Completeness of Modal Proofs in First-Order Predicate Logic

Abstract

Characterizing modal logic in first-order predicate logic is a hot research topic in mathematical logic. Van Benthem provided an elegant characterization in which the standard translation of modal formulas coincides with the class of first-order predicate formulas invariant for bisimulations. Whereas he characterized modal logic in first-order predicate logic at the level of formulas, we characterize modal logic in first-order predicate logic at the level of proofs. Specifically, we provide a complete translation from a term calculus based on intuitionistic modal logic into Barendregt's λP. This characterization, identified as the equality of proofs, is considered significant because a term calculus based on intuitionistic modal logic is expected to realize staged computation.