A Sahlqvist Theorem for Distributive Modal Logics
by
Hideo Nagahashi
New Mexico State University
Coauthors: Mai Gehrke (NMSU), Yde Venema (University of Amsterdam)

We present a generalization of Sahlqvist theorem within the framework of a generalized modal logic, which we call distributive modal logic. The algebraic counterpart of this logic is bounded distributive lattices with unary operators. The operators that preserve joins or meets of the lattice are generalized modal operators. The operators that flip joins to meets or vice versa are generalized negations that are weaker than the negation of Boolean algebras. We prove that canonicity holds in a large class of distributive modal logics using the theory of canonical extension of lattices that was recenty developed. We also give a proper Kripke-style semantics for these logics. In this setting Sahlqvist correspondence theorem readily follows from the classical correspondence theorem. Putting these results together, we get a generalized Sahlqvist completeness theorem. Along with the classical modal logic, many distributive lattice based logics such as positive modal logic can be captured in this framework.

Date received: April 4, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-95.