Page:Wittgenstein - Tractatus Logico-Philosophicus, 1922.djvu/19

he uses is $({\bar {p}},{\bar {\xi }},{\bar {N}}({\bar {\xi }}))$. The following is the explanation of this symbol:

${\bar {p}}$ stands for all atomic propositions,

${\bar {\xi }}$ stands for any set of propositions.

${\bar {N}}({\bar {\xi }})$ stands for the negation of all the propositions making up ${\bar {\xi }}$.

The whole symbol $({\bar {p}},{\bar {\xi }},{\bar {N}}({\bar {\xi }}))$ 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