Marcus Kracht and Oliver Kutz

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 modal-structures. 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. The resulting models are quite like cartesian metaframes. Our method therefore establishes a rather elementary completeness proof with respect to these types of models.