair of brackets,it the negation of these propositions.
and I call it the negation of those propositions.

5.501

5.502

5.503

Einen Klammerausdruck, dessen Glieder Sätze sind, deute ich – wenn die
Reihenfolge der Glieder in der Klammer
gleichgültig ist – durch ein Zeichen von derForm „(ξ)“ an. „ξ“ ist eine Variable, deren
Werte die Glieder des Klammerausdruckes
sind; und der Strich über der Variablen
deutet an, dass sie ihre sämtlichen Werte
in der Klammer vertritt.
(Hat also ξ etwa die 3 Werte P, Q, R, so
ist (ξ) = (P, Q, R).)
Die Werte der Variablen werden festgesetzt.
Die Festsetzung ist die Beschreibung
der Sätze, welche die Variable vertritt.

An expression in brackets whose terms
are propositions I indicate—if the order of
the terms in the bracket is indifferent—by
a sign of the form “(ξ)”. “ξ” is a variable
whose values are the terms of the expression in brackets, and the line over the variable indicates that it stands for all its values in the bracket.
(Thus if ξ has the 3 values P, Q, R, then(ξ) = (P, Q, R).)
The values of the variables must be determined.
The determination is the description of
the propositions which the variable standsfor.
How the description of the terms of
the expression in brackets takes place isunessential.
We may distinguish 3 kinds of description: 1. Direct enumeration. In this case
we can place simply its constant values instead of the variable. 2. Giving a function
f x, whose values for all values of x are thepropositions to be described. 3. Giving aformal law, according to which those propositions are constructed. In this case theterms of the expression in brackets are all
the terms of a formal series.

When a bracketed expression has
propositions as its terms—and the orderof the terms inside the brackets is indifferent—then I indicate it by a sign of theform ‘(ξ)’. ‘ξ’ is a variable whose values areterms of the bracketed expression and thebar over the variable indicates that it is
the representative of all its values in thebrackets.
(E.g. if ξ has the three values P, Q, R,
then (ξ) = (P, Q, R). )What the values of the variable are issomething that is stipulated.
The stipulation is a description of the
propositions that have the variable as theirrepresentative.
How the description of the terms of thebracketed expression is produced is not essential.
We can distinguish three kinds of description: 1. direct enumeration, in whichcase we can simply substitute for the variable the constants that are its values; 2.
giving a function f x 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 offorms.
So instead of ‘(− − − − −T) (ξ, . . . . .)’, I
write ‘N(ξ)’.

Wie die Beschreibung der Glieder desKlammerausdruckes geschieht, ist unwesentlich.
Wir k ö n n e n drei Arten der Beschreibung unterscheiden: 1. Die direkte Aufzählung. In diesem Fall können wir statt der
Variablen einfach ihre konstanten Werte
setzen. 2. Die Angabe einer Funktion f x,
deren Werte für alle Werte von x die zu
beschreibenden Sätze sind. 3. Die Angabe eines formalen Gesetzes, nach welchem
jene Sätze gebildet sind. In diesem Falle
sind die Glieder des Klammerausdrucks
sämtliche Glieder einer Formenreihe.
Ich schreibe also statt „(− − − − −W)
ThereforeI(ξ, . . . . .)“ „N(ξ)“.
write instead of “(− − − − −T) (ξ, . . . . .)”,
“N(ξ)”.
N(ξ) ist die Negation sämtlicher Werte
N(ξ) is the negation of all the values ofN(ξ) is the negation of all the values ofder Satzvariablen ξ.
the propositional variable ξ.
the propositional variable ξ.
Da sich offenbar leicht ausdrücken
As it is obviously easy to express how
It is obvious that we can easily express

5.51

5.511

5.512

5.513

lässt, wie mit dieser Operation Sätze gebildet werden können und wie Sätze mit
ihr nicht zu bilden sind, so muss dies aucheinen exakten Ausdruck finden können.
Hat ξ nur einen Wert, so ist N(ξ) = ∼ p(nicht p), hat es zwei Werte, so ist N(ξ) =∼ p . ∼ q (weder p noch q).
Wie kann die allumfassende, weltspiegelnde Logik so spezielle Haken und Manipulationen gebrauchen? Nur, indem sichalle diese zu einem unendlich feinen Netzwerk, zu dem großen Spiegel, verknüpfen.

propositions can be constructed by meansof this operation and how propositions are
not to be constructed by means of it, thismust be capable of exact expression.
If ξ has only one value, then N(ξ) = ∼ p(not p), if it has two values then N(ξ) = ∼ p .
∼ q (neither p nor q).
How can the all-embracing logic which
mirrors the world use such special catches
and manipulations? Only because all these
are connected into an infinitely fine network, to the great mirror.

how propositions may be constructed withthis operation, and how they may not be
constructed with it; so it must be possible
to find an exact expression for this.
If ξ has only one value, then N(ξ) = ∼ p(not p); if it has two values, then N(ξ) =∼ p . ∼ q (neither p nor q).
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, thegreat mirror.
„∼ p“ ist wahr, wenn „ p“ falsch ist. Al“∼ p” is true if “ p” is false. Therefore‘∼ p’ is true if ‘ p’ is false. Therefore, in
so in dem wahren Satz „∼ p“ ist „ p“ ein in the true proposition “∼ p” “ p” is a false the proposition ‘∼ p’, when it is true, ‘ p’ is a
falscher Satz. Wie kann ihn nun der Strich proposition. How then can the stroke “∼” false proposition. How then can the stroke„∼“ mit der Wirklichkeit zum Stimmen bring it into agreement with reality?
‘∼’ make it agree with reality?
bringen?
Das, was in „∼ p“ verneint, ist aberThat which denies in “∼ p” is however
But in ‘∼ p’ it is not ‘∼’ that negates, itnicht das „∼“, sondern dasjenige, was allen not “∼”, but that which all signs of this is rather what is common to all the signs
Zeichen dieser Notation, welche p vernei- notation, which deny p, have in common. of this notation that negate p.
nen, gemeinsam ist.
Also die gemeinsame Regel, nach welHence the common rule according toThat is to say the common rule thatcher „∼ p“, „∼∼∼ p“, „∼ p ∨ ∼ p“, „∼ p . ∼ p“, which “∼ p”, “∼∼∼ p”, “∼ p ∨ ∼ p”, “∼ p . ∼ p”, governs the construction of ‘∼ p’, ‘∼∼∼ p’,
etc. etc. (ad inf.) gebildet werden. Und dies etc. etc. (to infinity) are constructed. And ‘∼ p ∨ ∼ p’, ‘∼ p . ∼ p’, etc. etc. (ad inf.). And
Gemeinsame spiegelt die Verneinung wie- this which is common to them all mirrors this common factor mirrors negation.
der.
denial.
Man könnte sagen: Das Gemeinsame
We could say: What is common to all
We might say that what is common toaller Symbole, die sowohl p als q bejahen, symbols, which assert both p and q, is the all symbols that affirm both p and q is
ist der Satz „ p . q“. Das Gemeinsame aller proposition “ p . q”. What is common to all the proposition ‘ p . q’; and that what is
Symbole, die entweder p oder q bejahen, symbols, which asserts either p or q, is the common to all symbols that affirm either p
ist der Satz „ p ∨ q“.
proposition “ p ∨ q”.
or q is the proposition ‘ p ∨ q’.
Und so kann man sagen: Zwei Sätze
And similarly we can say: Two proposiAnd similarly we can say that twosind einander entgegengesetzt, wenn sie tions are opposed to one another when they propositions are opposed to one anothernichts miteinander gemein haben, und: Je- have nothing in common with one another; if they have nothing in common with one
der Satz hat nur ein Negativ, weil es nur and every proposition has only one nega- another, and that every proposition has
einen Satz gibt, der ganz außerhalb seiner tive, because there is only one proposition only one negative, since there is only one

5.5145.515

5.5151

liegt.
which lies altogether outside it.
Es zeigt sich so auch in Russells NotatiThus in Russell’s notation also it apon, dass „ q : p ∨ ∼ p“ dasselbe sagt wie „ q“; pears evident that “ q : p ∨ ∼ p” says thedass „ p ∨ ∼ p“ nichts sagt.
same thing as “ q”; that “ p ∨ ∼ p” says nothing.
Ist eine Notation festgelegt, so gibt es
If a notation is fixed, there is in it ain ihr eine Regel, nach der alle p vernei- rule according to which all the propositions
nenden Sätze gebildet werden, eine Regel, denying p are constructed, a rule accordnach der alle p bejahenden Sätze gebildet ing to which all the propositions assertwerden, eine Regel, nach der alle p oder q ing p are constructed, a rule according tobejahenden Sätze gebildet werden, u. s. f. which all the propositions asserting p orDiese Regeln sind den Symbolen äquiva- q are constructed, and so on. These rules
lent und in ihnen spiegelt sich ihr Sinn are equivalent to the symbols and in themwieder.
their sense is mirrored.
Es muss sich an unseren Symbolen zeiIt must be recognized in our symbols
gen, dass das, was durch „∨“, „.“, etc. mit- that what is connected by “∨”, “.”, etc.,
einander verbunden ist, Sätze sein müs- must be propositions.
sen.
Und dies ist auch der Fall, denn das
And this is the case, for the symbols “ p”
Symbol „ p“ und „ q“ setzt ja selbst das „∨“, and “ q” presuppose “∨”, “∼”, etc. If the sign„∼“, etc. voraus. Wenn das Zeichen „ p“ in “ p” in “ p ∨ q” does not stand for a complex„ p ∨ q“ nicht für ein komplexes Zeichen sign, then by itself it cannot have sense;
steht, dann kann es allein nicht Sinn ha- but then also the signs “ p ∨ p”, “ p . p”, etc.
ben; dann können aber auch die mit „ p“ which have the same sense as “ p” have no
gleichsinnigen Zeichen „ p ∨ p“, „ p . p“, etc. sense. If, however, “ p ∨ p” has no sense,
keinen Sinn haben. Wenn aber „ p ∨ p“ kei- then also “ p ∨ q” can have no sense.
nen Sinn hat, dann kann auch „ p ∨ q“ keinen Sinn haben.
Muss das Zeichen des negativen Satzes
Must the sign of the negative proposimit dem Zeichen des positiven gebildet wer- tion be constructed by means of the sign ofden? Warum sollte man den negativen Satz the positive? Why should one not be able tonicht durch eine negative Tatsache aus- express the negative proposition by means
drücken können. (Etwa: Wenn „a“ nicht in of a negative fact? (Like: if “a” does not
einer bestimmten Beziehung zu „ b“ steht, stand in a certain relation to “ b”, it could
könnte das ausdrücken, dass aRb nicht express that aRb is not the case.)
der Fall ist.)
Aber auch hier ist ja der negative Satz
But here also the negative proposition

proposition that lies completely outside it.
Thus in Russell’s notation too it is manifest that ‘ q : p ∨ ∼ p’ says the same thingas ‘ q’, that ‘ p ∨ ∼ p’ says nothing.
Once a notation has been established,
there will be in it a rule governing the construction of all propositions that negatep, 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 thatit can only be propositions that are combined with one another by ‘∨’, ‘.’, etc.
And this is indeed the case, since thesymbol in ‘ p’ and ‘ q’ itself presupposes ‘∨’,‘∼’, etc. If the sign ‘ p’ in ‘ p ∨ q’ does notstand for a complex sign, then it cannot
have sense by itself: but in that case thesigns ‘ p ∨ p’, ‘ p . p’, etc., which have thesame sense as p, must also lack sense. But
if ‘ p ∨ p’ has no sense, then ‘ p ∨ 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 meansof 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 nega-indirekt durch den positiven gebildet.

5.52

5.521

5.522

5.523
5.524

5.525

is indirectly constructed with the positive. tive proposition is constructed by an indirect use of the positive.
Der positive S a t z muss die Existenz
The positive proposition must presupThe positive proposition necessarily
des negativen S a t z e s voraussetzen und pose the existence of the negative proposi- presupposes the existence of the negativeumgekehrt.
tion and conversely.
proposition and vice versa.
Sind die Werte von ξ sämtliche Werte
If the values of ξ are the total values
If ξ has as its values all the values of aeiner Funktion f x für alle Werte von x, so of a function f x for all values of x, then function f x for all values of x, then N(ξ) =wird N(ξ) = ∼ (∃ x). f x.
∼ (∃ x). f x.
N(ξ) = ∼ (∃ x). f x.
Ich trenne den Begriff A l l e von der
I separate the concept all from theI dissociate the concept all from truthWahrheitsfunktion.
truth-function.
functions.
Frege und Russell haben die AllgemeinFrege and Russell have introduced genFrege and Russell introduced generalheit in Verbindung mit dem logischen Pro- erality in connexion with the logical prod- ity in association with logical product ordukt oder der logischen Summe eingeführt. uct or the logical sum. Then it would logical sum. This made it difficult to underSo wurde es schwer, die Sätze „(∃ x). f x“ be difficult to understand the propositions stand the propositions ‘(∃ x). f x’ and ‘( x). f x’,
und „( x). f x“, in welchen beide Ideen be- “(∃ x). f x” and “( x). f x” in which both ideas in which both ideas are embedded.
schlossen liegen, zu verstehen.
lie concealed.
Das Eigentümliche der AllgemeinheitsThat which is peculiar to theWhat is peculiar to the generality-signbezeichnung ist erstens, dass sie auf ein “symbolism of generality” is firstly, that it is first, that it indicates a logical prototype,logisches Urbild hinweist, und zweitens, refers to a logical prototype, and secondly, and secondly, that it gives prominence todass sie Konstante hervorhebt.
that it makes constants prominent.
constants.
Die Allgemeinheitsbezeichnung tritt
The generality symbol occurs as an arThe generality-sign occurs as an arguals Argument auf.
gument.
ment.
Wenn die Gegenstände gegeben sind,If the objects are given, therewith are
If objects are given, then at the same
so sind uns damit auch schon a l l e Ge- all objects also given.
time we are given all objects.
genstände gegeben.
Wenn die Elementarsätze gegeben sind,If the elementary propositions are
If elementary propositions are given,so sind damit auch a l l e Elementarsätze given, then therewith all elementary then at the same time all elementarygegeben.
propositions are also given.
propositions are given.
Es ist unrichtig, den Satz „(∃ x). f x“ –It is not correct to render the propoIt is incorrect to render the proposition
wie Russell dies tut – in Worten durch „ f x sition “(∃ x). f x”—as Russell does—in the ‘(∃ x). f x’ in the words, ‘ f x is possible’ as Rusist m ö g l i c h“ wiederzugeben.
words “ f x is possible”.
sell does.
Gewissheit, Möglichkeit oder UnmögCertainty, possibility or impossibility of
The certainty, possibility, or impossilichkeit einer Sachlage wird nicht durch a state of affairs are not expressed by a bility of a situation is not expressed by aeinen Satz ausgedrückt, sondern dadurch, proposition but by the fact that an expres- proposition, but by an expression’s being adass ein Ausdruck eine Tautologie, ein sion is a tautology, a significant proposition tautology, a proposition with a sense, or asinnvoller Satz oder eine Kontradiktion ist. or a contradiction.
contradiction.
Jener Präzedenzfall, auf den man sichThat precedent to which one would alThe precedent to which we are con-

5.5265.52615.5262

5.53

immer berufen möchte, muss schon im ways appeal, must be present in the symbol
Symbol selber liegen.
itself.
Man kann die Welt vollständig durch
One can describe the world completely
vollkommen verallgemeinerte Sätze be- by completely generalized propositions, i.e.
schreiben, das heißt also, ohne irgendeinen without from the outset co-ordinating any
Namen von vornherein einem bestimmten name with a definite object.
Gegenstand zuzuordnen.
Um dann auf die gewöhnliche AusIn order then to arrive at the customary
drucksweise zu kommen, muss man ein- way of expression we need simply say after
fach nach einem Ausdruck: „Es gibt ein an expression “there is one and only one x,
und nur ein x, welches . . . “ sagen: Und which . . . ”: and this x is a.
dies x ist a.
Ein vollkommen verallgemeinerter
A completely generalized proposition
Satz ist, wie jeder andere Satz, zusammen- is like every other proposition composgesetzt. (Dies zeigt sich daran, dass wir in ite. (This is shown by the fact that in„(∃ x, φ). φ x“ „φ“ und „ x“ getrennt erwähnen “(∃ x, φ). φ x” we must mention “φ” and “ x”
müssen. Beide stehen unabhängig in be- separately. Both stand independently inzeichnenden Beziehungen zur Welt, wie im signifying relations to the world as in theunverallgemeinerten Satz.)
ungeneralized proposition.)
Kennzeichen des zusammengesetzten
A characteristic of a composite symbol:
Symbols: Es hat etwas mit a n d e r e n it has something in common with other
Symbolen gemeinsam.
symbols.
Es verändert ja die Wahr- oder FalschThe truth or falsehood of every proposiheit j e d e s Satzes etwas am allgemeinen tion alters something in the general strucBau der Welt. Und der Spielraum, welcher ture of the world. And the range which
ihrem Bau durch die Gesamtheit der Ele- is allowed to its structure by the totality
mentarsätze gelassen wird, ist eben derje- of elementary propositions is exactly thatnige, welchen die ganz allgemeinen Sätze which the completely general propositions
begrenzen.
delimit.
(Wenn ein Elementarsatz wahr ist, so(If an elementary proposition is true,
ist damit doch jedenfalls Ein Elementar- then, at any rate, there is one more elesatz m e h r wahr.)mentary proposition true.)
Gleichheit des Gegenstandes drücke
Identity of the object I express by idenich durch Gleichheit des Zeichens aus, und tity of the sign and not by means of a sign
nicht mit Hilfe eines Gleichheitszeichens. of identity. Difference of the objects by difVerschiedenheit der Gegenstände durch ference of the signs.
Verschiedenheit der Zeichen.

stantly 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 witha particular object.
Then, in order to arrive at the customary mode of expression, we simply need toadd, after an expression like, ‘There is one
and only one x such that . . . ’, the words,‘and that x is a’.
A fully generalized proposition, like every other proposition, is composite. (Thisis shown by the fact that in ‘(∃ x, φ). φ x’ wehave to mention ‘φ’ and ‘ x’ separately. Theyboth, independently, stand in signifying relations to the world, just as is the case inungeneralized propositions.)
It is a mark of a composite symbol thatit has something in common with othersymbols.
The truth or falsity of every proposition
does make some alteration in the general
construction of the world. And the rangethat 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 identityof sign, and not by using a sign for identity.
Difference of objects I express by differenceof signs.

5.53015.53025.5303

5.531

5.532

5.5321

5.533

Dass die Identität keine Relation zwischen Gegenständen ist, leuchtet ein. Dies
wird sehr klar, wenn man z. B. den Satz„( x): f x .⊃. x = a“ betrachtet. Was dieser
Satz sagt, ist einfach, dass n u r a der
Funktion f genügt, und nicht, dass nur
solche Dinge der Funktion f genügen, welche eine gewisse Beziehung zu a haben.
Man könnte nun freilich sagen, dasseben n u r a diese Beziehung zu a habe,aber, um dies auszudrücken, brauchten wir
das Gleichheitszeichen selber.
Russells Definition von „=“ genügt
nicht; weil man nach ihr nicht sagen kann,dass zwei Gegenstände alle Eigenschaften gemeinsam haben. (Selbst wenn dieser
Satz nie richtig ist, hat er doch S i n n.)
Beiläufig gesprochen: Von z w e i Dingen zu sagen, sie seien identisch, ist ein
Unsinn, und von E i n e m zu sagen, es sei
identisch mit sich selbst, sagt gar nichts.
Ich schreibe also nicht „ f (a, b) . a = b“,sondern „ f (a, a)“ (oder „ f ( b, b)“). Und nicht„ f (a, b) . ∼a = b“, sondern „ f (a, b)“.
Und analog: Nicht „(∃ x, y). f ( x, y) . x =y“,sondern„(∃ x). f ( x, x)“;
und nicht „(∃ x, y). f ( x, y) . ∼ x = y“, sondern„(∃ x, y). f ( x, y)“.
(Also statt des Russell’schen „(∃ x, y).
f ( x, y)“: „(∃ x, y). f ( x, y) .∨. (∃ x). f ( x, x)“.)Statt „( x): f x ⊃ x = a“ schreiben wir alsoz. B. „(∃ x). f x .⊃. f a : ∼ (∃ x, y). f x . f y“.

That identity is not a relation between
objects is obvious. This becomes very clear
if, for example, one considers the proposition “( x): f x .⊃. x = a”. What this proposition says is simply that only a satisfies thefunction f , and not that only such things
satisfy the function f which have a certain
relation to a.
One could of course say that in fact only
a has this relation to a, but in order toexpress this we should need the sign ofidentity itself.
Russell’s definition of “=” won’t do; because according to it one cannot say thattwo objects have all their properties in common. (Even if this proposition is never true,
it is nevertheless significant.)
Roughly speaking: to say of two things
that they are identical is nonsense, and tosay of one thing that it is identical withitself is to say nothing.
I write therefore not “ f (a, b) . a = b” but“ f (a, a)” (or “ f ( b, b)”). And not “ f (a, b) .
∼a = b”, but “ f (a, b)”.
And
analogously: not “(∃ x, y). f ( x, y) . x = y”, but“(∃ x). f ( x, x)”; and not “(∃ x, y). f ( x, y) . ∼ x =y”, but “(∃ x, y). f ( x, y)”.
(Therefore instead of Russell’s “(∃ x, y).
f ( x, y)”: “(∃ x, y). f ( x, y) .∨. (∃ x). f ( x, x)”.)Instead of “( x): f x ⊃ x = a” we therefore
write e.g. “(∃ x). f x .⊃. f a : ∼ (∃ x, y). f x . f y”.

