The Semantics of Modal Predicate Logic I. Counterpart-Frames

We introduce a new semantics for modal predicate logic, with respect to which a rich class of first-order modal logics is complete, namely all normal first-order modal logics that are extensions of free quantified K. This logic is defined by combining positive free logic with equality PFL . = and the propositional modal logic K. We then uniformly construct—for each modal predicate logic L—a canonical model whose theory is exactly L. This proves completeness with respect to so-called modalstructures. We add some remarks on canonicity and frame-completeness and finally show that if suitable modal algebras of ‘admissible interpretations’ are added to modal predicate frames, general frame-completeness is gained.