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We revisit the duality between Kripke and algebraic semantics of intuitionistic and intuitionistic modal logic. We find that there is a certain mismatch between the two semantics, which means that not all algebraic models can be embedded into a Kripke model. This leads to an alternative proposal for a relational semantics, the stable semantics. Instead of an arbitrary partial order, the stable semantics requires a distributive lattice of worlds. We constructively show that the stable semantics is exactly as complete as the algebraic semantics. Categorifying these results leads to a 2-duality between two-dimensional stable semantics and categories of product-preserving presheaves, i.e. models of algebraic theories in the style of Lawvere.
G. A. Kavvos (Wed,) studied this question.
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