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
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matters stand with the definition of “” by “” and “.”, or of “” by “.” and “”? And how e.g. shall we introduce pq (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 pq 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 “pq”. 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, “pq” says something indefinable about the sense of those simple propositions which are contained in p and q.