If has only one value, then N() = ~p (not p); if it has two values then N() = ~p.~q (neither p, nor q).
5.511 How can the all-embracing logic which mirrors the world use such special catches and manipulations? Only because all these are connected into an infinitely fine network, to the great mirror.
5.512 "~p" is true if "p" is false. Therefore in the true proposition "~p" "p" is a false proposition. How then can the stroke "~" bring it into agreement with reality?
That which denies in "~p" is however not "~", but that which all signs of this notation, which deny p, have in common.
Hence the common rule according to which "~p", "~~~p", "~p v ~p", "~p . ~p", etc. etc. (to infinity) are constructed. And this which is common to them all mirrors denial.
5.513 We could say: What is common to all symbols, which assert both p and q, is the proposition "p . q". What is common to all symbols, which asserts either p or q, is the proposition "p v q".
And similarly we can say: Two propositions are opposed to one another when they have nothing in common with one another; and every proposition has only one negative, because there is only one proposition which lies altogether outside it.
Thus in Russell's notation also it appears evident that "q : p v ~p" says the same thing as "q"; that "p v ~p" says nothing.
5.514 If a notation is fixed, there is in it a rule according to which all the propositions denying p are constructed, a rule according to which all the propositions asserting p are constructed, a rule according to which all the propositions asserting p or q are constructed, and so on. These rules are equivalent to the symbols and in them their sense is mirrored.
5.515 (1) It must be recognized in our symbols that what is connected by "v", ".", etc., must be propositions.
And this is the case, for the symbols "p" and "q" presuppose "v", "~", etc. If the sign "p" in "p v q" does not stand for a complex sign, then by itself it cannot have sense; but then also the signs "p v p", "p. p", etc. which have the same sense as "p" have no sense. If, however, "p v p" has no sense, then also "p v q" can have no sense.