A Kripke semantics for the logic of Gelfand quantales

Studia Logica 68 (2):173-228 (2001)

Abstract

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 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

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References found in this work

Logics Without the Contraction Rule.Hiroakira Ono & Yuichi Komori - 1985 - Journal of Symbolic Logic 50 (1):169-201.
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Kripke Models for Linear Logic.Allwein Gerard & Dunn J. Michael - 1993 - Journal of Symbolic Logic 58 (2):514-545.
Relational Proof System for Relevant Logics.Ewa Orlowska - 1992 - Journal of Symbolic Logic 57 (4):1425-1440.

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