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. (shrink)

The picture of information acquisition as the elimination of possibilities has proven fruitful in many domains, serving as a foundation for formal models in philosophy, linguistics, computer science, and economics. While the picture appears simple, its formalization in dynamic epistemic logic reveals subtleties: given a valid principle of information dynamics in the language of dynamic epistemic logic, substituting complex epistemic sentences for its atomic sentences may result in an invalid principle. In this article, we explore such failures of uniform substitution. (...) First, we give epistemic examples inspired by Moore, Fitch, and Williamson. Second, we answer affirmatively a question posed by van Benthem: can we effectively decide when every substitution instance of a given dynamic epistemic principle is valid? In technical terms, we prove the decidability of this schematic validity problem for public announcement logic (PAL and PAL-RC) over models for finitely many fully introspective agents, as well as models for infinitely many arbitrary agents. The proof of this result illuminates the reasons for the failure of uniform substitution. (shrink)