5.513 We might say that what is common to all symbols that affirm both p
and q is the proposition 'p . q'; and that what is common to all symbols
that affirm either p or q is the proposition 'p C q'. And similarly we can
say that two propositions are opposed to one another if they have nothing
in common with one another, and that every proposition has only one
negative, since there is only one proposition that lies completely outside
it. Thus in Russell's notation too it is manifest that 'q : p C Pp' says
the same thing as 'q', that 'p C Pq' says nothing.


5.514 Once a notation has been established, there will be in it a rule
governing the construction of all propositions that negate p, a rule
governing the construction of all propositions that affirm p, and a rule
governing the construction of all propositions that affirm p or q; and so
on. These rules are equivalent to the symbols; and in them their sense is
mirrored.


5.515 It must be manifest in our symbols that it can only be propositions
that are combined with one another by 'C', '.', etc. And this is indeed the
case, since the symbol in 'p' and 'q' itself presupposes 'C', 'P', etc. If
the sign 'p' in 'p C q' does not stand for a complex sign, then it cannot
have sense by itself: but in that case the signs 'p C p', 'p . p', etc.,
which have the same sense as p, must also lack sense. But if 'p C p' has no
sense, then 'p C q' cannot have a sense either.


5.5151 Must the sign of a negative proposition be constructed with that of
the positive proposition? Why should it not be possible to express a
negative proposition by means of a negative fact? (E.g. suppose that "a'