It is self-evident that identity is not a
relation between objects. This becomesvery clear if one considers, for example,
the proposition ‘( x): f x .⊃. 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 thefunction f .
Of course, it might then be said that
only a did have this relation to a; but inorder to express that, we should need theidentity-sign itself.
Russell’s definition of ‘=’ is inadequate,
because according to it we cannot say thattwo 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 tosay of one thing that it is identical withitself 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) . ∼a =b’, but ‘ f (a, b)’.
And analogously I do not write‘(∃ x, y). f ( x, y) . x = y’, but ‘(∃ x). f ( x, x)’;and not ‘(∃ x, y). f ( x, y) . ∼ x = y’, but‘(∃ x, y). f ( x, y)’.
(So Russell’s ‘(∃ x, y). f ( x, y)’ becomes‘(∃ x, y). f ( x, y) .∨. (∃ x). f ( x, x)’.)Thus, for example, instead of ‘( x): f x ⊃x = a’ we write ‘(∃ x). f x .⊃. f a : ∼ (∃ x, y). f x .
f y’.
Und der Satz: „n u r Ein x befriedigt
And if the proposition “only one x satisAnd the proposition, ‘Only one x satf ( )“ lautet: „(∃ x). f x : ∼ (∃ x, y). f x . f y“.
fies f ( )” reads: “(∃ x). f x : ∼ (∃ x, y). f x . f y”. isfies f ( )’, will read ‘(∃ x). f x : ∼ (∃ x, y). f x .
f y’.
Das Gleichheitszeichen ist also kein weThe identity sign is therefore not an
The identity-sign, therefore, is not ansentlicher Bestandteil der Begriffsschrift.
5.534

5.535

5.5351essential constituent of logical notation.

Und nun sehen wir, dass Scheinsätze
And we see that the apparent proposiwie: „a = a“, „a = b . b = c .⊃ a = c“, „( x). x = tions like: “a = a”, “a = b . b = c .⊃ a = c”,
x“, „(∃ x). x = a“, etc. sich in einer richtigen “( x). x = x”. “(∃ x). x = a”, etc. cannot be writBegriffsschrift gar nicht hinschreiben las- ten in a correct logical notation at all.
sen.
Damit erledigen sich auch alle ProbleSo all problems disappear which areme, die an solche Scheinsätze geknüpft wa- connected with such pseudo-propositions.
ren.
Alle Probleme, die Russells „Axiom ofThis is the place to solve all the probInfinity“ mit sich bringt, sind schon hier zu lems with arise through Russell’s “Axiom
lösen.
of Infinity”.
Das, was das Axiom of Infinity sagen
What the axiom of infinity is meantsoll, würde sich in der Sprache dadurch to say would be expressed in language byausdrücken, dass es unendlich viele Na- the fact that there is an infinite number ofmen mit verschiedener Bedeutung gäbe.
names with different meanings.
Es gibt gewisse Fälle, wo man in VerThere are certain cases in which one
suchung gerät, Ausdrücke von der Form is tempted to use expressions of the form
„a = a“ oder „ p ⊃ p“ u. dgl. zu benützen. “a = a” or “ p ⊃ p”. As, for instance, when
Und zwar geschieht dies, wenn man von one would speak of the archetype Proposidem Urbild: Satz, Ding, etc. reden möch- tion, Thing, etc. So Russell in the Printe. So hat Russell in den „Principles of ciples of Mathematics has rendered theMathematics“ den Unsinn „ p ist ein Satz“ nonsense “ p is a proposition” in symbols
in Symbolen durch „ p ⊃ p“ wiedergegeben by “ p ⊃ p” and has put it as hypothesis
und als Hypothese vor gewisse Sätze ge- before certain propositions to show thatstellt, damit deren Argumentstellen nur their places for arguments could only be
von Sätzen besetzt werden könnten.
occupied by propositions.
(Es ist schon darum Unsinn, die Hypothese p ⊃ p vor einen Satz zu stellen,um ihm Argumente der richtigen Formzu sichern, weil die Hypothese für einen
Nicht-Satz als Argument nicht falsch, sondern unsinnig wird, und weil der Satzselbst durch die unrichtige Gattung vonArgumenten unsinnig wird, also sich selbst(It is nonsense to place the hypothesis p ⊃ p before a proposition in order toensure that its arguments have the rightform, because the hypotheses for a nonproposition as argument becomes not false
but meaningless, and because the proposition itself becomes senseless for argumentsof the wrong kind, and therefore it sur-essential constituent of conceptual notation.
And now we see that in a correct conceptual notation pseudo-propositions like‘a = a’, ‘a = b . b = c .⊃ a = c’, ‘( x). x = x’,‘(∃ x). x = a’, etc. cannot even be written
down.
This also disposes of all the problems
that were connected with such pseudopropositions.
All the problems that Russell’s ‘axiomof infinity’ brings with it can be solved at
this point.
What the axiom of infinity is intended
to say would express itself in language
through the existence of infinitely many
names with different meanings.
There are certain cases in which one
is tempted to use expressions of the form
‘a = a’ or ‘ p ⊃ p’ and the like. In fact, thishappens 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 ⊃ p’ and placed as an hypothesis in frontof certain propositions in order to exclude
from their argument-places everything but
propositions.
(It is nonsense to place the hypothesis‘ p ⊃ p’ in front of a proposition, in orderto ensure that its arguments shall havethe right form, if only because with a nonproposition as argument the hypothesis becomes not false but nonsensical, and because arguments of the wrong kind makethe proposition itself nonsensical, so that5.5352

5.54

5.541

5.542

5.5421ebenso gut, oder so schlecht, vor den un- vives the wrong arguments no better andrechten Argumenten bewahrt wie die zu no worse than the senseless hypothesis atdiesem Zweck angehängte sinnlose Hypo- tached for this purpose.)
these.)
Ebenso wollte man „Es gibt keine
Similarly it was proposed to express
D i n g e“ ausdrücken durch „∼ (∃ x). x = x“. “There are no things” by “∼ (∃ x). x = x”. But
Aber selbst wenn dies ein Satz wäre – wäre even if this were a proposition—would it
er nicht auch wahr, wenn es zwar „Dinge not be true if indeed “There were things”,
gäbe“, aber diese nicht mit sich selbst iden- but these were not identical with themtisch wären?
selves?
In der allgemeinen Satzform kommt
In the general propositional form,
der Satz im Satze nur als Basis der Wahr- propositions occur in a proposition only asheitsoperationen vor.
bases of the truth-operations.
Auf den ersten Blick scheint es, als könAt first sight it appears as if there werene ein Satz in einem anderen auch auf an- also a different way in which one proposidere Weise vorkommen.
tion could occur in another.
Besonders in gewissen Satzformen der
Especially in certain propositional
Psychologie, wie „A glaubt, dass p der Fall forms of psychology, like “A thinks, that
ist“, oder „A denkt p“, etc.
p is the case”, or “A thinks p”, etc.
Hier scheint es nämlich oberflächlich,
als stünde der Satz p zu einem GegenstandA in einer Art von Relation.
(Und in der modernen Erkenntnistheorie (Russell, Moore, etc.) sind jene Sätze
auch so aufgefasst worden.)Es ist aber klar, dass „A glaubt, dassp“, „A denkt p“, „A sagt p“ von der Form„‚ p‘ sagt p“ sind: Und hier handelt es sichnicht um eine Zuordnung von einer Tatsache und einem Gegenstand, sondern um
die Zuordnung von Tatsachen durch Zuordnung ihrer Gegenstände.
Dies zeigt auch, dass die Seele – das
Subjekt etc. – wie sie in der heutigen oberflächlichen Psychologie aufgefasst wird,
ein Unding ist.

Here it appears superficially as if the
proposition p stood to the object A in a kindof relation.
(And in modern epistemology (Russell,
Moore, etc.) those propositions have been
conceived in this way.)
But it is clear that “A believes that p”,“A thinks p”, “A says p”, are of the form “‘ p’says p”: and here we have no co-ordinationof a fact and an object, but a co-ordinationof facts by means of a co-ordination of theirobjects.

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 ‘∼ (∃ x). x = x’. But even if this were a
proposition, would it not be equally true
if in fact ‘there were things’ but they were
not identical with themselves?
In the general propositional form propositions occur in other propositions only asbases 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 ofproposition in psychology, such as ‘A believes that p is the case’ and A has thethought p’, etc.
For if these are considered superficially,
it looks as if the proposition p stood insome 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 thatp’, ‘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 anobject, but rather the correlation of facts
by means of the correlation of their objects.

This shows that there is no such thing
This shows too that there is no suchas the soul—the subject, etc.—as it is con- thing as the soul—the subject, etc.—as it
ceived in superficial psychology.
is conceived in the superficial psychologyof the present day.

5.54225.5423Eine zusammengesetzte Seele wäre
nämlich keine Seele mehr.
Die richtige Erklärung der Form desSatzes „A urteilt p“ muss zeigen, dass esunmöglich ist, einen Unsinn zu urteilen.
(Russells Theorie genügt dieser Bedingung
nicht.)
Einen Komplex wahrnehmen heißt
wahrnehmen, dass sich seine Bestandteile
so und so zu einander verhalten.
Dies erklärt wohl auch, dass man die
Figur

b

b
a

a

5.55

b

a

b

a

b
a

a
b

b
a

Indeed a composite soul would nolonger 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’stheory 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

b
a

a
b

b
a

A composite soul would not be a soul
any longer.
The correct explanation of the form of
the proposition “A judges p” must showthat it is impossible to judge a nonsense.
(Russell’s theory does not satisfy this condition.)
To perceive a complex means to perceive that its constituents are combinedin such and such a way.
This perhaps explains that the figure

b

b
a

a

auf zweierlei Art als Würfel sehen kann;und alle ähnlichen Erscheinungen. Denn
wir sehen eben wirklich zwei verschiedene
Tatsachen.
(Sehe ich erst auf die Ecken a und nur
flüchtig auf b, so erscheint a vorne; undumgekehrt.)

can be seen in two ways as a cube; and all as a cube; and all similar phenomena. For
similar phenomena. For we really see two we really see two different facts.
different facts.

Wir müssen nun die Frage nach allen
möglichen Formen der Elementarsätze apriori beantworten.
Der Elementarsatz besteht aus Namen.
Da wir aber die Anzahl der Namen vonverschiedener Bedeutung nicht angeben
können, so können wir auch nicht die Zusammensetzung des Elementarsatzes an-

We must now answer a priori the question as to all possible forms of the elementary propositions.
The elementary proposition consists ofnames. Since we cannot give the numberof names with different meanings, we cannot give the composition of the elementary
proposition.

(If I fix my eyes first on the corners aand only glance at b, a appears in front
and b behind, and vice versa.)(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 viceversa).
We now have to answer a priori thequestion about all the possible forms ofelementary propositions.
Elementary propositions consist ofnames. Since, however, we are unable togive the number of names with different
meanings, we are also unable to give thecomposition of elementary propositions.

5.551

5.552

5.5521

5.553

5.554
5.5541geben.
Unser Grundsatz ist, dass jede Frage,
die sich überhaupt durch die Logik entscheiden lässt, sich ohne weiteres entscheiden lassen muss.
(Und wenn wir in die Lage kommen,
ein solches Problem durch Ansehen der
Welt beantworten zu müssen, so zeigt dies,dass wir auf grundfalscher Fährte sind.)
Die „Erfahrung“, die wir zum Verstehen der Logik brauchen, ist nicht die, dasssich etwas so und so verhält, sondern, dassetwas i s t: aber das ist eben k e i n e Erfahrung.
Die Logik ist v o r jeder Erfahrung –dass etwas s o ist.
Sie ist vor dem Wie, nicht vor dem Was.
Und wenn dies nicht so wäre, wie könnten wir die Logik anwenden? Man könnte
sagen: Wenn es eine Logik gäbe, auch wennes keine Welt gäbe, wie könnte es danneine Logik geben, da es eine Welt gibt.
Russell sagte, es gäbe einfache Relationen zwischen verschiedenen Anzahlen
von Dingen (Individuals). Aber zwischen
welchen Anzahlen? Und wie soll sich das
entscheiden? – Durch die Erfahrung?
(Eine ausgezeichnete Zahl gibt es
nicht.)Die Angabe jeder speziellen Form wäre
vollkommen willkürlich.
Es soll sich a priori angeben lassen, ob
ich z. B. in die Lage kommen kann, etwas
mit dem Zeichen einer 27-stelligen Relation bezeichnen zu müssen.

Our fundamental principle is that evOur fundamental principle is that
ery question which can be decided at all by whenever a question can be decided bylogic can be decided off-hand.
logic at all it must be possible to decide
it without more ado.
(And if we get into a situation where we(And if we get into a position where weneed to answer such a problem by looking have to look at the world for an answer toat the world, this shows that we are on a such a problem, that shows that we are on
fundamentally wrong track.)
a completely wrong track.)
The “experience” which we need to unThe ‘experience’ that we need in orderderstand logic is not that such and such is to understand logic is not that something
the case, but that something is; but that is or other is the state of things, but thatno experience.
something is: that, however, is not an experience.
Logic precedes every experience—that
Logic is prior to every experience—that
something is so.
something is so.
It is before the How, not before the
It is prior to the question ‘How?’, notWhat.
prior to the question ‘What?’
And if this were not the case, how could
And if this were not so, how could we
we apply logic? We could say: if there were apply logic? We might put it in this way: if
a logic, even if there were no world, how there would be a logic even if there werethen could there be a logic, since there is a no world, how then could there be a logic
world?
given that there is a world?
Russell said that there were simple
Russell said that there were simple
relations between different numbers of relations between different numbers ofthings (individuals). But between what things (individuals). But between what
numbers? And how should this be de- numbers? And how is this supposed to be
cided—by experience?
decided?—By experience?
(There is no pre-eminent number.)(There is no privileged number.)
The enumeration of any special forms
would be entirely arbitrary.
How could we decide a priori whether,for example, I can get into a situation inwhich I need to symbolize with a sign of a27-termed relation?

It would be completely arbitrary to give
any specific form.
It is supposed to be possible to answer
a priori the question whether I can get
into a position in which I need the signfor a 27-termed relation in order to signify
something.

5.5542

5.555

5.556

5.55615.55625.5563Dürfen wir denn aber überhaupt so fraMay we then ask this at all? Can wegen? Können wir eine Zeichenform aufstel- set out a sign form and not know whether
len und nicht wissen, ob ihr etwas entspre- anything can correspond to it?
chen könne?
Hat die Frage einen Sinn: Was muss
Has the question sense: what musts e i n, damit etwas der-Fall-sein kann?
there be in order that anything can be thecase?
Es ist klar, wir haben vom ElementarIt is clear that we have a concept ofsatz einen Begriff, abgesehen von seiner the elementary proposition apart from its
besonderen logischen Form.
special logical form.
Wo man aber Symbole nach einem SyWhere, however, we can build symbols
stem bilden kann, dort ist dieses System according to a system, there this system is
das logisch wichtige und nicht die einzel- the logically important thing and not thenen Symbole.
single symbols.
Und wie wäre es auch möglich, dass ich
And how would it be possible that I
es in der Logik mit Formen zu tun hätte, should have to deal with forms in logic
die ich erfinden kann; sondern mit dem which I can invent: but I must have tomuss ich es zu tun haben, was es mir mög- deal with that which makes it possible forlich macht, sie zu erfinden.
me to invent them.
Eine Hierarchie der Formen der EleThere cannot be a hierarchy of thementarsätze kann es nicht geben. Nur was forms of the elementary propositions. Only
wir selbst konstruieren, können wir vor- that which we ourselves construct can we
aussehen.
foresee.
Die empirische Realität ist begrenzt
Empirical reality is limited by the totaldurch die Gesamtheit der Gegenstände. ity of objects. The boundary appears again
Die Grenze zeigt sich wieder in der Ge- in the totality of elementary propositions.
samtheit der Elementarsätze.
Die Hierarchien sind, und müssen unThe hierarchies are and must be indeabhängig von der Realität sein.
pendent of reality.
Wissen wir aus rein logischen Gründen,
If we know on purely logical grounds,
dass es Elementarsätze geben muss, dann that there must be elementary proposimuss es jeder wissen, der die Sätze in ihrer tions, then this must be known by everyone
unanalysierten Form versteht.
who understands propositions in their unanalysed form.
Alle Sätze unserer Umgangssprache
All propositions of our colloquial lansind tatsächlich, so wie sie sind, logisch guage are actually, just as they are, logvollkommen geordnet. – Jenes Einfachste, ically completely in order. That simple

But is it really legitimate even to ask
such a question? Can we set up a form ofsign without knowing whether anything
can correspond to it?
Does it make sense to ask what there
must be in order that something can be thecase?
Clearly we have some concept of elementary propositions quite apart fromtheir 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 thatin logic I should have to deal with forms
that I can invent? What I have to deal withmust be that which makes it possible forme to invent them.
There cannot be a hierarchy of theforms of elementary propositions. We canforesee only what we ourselves construct.
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 unanalyzed 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

5.5575.5571

5.6
5.61

was wir hier angeben sollen, ist nicht ein
Gleichnis der Wahrheit, sondern die volle
Wahrheit selbst.
(Unsere Probleme sind nicht abstrakt,sondern vielleicht die konkretesten, die es
gibt.)Die A n w e n d u n g der Logik entscheidet darüber, welche Elementarsätze
es gibt.
Was in der Anwendung liegt, kann die
Logik nicht vorausnehmen.
Das ist klar: Die Logik darf mit ihrer
Anwendung nicht kollidieren.
Aber die Logik muss sich mit ihrer Anwendung berühren.
Also dürfen die Logik und ihre Anwendung einander nicht übergreifen.
Wenn ich die Elementarsätze nicht apriori angeben kann, dann muss es zu offenbarem Unsinn führen, sie angeben zu
wollen.
Die Grenzen meiner Sprac h e bedeuten die Grenzen meiner Welt.
Die Logik erfüllt die Welt; die Grenzen
der Welt sind auch ihre Grenzen.
Wir können also in der Logik nicht sagen: Das und das gibt es in der Welt, jenes
nicht.
Das würde nämlich scheinbar voraussetzen, dass wir gewisse Möglichkeiten
ausschließen, und dies kann nicht der Fallsein, da sonst die Logik über die Grenzen
der Welt hinaus müsste; wenn sie nämlichdiese Grenzen auch von der anderen Seite
betrachten könnte.
Was wir nicht denken können, das können wir nicht denken; wir können alsothing which we ought to give here is not amodel of the truth but the complete truth
itself.
(Our problems are not abstract but perhaps the most concrete that there are.)thing, which we have to formulate here, is
not an image 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
The application of logic decides what
elementary propositions there are.
elementary propositions there are.
What lies in its application logic cannot
anticipate.
It is clear that logic may not conflictwith its application.
But logic must have contact with its
application.
Therefore logic and its application may
not overlap one another.
If I cannot give elementary propositions
a priori then it must lead to obvious nonsense to try to give them.

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 todo so must lead to obvious nonsense.

The limits of my language mean thelimits of my world.
Logic fills the world: the limits of theworld are also its limits.
We cannot therefore say in logic: Thisand this there is in the world, that there isnot.
For that would apparently presupposethat we exclude certain possibilities, andthis cannot be the case since otherwise
logic must get outside the limits of theworld: that is, if it could consider theselimits from the other side also.

The limits of my language mean thelimits 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 presupposethat we were excluding certain possibilities, and this cannot be the case, since itwould require that logic should go beyond
the limits of the world; for only in that way
could it view those limits from the otherside as well.
What we cannot think, that we cannot
We cannot think what we cannot think;
think: we cannot therefore say what we so what we cannot think we cannot say

5.62

5.621
5.63
5.631

5.632
5.633

5.6331auch nicht s a g e n, was wir nicht denken
können.
Diese Bemerkung gibt den Schlüssel
zur Entscheidung der Frage, inwieweit der
Solipsismus eine Wahrheit ist.
Was der Solipsismus nämlich m e i n t,
ist ganz richtig, nur lässt es sich nicht s a g e n, sondern es zeigt sich.
Dass die Welt m e i n e Welt ist, daszeigt sich darin, dass die Grenzen d e r
Sprache (der Sprache, die allein ich verstehe) die Grenzen m e i n e r Welt bedeuten.
Die Welt und das Leben sind Eins.
Ich bin meine Welt. (Der Mikrokosmos.)Das denkende, vorstellende, Subjekt
gibt es nicht.
Wenn ich ein Buch schriebe „Die Welt,
wie ich sie vorfand“, so wäre darin auchüber meinen Leib zu berichten und zu sagen, welche Glieder meinem Willen unterstehen und welche nicht, etc., dies ist nämlich eine Methode, das Subjekt zu isolieren, oder vielmehr zu zeigen, dass es ineinem wichtigen Sinne kein Subjekt gibt:
Von ihm allein nämlich könnte in diesem
Buche n i c h t die Rede sein.—
Das Subjekt gehört nicht zur Welt, sondern es ist eine Grenze der Welt.
Wo i n der Welt ist ein metaphysisches
Subjekt zu merken?
Du sagst, es verhält sich hier ganz wie
mit Auge und Gesichtsfeld. Aber das Augesiehst du wirklich n i c h t.
Und nichts a m G e s i c h t s f e l dlässt darauf schließen, dass es von einem
Auge gesehen wird.
Das Gesichtsfeld hat nämlich nicht et-cannot think.

either.

