The benefits of the extended Curry-Howard correspondence relating the simply typed lambda-calculus to proofs of intuitionistic propositional logic and to appropriate classes of categories that model the calculus are widely known. In this paper we show an analogous correspondence between a simple constructive modal logic CK (with both necessity and possibility ♦ operators) and a lambda-calculus with modality constructors. Then we investigate classes of categorical models for this logic. Parallel work for constructive S4 (CS4) has appeared before in [Bierman and de Paiva, 2000; Alechina et al., 2001]. The work on the basic system CK has appeared initially with co-authors Bellin and Ritter in the conference Methods for the Modalities [Bellin et al., 2001]. Since then the technical work has been improved by [Kakutani, 2007] and taken to a different, higher-order categorical setting by Ritter and myself. Here we expound on the logical significance of the earlier work.
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