A Sahlqvist Theorem for Distributive Modal Logic
Mai Gehrke, Hideo Nagahashi, Yde Venema
Abstract:
In this paper we consider distributive modal logic, a setting in which
we may add modalities, such as classical types of modalities as well
as weak forms of negation, to the fragment of classical propositional
logic given by conjunction, disjunction, true, and false. For these
logics we define both algebraic semantics, in the form of distributive
modal algebras, and relational semantics, in the form of ordered
Kripke structures. The main contributions of this paper lie in
extending the notion of Sahlqvist axioms to our generalized setting
and proving both a correspondence and a canonicity result for
distributive modal logics axiomatized by Sahlqvist axioms. Our proof
of the correspondence result relies on a reduction to the classical
case, but our canonicity proof departs from the traditional style and
uses the newly extended algebraic theory of canonical extensions.
Keywords: distributive modal logic, distributive modal algebras,
lattice expansions, canonical extensions, correspondence theory,
canonical logic, canonical varieties, duality