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An Introduction To Non-aristotelian Systems And General Semantics.

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740
SUPPLEMENT II
It is important to note that No. 1 states a formal implication, and that No. 3, No. 3A and No. 3B employ No. 1 as a particular assertion or specific argument to their functions. No. 3A is an instance of the implication expressed by No. 1, and is this instance because of the particular argument it does have. It states the fact that " 'all Cretanic statements are false' is a Cretanic Statement," implies that " 'all Cretanic statements are false' is false". Substitution of another argument would give a different instance; though of course of the same implication. The implication contained in its argument does not have instances. " 'Some Cretanic statements are false' is a Cretanic statement" or " 'This Cretanic statement is false' is a Cretanic statement" are not instances of " 'AH Cretanic statements are false' is a Cretanic statement," but of "P is a Cretanic statement". These three propositions have different subjects; they are different values of the same propositional function. That these subjects have relations to one another is of no moment. "My wife loves me" and "my mother-in-law is old (or loves me)" are two distinct and logically independent propositions, even though there is a relationship between the two subjects.
It is because any considered general proposition is at once an individual fact, and a formal implication or principle, with many possible arguments, that it is capable of being taken as an argument to itself. All propositions about words, logic, truth, meaning, ideas, etc., take arguments which fall in these same categories, and in so far as such a general proposition is stated in words, determined by logic, etc., it should, as such a fact, be an argument to itself as a formal implication. The principle must be false if this cannot be done, for it is sufficient, in order to overthrow a proposition of this kind, to produce one argument for which it does not hold. One may limit the principle by asserting that it holds for "all but...", in which case it is a restricted general proposition. Nominalism, association of ideas, scepticism, the theory of universal tautology, the denial of logic are defended in propositions which cannot take themselves as arguments, and which as facts are arguments to contradictory principles. Their contradictory principles therefore hold sometimes at least,