This remark provides a key to the quesThis remark provides the key to thetion, to what extent solipsism is a truth.
problem, how much truth there is in solipsism.
In fact what solipsism means, is quite
For what the solipsist means is quite
correct, only it cannot be said, but it shows correct; only it cannot be said, but makes
itself.
itself manifest.
That the world is my world, shows itself
The world is my world: this is manifest
in the fact that the limits of the language in the fact that the limits of language (of(the language which I understand) mean that language which alone I understand)the limits of my world.
mean the limits of my world.
The world and life are one.
The world and life are one.
I am my world. (The microcosm.)
I am my world. (The microcosm.)
The thinking, presenting subject; there
There is no such thing as the subject
is no such thing.
that thinks or entertains ideas.
If I wrote a book “The world as I foundIf I wrote a book called The World as Iit”, I should also have therein to report found it, I should have to include a report
on my body and say which members obey on my body, and should have to say which
my will and which do not, etc. This then parts were subordinate to my will, andwould be a method of isolating the subject which were not, etc., this being a methodor rather of showing that in an important of isolating the subject, or rather of showsense there is no subject: that is to say, of ing that in an important sense there is no
it alone in this book mention could not be subject; for it alone could not be mentioned
made.
in that book.—
The subject does not belong to the world
but it is a limit of the world.
Where in the world is a metaphysicalsubject to be noted?
You say that this case is altogether likethat of the eye and the field of sight. But
you do not really see the eye.
And from nothing in the field of sight
can it be concluded that it is seen from aneye.
For the field of sight has not a form like

The subject does not belong to theworld: rather, it is a limit of the world.
Where in the world is a metaphysicalsubject to be found?
You will say that this is exactly like thecase 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

wa eine solche Form:

Auge —
5.634

5.64

5.641

6

6.001

Das hängt damit zusammen, dass kein
Teil unserer Erfahrung auch a priori ist.

this:

not like thisEye —This is connected with the fact that nopart of our experience is also a priori.

Eye —This is connected with the fact that nopart of our experience is at the same timea priori.
Alles, was wir sehen, könnte auch anEverything we see could also be otherWhatever we see could be other than itders sein.
wise.
is.
Alles, was wir überhaupt beschreiben
Everything we describe at all could also
Whatever we can describe at all could
können, könnte auch anders sein.
be otherwise.
be other than it is.
Es gibt keine Ordnung der Dinge a prioThere is no order of things a priori.
There is no a priori order of things.
ri.
Hier sieht man, dass der Solipsismus,
Here we see that solipsism strictly carHere it can be seen that solipsism,
streng durchgeführt, mit dem reinen Rea- ried out coincides with pure realism. The I when its implications are followed out
lismus zusammenfällt. Das Ich des Solip- in solipsism shrinks to an extensionless strictly, coincides with pure realism. The
sismus schrumpft zum ausdehnungslosen point and there remains the reality co- self of solipsism shrinks to a point without
Punkt zusammen, und es bleibt die ihm ordinated with it.
extension, and there remains the reality
koordinierte Realität.
co-ordinated with it.
Es gibt also wirklich einen Sinn, in
There is therefore really a sense inThus there really is a sense in whichwelchem in der Philosophie nichtpsycho- which the philosophy we can talk of a non- philosophy can talk about the self in a nonlogisch vom Ich die Rede sein kann.
psychological I.
psychological way.
Das Ich tritt in die Philosophie dadurch
The I occurs in philosophy through theWhat brings the self into philosophy isein, dass „die Welt meine Welt ist“.
fact that the “world is my world”.
the fact that ‘the world is my world’.
Das philosophische Ich ist nicht der
The philosophical I is not the man, not
The philosophical self is not the huMensch, nicht der menschliche Körper, the human body or the human soul of man being, not the human body, or theoder die menschliche Seele, von der die which psychology treats, but the metaphys- human soul, with which psychology deals,
Psychologie handelt, sondern das metaphy- ical subject, the limit—not a part of the but rather the metaphysical subject, thesische Subjekt, die Grenze –nicht ein Teil world.
limit of the world—not a part of it.
– der Welt.
Die allgemeine Form der WahrheitsThe general form of truth-function is:
The general form of a truth-function is
funktion ist: [ p, ξ, N(ξ)].
[ p, ξ, N(ξ)].
[ p, ξ, N(ξ)].
Dies ist die allgemeine Form des SatThis is the general form of proposition.
This is the general form of a proposizes.
tion.
Dies sagt nichts anderes, als dass jederThis says nothing else than that every
What this says is just that every propo-

6.002

6.01

6.02

Satz ein Resultat der successiven Anwendung der Operation N ’(ξ) auf die Elementarsätze ist.
Ist die allgemeine Form gegeben, wie
ein Satz gebaut ist, so ist damit auch schon
die allgemeine Form davon gegeben, wie
aus einem Satz durch eine Operation ein
anderer erzeugt werden kann.

proposition is the result of successive applications of the operation N ’(ξ) to the elementary propositions.
If we are given the general form of theway in which a proposition is constructed,
then thereby we are also given the generalform of the way in which by an operationout of one proposition another can be created.
Die allgemeine Form der Operation
The general form of the operation Ω ’(η)
Ω ’(η) ist also: [ξ, N(ξ)]’(η) (= [η, ξ, N(ξ)]).
is therefore: [ξ, N(ξ)]’(η) (= [η, ξ, N(ξ)]).
Das ist die allgemeinste Form des ÜberThis is the most general form of transiganges von einem Satz zum anderen.
tion from one proposition to another.
Und s o kommen wir zu den Zahlen:
And thus we come to numbers: I define
Ich definiere

x = Ω0 ’ x Def. und

x = Ω0 ’ x Def. andsition is a result of successive applications
to elementary propositions of the operation
N(ξ).
If we are given the general form according to which propositions are constructed,
then with it we are also given the generalform according to which one proposition
can be generated out of another by meansof an operation.
Therefore the general form of an operation Ω ’(η) is [ξ, N(ξ)]’(η) (= [η, ξ, N(ξ)]).
This is the most general form of transition from one proposition to another.
And this is how we arrive at numbers.
I give the following definitionsx = Ω0 ’ x Def.

Ω ’ Ων ’ x = Ων+1 ’ x Def.
Ω ’ Ων ’ x = Ων+1 ’ x Def.
Ω ’ Ων ’ x = Ων+1 ’ x Def.
Nach diesen Zeichenregeln schreiben
According, then, to these symbolic rules
So, in accordance with these rules,
wir also die Reihe
we write the series
which deal with signs, we write the seriesx, Ω ’ x, Ω ’ Ω ’ x, Ω ’ Ω ’ Ω ’ x, . . . . .
so:x, Ω ’ x, Ω ’ Ω ’ x, Ω ’ Ω ’ Ω ’ x, . . . . .
as:x, Ω ’ x, Ω ’ Ω ’ x, Ω ’ Ω ’ Ω ’ x, . . . . .
in the following wayΩ0 ’ x, Ω0+1 ’ x, Ω0+1+1 ’ x, Ω0+1+1+1 ’ x, . . . . .
Also schreibe ich statt „[ x, ξ, Ω ’ ξ]“:Ω0 ’ x, Ω0+1 ’ x, Ω0+1+1 ’ x, Ω0+1+1+1 ’ x, . . . . .
Therefore I write in place of “[ x, ξ,Ω ’ ξ]”,Ω0 ’ x, Ω0+1 ’ x, Ω0+1+1 ’ x, Ω0+1+1+1 ’ x, . . . . .
Therefore, instead of ‘[ x, ξ, Ω ’ ξ]’,„[Ω0 ’ x, Ων ’ x, Ων+1 ’ x]“.
Und definiere:
0 + 1 = 1 Def.

6.021
6.022

“[Ω0 ’ x, Ων ’ x, Ων+1 ’ x]”.
And I define:0 + 1 = 1 Def.

0 + 1 + 1 = 2 Def.

0 + 1 + 1 = 2 Def.

0 + 1 + 1 + 1 = 3 Def.

0 + 1 + 1 + 1 = 3 Def.

I write‘[Ω0 ’ x, Ων ’ x, Ων+1 ’ x]’.
And I give the following definitions
0 + 1 = 1 Def.,
0 + 1 + 1 = 2 Def.,
0 + 1 + 1 + 1 = 3 Def.,(u. s. f.)and so on.
(and so on).
Die Zahl ist der Exponent einer OperaA number is the exponent of an operaA number is the exponent of an operation.
tion.
tion.
Der Zahlbegriff ist nichts anderes als
The concept number is nothing else
The concept of number is simply what
das Gemeinsame aller Zahlen, die allge- than that which is common to all numbers, is common to all numbers, the general form

meine Form der Zahl.
Der Zahlbegriff ist die variable Zahl.

6.03
6.031

6.1

Und der Begriff der Zahlengleichheit
ist die allgemeine Form aller speziellen
Zahlengleichheiten.
Die allgemeine Form der ganzen Zahl
ist: [0, ξ, ξ + 1].
Die Theorie der Klassen ist in der Mathematik ganz überflüssig.
Dies hängt damit zusammen, dass die
Allgemeinheit, welche wir in der Mathematik brauchen, nicht die z u f ä l l i g e
ist.
Die Sätze der Logik sind Tautologien.

6.11

Die Sätze der Logik sagen also Nichts.
(Sie sind die analytischen Sätze.)

6.111

Theorien, die einen Satz der Logik gehaltvoll erscheinen lassen, sind immer
falsch. Man könnte z. B. glauben, dass die
Worte „wahr“ und „falsch“ zwei Eigenschaften unter anderen Eigenschaften bezeichnen, und da erschiene es als eine merkwürdige Tatsache, dass jeder Satz eine dieser Eigenschaften besitzt. Das scheint nun
nichts weniger als selbstverständlich zu
sein, ebensowenig selbstverständlich, wie
etwa der Satz, „alle Rosen sind entweder
gelb oder rot“ klänge, auch wenn er wahr
wäre. Ja, jener Satz bekommt nun ganz
den Charakter eines naturwissenschaftlichen Satzes, und dies ist das sichere Anzeichen dafür, dass er falsch aufgefasst wurde.
Die richtige Erklärung der logischen

6.112

the general form of a number.
The concept number is the variable
number.
And the concept of equality of numbers
is the general form of all special equalities
of numbers.
The general form of the cardinal number is: [0, ξ, ξ + 1].
The theory of classes is altogether superfluous in mathematics.
This is connected with the fact that thegenerality which we need in mathematics
is not the accidental one.

of a number.
The concept of number is the variable
number.
And the concept of numerical equality
is the general form of all particular casesof numerical equality.
The general form of an integer is[0, ξ, ξ + 1].
The theory of classes is completely superfluous in mathematics.
This is connected with the fact that thegenerality required in mathematics is notaccidental generality.

The propositions of logic are tautologies.
The propositions of logic therefore say
nothing. (They are the analytical propositions.)
Theories which make a proposition oflogic appear substantial are always false.
One could e.g. believe that the words “true”
and “false” signify two properties among
other properties, and then it would appear
as a remarkable fact that every proposition
possesses one of these properties. This now
by no means appears self-evident, no moreso than the proposition “All roses are either
yellow or red” would seem even if it were
true. Indeed our proposition now gets quite
the character of a proposition of naturalscience and this is a certain symptom of its
being falsely understood.

The propositions of logic are tautologies.
Therefore the propositions of logic say
nothing. (They are the analytic propositions.)
All theories that make a propositionof logic appear to have content are false.
One might think, for example, that thewords ‘true’ and ‘false’ signified two properties among other properties, and then it
would seem to be a remarkable fact thatevery proposition possessed one of theseproperties. 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 itwere true. Indeed, the logical propositionacquires all the characteristics of a proposition of natural science and this is the suresign that it has been construed wrongly.

The correct explanation of logical propo-

The correct explanation of the proposi-6.1136.12

6.12016.1202

Sätze muss ihnen eine einzigartige Stellung unter allen Sätzen geben.
Es ist das besondere Merkmal der logischen Sätze, dass man am Symbol alleinerkennen kann, dass sie wahr sind, unddiese Tatsache schließt die ganze Philosophie der Logik in sich. Und so ist es aucheine der wichtigsten Tatsachen, dass sich
die Wahrheit oder Falschheit der nichtlogischen Sätze n i c h t am Satz allein erkennen lässt.
Dass die Sätze der Logik Tautologien
sind, das z e i g t die formalen – logischen– Eigenschaften der Sprache, der Welt.

sitions must give them a peculiar position
among all propositions.
It is the characteristic mark of logical
propositions that one can perceive in thesymbol alone that they are true; and thisfact contains in itself the whole philosophyof logic. And so also it is one of the mostimportant facts that the truth or falsehoodof non-logical propositions can not be recognized from the propositions alone.

tions of logic must assign to them a uniquestatus among all propositions.
It is the peculiar mark of logical propositions that one can recognize that they
are true from the symbol alone, and thisfact contains in itself the whole philosophyof 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 oflogic are tautologies shows the formal—logical—properties of language, of
the world.
Dass ihre Bestandteile s o verknüpft
That its constituent parts connected toeine Tautologie ergeben, das charakteri- gether in this way give a tautology characsiert die Logik ihrer Bestandteile.
terizes the logic of its constituent parts.

Damit Sätze, auf bestimmte Art und
Weise verknüpft, eine Tautologie ergeben,
dazu müssen sie bestimmte Eigenschaften der Struktur haben. Dass sie s o verbunden eine Tautologie ergeben, zeigt also,
dass sie diese Eigenschaften der Struktur
besitzen.
Dass z. B. die Sätze „ p“ und „∼ p“ inder Verbindung „∼( p . ∼ p)“ eine Tautologie
ergeben, zeigt, dass sie einander widersprechen. Dass die Sätze „ p ⊃ q“, „ p“ und „ q“
in der Form „( p ⊃ q) . ( p) :⊃: ( q)“ miteinander verbunden eine Tautologie ergeben,
zeigt, dass q aus p und p ⊃ q folgt. Dass„( x). f x :⊃: f a“ eine Tautologie ist, dass f a
aus ( x). f x folgt. etc. etc.

In order that propositions connected together in a definite way may give a tautology they must have definite propertiesof structure. That they give a tautology
when so connected shows therefore thatthey possess these properties of structure.

The fact that the propositions oflogic are tautologies shows the formal—logical—properties of language andthe world.
The fact that a tautology is yielded bythis 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 whencombined in this way shows that they possess these structural properties.

Es ist klar, dass man zu demselben

That e.g. the propositions “ p” and “∼ p”
in the connexion “∼( p . ∼ p)” give a tautology shows that they contradict one another. That the propositions “ p ⊃ q”, “ p”
and “ q” connected together in the form“( p ⊃ q) . ( p) :⊃: ( q)” give a tautology showsthat q follows from p and p ⊃ q. That“( x). f x :⊃: f a” is a tautology shows that f afollows from ( x). f x, etc. etc.

For example, the fact that the propositions ‘ p’ and ‘∼ p’ in the combination‘∼( p . ∼ p)’ yield a tautology shows thatthey contradict one another. The fact thatthe propositions ‘ p ⊃ q’, ‘ p’, and ‘ q’, combined with one another in the form ‘( p ⊃q) . ( p) :⊃: ( q)’, yield a tautology shows thatq follows from p and p ⊃ q. The fact that‘( x). f x :⊃: f a’ is a tautology shows that f afollows from ( x). f x. Etc. etc.
It is clear that we could have used for
It is clear that one could achieve the6.1203

Zweck statt der Tautologien auch die Kontradiktionen verwenden könnte.
Um eine Tautologie als solche zu erkennen, kann man sich, in den Fällen, in
welchen in der Tautologie keine Allgemeinheitsbezeichnung vorkommt, folgender anschaulichen Methode bedienen: Ich schreibe statt „ p“, „ q“, „ r “ etc. „W p F“, „W q F“,„W r F“ etc. Die Wahrheitskombinationen
drücke ich durch Klammern aus, z. B.:

W p F

W q F

this purpose contradictions instead of tautologies.
In order to recognize a tautology as
such, we can, in cases in which no signof generality occurs in the tautology, makeuse of the following intuitive method: I
write instead of “ p”, “ q”, “ r ”, etc., “T p F”,“T q F”, “T r F”, etc. The truth-combinations
I express by brackets, e.g.:

T p F

T q F

same purpose by using contradictions instead of tautologies.
In order to recognize an expression asa tautology, in cases where no generalitysign occurs in it, one can employ the following intuitive method: instead of ‘ p’, ‘ q’, ‘ r ’,etc. I write ‘T p F’, ‘T q F’, ‘T r F’, etc. Truthcombinations I express by means of brackets, e.g.

T p F

T q F,

und die Zuordnung der Wahr- oder Falschheit des ganzen Satzes und der Wahrheitskombinationen der Wahrheitsargumente
durch Striche auf folgende Weise:

and the co-ordination of the truth or falsityof the whole proposition with the truthcombinations of the truth-arguments bylines in the following way:and I use lines to express the correlation of
the truth or falsity of the whole proposition
with the truth-combinations of its trutharguments, in the following way

F

F

F

W p F

W q F

W
Dies Zeichen würde also z. B. den Satzp ⊃ q darstellen. Nun will ich z. B. den Satz∼( p . ∼ p) (Gesetz des Widerspruchs) daraufhin untersuchen, ob er eine Tautologie
ist. Die Form „∼ξ“ wird in unserer Notation

T p F

T q F

T
This sign, for example, would therefore
present the proposition p ⊃ q. Now I willproceed to inquire whether such a proposition as ∼( p . ∼ p) (The Law of Contradiction) is a tautology. The form “∼ξ” is written in our notation

T p F

T q F.

T
So this sign, for instance, would represent
the proposition p ⊃ q. Now, by way of example, I wish to examine the proposition∼( p . ∼ p) (the law of contradiction) in orderto determine whether it is a tautology. In
our notation the form ‘∼ξ’ is written as

W

T

T

„W ξ F““T ξ F”‘T ξ F’,F

F

F

geschrieben; die Form „ξ . η“ so:the form “ξ . η” thus:—W

and the form ‘ξ . η’ as

T

W ξ F

W η F

T

T ξ F

T η F

F

T η F.

F

Daher lautet der Satz ∼( p . ∼ q) so:F

Hence the proposition ∼( p . ∼ q) runs Hence the proposition ∼( p . ∼ q) reads as
thus:—
follows

F

F

W

F

T

W

T

T

W q F

T ξ F

W p F

F

T

T q F

T p F

F
F

Setzen wir hier statt „ q“ „ p“ ein und untersuchen die Verbindung der äußersten
W und F mit den innersten, so ergibt
sich, dass die Wahrheit des ganzen Satzes
a l l e n Wahrheitskombinationen seines
Argumentes, seine Falschheit keiner der

T p F.

F
F

W

T q F

F
T

If here we put “ p” instead of “ q” and examine the combination of the outermost T
and F with the innermost, it is seen thatthe truth of the whole proposition is coordinated with all the truth-combinationsof its argument, its falsity with none of theT
If we here substitute ‘ p’ for ‘ q’ and examine
how the outermost T and F are connectedwith the innermost ones, the result will bethat the truth of the whole proposition iscorrelated with all the truth-combinationsof its argument, and its falsity with none

6.121

6.122

6.1221

6.1222

6.1223Wahrheitskombinationen zugeordnet ist.
Die Sätze der Logik demonstrieren die
logischen Eigenschaften der Sätze, indem
sie sie zu nichtssagenden Sätzen verbinden.
Diese Methode könnte man auch eine
Nullmethode nennen. Im logischen Satzwerden Sätze miteinander ins Gleichgewicht gebracht und der Zustand des Gleichgewichts zeigt dann an, wie diese Sätze
logisch beschaffen sein müssen.
Daraus ergibt sich, dass wir auch ohne
die logischen Sätze auskommen können,
da wir ja in einer entsprechenden Notation die formalen Eigenschaften der Sätze
durch das bloße Ansehen dieser Sätze erkennen können.
Ergeben z. B. zwei Sätze „ p“ und „ q“ inder Verbindung „ p ⊃ q“ eine Tautologie, so
ist klar, dass q aus p folgt.

truth-combinations.
The propositions of logic demonstrate
the logical properties of propositions, by
combining them into propositions which
say nothing.
This method could be called a zeromethod. In a logical proposition propositions are brought into equilibrium with one
another, and the state of equilibrium thenshows how these propositions must be logically constructed.
Whence it follows that we can get on
without logical propositions, for we can recognize in an adequate notation the formal
properties of the propositions by mere inspection.
If for example two propositions “ p” and“ q” give a tautology in the connexion “ p ⊃q”, then it is clear that q follows from p.

Dass z. B. „ q“ aus „ p ⊃ q . p“ folgt, ersehen wir aus diesen beiden Sätzen selbst,
aber wir können es auch s o zeigen, indem
wir sie zu „ p ⊃ q . p :⊃: q“ verbinden und
nun zeigen, dass dies eine Tautologie ist.

E.g. that “ q” follows from “ p ⊃ q . p”
we see from these two propositions themselves, but we can also show it by combining them to “ p ⊃ q . p :⊃: q” and thenshowing that this is a tautology.

