On modal logics of model-theoretic relations

Abstract

Given a class $\mathcal{C}$ of models, a binary relation $\mathcal{R}$ between models, and a model-theoretic language $L$, we consider the modal logic and the modal algebra of the theory of $\mathcal{C}$ in $L$ where the modal operator is interpreted via $\mathcal{R}$. We discuss how modal theories of $\mathcal{C}$ and $\mathcal{R}$ 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.