Constructive Ck for Contexts

This note describes possible world semantics for a constructive modal logic CK. The system CK is weaker than other constructive modal logics K as it does not satisfy distribution of possibility over disjunctions, neither binary (3(A ∨ B) → 3A ∨ 3B) nor nullary (3⊥ → ⊥). Our intuition is that 3⊥ should not constructively imply ⊥, thus we dispense with both forms of distribution. We are interested in this version of constructive K for its application as contexts in AI [3]. However, logicians interested in contexts prefer their semantics possible worlds style, while system CK first had only a categorical semantics [2]. Wijesekera [4] investigated possible-worlds semantics of a system similar to CK, without the binary distribution, but satisfying the nullary one. We had hoped that an adaptation of Wisejekera’s results would work for us, but a flaw in his proofs meant that his results had to be reworked. We also wanted a possible-worlds semantics for CK because it can be seen as a natural generalization of our work [1] on a constructive and categorical version of modal S4.