Dies wirft ein Licht auf die Frage,
warum die logischen Sätze nicht durch die
Erfahrung bestätigt werden können, ebenso wenig, wie sie durch die Erfahrung widerlegt werden können. Nicht nur mussein Satz der Logik durch keine mögliche
Erfahrung widerlegt werden können, sondern er darf auch nicht durch eine solche
bestätigt werden können.
Nun wird klar, warum man oft fühlte,

This throws light on the question why
logical propositions can no more be empirically confirmed than they can be empirically refuted. Not only must a propositionof logic be incapable of being contradictedby any possible experience, but it mustalso be incapable of being confirmed by anysuch.
It now becomes clear why we often

of the truth-combinations.
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 zeromethod. In a logical proposition, propositions are brought into equilibrium with one
another, and the state of equilibrium thenindicates what the logical constitution ofthese propositions must be.
It follows from this that we can actually do without logical propositions; for ina suitable notation we can in fact recognize
the formal properties of propositions bymere inspection of the propositions themselves.
If, for example, two propositions ‘ p’ and‘ q’ in the combination ‘ p ⊃ q’ yield a tautology, then it is clear that q follows fromp.
For example, we see from the two propositions themselves that ‘ q’ follows from‘ p ⊃ q . p’, but it is also possible to show
it in this way: we combine them to form
‘ p ⊃ q . p :⊃: q’, and then show that this isa tautology.
This throws some light on the question why logical propositions cannot be
confirmed by experience any more thanthey can be refuted by it. Not only musta proposition of logic be irrefutable by any
possible experience, but it must also be
unconfirmable by any possible experience.

Now it becomes clear why people have

6.1224

6.123

6.1231

6.1232

6.1233

6.124

als wären die „logischen Wahrheiten“ von feel as though “logical truths” must be
uns zu „f o r d e r n“: Wir können sie näm- “postulated” by us. We can in fact postulich insofern fordern, als wir eine genügen- late them in so far as we can postulate ande Notation fordern können.
adequate notation.
Es wird jetzt auch klar, warum die LoIt also becomes clear why logic has beengik die Lehre von den Formen und vom called the theory of forms and of inference.
Schließen genannt wurde.
Es ist klar: Die logischen Gesetze dürIt is clear that the laws of logic cannot
fen nicht selbst wieder logischen Gesetzen themselves obey further logical laws.
unterstehen.
(Es gibt nicht, wie Russell meinte, für(There is not, as Russell supposed, forjede „Type“ ein eigenes Gesetz des Wider- every “type” a special law of contradiction;
spruches, sondern Eines genügt, da es auf but one is sufficient, since it is not appliedsich selbst nicht angewendet wird.)
to itself.)
Das Anzeichen des logischen Satzes ist
The mark of logical propositions is not
n i c h t die Allgemeingültigkeit.
their general validity.
Allgemein sein, heißt ja nur: ZufälliTo be general is only to be accidengerweise für alle Dinge gelten. Ein unver- tally valid for all things. An ungeneralized
allgemeinerter Satz kann ja ebensowohl proposition can be tautologous just as welltautologisch sein als ein verallgemeinerter. as a generalized one.
Die logische Allgemeingültigkeit könnLogical general validity, we could callte man wesentlich nennen, im Gegensatz essential as opposed to accidental generalzu jener zufälligen, etwa des Satzes: „alle validity, e.g. of the proposition “all men are
Menschen sind sterblich“. Sätze wie Rus- mortal”. Propositions like Russell’s “axiom
sells „Axiom of reducibility“ sind nicht logi- of reducibility” are not logical propositions,sche Sätze, und dies erklärt unser Gefühl: and this explains our feeling that, if true,
Dass sie, wenn wahr, so doch nur durch they can only be true by a happy chance.
einen günstigen Zufall wahr sein könnten.
Es lässt sich eine Welt denken, in derdas Axiom of reducibility nicht gilt. Es ist
aber klar, dass die Logik nichts mit der Frage zu schaffen hat, ob unsere Welt wirklichso ist oder nicht.
Die logischen Sätze beschreiben das Gerüst der Welt, oder vielmehr, sie stellen es
dar. Sie „handeln“ von nichts. Sie setzen

We can imagine a world in which theaxiom of reducibility is not valid. But it isclear that logic has nothing to do with thequestion whether our world is really of thiskind or not.
The logical propositions describe thescaffolding of the world, or rather theypresent it. They “treat” of nothing. They

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 canpostulate an adequate notation.
It also becomes clear now why logic wascalled the theory of forms and of inference.
Clearly the laws of logic cannot in theirturn be subject to laws of logic.
(There is not, as Russell thought, a special law of contradiction for each ‘type’; onelaw is enough, since it is not applied toitself.)
The mark of a logical proposition is not
general validity.
To be general means no more than tobe accidentally valid for all things. An ungeneralized proposition can be tautologicaljust as well as a generalized one.
The general validity of logic might becalled essential, in contrast with the accidental general validity of such propositions as ‘All men are mortal’. Propositions
like Russell’s ‘axiom of reducibility’ are notlogical propositions, and this explains our
feeling that, even if they were true, theirtruth could only be the result of a fortunate
accident.
It is possible to imagine a world inwhich 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 thescaffolding of the world, or rather they represent it. They have no ‘subject-matter’.

6.125

6.1251
6.126

voraus, dass Namen Bedeutung, und Elementarsätze Sinn haben: Und dies ist ihre
Verbindung mit der Welt. Es ist klar, dasses etwas über die Welt anzeigen muss, dassgewisse Verbindungen von Symbolen – welche wesentlich einen bestimmten Charakter haben – Tautologien sind. Hierin liegt
das Entscheidende. Wir sagten, manches
an den Symbolen, die wir gebrauchen, wäre willkürlich, manches nicht. In der Logik
drückt nur dieses aus: Das heißt aber, inder Logik drücken nicht w i r mit Hilfeder Zeichen aus, was wir wollen, sondern
in der Logik sagt die Natur der naturnotwendigen Zeichen selbst aus: Wenn wir die
logische Syntax irgendeiner Zeichensprache kennen, dann sind bereits alle Sätze
der Logik gegeben.

presuppose that names have meaning, andthat elementary propositions have sense.
And this is their connexion with the world.
It is clear that it must show something
about the world that certain combinationsof symbols—which essentially have a definite character—are tautologies. Hereinlies the decisive point. We said that in
the symbols which we use something isarbitrary, something not. In logic only thisexpresses: but this means that in logic it
is not we who express, by means of signs,
what we want, but in logic the nature of
the essentially necessary signs itself asserts. That is to say, if we know the logicalsyntax of any sign language, then all the
propositions of logic are already given.

Es ist möglich, und zwar auch nach deralten Auffassung der Logik, von vornherein eine Beschreibung aller „wahren“ logischen Sätze zu geben.
Darum kann es in der Logik auch n i eÜberraschungen geben.
Ob ein Satz der Logik angehört, kannman berechnen, indem man die logischen
Eigenschaften des S y m b o l s berechnet.
Und dies tun wir, wenn wir einen logischen Satz „beweisen“. Denn, ohne uns um
einen Sinn und eine Bedeutung zu kümmern, bilden wir den logischen Satz aus
anderen nach bloßen Z e i c h e n r e g e l n.
Der Beweis der logischen Sätze besteht
darin, dass wir sie aus anderen logischen
Sätzen durch successive Anwendung gewisser Operationen entstehen lassen, die

It is possible, also with the old conception of logic, to give at the outset a description of all “true” logical propositions.
Hence there can never be surprises inlogic.
Whether a proposition belongs to logic
can be calculated by calculating the logicalproperties of the symbol.
And this we do when we prove a logical proposition. For without troubling ourselves about a sense and a meaning, we
form the logical propositions out of others
by mere symbolic rules.
We prove a logical proposition by creating it out of other logical propositions byapplying in succession certain operations,which again generate tautologies out of the

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 worldmust be indicated by the fact that certain
combinations of symbols—whose essence
involves the possession of a determinatecharacter—are tautologies. This contains
the decisive point. We have said that somethings are arbitrary in the symbols that
we use and that some things are not. In
logic it is only the latter that express: butthat means that logic is not a field in which
we express what we wish with the help ofsigns, but rather one in which the natureof the natural and inevitable signs speaks
for itself. If we know the logical syntax ofany sign-language, then we have alreadybeen given all the propositions of logic.
It is possible—indeed possible even according to the old conception of logic—togive in advance a description of all ‘true’
logical propositions.
Hence there can never be surprises inlogic.
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

6.1261
6.1262

6.1263

6.1264

6.12656.127

aus den ersten immer wieder Tautologien
erzeugen. (Und zwar f o l g e n aus einer
Tautologie nur Tautologien.)
Natürlich ist diese Art zu zeigen, dassihre Sätze Tautologien sind, der Logik
durchaus unwesentlich. Schon darum, weil
die Sätze, von welchen der Beweis ausgeht,
ja ohne Beweis zeigen müssen, dass sie
Tautologien sind.
In der Logik sind Prozess und Resultatäquivalent. (Darum keine Überraschung.)
Der Beweis in der Logik ist nur ein mechanisches Hilfsmittel zum leichteren Erkennen der Tautologie, wo sie kompliziert
ist.
Es wäre ja auch zu merkwürdig, wenn
man einen sinnvollen Satz l o g i s c h aus
anderen beweisen könnte, und einen logischen Satz a u c h. Es ist von vornhereinklar, dass der logische Beweis eines sinnvollen Satzes und der Beweis i n der Logik
zwei ganz verschiedene Dinge sein müssen.
Der sinnvolle Satz sagt etwas aus, undsein Beweis zeigt, dass es so ist; in der Logik ist jeder Satz die Form eines Beweises.

first. (And from a tautology only tautolo- that always generate further tautologies
gies follow.)out of the initial ones. (And in fact only
tautologies follow from a tautology.)
Naturally this way of showing that
Of course this way of showing that theits propositions are tautologies is quite propositions of logic are tautologies is notunessential to logic. Because the propo- at all essential to logic, if only because thesitions, from which the proof starts, must propositions from which the proof starts
show without proof that they are tautolo- must show without any proof that they aregies.
tautologies.
In logic process and result are equivaIn logic process and result are equivalent. (Therefore no surprises.)lent. (Hence the absence of surprise.)
Proof in logic is only a mechanical exProof in logic is merely a mechanicalpedient to facilitate the recognition of tau- expedient to facilitate the recognition of
tology, where it is complicated.
tautologies in complicated cases.
It would be too remarkable, if one could
prove a significant proposition logicallyfrom another, and a logical proposition also.
It is clear from the beginning that the logical proof of a significant proposition andthe proof in logic must be two quite different things.

Indeed, it would be altogether too remarkable if a proposition that had sense
could be proved logically from others, andso too could a logical proposition. It isclear from the start that a logical proof ofa proposition that has sense and a proof inlogic must be two entirely different things.

The significant proposition asserts
something, and its proof shows that it isso; in logic every proposition is the form ofa proof.
Jeder Satz der Logik ist ein in Zeichen
Every proposition of logic is a modus
dargestellter modus ponens. (Und den mo- ponens presented in signs. (And the modus
dus ponens kann man nicht durch einen ponens can not be expressed by a proposiSatz ausdrücken.)tion.)
Immer kann man die Logik so auffasLogic can always be conceived to besen, dass jeder Satz sein eigener Beweis such that every proposition is its own proof.
ist.
Alle Sätze der Logik sind gleichberechAll propositions of logic are of equal
tigt, es gibt unter ihnen nicht wesentlich rank; there are not some which are essenGrundgesetze und abgeleitete Sätze.
tially primitive and others deduced from

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

6.1271

6.13

6.2

6.21
6.211

6.22

there.
Every tautology itself shows that it is atautology.
It is clear that the number of “primitive
propositions of logic” is arbitrary, for we
could deduce logic from one primitive
proposition by simply forming, for example, the logical produce of Frege’s primitive
propositions. (Frege would perhaps saythat this would no longer be immediately
self-evident. But it is remarkable that soexact a thinker as Frege should have appealed to the degree of self-evidence as thecriterion of a logical proposition.)

Jede Tautologie zeigt selbst, dass sieeine Tautologie ist.
Es ist klar, dass die Anzahl der „logischen Grundgesetze“ willkürlich ist, denn
man könnte die Logik ja aus Einem Grundgesetz ableiten, indem man einfach z. B.
aus Freges Grundgesetzen das logische
Produkt bildet. (Frege würde vielleicht sagen, dass dieses Grundgesetz nun nicht
mehr unmittelbar einleuchte. Aber es ist
merkwürdig, dass ein so exakter Denker
wie Frege sich auf den Grad des Einleuchtens als Kriterium des logischen Satzes
berufen hat.)
Die Logik ist keine Lehre, sondern ein
Logic is not a theory but a reflexion ofSpiegelbild der Welt.
the world.
Die Logik ist transcendental.
Logic is transcendental.
Die Mathematik ist eine logische MeMathematics is a logical method.
thode.
Die Sätze der Mathematik sind GleiThe propositions of mathematics arechungen, also Scheinsätze.
equations, and therefore pseudo-propositions.
Der Satz der Mathematik drückt keiMathematical propositions express nonen Gedanken aus.
thoughts.
Im Leben ist es ja nie der mathematiIn life it is never a mathematical proposche Satz, den wir brauchen, sondern wir sition which we need, but we use mathebenützen den mathematischen Satz n u r, matical propositions only in order to infer
um aus Sätzen, welche nicht der Mathe- from propositions which do not belong tomatik angehören, auf andere zu schließen, mathematics to others which equally dowelche gleichfalls nicht der Mathematik not belong to mathematics.
angehören.
(In der Philosophie führt die Frage: „wo(In philosophy the question “Why dozu gebrauchen wir eigentlich jenes Wort, we really use that word, that proposition?”
jenen Satz“ immer wieder zu wertvollen constantly leads to valuable results.)
Einsichten.)
Die Logik der Welt, die die Sätze der
The logic of the world which the propo-

Every tautology itself shows that it is atautology.
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 athinker as rigorous as Frege appealed tothe degree of self-evidence as the criterionof a logical proposition.)
Logic is not a body of doctrine, but amirror-image of the world.
Logic is transcendental.
Mathematics is a logical method.
The propositions of mathematics areequations, and therefore pseudo-propositions.
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

6.23

6.231

6.232

6.23216.2322

Logik in den Tautologien zeigen, zeigt die
Mathematik in den Gleichungen.
Wenn zwei Ausdrücke durch das
Gleichheitszeichen verbunden werden, so
heißt das, sie sind durch einander ersetzbar. Ob dies aber der Fall ist, muss sich an
den beiden Ausdrücken selbst zeigen.
Es charakterisiert die logische Formzweier Ausdrücke, dass sie durch einander
ersetzbar sind.
Es ist eine Eigenschaft der Bejahung,dass man sie als doppelte Verneinung auffassen kann.
Es ist eine Eigenschaft von „1 + 1 + 1 + 1“,dass man es als „(1 + 1) + (1 + 1)“ auffassenkann.
Frege sagt, die beiden Ausdrücke haben dieselbe Bedeutung, aber verschiedenen Sinn.
Das Wesentliche an der Gleichung ist
aber, dass sie nicht notwendig ist, um zu
zeigen, dass die beiden Ausdrücke, die das
Gleichheitszeichen verbindet, dieselbe Bedeutung haben, da sich dies aus den beiden
Ausdrücken selbst ersehen lässt.
Und, dass die Sätze der Mathematik
bewiesen werden können, heißt ja nichts
anderes, als dass ihre Richtigkeit einzusehen ist, ohne dass das, was sie ausdrücken,
selbst mit den Tatsachen auf seine Richtigkeit hin verglichen werden muss.
Die Identität der Bedeutung zweier
Ausdrücke lässt sich nicht b e h a u p t e n.
Denn, um etwas von ihrer Bedeutung behaupten zu können, muss ich ihre Bedeutung kennen: und indem ich ihre Bedeu-

sitions of logic show in tautologies, mathematics shows in equations.
If two expressions are connected by
the sign of equality, this means that they
can be substituted for one another. But
whether this is the case must show itself
in the two expressions themselves.

in tautologies by the propositions of logic,
is shown in equations by mathematics.
If two expressions are combined bymeans of the sign of equality, that meansthat they can be substituted for one another. But it must be manifest in the two
expressions themselves whether this is thecase or not.
It characterizes the logical form of two
When two expressions can be substiexpressions, that they can be substituted tuted for one another, that characterizes
for one another.
their logical form.
It is a property of affirmation that it
It is a property of affirmation that it
can be conceived as double denial.
can be construed as double negation.
It is a property of “1 + 1 + 1 + 1” that it
It is a property of ‘1 + 1 + 1 + 1’ that it
can be conceived as “(1 + 1) + (1 + 1)”.
can be construed as ‘(1 + 1) + (1 + 1)’.

Frege says that these expressions have
Frege says that the two expressions
the same meaning but different senses.
have the same meaning but different
senses.
But what is essential about equation
But the essential point about an equais that it is not necessary in order to show tion is that it is not necessary in order tothat both expressions, which are connected show that the two expressions connectedby the sign of equality, have the same by the sign of equality have the same meanmeaning: for this can be perceived from ing, since this can be seen from the two
the two expressions themselves.
expressions themselves.
And, that the propositions of matheAnd the possibility of proving the propomatics can be proved means nothing else sitions of mathematics means simply thatthan that their correctness can be seen their correctness can be perceived without
without our having to compare what they its being necessary that what they express
express with the facts as regards correct- should itself be compared with the facts inness.
order to determine its correctness.
The identity of the meaning of two exIt is impossible to assert the identity ofpressions cannot be asserted. For in order meaning of two expressions. For in order
to be able to assert anything about their to be able to assert anything about their
meaning, I must know their meaning, and meaning, I must know their meaning, and I
if I know their meaning, I know whether cannot know their meaning without know-

6.23236.233

6.2331

6.234
6.2341

6.24

6.241

tung kenne, weiß ich, ob sie dasselbe oderverschiedenes bedeuten.
Die Gleichung kennzeichnet nur den
Standpunkt, von welchem ich die beiden
Ausdrücke betrachte, nämlich vom Standpunkte ihrer Bedeutungsgleichheit.
Die Frage, ob man zur Lösung der mathematischen Probleme die Anschauung
brauche, muss dahin beantwortet werden,
dass eben die Sprache hier die nötige Anschauung liefert.
Der Vorgang des R e c h n e n s vermittelt eben diese Anschauung.
Die Rechnung ist kein Experiment.
Die Mathematik ist eine Methode derLogik.
Das Wesentliche der mathematischen
Methode ist es, mit Gleichungen zu arbeiten. Auf dieser Methode beruht es nämlich,dass jeder Satz der Mathematik sich vonselbst verstehen muss.
Die Methode der Mathematik, zu ihren
Gleichungen zu kommen, ist die Substitutionsmethode.
Denn die Gleichungen drücken die Ersetzbarkeit zweier Ausdrücke aus und wir
schreiten von einer Anzahl von Gleichungen zu neuen Gleichungen vor, indem wir,den Gleichungen entsprechend, Ausdrücke
durch andere ersetzen.
So lautet der Beweis des Satzes 2 × 2 =4:(Ων )µ ’ x = Ων×µ ’ x Def.
Ω ’ x = (Ω2 )2 ’ x = (Ω2 )1+1 ’ x = Ω2 ’ Ω2 ’ x= Ω1+1 ’ Ω1+1 ’ x = (Ω’Ω)’ (Ω’Ω)’ x= Ω ’ Ω ’ Ω ’ Ω ’ x = Ω1+1+1+1 ’ x = Ω4 ’ x.
2×2

they mean the same or something different.
The equation characterizes only thestandpoint from which I consider the two
expressions, that is to say the standpointof their equality of meaning.
To the question whether we need intuition for the solution of mathematical problems it must be answered that language
itself here supplies the necessary intuition.
The process of calculation brings about
just this intuition.
Calculation is not an experiment.
Mathematics is a method of logic.
The essential of mathematical method
is working with equations. On this method
depends the fact that every proposition ofmathematics must be self-evident.
The method by which mathematics arrives at its equations is the method of substitution.
For equations express the substitutability of two expressions, and we proceed froma number of equations to new equations,replacing expressions by others in accordance with the equations.

ing whether what they mean is the sameor different.
An equation merely marks the pointof view from which I consider the two expressions: it marks their equivalence inmeaning.
The question whether intuition isneeded for the solution of mathematicalproblems must be given the answer thatin this case language itself provides thenecessary intuition.
The process of calculating serves tobring about that intuition.
Calculation is not an experiment.
Mathematics is a method of logic.
It is the essential characteristic ofmathematical method that it employs
equations. For it is because of this methodthat every proposition of mathematics
must go without saying.
The method by which mathematics arrives at its equations is the method of substitution.
For equations express the substitutability of two expressions and, starting froma number of equations, we advance to newequations by substituting different expressions in accordance with the equations.

