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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 Lowenheim--Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.
Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of Godel logic. The calculi are used to establish completeness and complexity results for these fragments.
In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form $forall x_0 exists x_1 dots exists x_n bigwedge x_i R_lambda x_j$. We prove that many properties of these logics, such as finite axiom
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