4 A thought is a proposition with a sense.

4.001 The totality of propositions is language.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4.05 Reality is compared with propositions.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4.116 Everything that can be thought at all can be thought clearly. Everything that can be put into words can be put clearly.

4.12 Propositions can represent the whole of reality, but they cannot represent what they must have in common with reality in order to be able to represent it--logical form. In order to be able to represent logical form, we should have to be able to station ourselves with propositions somewhere outside logic, that is to say outside the world.

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

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

4.1212 What can be shown, cannot be said.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4.2 The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs.

4.21 The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs.

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

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

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

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

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

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

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

4.242 Expressions of the form 'a = b' are, therefore, mere representational devices. They state nothing about the meaning of the signs 'a' and 'b'.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4.53 The general propositional form is a variable.