If not-true and not being true are not identical, which one is false?

Jean-Pierre Marquis

pp. 79-94

in: Mathieu Marion, Robert S. Cohen (eds), Québec studies in the philosophy of science, part I, Berlin, Springer, 1995

Abstract

In classical logic, truth and falsity are highly symmetric and this symmetry is captured by the negation operator which "transforms' truth into falsehood and vice-versa. Moreover, the negation operation represented in the semantics by a unary operator on the two-element Boolean algebra is lifted to the higher-order operation of complementation on sets. However, if we abandon bivalence, then symmetry is problematic. For one thing, we are forced to distinguish between not-true, which is a truth-value, from not being true, the complement of the singleton set consisting of the truth, which is not the false and not even a truth-value. In classical logic, these two notions collapse into one. Once they are distinguished, their relationships have to be settled and in particular the links between complementation and negation have to be clarified. The purpose of this paper is to explore some of these relationships in a specific context, namely topos theory.