Kripke Completeness of Infinitary Predicate Multimodal Logics

Abstract

Kripke completeness of some infinitary predicate modal logics is presented. More precisely, we prove that if a normal modal logic $ \bf L$ above $ \bf K$ is $ \cal {D}$-persistent and universal, the infinitary and predicate extension of $ \bf L$ with BF_{$\scriptstyle \omega_{{1}}$} and BF is Kripke complete, where BF_{$\scriptstyle \omega_{{1}}$} and BF denote the formulas $ \bigwedge_{{{i\in\omega}}}^{{}}$ $ \Box$ p_i $ \supset$ $ \Box$ $ \bigwedge_{{{i\in\omega}}}^{{}}$p_i and $ \forall x \Box \varphi \supset \Box \forall x \varphi$, respectively. The results include the completeness of extensions of standard modal logics such as $ \bf K$, and its extensions by the schemata T, B, 4, 5, D, and their combinations. The proof is obtained by extending the correspondence between the representation of modal algebras and the completeness of propositional modal logic to infinite.