It is easy to suppose a contradiction in the fact that on the one hand
every possible complex proposition is a simple
ab-function of simple propositions, and that on the
other hand the repeated application of one
ab-function suffices to generate
all these propositions.
If e.g. an affirmation can be generated by double
negation, is negation in any sense contained in affirmation?
Does “p” deny
“not-p” or assert
“p”, or both?
And how do
15
matters stand with the definition of
“
⊃ ” by
“
⌵ ” and
“.”, or of
“
⌵ ” by “.” and
“
⊃ ”?
And how e.g. shall we introduce
p
❘q
(i.e.
~p
⌵
~q), if not by saying that this expression
says something indefinable about all arguments
p
and
q?
But the
ab-functions must be
introduced as follows: The function
p
❘q is
merely a mechanical instrument for constructing all possible
symbols of
ab-functions.
The symbols arising by repeated application of the symbol
“
|” do
not contain the symbol
“p
∣ q”.
We need a rule according to which we can form all symbols of
ab-functions, in order to be able to speak of the
class of them; and we now speak of them e.g. as
those symbols of functions which can be generated by repeated
application of the operation “
|”.
And we say now: For all p's and q's,
“p
∣ q” says something
indefinable about the sense of those simple propositions which are
contained in p and q.