An algebraic generalization of Kripke structures

Details An algebraic generalization of Kripke structures An algebraic generalization of Kripke structures

2007-04-14

arxiv.org/abs/0704.1886v1

Math. Proc. Cambridge Philos. Soc. 145 (2008) 549-577

Mathematics

Logic

39 pages

Scientific paper

10.1017/S0305004108001667

The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4, and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL, and the ramified temporal logic CTL.

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