Notes on Logic: Difference between revisions

There are no propositions containing real variables. Those symbols which are called propositions in which "variables occur are in reality not propositions at all, but only schemes of propositions, which do not become propositions unless we replace the variables by constants. There is no proposition which is expressed by "x = x", for "x" has no signification. But there is a proposition "(x).x = x", and propositions such as "Socrates = Socrates", etc. In books on logic no variables ought to occur, but only general propositions_ which justify the use of variables. It follows that the so-called definitions in logic are not definitions, but only schemes of definitions, and instead of these we ought to put general propositions. And similarly, the so-called primitive ideas (''Urzeichen'') of logic are not primitive ideas but schemes of them. The mistaken idea that there are ''things'' called facts or complexes and relations easily leads to the opinion that there must be a relation of questioning to the facts, and then the question arises whether a relation can hold between an arbitrary number of things, since a fact can follow from arbitrary cases. It is a fact that the proposition which, e.g., expresses that q follows from p and p ⊃ q is this: p. p ⊃ q. ⊃_p,q.q.

Cross-definability in the realm of general propositions leads to _quite similar questions to those in the realm of ab-functions. There is the same objection in the case of apparent variables_ to the usual indefinables as in the case of molecular functions. The application of the ab notation to apparent variable propositions becomes clear if we consider that, for instance, the proposition "for all x, φx" is to be true when φx is true for all x's, and false when φx is false for some x's. We see that ''some'' and ''all'' occur simultaneously in the proper apparent variable notation. The notation is