Generalized Kripke semantics for the Lambek-Grishin calculus

Kripke semantics plays a fundamental role for modal logics where it often provides modular semantics allowing a clear treatment of various logics related by the addition of axioms or connectives. Kripke semantics is also the “natural semantics” for classical modal logic in the sense that it is given by the representation theory for modal algebras. The algebraic structure corresponding to a traditional Kripke model is based on the powerset of a set of worlds. Powerset algebras are very special Boolean algebras, but they are general enough that every Boolean algebra can be embedded in one.