Thus the proof of the proposition 2 × 2 =Thus the proof of the proposition 2 × 2 =4 runs:4 runs as follows:(Ων )µ ’ x = Ων×µ ’ x Def.
Ω ’ x = (Ω2 )2 ’ x = (Ω2 )1+1 ’ x = Ω2 ’ Ω2 ’ x= Ω1+1 ’ Ω1+1 ’ x = (Ω’Ω)’ (Ω’Ω)’ x= Ω ’ Ω ’ Ω ’ Ω ’ x = Ω1+1+1+1 ’ x = Ω4 ’ x.
2×2(Ων )µ ’ x = Ων×µ ’ x Def.
Ω ’ x = (Ω2 )2 ’ x = (Ω2 )1+1 ’ x = Ω2 ’ Ω2 ’ x= Ω1+1 ’ Ω1+1 ’ x = (Ω’Ω)’ (Ω’Ω)’ x= Ω ’ Ω ’ Ω ’ Ω ’ x = Ω1+1+1+1 ’ x = Ω4 ’ x.
2×2

6.3

6.31

6.32
6.321

6.3211

6.33

6.34

6.341

Die Erforschung der Logik bedeutet die
Erforschung a l l e r G e s e t z m ä ß i g k e i t. Und außerhalb der Logik ist alles
Zufall.
Das sogenannte Gesetz der Induktion
kann jedenfalls kein logisches Gesetz sein,denn es ist offenbar ein sinnvoller Satz. –Und darum kann es auch kein Gesetz apriori sein.
Das Kausalitätsgesetz ist kein Gesetz,
sondern die Form eines Gesetzes.
„Kausalitätsgesetz“, das ist ein Gattungsname. Und wie es in der Mechanik,
sagen wir, Minimum-Gesetze gibt – etwa
der kleinsten Wirkung – so gibt es in der
Physik Kausalitätsgesetze, Gesetze vonder Kausalitätsform.
Man hat ja auch davon eine Ahnung
gehabt, dass es e i n „Gesetz der kleinsten
Wirkung“ geben müsse, ehe man genau
wusste, wie es lautete. (Hier, wie immer,
stellt sich das a priori Gewisse als etwas
rein Logisches heraus.)
Wir g l a u b e n nicht a priori an ein
Erhaltungsgesetz, sondern wir w i s s e n
a priori die Möglichkeit einer logischen
Form.
Alle jene Sätze, wie der Satz vom Grunde, von der Kontinuität in der Natur, vomkleinsten Aufwande in der Natur etc. etc.,
alle diese sind Einsichten a priori über diemögliche Formgebung der Sätze der Wissenschaft.
Die Newtonsche Mechanik z. B. bringt
die Weltbeschreibung auf eine einheitlicheForm. Denken wir uns eine weiße Fläche,Logical research means the investigaThe exploration of logic means the extion of all regularity. And outside logic all ploration of everything that is subject tois accident.
law. And outside logic everything is accidental.
The so-called law of induction cannot in
The so-called law of induction cannot
any case be a logical law, for it is obviously possibly be a law of logic, since it is obvia significant proposition.—And therefore ously a proposition with sense.—Nor, thereit cannot be a law a priori either.
fore, can it be an a priori law.
The law of causality is not a law but
the form of a law.*“Law of Causality” is a class name. And
as in mechanics there are, for instance,
minimum-laws, such as that of least actions, so in physics there are causal laws,
laws of the causality form.
Men had indeed an idea that there
must be a “law of least action”, before theyknew exactly how it ran. (Here, as always,
the a priori certain proves to be something
purely logical.)

The law of causality is not a law but
the form of a law.
‘Law of causality’—that is a generalname. And just as in mechanics, for example, there are ‘minimum-principles’, suchas the law of least action, so too in physics
there are causal laws, laws of the causal
form.
Indeed people even surmised that there
must be a ‘law of least action’ before theyknew exactly how it went. (Here, as always, what is certain a priori proves to besomething purely logical.)We do not believe a priori in a law of
We do not have an a priori belief inconservation, but we know a priori the pos- a law of conservation, but rather a prisibility of a logical form.
ori knowledge of the possibility of a logicalform.
All propositions, such as the law of cauAll such propositions, including thesation, the law of continuity in nature, the principle of sufficient reason, the laws oflaw of least expenditure in nature, etc. etc., continuity in nature and of least effort inall these are a priori intuitions of possible nature, etc. etc.—all these are a priori informs of the propositions of science.
sights about the forms in which the propositions of science can be cast.
Newtonian mechanics, for example,
Newtonian mechanics, for example, imbrings the description of the universe to poses a unified form on the description ofa unified form. Let us imagine a white the world. Let us imagine a white surface* [Ogden only] I.e. not the form of one particular law, but of any law of a certain sort (B. R.).

auf der unregelmäßige schwarze Flecken
wären. Wir sagen nun: Was für ein Bildimmer hierdurch entsteht, immer kannich seiner Beschreibung beliebig nahe kommen, indem ich die Fläche mit einem entsprechend feinen quadratischen Netzwerk
bedecke und nun von jedem Quadrat sage,
dass es weiß oder schwarz ist. Ich werde
auf diese Weise die Beschreibung der Fläche auf eine einheitliche Form gebracht haben. Diese Form ist beliebig, denn ich hätte mit dem gleichen Erfolge ein Netz aus
dreieckigen oder sechseckigen Maschen
verwenden können. Es kann sein, dassdie Beschreibung mit Hilfe eines DreiecksNetzes einfacher geworden wäre; das heißt,
dass wir die Fläche mit einem gröberen
Dreiecks-Netz genauer beschreiben könnten als mit einem feineren quadratischen(oder umgekehrt) usw. Den verschiedenen
Netzen entsprechen verschiedene Systeme
der Weltbeschreibung. Die Mechanik bestimmt eine Form der Weltbeschreibung,indem sie sagt: Alle Sätze der Weltbeschreibung müssen aus einer Anzahl gegebener
Sätze – den mechanischen Axiomen – aufeine gegebene Art und Weise erhalten werden. Hierdurch liefert sie die Bausteine
zum Bau des wissenschaftlichen Gebäudes und sagt: Welches Gebäude immer duaufführen willst, jedes musst du irgendwie mit diesen und nur diesen Bausteinenzusammenbringen.
(Wie man mit dem Zahlensystem jede
beliebige Anzahl, so muss man mit dem System der Mechanik jeden beliebigen Satz
der Physik hinschreiben können.)surface with irregular black spots. We
now say: Whatever kind of picture thesemake I can always get as near as I liketo its description, if I cover the surface
with a sufficiently fine square network andnow say of every square that it is whiteor black. In this way I shall have brought
the description of the surface to a unified
form. This form is arbitrary, because I
could have applied with equal success a
net with a triangular or hexagonal mesh.
It can happen that the description would
have been simpler with the aid of a triangular mesh; that is to say we might havedescribed the surface more accurately witha triangular, and coarser, than with thefiner square mesh, or vice versa, and soon. To the different networks corresponddifferent systems of describing the world.
Mechanics determine a form of description
by saying: All propositions in the description of the world must be obtained in agiven way from a number of given propositions—the mechanical axioms. It thus
provides the bricks for building the edificeof science, and says: Whatever building
thou wouldst erect, thou shalt construct
it in some manner with these bricks andthese alone.

with irregular black spots on it. We thensay that whatever kind of picture thesemake, I can always approximate as closely
as I wish to the description of it by covering the surface with a sufficiently finesquare mesh, and then saying of every
square whether it is black or white. In thisway 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 theuse of a triangular mesh would have madethe description simpler: that is to say, it
might be that we could describe the surface
more accurately with a coarse triangularmesh 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 formof description of the world by saying thatall 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 thebricks for building the edifice of science,and it says, ‘Any building that you want toerect, whatever it may be, must somehow
be constructed with these bricks, and withthese alone.’(As with the system of numbers one(Just as with the number-system wemust be able to write down any arbitrary must be able to write down any number we
number, so with the system of mechanics wish, so with the system of mechanics weone must be able to write down any arbi- must be able to write down any proposition

6.342

Und nun sehen wir die gegenseitige
Stellung von Logik und Mechanik. (Mankönnte das Netz auch aus verschiedenartigen Figuren etwa aus Dreiecken und Sechsecken bestehen lassen.) Dass sich ein Bild,wie das vorhin erwähnte, durch ein Netz
von gegebener Form beschreiben lässt, sagtüber das Bild n i c h t s aus. (Denn dies gilt
für jedes Bild dieser Art.) Das aber charakterisiert das Bild, dass es sich durch ein bestimmtes Netz von b e s t i m m t e r Feinheit v o l l s t ä n d i g beschreiben lässt.
So auch sagt es nichts über die Welt
aus, dass sie sich durch die Newtonsche
Mechanik beschreiben lässt; wohl aber,dass sie sich s o durch jene beschreiben lässt, wie dies eben der Fall ist. Auchdas sagt etwas über die Welt, dass sie
sich durch die eine Mechanik einfacher beschreiben lässt als durch die andere.

6.343

Die Mechanik ist ein Versuch, allew a h r e n Sätze, die wir zur Weltbeschreibung brauchen, nach Einem Plane zu konstruieren.
Durch den ganzen logischen Apparathindurch sprechen die physikalischen Gesetze doch von den Gegenständen der Welt.
Wir dürfen nicht vergessen, dass die
Weltbeschreibung durch die Mechanik immer die ganz allgemeine ist. Es ist in ihrz. B. nie von b e s t i m m t e n materiellen
Punkten die Rede, sondern immer nur voni r g e n d w e l c h e n.
Obwohl die Flecke in unserem Bildgeometrische Figuren sind, so kann

6.3431

6.3432

6.35

trary physical proposition.)
And now we see the relative positionof logic and mechanics. (We could construct the network out of figures of different kinds, as out of triangles and hexagons
together.) That a picture like that instanced above can be described by a network of a given form asserts nothing about
the picture. (For this holds of every pictureof this kind.) But this does characterize
the picture, the fact, namely, that it canbe completely described by a definite net ofdefinite fineness.
So too the fact that it can be describedby Newtonian mechanics asserts nothing
about the world; but this asserts something, namely, that it can be described inthat particular way in which as a matter offact it is described. The fact, too, that it canbe described more simply by one system of
mechanics than by another says something
about the world.
Mechanics is an attempt to constructaccording to a single plan all true propositions which we need for the description of
the world.
Through their whole logical apparatus
the physical laws still speak of the objectsof the world.
We must not forget that the descriptionof the world by mechanics is always quite
general. There is, for example, never any
mention of particular material points in it,
but always only of some points or other.

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 kindof 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 particularsize of mesh.
Similarly the possibility of describing
the world by means of Newtonian mechanics tells us nothing about the world: butwhat does tell us something about it is theprecise way in which it is possible to describe it by these means. We are also told
something about the world by the fact thatit can be described more simply with one
system of mechanics than with another.
Mechanics is an attempt to constructaccording 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 world.
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 particularpoint-masses: it will only talk about any
point-masses whatsoever.
Although the spots in our picture are
Although the spots in our picture aregeometrical figures, geometry can obvi- geometrical figures, nevertheless geome-

6.36

6.361

6.3611doch selbstverständlich die Geometrie gar
nichts über ihre tatsächliche Form und Lage sagen. Das Netz aber ist r e i n geometrisch, alle seine Eigenschaften können apriori angegeben werden.
Gesetze wie der Satz vom Grunde, etc.
handeln vom Netz, nicht von dem, was das
Netz beschreibt.
Wenn es ein Kausalitätsgesetz gäbe, so
könnte es lauten: „Es gibt Naturgesetze“.
Aber freilich kann man das nicht sagen:es zeigt sich.
In der Ausdrucksweise Hertz’s könnte
man sagen: Nur g e s e t z m ä ß i g e Zusammenhänge sind d e n k b a r.
Wir können keinen Vorgang mit dem„Ablauf der Zeit“ vergleichen – diesen gibt
es nicht –, sondern nur mit einem anderen
Vorgang (etwa mit dem Gang des Chronometers).
Daher ist die Beschreibung des zeitlichen Verlaufs nur so möglich, dass wir uns
auf einen anderen Vorgang stützen.
Ganz Analoges gilt für den Raum. Wo
man z. B. sagt, es könne keines von zwei
Ereignissen (die sich gegenseitig ausschließen) eintreten, weil k e i n e U r s a c h evorhanden sei, warum das eine eher als
das andere eintreten solle, da handelt essich in Wirklichkeit darum, dass man gar
nicht e i n e s der beiden Ereignisse beschreiben kann, wenn nicht irgend eine
Asymmetrie vorhanden ist. Und w e n neine solche Asymmetrie vorhanden i s t, so
können wir diese als U r s a c h e des Eintreffens des einen und Nicht-Eintreffens

ously say nothing about their actual form try can obviously say nothing at all about
and position. But the network is purely their actual form and position. The netgeometrical, and all its properties can be work, however, is purely geometrical; allgiven a priori.
its properties can be given a priori.
Laws, like the law of causation, etc.,
Laws like the principle of sufficient reatreat of the network and not what the net- son, etc. are about the net and not about
work describes.
what the net describes.
If there were a law of causality, it might
If there were a law of causality, it might
run: “There are natural laws”.
be put in the following way: There are lawsof nature.
But that can clearly not be said: it
But of course that cannot be said: itshows itself.
makes itself manifest.
In the terminology of Hertz we might
One might say, using Hertz’s terminolsay: Only uniform connections are think- ogy, that only connexions that are subject
able.
to law are thinkable.
We cannot compare any process withWe cannot compare a process withthe “passage of time”—there is no such ‘the passage of time’—there is no suchthing—but only with another process (say, thing—but only with another process (suchwith the movement of the chronometer).
as the working of a chronometer).
Hence the description of the temporalsequence of events is only possible if we
support ourselves on another process.
It is exactly analogous for space. When,for example, we say that neither of two
events (which mutually exclude one another) can occur, because there is no cause
why the one should occur rather than theother, it is really a matter of our being unable to describe one of the two events unless there is some sort of asymmetry. And if
there is such an asymmetry, we can regard
this as the cause of the occurrence of theone and of the non-occurrence of the other.

Hence we can describe the lapse of time
only by relying on some other process.
Something exactly analogous applies tospace: e.g. when people say that neitherof 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 todescribe 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 occurrenceof the one and the non-occurrence of theother.

des anderen auffassen.
6.36111
Das Kant’sche Problem von der rechten und linken Hand, die man nicht zur
Deckung bringen kann, besteht schon inder Ebene, ja im eindimensionalen Raum,
wo die beiden kongruenten Figuren a und
b auch nicht zur Deckung gebracht werden
können, ohne aus diesem Raum
a

× ×

b

6.363

6.3631

6.36311

a

× ×

b

a

× ×

b

in which the two congruent figures, a andb, cannot be made to coincide unless they
are moved out of this space. The right hand
and the left hand are in fact completelycongruent. It is quite irrelevant that theycannot be made to coincide.
Den rechten Handschuh könnte man
A right-hand glove could be put on a
A right-hand glove could be put on the
an die linke Hand ziehen, wenn man ihn left hand if it could be turned round in left hand, if it could be turned round inim vierdimensionalen Raum umdrehen four-dimensional space.
four-dimensional space.
könnte.
Was sich beschreiben lässt, das kannWhat can be described can happen too,
What can be described can happen too:auch geschehen, und was das Kausalitäts- and what is excluded by the law of causal- and what the law of causality is meant togesetz ausschließen soll, das lässt sich ity cannot be described.
exclude cannot even be described.
auch nicht beschreiben.
Der Vorgang der Induktion besteht darThe process of induction is the process
The procedure of induction consists in
in, dass wir das e i n f a c h s t e Gesetz of assuming the simplest law that can be accepting as true the simplest law that canannehmen, das mit unseren Erfahrungen made to harmonize with our experience.
be reconciled with our experiences.
in Einklang zu bringen ist.
Dieser Vorgang hat aber keine logische,This process, however, has no logicalThis procedure, however, has no logicalsondern nur eine psychologische Begrün- foundation but only a psychological one.
justification but only a psychological one.
dung.
Es ist klar, dass kein Grund vorhanden
It is clear that there are no grounds for
It is clear that there are no grounds for
ist, zu glauben, es werde nun auch wirklich believing that the simplest course of events believing that the simplest eventuality will
der einfachste Fall eintreten.
will really happen.
in fact be realized.
Dass die Sonne morgen aufgehen wird,
That the sun will rise to-morrow, is an
It is an hypothesis that the sun will rise
ist eine Hypothese; und das heißt: wir hypothesis; and that means that we do not tomorrow: and this means that we do notherausbewegt zu werden. Rechte und linke
Hand sind tatsächlich vollkommen kongruent. Und dass man sie nicht zur Deckungbringen kann, hat damit nichts zu tun.

6.362

The Kantian problem of the right andKant’s problem about the right handleft hand which cannot be made to cover and the left hand, which cannot be madeone another already exists in the plane, to coincide, exists even in two dimensions.
and even in one-dimensional space; where Indeed, it exists in one-dimensional space
the two congruent figures a and b cannot
be made to cover one another without

moving them out of this space. The rightand left hand are in fact completely congruent. And the fact that they cannot bemade to cover one another has nothing to
do with it.

6.37

6.371

6.372

w i s s e n nicht, ob sie aufgehen wird.
Einen Zwang, nach dem Eines geschehen müsste, weil etwas anderes geschehenist, gibt es nicht. Es gibt nur eine l o g i s c h e Notwendigkeit.
Der ganzen modernen Weltanschauung
liegt die Täuschung zugrunde, dass die sogenannten Naturgesetze die Erklärungender Naturerscheinungen seien.
So bleiben sie bei den Naturgesetzen
als bei etwas Unantastbarem stehen, wie
die Älteren bei Gott und dem Schicksal.
Und sie haben ja beide Recht, und Unrecht. Die Alten sind allerdings insofernklarer, als sie einen klaren Abschluss anerkennen, während es bei dem neuen Systemscheinen soll, als sei a l l e s erklärt.

6.373
6.374

6.375

6.3751

Die Welt ist unabhängig von meinem
Willen.
Auch wenn alles, was wir wünschen,geschähe, so wäre dies doch nur, sozusagen, eine Gnade des Schicksals, denn es ist
kein l o g i s c h e r Zusammenhang zwischen Willen und Welt, der dies verbürgte,
und den angenommenen physikalischen
Zusammenhang könnten wir doch nicht
selbst wieder wollen.
Wie es nur eine l o g i s c h e Notwendigkeit gibt, so gibt es auch nur eine l o g i s c h e Unmöglichkeit.
Dass z. B. zwei Farben zugleich an einem Ort des Gesichtsfeldes sind, ist unmöglich, und zwar logisch unmöglich, denn
es ist durch die logische Struktur der Farbe
ausgeschlossen.

know whether it will rise.
know whether it will rise.
A necessity for one thing to happen beThere is no compulsion making one
cause another has happened does not exist. thing happen because another has hapThere is only logical necessity.
pened. The only necessity that exists islogical necessity
At the basis of the whole modern view
The whole modern conception of theof the world lies the illusion that the so- world is founded on the illusion that the socalled laws of nature are the explanations called laws of nature are the explanationsof natural phenomena.
of natural phenomena.
So people stop short at natural laws
Thus people today stop at the laws of
as something unassailable, as did the an- nature, treating them as something inviocients at God and Fate.
lable, just as God and Fate were treated inpast ages.
And they are both right and wrong. but
And in fact both are right and boththe ancients were clearer, in so far as they wrong: though the view of the ancientsrecognized one clear terminus, whereas the is clearer in so far as they have a clear andmodern system makes it appear as though acknowledged terminus, while the modern
everything were explained.
system tries to make it look as if everything
were explained.
The world is independent of my will.
The world is independent of my will.
Even if everything we wished were tohappen, this would only be, so to speak, afavour of fate, for there is no logical connexion between will and world, which would
guarantee this, and the assumed physicalconnexion itself we could not again will.

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

As there is only a logical necessity, so
Just as the only necessity that exists isthere is only a logical impossibility.
logical necessity, so too the only impossibility that exists is logical impossibility.
For two colours, e.g. to be at one place
For example, the simultaneous presin the visual field, is impossible, logically ence of two colours at the same place in
impossible, for it is excluded by the logical the visual field is impossible, in fact logistructure of colour.
cally impossible, since it is ruled out by the
logical structure of colour.

6.4
6.41

6.42

6.421

Denken wir daran, wie sich dieser Widerspruch in der Physik darstellt: Ungefähr so, dass ein Teilchen nicht zu gleicher
Zeit zwei Geschwindigkeiten haben kann;das heißt, dass es nicht zu gleicher Zeitan zwei Orten sein kann; das heißt, dassTeilchen an verschiedenen Orten zu Einer
Zeit nicht identisch sein können.
(Es ist klar, dass das logische Produkt
zweier Elementarsätze weder eine Tautologie noch eine Kontradiktion sein kann. Die
Aussage, dass ein Punkt des Gesichtsfeldes
zu gleicher Zeit zwei verschiedene Farben
hat, ist eine Kontradiktion.)
Alle Sätze sind gleichwertig.
Der Sinn der Welt muss außerhalb ihrer liegen. In der Welt ist alles, wie es ist,
und geschieht alles, wie es geschieht; esgibt i n ihr keinen Wert – und wenn es
ihn gäbe, so hätte er keinen Wert.
Wenn es einen Wert gibt, der Wert hat,so muss er außerhalb alles Geschehens
und So-Seins liegen. Denn alles Geschehen
und So-Sein ist zufällig.
Was es nichtzufällig macht, kann nicht
i n der Welt liegen, denn sonst wäre dieswieder zufällig.
Es muss außerhalb der Welt liegen.
Darum kann es auch keine Sätze derEthik geben.
Sätze können nichts Höheres ausdrücken.
Es ist klar, dass sich die Ethik nicht
aussprechen lässt.
Die Ethik ist transcendental.
(Ethik und Ästhetik sind Eins.)

