2018/04/25 by Saveliev, Denis I., Shapirovsky, Ilya B.
#03B45 #03C30 #03C52 (Primary) #03C75 #03C85 #03E40 #03F45 (Secondary) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1804.09810
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.