Section 10:
Propositions 6-6.241

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

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

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

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

6.02 And this is how we arrive at numbers. I give the following definitions x = /0x Def., /'/v'x = /v+1'x Def. So, in accordance with these rules, which deal with signs, we write the series x, /'x, /'/'x, /'/'/'x, ... , in the following way /0'x, /0+1'x, /0+1+1'x, /0+1+1+1'x, ... . Therefore, instead of '[x, E, /'E]', I write '[/0'x, /v'x, /v+1'x]'. And I give the following definitions 0 + 1 = 1 Def., 0 + 1 + 1 = 2 Def., 0 + 1 + 1 +1 = 3 Def., (and so on).

6.021 A number is the exponent of an operation.

6.022 The concept of number is simply what is common to all numbers, the general form of a number. The concept of number is the variable number. And the concept of numerical equality is the general form of all particular cases of numerical equality.

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

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

6.1 The propositions of logic are tautologies.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

6.1251 Hence there can never be surprises in logic.

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

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

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

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

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

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

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

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

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

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

6.21 A proposition of mathematics does not express a thought.

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

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

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

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

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

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

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

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

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

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

6.234 Mathematics is a method of logic.

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

6.24 The method by which mathematics arrives at its equations is the method of substitution. For equations express the substitutability of two expressions and, starting from a number of equations, we advance to new equations by substituting different expressions in accordance with the equations.

6.241 Thus the proof of the proposition 2 t 2 = 4 runs as follows: (/v)n'x = /v x u'x Def., /2 x 2'x = (/2)2'x = (/2)1 + 1'x = /2' /2'x = /1 + 1'/1 + 1'x = (/'/)'(/'/)'x =/'/'/'/'x = /1 + 1 + 1 + 1'x = /4'x.