A Logical Treatment of Constructive Duality

We present an investigation of duality in the traditional logical manner. We extend Nelson's symmetrization of intuitionistic logic, constructible falsity, to a self-dual logic; constructible duality. We develop a self-dual model by considering an interval of worlds in an intuitionistic Kripke model. The duality arises through how we judge truth and falsity. Truth is judged forward in the Kripke model, as in intuitionistic logic, while falsity is judged backwards, that is forward in the dual model. We deene a symmetrization of the Beth-Fitting construction which transforms an interval Kripke model in a self-dual algebra for the logic of constructible duality and back. We then show that every point in the algebra is representable by some formula in the logic. This algebra arises as an instantiation of a pseudo-Boolean algebra into several categorical constructions. In particular, we show that this algebra is an instantiation of the Chu construction applied to a pseudo-Boolean algebra, the second Dialectica construction applied to a pseudo-Boolean algebra, and as posetal case of a suggestion by Pitts for the study of recursion and corecursion. Thus the algebra provides a common base for these constructions, and suggests itself as a necessary part of any logical treatment of constructive duality.