Dualities for Some Intuitionistic Modal Logics

We present a duality for the intuitionistic modal logic IK introduced by Fischer Servi in [10, 11]. Unlike other dualities for IK, the dual structures of the duality presented here are ordered topological spaces endowed with just one extra relation, that is used to define the set-theoretic representation of both 2 and 3. We also give a parallel presentation of dualities for the intuitionistic modal logics I2 and I3. Finally, we turn to the intuitionistic modal logic MIPC, and give a very natural characterization of the dual spaces for MIPC introduced in [2] as a subcategory of the category of the dual spaces for IK introduced here.

This paper is devoted to study an extension of intuitionistic modal logic introduced by Fischer-Servi [6] by means of Lemmon-Scott axiom. We shall prove that this logic is canonical. Key words and phrases: modal logic, intuitionistic logic, intuitionistic modal Logic.

Publications of the Research Institute for Mathematical Sciences, 1977

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

2000

In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4—our formulation has several important metatheoretic properties. In addition, we study models of IS4—not in the framework of Kirpke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability.

Journal of Logic and Computation, 2020

The system of intuitionistic modal logic $\textbf{IEL}^{-}$ 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 $\textbf{IEL}^{-}$ 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 complete Kripke–Joyal-style semantics for predicate extensions of $\textbf{IEL}^{-}$ 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].

Journal of Philosophical Logic

We define a family of intuitionistic non-normal modal logics; they can bee seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only one between Necessity and Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then give a semantic characterisation of our logics in terms of neighbourhood models. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera's Constructive Concurrent Dynamic Logic. * Preliminary version. This work was partially supported by the Project TICAMORE ANR-16-CE91-0002-01. 1 For a recent survey see Stewart et al. [25] and references therein.

The series of workshops on Intuitionistic Modal Logic and Applications (IMLA) owes its existence to the hope that philosophers, mathematical logicians and computer scientists would share information and tools when investigating intuitionistic modal logics and modal type theories, if they knew of each other's work. [...]

Studia Logica, 1985