On modal logics of model-theoretic relations

Given a class of models, a binary relation between models, and a model-theoretic language, we consider the modal logic and the modal algebra of the theory of the class in the language interpreting the modal operator via this relation. We discuss how modal theories of given classes and relations depend on model-theoretic languages, their Kripke completeness, the expressibility and non-expressibility of the modality inside various languages, and prove a downward Löwenheim – Skolem theorem for first-order language expanded with the modal operator for the extension relation between models. We define the robust modal theory of a relation on a class as the theory that does not change under increasing expressibility of the model-theoretic language, and calculate robust theories for the submodel and the quotient relations on various classes of models.

References

1. Denis I. Saveliev, Ilya B. Shapirovsky, “On modal logics of model-theoretic relations”, Studia Logica, 108 (2020), 989–1017