5 Model theory of modal logic

Studia Logica - An International Journal for Symbolic Logic, 1995

Abstract.

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2017

The aim of this paper is to establish a connection between modal logics and labeled graphs, which is useful in solving the problem of undeterminism.

MISCELLANEA LOGICA

2022

In this paper, we deal with the problem of putting together modal worlds that operate in different logic systems. When evaluating a modal sentence 2φ, we argue that it is not sufficient to inspect the truth of φ in accessed worlds (possibly in different logics). Instead, ways of transferring more subtle semantic information between logical systems must be established. Thus, we will introduce modal structures that accommodate communication between logic systems by fixing a common lattice L where different logics build their semantics. The semantics of each logic being considered in the modal structure is a sublattice of L. In this system, necessity and possibility of a statement should not solely rely on the satisfaction relation in each world and the accessibility relation. The value of a formula 2φ will be defined in terms of a comparison between the values of φ in accessible worlds and the common lattice L. We will investigate natural instances where formulas φ can be said to be n...

Advances in Modal Logic, 2024

Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal logics via algebraic representation theorems. We begin with complete lattices L equipped with an antitone operation ¬ sending 1 to 0, a completely multiplicative operation □, and a completely additive operation ◊. Such lattice expansions can be represented by means of a set X together with binary relations ◁, R, and Q, satisfying some first-order conditions, used to represent (L, ¬), □, and ◊, respectively. Indeed, any lattice L equipped with such a ¬, a multiplicative □, and an additive ◊ embeds into the lattice of propositions of a frame (X, ◁, R, Q). Building on our recent study of fundamental logic, we focus on the case where ¬ is dually self-adjoint (a ≤ ¬b implies b ≤ ¬a) and ◊¬a ≤ ¬□a. In this case, the representations can be constrained so that R = Q, i.e., we need only add a single relation to (X, ◁) to represent both □ and ◊. Using these results, we prove that a system of fundamental modal logic is sound and complete with respect to an elementary class of bi-relational structures (X, ◁, R).

Logic Journal of The Igpl / Bulletin of The Igpl, 1995

arXiv (Cornell University), 2018

Given a class C of models, a binary relation R between models, and a model-theoretic 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 Löwenheim-Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.

The discursiveness in the process of (logical) argumentation must take into consideration a series of aspects connected to the concepts used, the definitions resulted with the help of these concepts, the theories analyzed. From this perspective, the modal logic and its specific forms involve a reevaluation of reasoning. Moreover, through pertinent discursive argumentations are visible problematical approaches which have certain philosophical significances. Therefore, the new logics reflect at the level of discursiveness a certain meaning which participates in its turn in assuming a certain logical paradigm. The modal logic (founded in fact by Aristotle 1) represents that part of logic where the modal propositions are studied. The modal propositions are formulated like this:-It is necessary to learn the modal logic‖,-It is possible for us to go on a trip tomorrow‖,-It is forbidden to walk on this road‖ etc. In these sentences one can notice expressions such as:-it is necessary to‖,-i...

Journal of Symbolic Logic, 1986

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