5.44 Truth-functions are not material functions. For example, an
affirmation can be produced by double negation: in such a case does it
follow that in some sense negation is contained in affirmation? Does 'PPp'
negate Pp, or does it affirm p--or both? The proposition 'PPp' is not about
negation, as if negation were an object: on the other hand, the possibility
of negation is already written into affirmation. And if there were an
object called 'P', it would follow that 'PPp' said something different from
what 'p' said, just because the one proposition would then be about P and
the other would not.


5.441 This vanishing of the apparent logical constants also occurs in the
case of 'P(dx) . Pfx', which says the same as '(x) . fx', and in the case
of '(dx) . fx . x = a', which says the same as 'fa'.


5.442 If we are given a proposition, then with it we are also given the
results of all truth-operations that have it as their base.


5.45 If there are primitive logical signs, then any logic that fails to
show clearly how they are placed relatively to one another and to justify
their existence will be incorrect. The construction of logic out of its
primitive signs must be made clear.


5.451 If logic has primitive ideas, they must be independent of one
another. If a primitive idea has been introduced, it must have been
introduced in all the combinations in which it ever occurs. It cannot,