How is this useful?

How can such specialized rigs and catches be used so that logic acts as the all inclusive the mirror of the world? Well, in that they all hook up in an infinitely fine grid to form that great mirror.

(~p) is true if 'p' is false. Therefore, in the case when the proposition (~p) is true, proposition (p) is false. How then can the prefixed tilde '~' make it agree with reality? In (~p) it is not '~' that negates, but rather what it is that all the signs in this notation that negate p have in common. And that is the common rule that governs the construction of (~p), (~~~p), (~p ∨ ~p), (~p.~p), etc. etc. ad inf. It is this common rule that mirrors negation.

We might say: What is common to all symbols that affirm both p and q is the proposition (p.q). What is common to all symbols that affirm either p or q is the proposition (p∨q). Similarly, we can say that two propositions are opposed to one another if they have nothing in common with one another. And we can also say that every proposition has only one negative, since there is only one proposition that lies completely outside if it. Thus, in Russell's notation as well, we see that (q:p∨~p) says the same thing as 'q', and that (p∨~p) says nothing.

Once a notation has been established, it will contain a rule governing the construction of all propositions that negate p, affirm p, affirm p or q, and so on. These rules are equivalent to the symbols and reflect their sense.

Our symbols themselves must show that compounds created by using '∨', '.', etc. can only be propositions. This is indeed the case, since the symbol in (p) and (q) itself presupposes '∨', '~', etc. If the sign 'p' in (p∨q) could not stand for a complex sign, then it would not make sense in isolation. But in that case, the signs (p∨p), (p.p), etc., which have the same sense as (p), could not make sense either. But if (p∨p) makes no sense, then neither can (p∨q).

Must the sign of a negative proposition be constructed from that of the positive proposition? (Such as: when 'a' does not stand in a certain relation to 'b' then this could say that (aRb) was not the case.) The positive proposition necessarily presupposes the existence of the negative proposition and vice versa.