Notes on Logic: Difference between revisions

To every molecular function a TF (or ab) scheme corresponds. Therefore we may use the TF scheme itself instead of the function. Now what the TF scheme does is that it correlates the letters T and F with each proposition. These two letters are the poles of atomic propositions. Then the scheme correlates another T and F to these poles. In this notation all that matters is the correlation of the outside poles to the poles of the atomic propositions. Therefore not-not-p is the same symbol as p. And therefore we shall never get two symbols for the same molecular function. As the ab (TF)-functions of atomic propositions are bi-polar propositions again, we can perform ab operations on them. We shall, by doing so, correlate two new outside poles via the old outside poles to the poles of the atomic propositions. The symbolizing fact in a-p-b is that ''say'' a is on the left of p and b on the right of p. [This is quite arbitrary, but if we once have fixed on which order the poles have to stand in, we must of course stick to our convention. If, for instance, "apb" says p, then bpa says ''nothing'' (it does ''not'' say ~p). But a-apb-b is the same symbol as apb (here the ab-function vanishes automatically) for here the new poles are related to the same side of p as the old ones. The question is always: how are the new poles correlated to p compared with the way the old poles are correlated to p?] Then, given apb, the correlation of new poles is to be transitive, so that, for instance, if a new pole a in what­ ever way, i.e. via whatever poles, is correlated to the inside a, the symbol is not changed thereby. It is therefore possible to construct all possible ab-functions by performing one ab-operation repeatedly, and we can therefore talk of all ab-functions as of all those functions which can be obtained by performing this ab-operation repeatedly (''cf.'' Sheffer's work).