Let us consider how this contradiction
presents itself in physics. Somewhat asfollows: That a particle cannot at the sametime have two velocities, i.e. that at thesame time it cannot be in two places, i.e.
that particles in different places at thesame time cannot be identical.

Let us think how this contradiction appears in physics: more or less as follows—aparticle cannot have two velocities at thesame 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 thesame time cannot be identical.

It is clear that the logical product of
two elementary propositions can neither
be a tautology nor a contradiction. The
assertion that a point in the visual field
has two different colours at the same time,
is a contradiction.
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 happens as it does happen. In
it there is no value—and if there were, itwould be of no value.
If there is a value which is of value, it
must lie outside all happening and beingso. For all happening and being-so is accidental.
What makes it non-accidental cannot
lie in the world, for otherwise this would
again be accidental.
It must lie outside the world.
Hence also there can be no ethicalpropositions.
Propositions cannot express anythinghigher.
It is clear that ethics cannot be expressed.
Ethics is transcendental.
(Ethics and æsthetics are one.)(It is clear that the logical product of
two elementary propositions can neither
be a tautology nor a contradiction. The
statement that a point in the visual field
has two different colours at the same time
is a contradiction.)
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 havevalue, it must lie outside the whole sphereof what happens and is the case. For allthat 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 thatis higher.
It is clear that ethics cannot be put intowords.
Ethics is transcendental.
(Ethics and aesthetics are one and thesame.)

6.422

6.423

6.43

6.431
6.4311

Der erste Gedanke bei der Aufstellungeines ethischen Gesetzes von der Form „dusollst . . . .“ ist: Und was dann, wenn ich es
nicht tue? Es ist aber klar, dass die Ethik
nichts mit Strafe und Lohn im gewöhnlichen Sinne zu tun hat. Also muss diese
Frage nach den F o l g e n einer Handlung
belanglos sein. – Zum Mindesten dürfen
diese Folgen nicht Ereignisse sein. Dennetwas muss doch an jener Fragestellung
richtig sein. Es muss zwar eine Art von
ethischem Lohn und ethischer Strafe geben, aber diese müssen in der Handlung
selbst liegen.
(Und das ist auch klar, dass der Lohn
etwas Angenehmes, die Strafe etwas Unangenehmes sein muss.)
Vom Willen als dem Träger des Ethischen kann nicht gesprochen werden.
Und der Wille als Phänomen interessiert nur die Psychologie.
Wenn das gute oder böse Wollen die
Welt ändert, so kann es nur die Grenzen
der Welt ändern, nicht die Tatsachen; nichtdas, was durch die Sprache ausgedrückt
werden kann.
Kurz, die Welt muss dann dadurchüberhaupt eine andere werden. Sie musssozusagen als Ganzes abnehmen oder zunehmen.
Die Welt des Glücklichen ist eine andere als die des Unglücklichen.
Wie auch beim Tod die Welt sich nichtändert, sondern aufhört.
Der Tod ist kein Ereignis des Lebens.
Den Tod erlebt man nicht.

The first thought in setting up an ethical law of the form “thou shalt . . . ” is: And
what if I do not do it? But it is clear that
ethics has nothing to do with punishment
and reward in the ordinary sense. This
question as to the consequences of an action
must therefore be irrelevant. At least these
consequences will not be events. For there
must be something right in that formulation of the question. There must be some
sort of ethical reward and ethical punishment, but this must lie in the action itself.

When an ethical law of the form, ‘Thou
shalt . . . ’ is laid down, one’s first thoughtis, ‘And what if I do not do it?’ It is clear,
however, that ethics has nothing to do withpunishment and reward in the usual senseof the terms. So our question about theconsequences 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 ethicalreward and ethical punishment, but theymust reside in the action itself.

(And this is clear also that the reward(And it is also clear that the reward
must be something acceptable, and the must be something pleasant and the punpunishment something unacceptable.)ishment something unpleasant.)
Of the will as the subject of the ethical
It is impossible to speak about the will
we cannot speak.
in so far as it is the subject of ethical attributes.
And the will as a phenomenon is only
And the will as a phenomenon is of inof interest to psychology.
terest only to psychology.
If good or bad willing changes the
If the good or bad exercise of the will
world, it can only change the limits of the does alter the world, it can alter only theworld, not the facts; not the things that can limits of the world, not the facts—not what
be expressed in language.
can be expressed by means of language.
In brief, the world must thereby become
In short the effect must be that it bequite another, it must so to speak wax or comes an altogether different world. It
wane as a whole.
must, so to speak, wax and wane as awhole.
The world of the happy is quite another
The world of the happy man is a differthan that of the unhappy.
ent one from that of the unhappy man.
As in death, too, the world does not
So too at death the world does not alter,change, but ceases.
but comes to an end.
Death is not an event of life. Death is
Death is not an event in life: we do not
not lived through.
live to experience death.

6.43126.432

6.4321
6.44
6.45

6.5

Wenn man unter Ewigkeit nicht unendliche Zeitdauer, sondern Unzeitlichkeit versteht, dann lebt der ewig, der in der Gegenwart lebt.
Unser Leben ist ebenso endlos, wie unser Gesichtsfeld grenzenlos ist.
Die zeitliche Unsterblichkeit der Seele
des Menschen, das heißt also ihr ewiges
Fortleben auch nach dem Tode, ist nicht
nur auf keine Weise verbürgt, sondern vor
allem leistet diese Annahme gar nicht das,
was man immer mit ihr erreichen wollte.
Wird denn dadurch ein Rätsel gelöst, dassich ewig fortlebe? Ist denn dieses ewige
Leben dann nicht ebenso rätselhaft wie
das gegenwärtige? Die Lösung des Rätsels
des Lebens in Raum und Zeit liegt a u ß e r h a l b von Raum und Zeit.
(Nicht Probleme der Naturwissenschaft sind ja zu lösen.)W i e die Welt ist, ist für das Höhere vollkommen gleichgültig. Gott offenbart
sich nicht i n der Welt.
Die Tatsachen gehören alle nur zur Aufgabe, nicht zur Lösung.
Nicht w i e die Welt ist, ist das Mystische, sondern d a s s sie ist.
Die Anschauung der Welt sub specie aeterni ist ihre Anschauung als – begrenztes– Ganzes.
Das Gefühl der Welt als begrenztes
Ganzes ist das mystische.
Zu einer Antwort, die man nicht aussprechen kann, kann man auch die Frage
nicht aussprechen.
D a s R ä t s e l gibt es nicht.

If by eternity is understood not endlessIf we take eternity to mean not infinite
temporal duration but timelessness, then temporal duration but timelessness, thenhe lives eternally who lives in the present. eternal life belongs to those who live in thepresent.
Our life is endless in the way that our
Our life has no end in just the way invisual field is without limit.
which our visual field has no limits.
The temporal immortality of the huNot only is there no guarantee of theman soul, that is to say, its eternal survival temporal immortality of the human soul,also after death, is not only in no way guar- that is to say of its eternal survival after
anteed, but this assumption in the first death; but, in any case, this assumption
place will not do for us what we always completely fails to accomplish the purpose
tried to make it do. Is a riddle solved by for which it has always been intended. Or
the fact that I survive for ever? Is this is some riddle solved by my surviving for
eternal life not as enigmatic as our present ever? Is not this eternal life itself as muchone? The solution of the riddle of life in of a riddle as our present life? The solution
space and time lies outside space and time. of the riddle of life in space and time lies
outside space and time.
(It is not problems of natural science(It is certainly not the solution of any
which have to be solved.)
problems of natural science that is required.)
How the world is, is completely indifferHow things are in the world is a matterent for what is higher. God does not reveal of complete indifference for what is higher.
himself in the world.
God does not reveal himself in the world.
The facts all belong only to the task and
The facts all contribute only to setting
not to its performance.
the problem, not to its solution.
Not how the world is, is the mystical,
It is not how things are in the world
but that it is.
that is mystical, but that it exists.
The contemplation of the world sub
To view the world sub specie aeterni is
specie aeterni is its contemplation as a lim- to view it as a whole—a limited whole.
ited whole.
The feeling that the world is a limited
Feeling the world as a limited whole—it
whole is the mystical feeling.
is this that is mystical.
For an answer which cannot be exWhen the answer cannot be put into
pressed the question too cannot be ex- words, neither can the question be put into
pressed.
words.
The riddle does not exist.
The riddle does not exist.

6.51

6.52

6.521

6.522

6.53

Wenn sich eine Frage überhaupt stellen lässt, so k a n n sie auch beantwortet
werden.
Skeptizismus ist n i c h t unwiderleglich, sondern offenbar unsinnig, wenn er
bezweifeln will, wo nicht gefragt werdenkann.
Denn Zweifel kann nur bestehen, wo
eine Frage besteht; eine Frage nur, wo eine
Antwort besteht, und diese nur, wo etwas
g e s a g t werden k a n n.
Wir fühlen, daß selbst, wenn allem ö g l i c h e n wissenschaftlichen Fragen
beantwortet sind, unsere Lebensprobleme
noch gar nicht berührt sind. Freilich bleibt
dann eben keine Frage mehr; und eben
dies ist die Antwort.
Die Lösung des Problems des Lebens
merkt man am Verschwinden dieses Problems.
(Ist nicht dies der Grund, warum Menschen, denen der Sinn des Lebens nach
langen Zweifeln klar wurde, warum diesedann nicht sagen konnten, worin dieser
Sinn bestand.)Es gibt allerdings Unaussprechliches.
Dies z e i g t sich, es ist das Mystische.
Die richtige Methode der Philosophie
wäre eigentlich die: Nichts zu sagen, als
was sich sagen lässt, also Sätze der Naturwissenschaft – also etwas, was mit Philosophie nichts zu tun hat – , und dann immer,
wenn ein anderer etwas Metaphysisches
sagen wollte, ihm nachzuweisen, dass er
gewissen Zeichen in seinen Sätzen keine
Bedeutung gegeben hat. Diese MethodeIf a question can be put at all, then it
If a question can be framed at all, it iscan also be answered.
also possible to answer it.
Scepticism is not irrefutable, but palScepticism is not irrefutable, but obvipably senseless, if it would doubt where a ously nonsensical, when it tries to raisequestion cannot be asked.
doubts where no questions can be asked.
For doubt can only exist where there isa question; a question only where there isan answer, and this only where somethingcan be said.
We feel that even if all possible scientific questions be answered, the problemsof life have still not been touched at all. Of
course there is then no question left, andjust this is the answer.

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
The solution of the problem of life isseen in the vanishing of this problem.
seen in the vanishing of the problem.
(Is not this the reason why men to(Is not this the reason why those who
whom after long doubting the sense of life have found after a long period of doubt that
became clear, could not then say wherein the sense of life became clear to them havethis sense consisted?)then been unable to say what constitutedthat sense?)
There is indeed the inexpressible. This
There are, indeed, things that cannot
shows itself; it is the mystical.
be put into words. They make themselves
manifest. They are what is mystical.
The right method of philosophy would
The correct method in philosophy
be this: To say nothing except what can be would really be the following: to say
said, i.e. the propositions of natural science, nothing except what can be said, i.e.
i.e. something that has nothing to do with propositions of natural science—i.e. somephilosophy: and then always, when some- thing that has nothing to do with philosone else wished to say something meta- ophy—and then, whenever someone else
physical, to demonstrate to him that he wanted to say something metaphysical, to
had given no meaning to certain signs in demonstrate to him that he had failed
his propositions. This method would be un- to give a meaning to certain signs in his

6.54

7

wäre für den anderen unbefriedigend – er
hätte nicht das Gefühl, dass wir ihn Philosophie lehrten – aber s i e wäre die einzig
streng richtige.

satisfying to the other—he would not havethe feeling that we were teaching him philosophy—but it would be the only strictly
correct method.

Meine Sätze erläutern dadurch, dasssie der, welcher mich versteht, am Endeals unsinnig erkennt, wenn er durch sie –auf ihnen – über sie hinausgestiegen ist.
(Er muss sozusagen die Leiter wegwerfen,
nachdem er auf ihr hinaufgestiegen ist.)

My propositions are elucidatory in thisway: he who understands me finally recognizes them as senseless, when he has
climbed out through them, on them, overthem. (He must so to speak throw away
the ladder, after he has climbed up on it.)Er muss diese Sätze überwinden, dannHe must surmount these propositions;sieht er die Welt richtig.
then he sees the world rightly.
Wovon man nicht sprechen kann, darWhereof one cannot speak, thereof oneüber muss man schweigen.
must be silent.

propositions. Although it would not be satisfying to the other person—he would nothave the feeling that we were teaching him
philosophy—this method would be the only
strictly correct one.
My propositions serve as elucidationsin the following way: anyone who understands me eventually recognizes them as
nonsensical, when he has used them—assteps—to climb up beyond them. (He must,so to speak, throw away the ladder after
he has climbed up it.)He must transcend these propositions,and then he will see the world aright.
What we cannot speak about we must
pass over in silence.

Index (Pears/McGuinness)[Original note by Pears and McGuinness.]
The translators’ aim has been to include all
the more interesting words, and, in each case,either to give all the occurrences of a word,or else to omit only a few unimportant ones.
Paragraphs in the preface are referred to as
P1, P2, etc. Propositions are indicated bynumbers without points [—the points havebeen restored for the side-by-side-by-side edition—]; more than two consecutive propositions, by two numbers joined by an en-rule, as
202–2021.
In the translation it has sometimes beennecessary to use different English expressions
for the same German expression or the same
English expression for different German expressions. The index contains various devices
designed to make it an informative guide to
the German terminology and, in particular, todraw attention to some important connexions
between ideas that are more difficult to bringout in English than in German.
First, when a German expression is of any
interest in itself, it is given in brackets after
the English expression that translates it, e.g.
situation [Sachlage]; also, whenever an English expression is used to translate more thanone German expression, each of the German
expressions is given separately in numbered
brackets, and is followed by the list of passages
in which it is translated by the English expression, e.g. reality 1. [Realität], 55561, etc. 2.
[Wirklichkeit], 206, etc.
Secondly, the German expressions given inthis way sometimes have two or more English

translations in the text; and when this is so,
if the alternative English translations are ofinterest, they follow the German expression
inside the brackets, e.g. proposition [Satz:law; principle].
The alternative translations recorded bythese two devices are sometimes given in an
abbreviated way. For a German expressionneed not actually be translated by the English
expressions that it follows or precedes, as it isin the examples above. The relationship may
be more complicated. For instance, the German expression may be only part of a phrasethat is translated by the English expression,e.g. stand in a relation to one another; arerelated [sich verhalten: stand, how things;
state of things].
Thirdly, cross-references have been used to
draw attention to other important connexions
between ideas, e.g. true, cf. correct; right: anda priori, cf. advance, in.
In subordinate entries and cross-references
the catchword is indicated by ∼, unless thecatchword contains /, in which case the partpreceding / is so indicated, e.g. accident; ∼alfor accident; accidental, and state of /affairs; ∼ things for state of affairs; state ofthings. Cross-references relate to the last preceding entry or numbered bracket. When references are given both for a word in its own rightand for a phrase containing it, occurrences ofthe latter are generally not also counted as
occurrences of the former, so that both entries
should be consulted.

about [von etwas handeln: concerned with;deal with; subject-matter], 3.24, 5.44, 6.35;
cf. mention; speak; talk.
abstract, 5.5563
accident; ∼al [Zufall], 2.012, 2.0121, 3.34,
5.4733, 6.031, 6.1231, 6.1232, 6.3, 6.41
action, 5.1362, 6.422
activity, 4.112
addition, cf. logical.
adjectiv/e; ∼al, 3.323, 5.4733
advance, in [von vornherein], 5.47, 6.125; cf. apriori.
aesthetics, 6.421
affirmation [Bejahung], 4.064, 5.124, 5.1241,
5.44, 5.513, 5.514, 6.231
affix, [Index], 4.0411, 5.02
agreement
1. [stimmmen: right; true], 5.512
2. [Übereinstimmmung], 2.21, 2.222, 4.2,
4.4, 4.42–4.431, 4.462
analysis [Analyse], 3.201, 3.25, 3.3442, 4.1274,
4.221, 5.5562; cf. anatomize; dissect;resolve.
analytic, 6.11
anatomize [auseinanderlegen], 3.261; cf.
analysis.
answer, 4.003, 4.1274, 5.4541, 5.55, 5.551,
6.5–6.52
apparent, 4.0031, 5.441, 5.461; cf. pseudo-.
application [Anwendung: employment], 3.262,
3.5, 5.2521, 5.2523, 5.32, 5.5, 5.5521,
5.557, 6.001, 6.123, 6.126
a priori, 2.225, 3.04, 3.05, 5.133, 5.4541,
5.4731, 5.55, 5.5541, 5.5571, 5.634, 6.31,
6.3211, 6.33, 6.34, 6.35; cf. advance, in.

arbitrary, 3.315, 3.322, 3.342, 3.3442, 5.02,
5.473, 5.47321, 5.554, 6.124, 6.1271
argument, 3.333, 4.431, 5.02, 5.251, 5.47,
5.523, 5.5351; cf. truth-argument.
∼-place, 2.0131, 4.0411, 5.5351
arithmetic, 4.4611, 5.451
arrow, 3.144, 4.461
articulate [artikuliert], 3.141, 3.251∼d [gegliedert], 4.032
ascribe [aussagen: speak; state; statement;tell], 4.1241
assert
1. [behaupten], 4.122, 4.21, 6.2322
2. [zusprechen], 4.124
asymmetry, 6.3611
axiom, 6.341∼ of infinity, 5.535∼ of reducibility, 6.1232, 6.1233
bad, 6.43
basis, 5.21, 5.22, 5.234, 5.24, 5.25, 5.251,
5.442, 5.54
beautiful, 4.003
belief, 5.1361, 5.1363, 5.541, 5.542, 6.33,
6.3631bound; ∼ary [Grenze: delimit; limit], 4.112,
4.463
brackets, 4.441, 5.46, 5.461
build [Bau: construction], 6.341
calculation, 6.126, 6.2331
cardinal, cf. number.
case, be the1. [der Fall sein], 1, 1.12, 1.21, 2, 2.024,
3.342, 4.024, 5.1362, 5.5151, 5.541,
5.5542, 6.23
2. [So-Sein], 6.41

causality, 5.136–5.1362, 6.32, 6.321, 6.36,
6.3611, 6.362; cf. law.
certainty [Gewissheit], 4.464, 5.152, 5.156,
5.525, 6.3211
chain, 2.03; cf. concatenation.
clarification, 4.112
class [Klasse: set], 3.311, 3.315, 4.1272, 6.031
clear, P2, 3.251, 4.112, 4.115, 4.116
make ∼ [erklären: definition; explanation],
5.452
colour, 2.0131, 2.0232, 2.0251, 2.171, 4.123,
6.3751∼-space, 2.0131
combination
1. [Kombination], 4.27, 4.28, 5.46; cf. rule,
combinatory; truth-∼.
2. [Verbindung: connexion], 2.01, 2.0121,
4.0311, 4.221, 4.466, 4.4661, 5.131,
5.451, 5.515, 6.12, 6.1201, 6.121,
6.1221, 6.124, 6.23, 6.232; cf. sign.
common, 2.022, 2.16, 2.17, 2.18, 2.2, 3.31,
3.311, 3.317, 3.321, 3.322, 3.333, 3.341,
3.3411, 3.343–3.3441, 4.014, 4.12, 5.11,
5.143, 5.152, 5.24, 5.47, 5.4733, 5.512,
5.513, 5.5261, 6.022
comparison, 2.223, 3.05, 4.05, 6.2321, 6.3611
complete
1. [vollkommen: folly], 5.156
2. [vollstädig], 5.156;
analyse ∼ly, 3.201, 3.25;
describe ∼ly, 2.0201, 4.023, 4.26, 5.526,
6.342
complex, 2.0201, 3.1432, 3.24, 3.3442, 4.1272,
4.2211, 4.441, 5.515, 5.5423
composite [zummmengesetzt], 2.021, 3.143,
3.1431, 3.3411, 4.032, 4.2211, 5.47, 5.5261,
5.5421, 5.55

compulsion, 6.37
concatenation [Verkettung], 4.022; cf. chain.
concept [Begriff : primitive idea], 4.063,
4.126–4.1274, 4.431, 5.2523, 5.521, 5.555,
6.022; cf. formal ∼; pseudo-∼.
∼ual notation [Begriffsschrift], 3.325,
4.1272, 4.1273, 4.431, 5.533, 5.534∼-word, 4.1272
concerned with [von etwas handeln: about;deal with; subject-matter], 4.011, 4.122
concrete, 5.5563
condition, 4.41, 4.461, 4.462; cf. truth-∼.
configuration, 2.0231, 2.0271, 2.0272, 3.21
connexion
1. [Verbindung: combination], 6.124, 6.232
2. [Zusammenhang: nexus], 2.0122, 2.032,
2.15, 4.03, 5.1311, 5.1362, 6.361, 6.374
consequences, 6.422
conservation, cf. law.
constant, 3.312, 3.313, 4.126, 4.1271, 5.501,
5.522; cf. logical ∼.
constituent [Bestandteil], 2.011, 2.0201, 3.24,
3.315, 3.4, 4.024, 4.025, 5.4733, 5.533,
5.5423, 6.12
construct [bilden], 4.51, 5.4733, 5.475, 5.501,
5.503, 5.512, 5.514, 5.5151, 6.126, 6.1271
construction
1. [Bau: build], 4.002, 4.014, 5.45, 5.5262,
6.002
2. [Konstruktion], 4.023, 4.5, 5.233, 5.556,
6.343
contain [enthalten], 2.014, 2.203, 3.02, 3.13,
3.24, 3.332, 3.333, 5.121, 5.122, 5.44, 5.47
content
1. [Gehalt], 6.111
2. [Inhalt], 2.025, 3.13, 3.31
continuity, cf. law.

