Given a class \ of models, a binary relation \ between models, and a model-theoretic language L, we consider the modal logic and the modal algebra of the theory of \ in L where the modal operator is interpreted via \. We discuss how modal theories of \ and \ depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside L. We calculate such theories for the submodel and the quotient relations. We prove a downward Löwenheim–Skolem (...) theorem for first-order language expanded with the modal operator for the extension relation between models. (shrink)

In this paper we study modal logics of closed domains on the real plane ordered by the chronological future relation. For the modal logic determined by an arbitrary closed convex domain with a smooth bound, we present a finite axiom system and prove the finite modal property.