Ultraproducts of Admissible Models for Quantified Modal Logic

Admissible models for quantified modal logic have a restriction on which sets of worlds are admissible as propositions. They give an actualist interpretation of quantifiers that leads to very general completeness results: for any propositional modal logic S there is a quantificational proof system QS that is complete for validity in models whose algebra of admissible propositions validates S. In this paper, we construct ultraproducts of admissible models and use them to derive compactness theorems that combine with completeness to yield strong completeness: any QS-consistent set of formulas is satisfiable in a model whose admissible propositions validate S. The Barcan Formula is analysed separately and shown to axiomatise certain logics that are strongly complete over admissible models in which the quantifiers are given their standard Kripkean interpretation.