contradiction
1. [Kontradiktion], 4.46–4.4661, 5.101,
5.143, 5.152, 5.525, 6.1202, 6.3751
2. [Widerspruch], 3.032, 4.1211, 4.211,
5.1241, 6.1201, 6.3751; cf. law of ∼.
convention
1. [Abmachung], 4.002
2. [Übereinkunft], 3.315, 5.02
co-ordinate, 3.032, 3.41, 3.42, 5.64
copula, 3.323
correct [richtig], 2.17, 2.173, 2.18, 2.21, 3.04,
5.5302, 5.62, 6.2321; cf. incorrect; true.
correlate [zuordnen], 2.1514, 2.1515, 4.43,
4.44, 5.526, 5.542, 6.1203
correspond [entsprechen], 2.13, 3.2, 3.21,
3.315, 4.0621, 4.063, 4.28, 4.441, 4.466,
5.5542
creation, 3.031, 5.123
critique of language, 4.0031
cube, 5.5423

description [Beschreibung], 2.0201, 2.02331,
3.144, 3.24, 3.317, 3.33, 4.016, 4.023,
4.0641, 4.26, 4.5, 5.02
∼ of the world [Weltb.], 6.341, 6.343, 6.3432
designate [bezeichnen: sign; signify], 4.063
determin/ate [bestimmt], 2.031, 2.032, 2.14,
2.15, 3.14, 3.23, 3.251, 4.466, 6.124; cf.
indeterminateness; undetermined.
∼e, 1.11, 1.12, 2.0231, 2.05, 3.327, 3.4, 3.42,
4.063, 4.0641, 4.431, 4.463
difference [Verschiedenheit], 2.0233, 5.135,
5.53, 6.232, 6.3751
display [aufweisen], 2.172, 4.121; cf. show.
dissect [zerlegen], 3.26; cf. analysis.
doctrine [Lehre: theory], 4.112, 6.13
doubt, 6.51, 6.521
dualism, 4.128
duration, 6.4311
dynamical model, 4.04

effort, least, cf. law.
Darwin, 4.1122
element, 2.13–2.14, 2.15, 2.151, 2.1514,
deal with [von etwas handeln: about;2.1515, 3.14, 3.2, 3.201, 3.24, 3.42
concerned with; subject-matter], 2.0121∼ary proposition [Elementarsatz],
death, 6.431–6.4312
4.21–4.221, 4.23, 4.24, 4.243–4.26,
deduce [folgern], 5.132–5.134; cf. infer.
4.28–4.42, 4.431, 4.45, 4.46, 4.51, 4.52,
definition
5, 5.01, 5.101, 5.134, 5.152, 5.234,
1. [Definition], 3.24, 3.26–3.262, 3.343,
5.3–5.32, 5.41, 5.47, 5.5, 5.524, 5.5262,
4.241, 5.42, 5.451, 5.452, 5.5302, 6.02
5.55, 5.555–5.5571, 6.001, 6.124,
2. [Erklärung: clear, make; explanation],
6.3751
5.154
elucidation [Erläuterung], 3.263, 4.112, 6.54
delimit [begrenzen: bound; limit], 5.5262
empirical, 5.5561
depiction [Abbildung: form, logico-pictorial;
employment
form, pictorial; pictorial], 2.16–2.172, 2.18,
1. [Anwendung: application], 3.202, 3.323,
5.452
2.19, 2.2, 2.201, 4.013, 4.014, 4.015, 4.016,
4.041
2. [Verwendung: use], 3.327
derive [ableiten], 4.0141, 4.243, 6.127, 6.1271; enumeration, 5.501
cf. infer.
equal value, of [gleichwertig], 6.4

equality/, numerical [Zahlengleichheit], 6.022
sign of ∼ [Gleichheitszeichen: identity, signfor], 6.23, 6.232
equation [Gleichung], 4.241, 6.2, 6.22, 6.232,
6.2323, 6.2341, 6.24
equivalent, cf. meaning, ∼ n. [äquivalent],
5.232, 5.2523, 5.47321, 5.514, 6.1261
essence [Wesen], 2.011, 3.143, 3.1431, 3.31,
3.317, 3.34–3.3421, 4.013, 4.016, 4.027,
4.03, 4.112, 4.1121, 4.465, 4.4661, 4.5, 5.3,
5.471, 5.4711, 5.501, 5.533, 6.1232, 6.124,
6.126, 6.127, 6.232, 6.2341
eternity, 6.4311, 6.4312; cf. sub specie aeterni.
ethics, 6.42–6.423
everyday language [Umgangssprache], 3.323,
4.002, 5.5563
existence
1. [Bestehen: hold; subsist], 2, 2.0121,
2.04–2.06, 2.062, 2.11, 2.201, 4.1, 4.122,
4.124, 4.125, 4.2, 4.21, 4.25, 4.27, 4.3,
5.131, 5.135
2. [Existenz], 3.032, 3.24, 3.323, 3.4, 3.411,
4.1274, 5.5151
experience [Erfahrung], 5.552, 5.553, 5.634,
6.1222, 6.363
explanation [Erklärung: clear, make;definition], 3.263, 4.02, 4.021, 4.026, 4.431,
5.5422, 6.371, 6.372
exponent, 6.021
expression [Ausdruck: say], P3, 3.1, 3.12, 3.13,
3.142, 3.1431, 3.2, 3.24, 3.251, 3.262,
3.31–3.314, 3.318, 3.323, 3.33, 3.34, 3.341,
3.3441, 4.002, 4.013, 4.03, 4.0411, 4.121,
4.124, 4.125, 4.126, 4.1272, 4.1273, 4.241,
4.4, 4.43, 4.431, 4.441, 4.442, 4.5, 5.131,
5.22, 5.24, 5.242, 5.31, 5.476, 5.503, 5.5151,
5.525, 5.53, 5.5301, 5.535, 5.5352, 6.124,

6.1264, 6.21, 6.23, 6.232–6.2323, 6.24
mode of ∼ [Ausdrucksweise], 4.015, 5.21,
5.526
external, 2.01231, 2.0233, 4.023, 4.122, 4.1251

pictorial ∼ [Form der Abbildung: depiction;
pictorial], 2.15, 2.151, 2.17, 2.172,
2.181, 2.22
representational ∼ [Form der Darstellung:present; represent], 2.173, 2.174
formal [formal], 4.122, 5.501∼ concept, 4.126–4.1273∼ property, 4.122, 4.124, 4.126, 4.1271,
5.231, 6.12, 6.122∼ relation [Relation], 4.122, 5.242
formulate [angeben: give; say], 5.5563
free will, 5.1362
Frege, P6, 3.143, 3.318, 3.325, 4.063, 4.1272,
4.1273, 4.431, 4.442, 5.02, 5.132, 5.4, 5.42,
5.451, 5.4733, 5.521, 6.1271, 6.232
fully [vollkommen: complete], ∼ generalized,
5.526, 5.5261
function [Funktion], 3.318, 3.333, 4.126,
4.1272, 4.12721, 4.24, 5.02, 5.2341, 5.25,
5.251, 5.44, 5.47, 5.501, 5.52, 5.5301; cf.
truth-∼.
Fundamental Laws of Arithmetic[Grundgesetze der Arithmetik], 5.451; cf.
primitive proposition.
future, 5.1361, 5.1362

fact [Tatsache], 1.1–1.2, 2, 2.0121, 2.034, 2.06,
2.1, 2.141, 2.16, 3, 3.14, 3.142, 3.143, 4.016,
4.0312, 4.061, 4.063, 4.122, 4.1221, 4.1272,
4.2211, 4.463, 5.156, 5.43, 5.5151, 5.542,
5.5423, 6.2321, 6.43, 6.4321; cf. negative ∼.
fairy tale, 4.014
false [falsch: incorrect], 2.0212, 2.21, 2.22,
2.222–2.224, 3.24, 4.003, 4.023,
4.06–4.063, 4.25, 4.26, 4.28, 4.31, 4.41,
4.431, 4.46, 5.512, 5.5262, 5.5351, 6.111,
6.113, 6.1203; cf. wrong.
fate, 6.372, 6.374
feature [Zug], 3.34, 4.1221, 4.126
feeling, 4.122, 6.1232, 6.45
finite, 5.32
follow, 4.1211, 4.52, 5.11–5.132, 5.1363–5.142,
5.152, 5.43, 6.1201, 6.1221, 6.126
foresee, 4.5, 5.556
form [Form], 2.0122, 2.0141, 2.022–2.0231,
2.025–2.026, 2.033, 2.18, 3.13, 3.31, 3.312,
3.333, 4.002, 4.0031, 4.012, 4.063, 4.1241,
4.1271, 4.241, 4.242, 4.5, 5.131, 5.156,
general [allgemein], 3.3441, 4.0141, 4.1273,
5.231, 5.24, 5.241, 5.2522, 5.451, 5.46, 5.47,
4.411, 5.1311, 5.156, 5.242, 5.2522, 5.454,
5.501, 5.5351, 5.542, 5.5422, 5.55, 5.554,
5.46, 5.472, 5.521, 5.5262, 6.031, 6.1231,
5.5542, 5.555, 5.556, 5.6331, 6, 6.002, 6.01,
6.3432
6.022, 6.03, 6.1201, 6.1203, 6.1224, 6.1264,∼ form, 3.312, 4.1273, 4.5, 4.53, 5.46, 5.47,
6.32, 6.34–6.342, 6.35, 6.422; cf. ∼al;
5.471, 5.472, 5.54, 6, 6.002, 6.01, 6.022,
general ∼; propositional ∼; series of ∼s.
6.03
logical ∼, 2.0233, 2.18, 2.181, 2.2, 3.315,∼-ity-sign, 3.24, 4.0411, 5.522, 5.523, 6.1203
3.327, 4.12, 4.121, 4.128, 5.555, 6.23,∼ validity, 6.1231, 6.1232
generalization [verallgemeinerung], 4.0411,
6.33
logico-pictorial ∼ [logische Form der
4.52, 5.156, 5.526, 5.5261, 6.1231; cf. fully.
Abbildung], 2.2
geometry, 3.032, 3.0321, 3.411, 6.35

give [angeben: formulate; say], 3.317, 4.5,
5.4711, 5.55, 5.554, 6.35
given [gegeben], 2.0124, 3.42, 4.12721, 4.51,
5.442, 5.524, 6.002, 6.124
God, 3.031, 5.123, 6.372, 6.432
good, 4.003, 6.43
grammar, cf. logical.
happy, 6.374
Hertz, 4.04, 6.361
hierarchy, 5.252, 5.556, 5.5561
hieroglyphic script, 4.016
higher, 6.42, 6.432
hold [bestehen: existence; subsist], 4.014
how [wie], 6.432, 6.44; cf. stand, ∼ things.
∼) (what, 3.221, 5.552
hypothesis, 4.1122, 5.5351, 6.36311
idea, cf. primitive ∼.
1. [Gedanke: thought], musical ∼, 4.014
2. [Vorstellung: present; represent], 5.631
idealist, 4.0412
identical [identisch], 3.323, 4.003, 4.0411,
5.473, 5.4733, 5.5303, 5.5352, 6.3751; cf.
difference.
identity [Gleichheit], 5.53
sign for ∼ [Gleichheitszeichen: equality, signof], 3.323, 5.4733, 5.53, 5.5301, 5.533;
cf. equation.
illogical [unlogisch], 3.03, 3.031, 5.4731
imagine [sich etwas denken: think], 2.0121,
2.022, 4.01, 6.1233
immortality, 6.4312
impossibility [Unmöglichkeit], 4.464, 5.525,
5.5422, 6.375, 6.3751
incorrect
1. [falsch: false], 2.17, 2.173, 2.18
2. [unrichtig], 2.21

independence [Selbständigkeit], 2.0122, 3.261 language [Sprache], P2, P4, 3.032, 3.343,
independent [unabhängig], 2.024, 2.061, 2.22,
4.001–4.0031, 4.014, 4.0141, 4.025, 4.121,
4.061, 5.152, 5.154, 5.451, 5.5261, 5.5561,
4.125, 5.4731, 5.535, 5.6, 5.62, 6.12, 6.233,
6.373
6.43; cf. critique of ∼; everyday ∼; sign-∼.
indeterminateness [Unbestimmtheit], 3.24
law
indicate
1. [Gesetz: minimum-principle; primitive
1. [anzeigen], 3.322, 6.121, 6.124
proposition], 3.031, 3.032, 3.0321,
2. [auf etwas zeigen: manifest; show],
4.0141, 5.501, 6.123, 6.3–6.3211,
2.02331, 4.063
6.3431, 6.35, 6.361, 6.363, 6.422;
individuals, 5.553∼ of causality [Kausalitätsg.], 6.32,
induction, 6.31, 6.363
6.321;
infer [schließen], 2.062, 4.023, 5.1311, 5.132,∼ of conservation [Erhaltungsg.], 6.33;
5.135, 5.1361, 5.152, 5.633, 6.1224, 6.211;∼ of contradiction [G. des Widerspruchs],
cf. deduce; derive.
6.1203, 6.123;
infinite, 2.0131, 4.2211, 4.463, 5.43, 5.535,∼ of least action [G. der kleinsten
6.4311
Wirkung], 6.321, 6.3211;
infinity, cf. axiom.
∼ of nature [Naturg.], 5.154, 6.34, 6.36,
inner, 4.0141, 5.1311, 5.1362
6.371, 6.372
internal, 2.01231, 3.24, 4.014, 4.023,
2. [Satz: principle of sufficient reason;
4.122–4.1252, 5.131, 5.2, 5.21, 5.231, 5.232
proposition], 6.34;
intuition [Anschauung], 6.233, 6.2331∼ of continuity [S. von der Kontinuität],
intuitive [anschaulich], 6.1203
6.34;∼ of least effort [S. von kleinsten
judgement [Urteil], 4.063, 5.5422
Aufwande], 6.34∼-stroke [Urteilstrich], 4.442
life, 5.621, 6.4311, 6.4312, 6.52, 6.521
Julius Caesar, 5.02
limit [Grenze: bound; delimit], P3, P4, 4.113,
4.114, 4.51, 5.143, 5.5561, 5.6–5.62, 5.632,
Kant, 6.36111
5.641, 6.4311, 6.45
knowlogic; ∼al, 2.012, 2.0121, 3.031, 3.032, 3.315,
1. [kennen], 2.0123, 2.01231, 3.263, 4.021,
3.41, 3.42, 4.014, 4.015, 4.023, 4.0312,
4.243, 6.2322; cf. theory of knowledge.
4.032, 4.112, 4.1121, 4.1213, 4.126, 4.128,
2. [wissen], 3.05, 3.24, 4.024, 4.461, 5.1362,
4.466, 5.02, 5.1362, 5.152, 5.233, 5.42,
5.156, 5.5562 [—the previous entry
5.43, 5.45–5.47, 5.472–5.4731, 5.47321,
was mistakenly listed as 5.562 in Pears
5.522, 5.551–5.5521, 5.555, 5.5562–5.557,
and McGuinness’s original index—],5.61, 6.1–6.12, 6.121, 6.122, 6.1222–6.2,
6.3211, 6.33, 6.36311
6.22, 6.234, 6.3, 6.31, 6.3211, 6.342,
6.3431, 6.3631, 6.37, 6.374–6.3751; cf.

form, ∼al; illogical.
∼al addition, 5.2341∼al constant, 4.0312, 5.4, 5.441, 5.47
∼al grammar, 3.325∼al multiplication, 5.2341∼al object, 4.441, 5.4∼al picture, 2.18–2.19, 3, 4.03
∼al place, 3.41–3.42, 4.0641∼al product, 3.42, 4.465, 5.521, 6.1271,
6.3751∼al space, 1.13, 2.11, 2.202, 3.4, 3.42, 4.463∼al sum, 3.42, 5.521∼al syntax, 3.325, 3.33, 3.334, 3.344, 6.124∼o-pictorial, cf. form.
∼o-syntactical, 3.327
manifest [sich zeigen: indicate; show], 4.122,
5.24, 5.4, 5.513, 5.515, 5.5561, 5.62, 6.23,
6.36, 6.522
material, 2.0231, 5.44
mathematics, 4.04–4.0411, 5.154, 5.43, 5.475,
6.031, 6.2–6.22, 6.2321, 6.233, 6.234–6.24
Mauthner, 4.0031
mean [meinen], 3.315, 4.062, 5.62
meaning [Bedeutung: signify], 3.203, 3.261,
3.263, 3.3, 3.314, 3.315, 3.317, 3.323,
3.328–3.331, 3.333, 4.002, 4.026, 4.126,
4.241–4.243, 4.466, 4.5, 5.02, 5.31, 5.451,
5.461, 5.47321, 5.4733, 5.535, 5.55, 5.6,
5.62, 6.124, 6.126, 6.232, 6.2322, 6.53
equivalent in ∼ [Bedeutungsgleichheit],
4.243, 6.2323∼ful [bedeutungsvoll], 5.233∼less [bedeutungslos], 3.328, 4.442, 4.4661,
5.47321mechanics, 4.04, 6.321, 6.341–6.343, 6.3432
mention [von etwas reden: talk about], 3.24,
3.33, 4.1211, 5.631, 6.3432; cf. about.

metaphysical, 5.633, 5.641, 6.53
method, 3.11, 4.1121, 6.121, 6.2, 6.234–6.24,
6.53; cf. projection, ∼ of; zero-∼.
microcosm, 5.63
minimum-principle [Minimum-Gesetz: law],
6.321
mirror, 4.121, 5.511, 5.512, 5.514, 6.13∼-image [Spiegelbild: picture], 6.13
misunderstanding, P2
mode, cf. expression; signification.
model, 2.12, 4.01, 4.463; cf. dynamical ∼.
modus ponens, 6.1264
monism, 4.128
Moore, 5.541
multiplicity, 4.04–4.0412, 5.475
music, 3.141, 4.011, 4.014, 4.0141
mystical, 6.44, 6.45, 6.522

