Algebras and relational frames for G"{o}del modal logic and some of its extensions (original) (raw)
Related papers
2009
In this paper we consider an approach where both propositions and the accessibility relation are infinitely many-valued over G\"{o}del algebras. In particular, we consider separately the Box\Box Box-fragment and the Diamond\Diamond Diamond-fragment of our G\"{o}del modal logic and prove that both logics are complete with respect to the class of models with values in the linear Hetying algebra [0,1]. In addition, we show that the first fragment is uniquely determined by the class of models having crisp accessibility relation but it has not the finite model property. On the contrary, the second fragment is not characterized by crisp accessibility models alone but it has the finite model property. Finally, we show that the approach can be extended to include finitely many rational truth-values \`{a} la Pavelka.
On modal and intuitionistic logics
Metascience, 2014
The volume under review contains work dedicated to the memory of Leo Esakia, who died in 2010, after having worked for over 40 years towards developing duality theory for modal and intuitionistic logics. The collection comprises ten technical contributions that follow the first chapter, in which the reader can find information on Esakia's studies and career, as well as a complete list of his research publications. In the sequel, we will refer briefly to each of these ten chapters, following the order in the list of contents. B. Jónsson and A. Tarski, in two papers they published in the early 1950s in the American Journal of Mathematics, initiated the study of duality for Boolean algebras with additional operations, via the theory of canonical extensions. Esakia was among the first researchers who studied duality for lattices with additional operations [Topological Kripke models. Soviet Math. Dokl. 15 (1974), 147-151], in particular for Heyting algebras and S4 modal algebras. M. Gehrke, author of the second chapter, shows how distributive lattices, Heyting algebras and S4 modal algebras can be viewed as certain maps between distributive lattices and Boolean algebras. Furthermore, he shows how Stone duality follows from the canonical extension results and how both Priestley and Esakia duality can be derived from Stone duality. In the third chapter, N. Bezhanishvili, S. Ghilardi and M. Jibladze discuss the step-by-step method, i.e. how duality theory can be used to arrive at descriptions of finitely generated free algebras, thus shedding light on issues concerning modal propositional logics. The authors begin by recalling how this method works for free rank one modal logics and then, exploiting the method developed by D. Coumans and S. Van Gool [On generalizing free algebras for a functor. J. Logic Comput. 23 (2012), 645-672], show how it can be extended to work for logics of rank greater than one, such as T, K4 and S4. The paper ends with
Studia Logica - An International Journal for Symbolic Logic, 2010
We prove strong completeness of the □-version and the ◊-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.
Modal Logic and Universal Algebra I: Modal Axiomatizations of Structures
A non-empty universe with a collection of functions and predicates of finite arity, which is simply called a structure, induces a collection of corresponding accessibility relations constituting a generalized Kripke frame, so that a multimodal logic is introduced by a structure via the induced generalized Kripke frame. In this paper the authors discuss the problem of axiomatizing the multimodal logics thus introduced by structures in such a general setting as to cover many important classes of structures including, for instance, all expansions of implicative lattices, those of groups, and so on. First the least multimodal logic of the class of all structures with an arbitrarily fixed type is axiomatized featuring the “difference” modal operator, which is adjoined to those induced by the functions and predicates of given type. Then, on the basis of the axiomatization of the least multimodal logic, an axiomatization is given for the logic determined by a universal class of structures ...
Axiomatization of Crisp Gödel Modal Logic
Studia Logica, 2020
In this paper we consider the modal logic with both ✷ and ✸ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Gödel algebra [0, 1]G. We provide an axiomatic system extending the one from [3] for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given too. We also prove that in the studied logic it is not possible to get ✸ as an abbreviation of ✷, nor vice-versa, showing that indeed the axiomatic system we present does not coincide with any of the mono-modal fragments previously axiomatized in the literature.
Priestley Duality, a Sahlqvist Theorem and a Goldblatt-Thomason Theorem for Positive Modal Logic
Logic Journal of The Igpl / Bulletin of The Igpl, 1999
In the study of Positive Modal Logic (PML) is initiated using standard Kripke semantics and the positive modal algebras (a class of bounded distributive lattices with modal operators) are introduced. The minimum system of Positive Modal Logic is the (∧, ∨, 2, 3, ⊥, )-fragment of the local consequence relation defined by the class of all Kripke models. It can be axiomatized by a sequent calculus and extensions of it can be obtained by adding sequents as new axioms. In [6] a new semantics for PML is proposed to overcome some frame incompleteness problems discussed in . The frames of this semantics consists of a set of indexes, a quasi-order on them and an accessibility relation. The models are obtained by using increasing valuations relatively to the quasiorder of the frame. This semantics is coherent with the dual structures obtained by developing the Priestley duality for positive modal algebras, one of the topics or the present paper, and can be seen also as arising from the Kripke semantics for a suitable intuitionistic modal logic. The present paper is devoted to the study of the mentioned duality as well as to proving some d-persistency results as well as a Sahlqvist Theorem for sequents and the semantics proposed in . Also a Goldblatt-Thomason theorem that characterizes the elementary classes of frames of that semantics that are definable by sets of sequents is proved.
Subordination algebras in modal logic
arXiv: Logic, 2020
The aim of this paper is to show that even if the natural algebraic semantic for modal (normal) logic is modal algebra, the more general class of subordination algebras (roughly speaking, the non symmetric contact algebras) is adequate too - so leading to completeness results. This motivates for an algebraic (in the sense of universal algebra) study of those relational structures that are subordinate algebras.
Journal of Logic and Computation, 2020
The system of intuitionistic modal logic textbfIEL−\textbf{IEL}^{-}textbfIEL− was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic (S. Artemov and T. Protopopescu. Intuitionistic epistemic logic. The Review of Symbolic Logic, 9, 266–298, 2016). We construct the modal lambda calculus, which is Curry–Howard isomorphic to textbfIEL−\textbf{IEL}^{-}textbfIEL− as the type-theoretical representation of applicative computation widely known in functional programming.We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a Cartesian closed category. Finally, we study Heyting algebras and locales with corresponding operators. Such operators are used in point-free topology as well. We study complete Kripke–Joyal-style semantics for predicate extensions of textbfIEL−\textbf{IEL}^{-}textbfIEL− and related logics using Dedekind–MacNeille completions and modal cover systems introduced by Goldblatt (R. Goldblatt. Cover semantics for...
arXiv (Cornell University), 2020
The system of intuitionistic modal logic IEL´was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic [3]. We construct the modal lambda calculus which is Curry-Howard isomorphic to IEL´as the type-theoretical representation of applicative computation widely known in functional programming. We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a cartesian closed category. Finally, we study Heyting algebras and locales with corresponding operators. Such operators are used in point-free topology as well. We study compelete Kripke-Joyal-style semantics for predicate extensions of IEL´and related logics using Dedekind-MacNeille completions and modal cover systems introduced by Goldblatt [26]. The paper extends the conference paper published in the LFCS'20 volume [55].