A Kripke Semantics for the Logic of Gelfand Quantales

Gelfand quantales are complete unital quantales with an involution, *, satisfying the property that for any element a, if a ⊙ b ≤ a for all b, then a ⊙ a* ⊙ a = a. A Hilbert-style axiom system is given for a propositional logic, called Gelfand Logic, which is sound and complete with respect to Gelfand quantales. A Kripke semantics is presented for which the soundness and completeness of Gelfand logic is shown. The completeness theorem relies on …