Finite Kripke Models and Predicate Logics of Provability

Abstract

The paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic: If a closed modal predicate-logical formula $R$ is not valid in some finite Kripke model, then there exists an arithmetical interpretation $f$ such that $PA \nvdash fR$. This result implies the arithmetical completeness of arithmetically correct modal predicate logics with the finite model property (including the one-variable fragments of $QGL$ and $QS$). The proof was obtained by adding "the predicate part" as a specific addition to the standard Solovay construction.