Axiomatic classes in propositional modal logic

Abstract

In his review (Kaplan [1966]) of the article in which Kripke first proposed his relational semantics for modal logic, David Kaplan posed the question: which properties of a binary relation are expressible by formulas of propositional modal logic? A class of Kripke frames is said to be modal-axiomatic if it comprises exactly the frames on which every one of some set of formulas of propositional modal logic is valid. This work is addressed to the problem, suggested by Kaplan's question, of characterizing the modal-axiomatic classes of Kripke frames. In §i we obtain such a characterization, in terms of closure under certain constructions. In §2 we show that, in the case of classes closed under elementary equivalence, much simpler constructions suffice.