Tractatus Logico-Philosophicus

ments to our truth-function are given by enumeration. Wittgenstein, however, by a very interesting analysis succeeds in extending the process to general propositions, i.e. to cases where the propositions which are arguments to our truth-function are not given by enumeration but are given as all those satisfying some condition. For example, let fx a propositional function be a propositional function (i.e. a function whose values are propositions), such as “x is human”—then the various values of fx form a set of propositions. We may extend the idea “not-p and not-q” so as to apply to the simultaneous denial of all the propositions which are values of fx. In this way we arrive at the proposition which is ordinarily represented in mathematical logic by the words “fx is false for all values of x”. The negation of this would be the proposition “there is at least one x for which fx is true” which is represented by “(∃x). fx” there exists an x such that fx. If we had started with not-fx instead of fx we should have arrived at the proposition “fx is true for all values of x” which is represented by “(x). fx” for all x, fx. Wittgenstein's method of dealing with general propositions [i.e. “(x). fx” and “(∃x). fx”] differs from previous methods by the fact that the generality comes only in specifying the set of propositions concerned, and when this has been done the building up of truth-functions proceeds exactly as it would in the case of a finite number of enumerated arguments p, q, r, ...

Mr. Wittgenstein's explanation of his symbolism at this point is not quite fully given in the text. The symbol he uses is [p̅, ξ̅, N(ξ̅)] the general form of a proposition. The following is the explanation of this symbol:

p̅ the set of all atomic propositions stands for all atomic propositions.
ξ̅ any set of propositions stands for any set of propositions.
N(ξ̅) negation of all propositions in set ξ stands for the negation of all the propositions making up ξ̅.

The whole symbol [p̅, ξ̅, N(ξ̅)] means whatever can be obtained by taking any selection of atomic propositions, negating them all, then taking any selection of the set of propositions now obtained, together with any of the originals—and so on indefinitely. This is, he says, the general truth-function and also the general form of proposition. What is meant is somewhat less complicated than it sounds. The symbol is intended to describe a process by the help of which, given the atomic propositions, all others can be manufactured. The process depends upon:

(a). Sheffer's proof that all truth-functions can be obtained out of simultaneous negation, i.e. out of “not-p and not-q”;

(b). Mr. Wittgenstein's theory of the derivation of general propositions from conjunctions and disjunctions;

(c). The assertion that a proposition can only occur in another proposition as argument to a truth-function. Given these three foundations, it follows that all propositions which are not atomic can be derived from such as are, by a uniform process, and it is this process which is indicated by Mr. Wittgenstein's symbol.

From this uniform method of construction we arrive at an amazing simplification of the theory of inference, as well as a definition of the sort of propositions that belong to logic. The method of generation which has just been described, enables Wittgenstein to say that all propositions can be constructed in the above manner from atomic propositions, and in this way the totality of propositions is defined. (The apparent exceptions which we mentioned above are dealt with in a manner which we shall consider later.) Wittgenstein is enabled to assert that propositions are all that follows from the totality of atomic propositions (together with the fact that it is the totality of them); that a proposition is always a truth-function of atomic propositions; and that if p follows from q the meaning of p is contained in the meaning of q, from which of course it results that nothing can be deduced from an atomic proposition. All the propositions of logic, he maintains, are tautologies, such, for example, as “p or not p”.

The fact that nothing can be deduced from an atomic proposition has interesting applications, for example, to causality. There cannot, in Wittgenstein's logic, be any such thing as a causal nexus. “The events 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 to-morrow is a hypothesis. We do not in fact know whether it will rise, since there is no compulsion according to which one thing must happen because another happens.

Let us now take up another subject—that of names. In Wittgenstein's theoretical logical language, names are only given to simples. We do not give two names to one thing, or one name to two things. There is no way whatever, according to him, by which we can describe the totality of things that can be named, in other words, the totality of