A Lindström theorem for modal logic

Abstract

A modal analogue of Lindstrom's characterization of first-order logic is proved.Basic modal logics are characterized as the only modal logics that have a notion of finite rank, or, equivalently, as the strongest modal logic whose formulas are preserved under ultraproducts over w.Also, basic modal logic is the strongest classical logic whose formulas are preserved under bisimulations and ultraproducts over w.