Section 6:
Propositions 4.2-5.156

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 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 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 . 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 of the elementary propositions. We say that the are tautological. In the second case the proposition is false for all the truth-possibilities: the truth-conditions are contradictory . In the first case we call the proposition a tautology; in the second, a contradiction.

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

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

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

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

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

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

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

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

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

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

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

4.53 The general propositional form is a variable.

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

5.01 Elementary propositions are the truth-arguments of propositions.

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

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

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

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

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

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

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

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

5.124 A proposition affirms every proposition that follows from it.

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

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

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

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

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

5.133 All deductions are made a priori.

5.134 One elementary proposition cannot be deduced form another.

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

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

5.1361 We cannot infer the events of the future from those of the present. THIS PARAGRAPH COULD NOT BE READ IN FROM DISK

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

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

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

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

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

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

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

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

5.1511 There is no special object peculiar to probability propositions.

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

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

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

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

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