Paradoxes and many-valued set theory

Abstract

It is a common misconception that the paradoxes of Naive Set Theory (NST) can be dodged by simply giving up the Principle of Bivalence, and by shifting thereby from the classical two-valued logic to a many-valued logic. This dodge, if successful, would result in a many-valued set theory. The basis for this misconception, presumably, is the belief that a contradiction can be avoided by assigning the same intermediate truth-value to otherwise incompatible statements. Consider, for example, the following proposal for dodging Russell’s Paradox. Let L be a logic with true, false, and middle as its truth-values. Suppose that the truth-rules of L are such that a statement is middle just in case its negation is middle, and that a biconditional is true just in case its immediate components have the same truth-value. Now Russell’s Paradox shows up in NST just at the point where it is derived that Russell’s class is a member of itself if and only if it is not a member of itself, since such a biconditional cannot be true in the classical two-valued logic. But the latter biconditional can be true in L. We simply force the statement that Russell’s class is a member of itself to be middle. The dodging of paradoxes is not nearly as simple, however, as the above example might seem to suggest. It will be shown in this paper that the axioms of NST are inconsistent in sundry well known many-valued logics, infinite as well as finite-valued. A general method will also be developed for generating set-theoretic paradoxes in many-valued logics. In fact, only three of the well known many-valued logics described below even prima facie stand a chance of basing a set theory which has axioms at least as strong as those of NST, and all three of these logics are indenumerablevalued.