5.515
It must be manifest in our symbols that it can only be
propositions that are combined with one another by 'V', '.', etc.
And this is indeed the case, since the symbol in 'p' and 'q' itself
presupposes 'V', '~', etc. If the sign 'p' in 'p V q' does not stand for
a complex sign, then it cannot have sense by itself: but in that case
the signs 'p V p', 'p. p', etc., which have the same sense as p, must
also lack sense. But if 'p V p' has no sense, then 'p V q' cannot have a
sense either.
It does not depend on being "manifest in our symbols" what "V" (the logical constant for disjunction) combines with: This depends on the specification of one's formal language. Likewise it is not true that "the symbol in 'p' and 'q' itself
presupposes 'V', '~', etc." but it is again true these things are settled by the assumptions that set up the syntax of one's formal language. This is along llnes like these (only sketched here):
(1) We suppose “p“, "q" and "r" to be variables for propositions.
(2) We suppose that if X is a proposition, so is (~X)
(3) We suppose that if X and Y are propopstions, so are (X & Y) and (X V Y)
From this one can infer that e.g. ((p&q) V (~r)) is a proposition.
