On an intuitionistic modal logic

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

… of the Conference on Advances in …, 1998

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. [...]

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...

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, 1977

A definition of the concept of "Intuitionist Modal Analogue" is presented and motivated through the existenct, of a theorem preservin~ translatiott from J)IlPC (see [2]) to a bimodal $4-,~ 5 calculus. w This paper is devoted tc~ answerilw' tile following formal question: can we find a o'eneral criterion tha.t will oive us "tile" intuitionistie ,~m~logue of some of the nmst usual modal systenls? The problenl as stated is of a technical nature and therefore the philosophictfi issues relating to the plausibility of an intnitionistic logic of nlod~flity will be in this context ignored. R.. A. Bt-LL in [1], following a suggestion of PRIOR, proposes an extension of the intuitionist propositional calculus (I(1) with the following rules: R I L a-+ fl ; R ., a-+ M fl ; a->fi R3 a->Lfl' if a i,~ flflly modalized; a-+ fl R~-Ma-~fl' if fl is fully modalized. This system-which I call $5-ICturns out to be analogous to LEWIS' S 5 ill the sense that: (1) adding the excluded middle to S~.I(' gives a logic equivalent to S 5 ; (2) collapsing the nlod~l ol)erator,~ yields Heyting's c~lculus. Conditions (t) a, nd (2) a.rc necess~ry but not sufficient to single out lmambiguously the "correct S; analo~le", for it is easy to find non equivalent systems which satisfy both conditions. T~ke~ for inst~mee, a,n SS-thesis which is not deriw~ble in S~-IC, e.g.

Studia Logica, 1985

Logic-Based Program Synthesis and Transformation, 2021

We propose a mechanism for automating discovery of definitions, that, when added to a logic system for which we have a theorem prover, extends it to support an embedding of a new logic system into it. As a result, the synthesized definitions, when added to the prover, implement a prover for the new logic. As an instance of the proposed mechanism, we derive a Prolog theorem prover for an interesting but unconventional epistemic Logic by starting from the sequent calculus G4IP that we extend with operator definitions to obtain an embedding in intuitionistic propositional logic (IPC). With help of a candidate definition formula generator, we discover epistemic operators for which axioms and theorems of Artemov and Protopopescu's Intuitionistic Epistemic Logic (IEL) hold and formulas expected to be non-theorems fail. We compare the embedding of IEL in IPC with a similarly discovered successful embedding of Dosen's double negation modality, judged inadequate as an epistemic operator. Finally, we discuss the failure of the necessitation rule for an otherwise successful S4 embedding and share our thoughts about the intuitions explaining these differences between epistemic and alethic modalities in the context of the Brouwer-Heyting-Kolmogorov semantics of intuitionistic reasoning and knowledge acquisition.