Modal logics of structures

Studia Scientiarum Mathematicarum Hungarica, 2019

We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an N P complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.

The Review of Symbolic Logic, 2011

A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra-Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics. §1. Yet another semantics for modal logic? Besides its indisputable success, there may be reasons to doubt about the universal acceptability of the possible-worlds semantics (often called relational semantics or Kripke semantics) for modal logics, basically because they reduce (or equate) meaning to extensions of worlds. Indeed, criticisms against possible-worlds semantics abound. For instance, Kuczynski (2007) purports to show that the very assumptions of possible-worlds semantics lead to the absurd conclusion that all propositions are necessarily true. Even if this critique would be perhaps circumscribed to uses of possible-worlds semantics in quantified modal logic, it is not easy to rebut the well-known Quine's criticisms about the difficulties in separating the notion of possibility within a world from the notion of consistency of the world's description (cf., Section 1 of Quine, 1953). But there are other issues with regard to the possible-worlds account of modal notions. As argued in , the mathematics associated with such semantics may be quite complicated, besides the inappropriateness of the possible-worlds to formalizing knowledge, belief, and other nonalethic modal concepts. Other interesting semantical approaches like algebraic semantics, neighborhood semantics, and topological semantics have been employed in the characterization of modal logics (see Blackburn &, which by itself suggests that the

2021

Gödel modal logics can be seen as extenions of intutionistic modal logics with the prelinearity axiom. In this paper we focus on the algebraic and relational semantics for Gödel modal logics that leverages on the duality between finite Gödel algebras and finite forests, i.e. finite posets whose principal downsets are totally ordered. We consider different subvarieties of the basic variety GAO of Gödel algebras with two modal operators (GAOs for short) and their corresponding classes of forest frames, either with one or two accessibility relations. These relational structures can be considered as prelinear versions of the usual relational semantics of intuitionistic modal logic. More precisely we consider two main extensions of finite Gödel algebras with operators: the one obtained by adding Dunn axioms, typically studied in the fragment of positive classical (and intuitionistic) logic, and the one determined by adding Fischer Servi axioms. We present Jónsson-Tarski like representati...

2006

Artemov introduced the Logic of Proofs (LP) as a logic of explicit proofs. We can also offer an epistemic reading of this formula: "t is a possible justification of φ". Motivated, in part, by this epistemic reading, Fitting introduced a Kripke style semantics for LP in . In this note, we prove soundness and completeness of some axiom systems which are not covered in [8].

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).

2013

We prove completeness, interpolation, decidability and an omitting types theorem for certain multi dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing (varieties generated by) complex algebras of Kripke semantics for such logic. Those algebras, whose elements are sets of states are common reducts of cylindric and polyadic algebras.

Journal of Applied Non-Classical Logics, 1992

We enrich propositional modal logic with operators >n (n ∈ N) which are interpreted on Kripke structures as "there are more than n accessible worlds for which ...", thus obtaining a basic graded modal logic GrK. We show how some familiar concepts (such as subframes, p-morphisms, disjoint unions and filtrations) and techniques from modal model theory can be used to obtain results about expressiveness (like graded modal equivalence, correspondence and definability) for this language. On the basis of the class of linear frames we demonstrate that the expressive power of the language is considerably stronger than that of classical modal logic. We give a class of formulas for which a first-order equivalent can be systematically obtained, but also show that the set of formulas for which such an equivalence exists is in some sense a proper subset of the set of so called Sahlqvist formulas, a syntactically defined set of modal formulas for which a corresponding formula is guaranteed to exist. Finally we show how, combining the technique of 'filtration' with a notion of 'copying worlds' -in view of the "more than n" interpretation, one cannot simply collapse worlds -for some graded modal logics (GrK, GrT, . . . ), the finite model property (and also decidability) is obtained.

Electronic Notes in Theoretical Computer Science, 2007

This paper discusses a bimodal hybrid language with a sub-modality (called the irreflexive modality) associated with the intersection of the accessibility relation R and the inequality =. First, we provide the Hilbert-style axiomatizations (with and without the COV-rule) for logics of our language, and prove the Kripke completeness and the finite frame property for them. Second, with respect to the frame expressive power, we compare our language containing the irreflexive modality with the hybrid languages H and H(E). Finally, we establish the Goldblatt-Thomason-style characterization for our language.

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.