The inconsistency of certain formal logics

Abstract

A proof that certain systems of formal logic are inconsistent, in the sense that every formula expressible in them is provable, was published by Kleene and Rosser under the above title in 1935. For the case where the underlying system satisfies an additional condition—viz. the possession of an operator analogous to Schönfinkel's K—I gave a simpler derivation of this condition in a previous paper. That argument, like the original one of Kleene and Rosser, was a refinement of the Richard paradox. The object of the present paper is to show that, if we use other paradoxes, an inconsistency will result from a very much simpler argument and on much less restrictive hypotheses. The contradiction can no longer be called “the paradox of Kleene and Rosser,” because it is based on an entirely different principle; but, in deference to the work of the original discoverers of the inconsistency, the paper is given the same title as that which their paper bears. The central idea of the new derivation was suggested by some work of R. Carnap.This paper is based on the one above cited, which will be referred to as PKR. However, acquaintance with that paper will be presupposed only through 3.4, i.e., through the statement of the basic hypotheses and conventions for a combinatorially complete system—except that for the second (alternative) method of construction given below the substance of PKR through 9.6 is needed.