On Modal Logics of Model-Theoretic Relations
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On Modal Logics of Model-Theoretic Relations
Abstract. Given a class C of models, a binary relation R between models, and a modeltheoretic language L, we consider the modal logic and the modal algebra of the theory of C in L where the modal operator is interpreted via R. We discuss how modal theories of C and 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 L¨ owenheim–Skolem theorem for first-order language expanded with the modal operator for the extension relation between models. Keywords: Modal logic, Modal algebra, Robust modal theory, Logic of submodels, Logic of quotients, Logic of forcing, Provability logic, Model-theoretic logic.
Introduction We consider modal systems in which the modal operator is interpreted via a binary relation on a class of models. Many instances of such systems can be found in the literature. During the last years, modal logics of various relations between models of set theory have been studied, see, e.g., [7,15,17,18,21]. A well established area in provability logic deals with modal axiomatizations of relations between models of arithmetic (and between arithmetic theories), see, e.g., [5,16,19,20,27,34,35]. In another extensively studied area, modalities are interpreted by relations between Kripke and temporal models, see, e.g., [4,11,33] or the monograph [32]. In [3], the consequence along an abstract relation between models is studied, which is closely related to our consideration. Let f be a unary operation on sentences of a model-theoretic language L, and T a set of sentences of L (e.g., the set of theorems in a given calculus, or the set of sentences valid in a given class of models). Using the propositional modal language, one can consider the following “fragment” of T : variables are evaluated by sentences of L, and f interprets the modal operator; the modal theory of f on T , or just the f -fragment of T , is defined as the set of those modal formulas which are in T under every valuation. A well-known example of this approach is a complete modal axiomatization
Presented by Yde Venema; Received April 23, 2018
Studia Logica https://doi.org/10.1007/s11225-019-09885-y
c Springer Nature B.V. 2019
D. I. Saveliev, I. B. Shapirovsky
of formal provability in Peano arithmetic given by Solovay [31]. Another important example is the theorem by Hamkins and L¨ owe axiomatizing the modal logic of forcing (introduced earlier by Hamkins in [15]) where the modal operator expresses the satisfiability in forcing extensions [18]. Both these modal systems have good semantic and algorithmic properties; in particular, they have the finite model property, are finitely axiomatizable, and hence decidable. These examples inspire the following observation. Let C be an arbitrary class of models of the same signature, T = Th L (C) the theory of C in a modeltheoretic language L, and R a binary relation on C. Assuming that the satisfiability in R-im
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