Modal Multilattice Logic

Abstract

A modal extension of multilattice logic, called modal multilattice logic, is introduced as a Gentzen-type sequent calculus $$\hbox {MML}_n$$ . Theorems for embedding $$\hbox {MML}_n$$ into a Gentzen-type sequent calculus S4C (an extended S4-modal logic) and vice versa are proved. The cut-elimination theorem for $$\hbox {MML}_n$$ is shown. A Kripke semantics for $$\hbox {MML}_n$$ is introduced, and the completeness theorem with respect to this semantics is proved. Moreover, the duality principle is proved as a characteristic property of $$\hbox {MML}_n$$ .