5.512, 5.514, 6.231
negative [negativ], 4.463, 5.513, 5.5151∼ fact, 2.06, 4.063, 5.5151
network, 5.511, 6.341, 6.342, 6.35
Newton, 6.341, 6.342
nexus
1. [Nexus], 5.136, 5.1361
2. [Zusammenhang: connexion], 3.3, 4.22,
4.23
non-proposition, 5.5351
nonsense [Unsinn], P4, 3.24, 4.003, 4.124,
4.1272, 4.1274, 4.4611, 5.473, 5.5303,
5.5351, 5.5422, 5.5571, 6.51, 6.54; cf. sense,
have no.
notation, 3.342, 3.3441, 5.474, 5.512–5.514,
6.1203, 6.122, 6.1223; cf. conceptual ∼.
number
1. [Anzahl], 4.1272, 5.474–5.476, 5.55, 5.553,
name
6.1271
1. [Name], 3.142, 3.143, 3.144, 3.202, 3.203,
2. [Zahl: integer], 4.1252, 4.126, 4.1272,
3.22, 3.26, 3.261, 3.3, 3.314, 3.3411,
4.12721, 4.128, 5.453, 5.553, 6.02,
4.0311, 4.126, 4.1272, 4.22, 4.221, 4.23,
6.022; cf. equality, numerical;
4.24, 4.243, 4.5, 5.02, 5.526, 5.535, 5.55,
privileged ∼s; series of ∼s; variable ∼.
6.124; cf. variable ∼.
cardinal ∼, 5.02;
general ∼ [Gattungsn.], 6.321∼-system, 6.341
proper ∼ of a person [Personenn.], 3.323
2. [benennen; nennen], 3.144, 3.221
object [Gegenstand], 2.01, 2.0121,
natur/e, 2.0123, 3.315, 5.47, 6.124; cf. law of
2.0123–2.0124, 2.0131–2.02, 2.021,∼e.
2.023–2.0233, 2.0251–2.032, 2.13, 2.15121,∼al phenomena, 6.371
3.1431, 3.2, 3.203–3.221, 3.322, 3.3411,∼al science, 4.11, 4.111, 4.1121–4.113,
4.023, 4.0312, 4.1211, 4.122, 4.123, 4.126,
4.127, 4.1272, 4.12721, 4.2211, 4.431,
6.111, 6.4312, 6.53
4.441, 4.466, 5.02, 5.123, 5.1511, 5.4, 5.44,
necessary, 4.041, 5.452, 5.474, 6.124; cf.
unnecessary.
5.524, 5.526, 5.53–5.5302, 5.541, 5.542,
negation
5.5561, 6.3431; cf. thing.
1. [Negation], 5.5, 5.502
obvious [sich von selbst verstehen: say;
2. [Verneinung], 3.42, 4.0621, 4.064, 4.0641,
understand], 6.111; cf. self-evidence.
5.1241, 5.2341, 5.254, 5.44, 5.451, 5.5, Occam, 3.328, 5.47321

occur [vorkommen], 2.012–2.0123, 2.0141,
3.24, 3.311, 4.0621, 4.1211, 4.23, 4.243,
5.25, 5.451, 5.54, 5.541, 6.1203
operation, 4.1273, 5.21–5.254, 5.4611, 5.47,
5.5, 5.503, 6.001–6.01, 6.021, 6.126; cf.
sign for a logical ∼; truth-∼.
oppos/ed; ∼ite [entgegengesetzt], 4.0621, 4.461,
5.1241, 5.513
order, 4.1252, 5.5563, 5.634
paradox, Russell’s, 3.333
particle, 6.3751
perceive, 3.1, 3.11, 3.32, 5.5423
phenomenon, 6.423; cf. natural ∼.
philosophy, P2, P5, 3.324, 3.3421, 4.003,
4.0031, 4.111–4.115, 4.122, 4.128, 5.641,
6.113, 6.211, 6.53
physics, 3.0321, 6.321, 6.341, 6.3751
pictorial
1. [abbilden: depict; form, logico-∼], 2.15,
2.151, 2.1513, 2.1514, 2.17, 2.172,
2.181, 2.22; cf. form, ∼.
2. [bildhaftig], 4.013, 4.015
picture [Bild: mirror-image; tableau vivant],
2.0212, 2.1–2.1512, 2.1513–3.01, 3.42,
4.01–4.012, 4.021, 4.03, 4.032, 4.06, 4.462,
4.463, 5.156, 6.341, 6.342, 6.35; cf. logical∼; prototype.
place [Ort], 3.411, 6.3751; cf. logical ∼.
point-mass [materieller Punkt], 6.3432
positive, 2.06, 4.063, 4.463, 5.5151
possible, 2.012, 2.0121, 2.0123–2.0141, 2.033,
2.15, 2.151, 2.201–2.203, 3.02, 3.04, 3.11,
3.13, 3.23, 3.3421, 3.3441, 3.411, 4.015,
4.0312, 4.124, 4.125, 4.2, 4.27–4.3, 4.42,
4.45, 4.46, 4.462, 4.464, 4.5, 5.252, 5.42,
5.44, 5.46, 5.473, 5.4733, 5.525, 5.55, 5.61,
6.1222, 6.33, 6.34, 6.52; cf. impossibility;truth-possibility.
postulate [Forderung: requirement], 6.1223
predicate, cf. subject.
present
1. [darstellen: represent], 3.312, 3.313, 4.115
2. [vorstellen: Idea; represent], 2.11, 4.0311
presuppose [voraussetzen], 3.31, 3.33, 4.1241,
5.515, 5.5151, 5.61, 6.124
primitive idea [Grundbegriff ], 4.12721, 5.451,
5.476
primitive proposition [Grundgesetz], 5.43,
5.452, 6.127, 6.1271; cf. Fundamental
Laws of Arithmetic; law.
primitive sign [Urzeichen], 3.26, 3.261, 3.263,
5.42, 5.45, 5.451, 5.46, 5.461, 5.472
Principia Mathematica, 5.452
principle of sufficient reason [Satz vom
Grunde: law; proposition], 6.34, 6.35
Principles of Mathematics, 5.5351
privileged [ausgezeichnet], ∼ numbers, 4.128,
5.453, 5.553
probability, 4.464, 5.15–5.156
problem
1. [Fragestellung: question], P2, 5.62
2. [Problem], P2, 4.003, 5.4541, 5.535, 5.551,
5.5563, 6.4312, 6.521
product, cf. logical.
project/ion; ∼ive, 3.11–3.13, 4.0141
method of ∼ion, 3.11
proof [Beweis], 6.126, 6.1262, 6.1263–6.1265,
6.2321, 6.241
proper, cf. name.
property [Eigenschaft], 2.01231, 2.0231,
2.0233, 2.02331, 4.023, 4.063,
4.122–4.1241, 5.473, 5.5302, 6.111, 6.12,
6.121, 6.126, 6.231, 6.35; cf. formal ∼.
proposition [Satz: law; principle], 2.0122,

2.0201, 2.0211, 2.0231, 3.1 (& passim
4.4661, 5.131, 5.1311, 5.2–5.22, 5.42,
thereafter); cf. non-∼; primitive ∼;5.461, 5.4733, 5.5151, 5.5261, 5.5301;
pseudo-∼; variable, ∼al; variable ∼.
cf. pseudo-.
∼al form, 3.312, 4.0031, 4.012, 4.5, 4.53,
2. [Relation], 4.122, 4.123, 4.125, 4.1251,
5.131, 5.1311, 5.156, 5.231, 5.24, 5.241,
5.232, 5.42, 5.5301, 5.541, 5.553,
5.451, 5.47, 5.471, 5.472, 5.54–5.542,
5.5541; cf. formal.
5.5422, 5.55, 5.554, 5.555, 5.556, 6,
3. stand in a ∼ to one another; are related
6.002[sich verhalten: stand, how things;∼al sign, 3.12, 3.14, 3.143, 3.1431, 3.2, 3.21,
state of things], 2.03, 2.14, 2.15, 2.151,
3.332, 3.34, 3.41, 3.5, 4.02, 4.44, 4.442,
3.14, 5.5423
5.31
represent
prototype [Urbild], 3.24, 3.315, 3.333, 5.522,
1. [darstellen: present], 2.0231, 2.173, 2.174,
5.5351; cf. picture.
2.201–2.203, 2.22, 2.221, 3.032, 3.0321,
pseudo-, cf. apparent.
4.011, 4.021, 4.031, 4.04, 4.1, 4.12,∼-concept, 4.1272
4.121, 4.122, 4.124, 4.125, 4.126,∼-proposition, 4.1272, 5.534, 5.535, 6.2
4.1271, 4.1272, 4.24, 4.31, 4.462, 5.21,∼relation, 5.461
6.1203, 6.124, 6.1264; cf. form,
psychology, 4.1121, 5.541, 5.5421, 5.641,∼ational.
6.3631, 6.423
2. [vorstellen: idea; present], 2.15
punishment, 6.422
representative, be the ∼ of [vertreten], 2.131,
3.22, 3.221, 4.0312, 5.501
question [Frage: problem], 4.003, 4.1274,
requirement [Forderung: postulate], 3.23
5.4541, 5.55, 5.551, 5.5542, 6.5–6.52
resolve, cf. analysis.
1. [auflösen], 3.3442
range [Spielraum], 4.463, 5.5262; cf. space.
2. [zerlegen], 2.0201
real [wirklich], 2.022, 4.0031, 5.461
reward,6.422
realism, 5.64
riddle, 6.4312, 6.5
realityright [stimmen: agreement; true], 3.24
1. [Realität], 5.5561, 5.64
rule [Regel], 3.334, 3.343, 3.344, 4.0141,
2. [Wirklichkeit], 2.06, 2.063, 2.12, 2.1511,
5.47321, 5.476, 5.512, 5.514
2.1512, 2.1515, 2.17, 2.171, 2.18, 2.201,
combinatory∼ [Kombinationsr.], 4.442
2.21, 2.222, 2.223, 4.01, 4.011, 4.021,∼ dealing with signs [Zeichenr.], 3.331,
4.023, 4.05, 4.06, 4.0621, 4.12, 4.121,
4.241, 6.02, 6.126
4.462, 4.463, 5.512
reducibility, cf. axiom.
Russell, P6, 3.318, 3.325, 3.331, 3.333, 4.0031,
4.1272–4.1273, 4.241, 4.442, 5.02, 5.132,
relation
1. [Beziehung], 2.1513, 2.1514, 3.12, 3.1432,
5.252, 5.4, 5.42, 5.452, 5.4731, 5.513,
3.24, 4.0412, 4.061, 4.0641, 4.462,
5.521, 5.525, 5.5302, 5.532, 5.535, 5.5351,

5.541, 5.5422, 5.553, 6.123, 6.1232

have no ∼; lack ∼; without ∼ [sinnlos], 4.461,
5.132, 5.1362, 5.5351; cf. nonsense.
say∼ of touch [Tastsinn], 2.0131
1. [angeben: give], 5.5571
series [Reihe], 4.1252, 4.45, 5.1, 5.232, 6.02
2. [audrücksen: expression], 5.5151∼ of forms [Formenr.], 4.1252, 4.1273, 5.252,
3. [aussprechen: words, put into], ∼ clearly,
5.2522, 5.501
3.262∼ of numbers [Zahlenr.], 4.1252
4. [sagen], can be said, P3, 3.031, 4.115,
set [Klasse: class], 3.142
4.1212, 5.61, 5.62, 6.36, 6.51, 6.53;said) (shown, 4.022, 4.1212, 5.535, 5.62, show [zeigen: indicate; manifest], 3.262, 4.022,
4.0621, 4.0641, 4.121–4.1212, 4.126, 4.461,
6.36;
5.1311, 5.24, 5.42, 5.5261, 5.5421, 5.5422,∼ nothing, 4.461, 5.142, 5.43, 5.4733,
5.631, 6.12, 6.1201, 6.1221, 6.126, 6.127,
5.513, 5.5303, 6.11, 6.121, 6.342,
6.22, 6.232; cf. display; say.
6.35
sign
[Zeichen], 3.11, 3.12, 3.1432, 3.201–3.203,
5. [sich von selbst verstehen: obvious;
3.21, 3.221, 3.23, 3.261–3.263, 3.315,
understand], ∼ing, go without, 3.334,
3.32–3.322, 3.325–3.334, 3.3442, 4.012,
6.23414.026, 4.0312, 4.061, 4.0621, 4.126, 4.1271,
scaffolding, 3.42, 4.023, 6.124
4.1272, 4.241–4.243, 4.431–4.441, 4.466,
scepticism, 6.51
4.4661, 5.02, 5.451, 5.46, 5.473,
schema, 4.31, 4.43, 4.441, 4.442, 5.101, 5.151,
5.4732–5.4733, 5.475, 5.501, 5.512, 5.515,
5.31
5.5151, 5.53, 5.5541, 5.5542, 6.02, 6.1203,
science, 6.34, 6.341, 6.52; cf. natural ∼.
6.124, 6.126, 6.1264, 6.53; cf. primitive ∼;scope, 4.0411
propositional ∼; rule dealing with ∼s;self, the [das Ich], 5.64, 5.641
simple ∼.
self-evidence [Einleuchten], 5.1363, 5.42,
be a ∼ for [bezeichnen: designate; signify],
5.4731, 5.5301, 6.1271; cf. obvious.
5.42
sense [Sinn; sinnvoll], P2, 2.0211, 2.221,
combinationof ∼s [Zeichenverbindung],
2.222, 3.11, 3.13, 3.142, 3.1431, 3.144,
4.466, 5.451
3.23, 3.3, 3.31, 3.326, 3.34, 3.341, 3.4,
∼ for a logical operation [logisches
4.002, 4.011, 4.014, 4.02–4.022,
Operationsz.], 5.4611
4.027–4.031, 4.032, 4.061, 4.0621–4.064,∼-language[Zeichensprache], 3.325, 3.343,
4.1211, 4.122, 4.1221, 4.1241, 4.126, 4.2,
4.011, 4.1121, 4.1213, 4.5, 6.124
4.243, 4.431, 4.465, 4.52, 5.02, 5.122,
signif/y
5.1241, 5.2341, 5.25, 5.2521, 5.4, 5.42,
1. [bedeuten: meaning], 4.115
5.44, 5.46, 5.4732, 5.4733, 5.514, 5.515,
2. [bezeichnen: designate: sign], 3.24, 3.261,
5.5302, 5.5542, 5.631, 5.641, 6.124, 6.126,
3.317, 3.321, 3.322, 3.333, 3.334,
6.232, 6.41, 6.422, 6.521
3.3411, 3.344, 4.012, 4.061, 4.126,
have the same ∼ [gleichsinnig], 5.515

4.127, 4.1272, 4.243, 5.473, 5.4733,
5.476, 5.5261, 5.5541, 6.111;
mode of ∼ication [Bezeichnungsweise],
3.322, 3.323, 3.325, 3.3421, 4.0411,
5.1311
similarity, 4.0141, 5.231
simple, 2.02, 3.24, 4.21, 4.24, 4.51, 5.02,
5.4541, 5.553, 5.5563, 6.341, 6.342, 6.363,
6.3631;∼ sign, 3.201, 3.202, 3.21, 3.23, 4.026
simplex sigillum veri, 5.4541
situation [Sachlage], 2.0121, 2.014, 2.11,
2.202, 2.203, 3.02, 3.11, 3.144, 3.21, 4.021,
4.03, 4.031, 4.032, 4.04, 4.124, 4.125,
4.462, 4.466, 5.135, 5.156, 5.525
Socrates, 5.473, 5.4733
solipsism, 5.62, 5.64
solution, P8, 5.4541, 5.535, 6.4312, 6.4321,
6.521
soul, 5.5421, 5.641, 6.4312
space [Raum], 2.0121, 2.013, 2.0131, 2.0251,
2.11, 2.171, 2.182, 2.202, 3.032–3.0321,
3.1431, 4.0412, 4.463, 6.3611, 6.36111,
6.4312; cf. colour-∼; logical ∼; range.
speak/ about [von etwas sprechen], 3.221,
6.3431, 6.423, 7; cf. about.
∼ for itself [aussagen: ascribe; state;statement; tell], 6.124
stand/, how things [sich verhalten: relation;state of things], 4.022, 4.023, 4.062, 4.5∼ for [für etwas stehen], 4.0311, 5.515
state [aussagen: ascribe; speak; statement;tell], 3.317, 4.03, 4.242, 4.442, 6.1264
statement [Aussage], 2.0201, 6.3751
make a ∼ [aussagen: ascribe; speak; state;tell], 3.332, 5.25
state of/ affairs [Sachverhalt: ∼ things],

2–2.013, 2.014, 2.0272–2.062, 2.11, 2.201,
3.001, 3.0321, 4.023, 4.0311, 4.1, 4.122,
4.2, 4.21, 4.2211, 4.25, 4.27, 4.3∼ things
1. [Sachverhalt: ∼ affairs], 2.01
2. [sich verhalten: relation; stand, how
things], 5.552
stipulate [festsetzen], 3.316, 3.317, 5.501
structure [Struktur], 2.032–2.034, 2.15,
4.1211, 4.122, 5.13, 5.2, 5.22, 6.12, 6.3751
subject
1. [Subjekt], 5.5421, 5.631–5.633, 5.641;∼-predicate propositions, 4.1274
2. [Träger], 6.423
3. ∼-matter [von etwas handeln: about;concerned with; deal with], 6.124
subsistent [bestehen: existence; hold], 2.024,
2.027, 2.0271
sub specie aeterni, 6.45; cf. eternity.
substance [Substanz], 2.021, 2.0211, 2.0231,
2.04
substitut/e, 3.344, 3.3441, 4.241, 6.23, 6.24
∼ion, method of, 6.24
successor [Nachfolger], 4.1252, 4.1273
sum, cf. logical.
sum-total [gesamt: totality; whole], 2.063
superstition, 5.1361
supposition [Annahme], 4.063
survival [Fortleben], 6.4312
symbol [Symbol], 3.24, 3.31, 3.317, 3.32, 3.321,
3.323, 3.325, 3.326, 3.341, 3.3411, 3.344,
4.126, 4.24, 4.31, 4.465, 4.4661, 4.5,
5.1311, 5.473, 5.4733, 5.513–5.515, 5.525,
5.5351, 5.555, 6.113, 6.124, 6.126∼ism [Symbolismus], 4.461, 5.451
syntax, cf. logical.
system, 5.475, 5.555, 6.341, 6.372; cf.

number-∼.

1.12, 2.04, 2.05, 3.01, 4.001, 4.11, 4.52,
5.5262, 5.5561
tableau vivant [lebendes Bild: picture], 4.0311
transcendental, 6.13, 6.421
talk about [von etwas reden: mention], P2,
translation, 3.343, 4.0141, 4.025, 4.243
5.641, 6.3432; cf. about.
tru/etautology, 4.46–4.4661, 5.101, 5.1362, 5.142,
1. [Faktum], 5.154
5.143, 5.152, 5.525, 6.1, 6.12–6.1203,
2. [wahr], 2.0211, 2.0212, 2.21, 2.22,
6.1221, 6.1231, 6.124, 6.126, 6.1262, 6.127,
2.222–2.225, 3.01, 3.04, 3.05,
6.22, 6.3751
4.022–4.024, 4.06–4.063, 4.11, 4.25,
tell [aussagen: ascribe; speak; state;
4.26, 4.28, 4.31, 4.41, 4.43, 4.431, 4.442,
statement], 6.342
4.46, 4.461, 4.464, 4.466, 5.11, 5.12,
term [Glied], 4.1273, 4.442, 5.232, 5.252,
5.123, 5.13, 5.131, 5.1363, 5.512,
5.2522, 5.501
5.5262, 5.5352, 5.5563, 5.62, 6.111,
theory6.113, 6.1203, 6.1223, 6.1232, 6.125,
1. [Lehre: doctrine], 6.1224;
6.343; cf. correct; right.
∼ of probability, 4.464
come ∼e [stimmmen: agreement; right],
2. [Theorie], 4.1122, 5.5422, 6.111;
5.123∼ of classes, 6.031;∼th-argument, 5.01, 5.101, 5.152, 6.1203∼ of knowledge, 4.1121, 5.541;∼th-combination, 6.1203∼ of types, 3.331, 3.332∼th-condition, 4.431, 4.442, 4.45–4.461,
thing, cf. object; state of affairs; state of ∼s.
4.463
1. [Ding], 1.1, 2.01–2.0122, 2.013, 2.02331,∼th-function, 3.3441, 5, 5.1, 5.101, 5.234,
2.151, 3.1431, 4.0311, 4.063, 4.1272,
5.2341, 5.3, 5.31, 5.41, 5.44, 5.5, 5.521,
4.243, 5.5301, 5.5303, 5.5351, 5.5352,
65.553, 5.634, 6.1231∼th-ground, 5.101–5.121, 5.15
2. [Sache], 2.01, 2.15, 2.1514, 4.1272∼th-operation, 5.234, 5.3, 5.32, 5.41, 5.442,
think [denken: imagine], P3, 3.02, 3.03, 3.11,
5.54
3.5, 4.114, 4.116, 5.4731, 5.541, 5.542,∼th-possibility, 4.3–4.44, 4.442, 4.45, 4.46,
5.61, 5.631
5.101∼able [denkbar], P3, 3.001, 3.02, 6.361; cf.
∼th-value, 4.063
unthinkable.
type, 3.331, 3.332, 5.252, 6.123; cf. prototype.
thought [Gedanke: idea], P3, 3, 3.01, 3.02,
unalterable [fest], 2.023, 2.026–2.0271
3.04–3.1, 3.12, 3.2, 3.5, 4, 4.002, 4.112,
understand [verstehen: obvious; say], 3.263,
6.21∼-process [Denkprozess], 4.1121
4.002, 4.003, 4.02, 4.021, 4.024, 4.026,
4.243, 4.411, 5.02, 5.451, 5.521, 5.552,
time, 2.0121, 2.0251, 6.3611, 6.3751, 6.4311,
5.5562, 5.62; cf. misunderstanding.
6.4312
make oneself understood [sich verständigen],totality [Gesamtheit: sum-total; whole], 1.1,

4.026, 4.062
variable, 3.312–3.317, 4.0411, 4.1271, 4.1272,
undetermined [nicht bestimmt], 3.24, 4.431
4.1273, 4.53, 5.24, 5.242, 5.2522, 5.501,
unit, 5.155, 5.47321
6.022
unnecessary, 5.47321
propositional ∼ [Satzvariable], 3.313, 3.317,
unthinkable, 4.123
4.126, 4.127, 5.502
use∼ name, 3.314, 4.1272
1. [Gebrauch], 3.326, 4.123, 4.1272, 4.241,∼ number, 6.022
6.211;∼ proposition [variabler Satz], 3.315∼less [nicht gebraucht], 3.328
visual field, 2.0131, 5.633, 5.6331, 6.3751,
2. [Verwendung: employment], 3.325, 4.013,
6.43116.1202Whitehead, 5.252, 5.452
validity, 6.1233; cf. general ∼.
whole [gesamt: sum-total; totality], 4.11, 4.12
value [Wert], 6.4, 6.41; cf. truth-∼.
will [Wille; wollen], 5.1362, 5.631, 6.373,∼ of a variable, 3.313, 3.315–3.317, 4.127,
6.374, 6.423, 6.43
4.1271, 5.501, 5.51, 5.52
wish [wünschen], 6.374

word [Wort], 2.0122, 3.14, 3.143, 3.323, 4.002,
4.026, 4.243, 6.211; cf. concept-∼.
put into ∼s [aussprechen; unausprechlich:say], 3.221, 4.116, 6.421, 6.5, 6.522
world, 1–1.11, 1.13, 1.2, 2.021–2.022, 2.0231,
2.026, 2.063, 3.01, 3.12, 3.3421, 4.014,
4.023, 4.12, 4.2211, 4.26, 4.462, 5.123,
5.4711, 5.511, 5.526–5.5262, 5.551, 5.5521,
5.6–5.633, 5.641, 6.12, 6.1233, 6.124, 6.22,
6.342, 6.3431, 6.371, 6.373, 6.374, 6.41,
6.43, 6.431, 6.432, 6.44, 6.45, 6.54; cf.
description of the ∼.
wrong [nicht stimmen: agreement; true], 3.24;
cf. false.
zero-method, 6.121

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