, Volume 68, Issue 2, pp 173-228

A Kripke Semantics for the Logic of Gelfand Quantales

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Abstract

Gelfand quantales are complete unital quantales with an involution, *, satisfying the property that for any element a, if aba for all b, then aa* ⊙ 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 a Stone style representation theorem for complete lattices. A Rasiowa/Sikorski style semantic tableau system is also presented with the property that if all branches of a tableau are closed, then the formula in question is a theorem of Gelfand Logic. An open branch in a completed tableaux guarantees the existence of an Kripke model in which the formula is not valid; hence it is not a theorem of Gelfand Logic.