Page:Wittgenstein - Tractatus Logico-Philosophicus, 1922.djvu/18

Rh of Dr Sheffer's work. The manner in which other truth-functions are constructed out of "not-p and not-q" is easy to see. ** Not-p and not-p" is equivalent to "not-p," hence we obtain a definition of negation in qerms of our primitive function: hence we can define p or q," since this is the negation of "not-p and not-q," i.e. of our primitive function. The development of other truth-functions out of "not-p" and "p or q" is given in detail at the beginning of Principia Mathematica, This gives all that is wanted when the propositions which are arguments 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 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 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 "$$(\exists x).fx$$." If we had started with not-fx instead oi fx we should have arrived at the proposition "fx is true for all values of x" which is represented by $$(x).fx$$." Wittgenstein's method of dealing with general propositions [i.e. "$$(x).fx$$" and "$$(\exists x).fx$$"] 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 Rh