Intuitionistic Epistemic Logic Research Articles

Bitopological models of intuitionistic epistemic logic

Bitopological models of intuitionistic epistemic logic

Bitopological models of intuitionistic epistemic logic

Bitopological models of intuitionistic epistemic logic

Linear Depth Deduction with Subformula Property for Intuitionistic Epistemic Logic

In their seminal paper Artemov and Protopopescu provide Hilbert formal systems, Brower–Heyting–Kolmogorov and Kripke semantics for the logics of intuitionistic belief and knowledge. Subsequently Krupski has proved that the logic of intuitionistic knowledge is PSPACE-complete and Su and Sano have provided calculi enjoying the subformula property. This paper continues the investigations around sequent calculi for Intuitionistic Epistemic Logics by providing sequent calculi that have the subformula property and that are terminating in linear depth. Our calculi allow us to design a procedure that for invalid formulas returns a Kripke model of minimal depth. Finally we also discuss refutational sequent calculi, that is sequent calculi to prove that formulas are invalid.

Constructive and mechanised meta-theory of IEL and similar modal logics

Abstract Artemov and Protopopescu proposed intuitionistic epistemic logic (IEL) to capture an intuitionistic conception of knowledge. By establishing completeness, they provided the base for a meta-theoretic investigation of IEL, which was continued by Krupski with a proof of cut-elimination, and Su and Sano establishing semantic cut-elimination and the finite model property. However, no analysis of these results in a constructive meta-logic has been conducted, arguably impeding the intuitionistic justification of IEL. We aim to close this gap and investigate IEL in the constructive-type theory of the Coq proof assistant. Concretely, we present a constructive and mechanised completeness proof for IEL, employing a syntactic decidability proof based on cut-elimination to constructivise the ideas from the literature. Following Su and Sano, we then also give constructive versions of semantic cut-elimination and the finite model property. Given our constructive and mechanised setting, all these results now bear executable algorithms. Our particular strategy to establish constructive completeness exploiting syntactic decidability can be used for similar modal logics, which we illustrate with the examples of the classical modal logics K, D and T. For modal logics including the four axioms, however, the method seems not to apply immediately.

Intuitionistic Epistemic Logic with Distributed Knowledge

We develop intuitionistic epistemic logicswith distributed knowledge, which is more general thana logic proposed by (J¨ager & Marti 2016) in thata distributed knowledge operator is parameterized bya group of agents.Specifically, we present Hilbertsystems of intuitionistic K, KT, KD, K4, K4D, and S4 withdistributed knowledge. The semantic completeness ofthe logics with regard to suitable Kripke frames is shownby modifying the standard argument of the semanticcompleteness of classical distributed knowledge logicsvia the concept of pseudo-model.We also presentcut-free sequent calculi for the logics, based on which weestablish Craig interpolation theorem and decidability.

Categorical and algebraic aspects of the intuitionistic modal logic IEL― and its predicate extensions

Abstract 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 quantified lax logic. Journal of Logic and Computation, 21, 1035–1063, 2011). The paper extends the conference paper published in the LFCS’20 volume (D. Rogozin. Modal type theory based on the intuitionistic modal logic IEL. In International Symposium on Logical Foundations of Computer Science, pp. 236–248. Springer, 2020).

Cut elimination and complexity bounds for intuitionistic epistemic logic

Abstract The formal system of intuitionistic epistemic logic (IEL) was proposed by S. Artemov and T. Protopopescu. It provides the formal foundation for the study of knowledge from an intuitionistic point of view based on Brouwer–Heyting–Kolmogorov semantics of intuitionism. We construct a cut-free sequent calculus for IEL and establish that polynomial space is sufficient for the proof search in it. We prove that IEL is PSPACE-complete.

An arithmetic interpretation of intuitionistic verification

Abstract Intuitionistic epistemic logic introduces an epistemic operator to intuitionistic logic, which reflects the intended Brouwer–Heyting–Kolmogorov (BHK) semantics of intuitionism. The fundamental assumption concerning intuitionistic knowledge and belief is that it is the product of verification. The BHK interpretation of intuitionistic logic has a precise formulation in the logic of proofs and its arithmetical semantics. We show here that this interpretation can be extended to the notion of verification upon which intuitionistic knowledge is based, providing the systems of intuitionistic epistemic logic based on verification with an arithmetical semantics too. This confirms that the conception of verification incorporated in these systems reflects the BHK interpretation.

Reasoning about proof and knowledge

In previous work [15], we presented a hierarchy of classical modal systems, along with algebraic semantics, for the reasoning about intuitionistic truth, belief and knowledge. Deviating from Gödel's interpretation of IPC in S4, our modal systems contain IPC in the way established in [13]. The modal operator can be viewed as a predicate for intuitionistic truth, i.e. proof. Epistemic principles are partially adopted from Intuitionistic Epistemic Logic IEL [4]. In the present paper, we show that the S5-style systems of our hierarchy correspond to an extended Brouwer–Heyting–Kolmogorov interpretation and are complete w.r.t. a relational semantics based on intuitionistic general frames. In this sense, our S5-style logics are adequate and complete systems for the reasoning about proof combined with belief or knowledge. The proposed relational semantics is a uniform framework in which also IEL can be modeled. Verification-based intuitionistic knowledge formalized in IEL turns out to be a special case of the kind of knowledge described by our S5-style systems.

Epistemic extensions of combined classical and intuitionistic propositional logic

Logic $L$ was introduced by Lewitzka [7] as a modal system that combines intuitionistic and classical logic: $L$ is a conservative extension of CPC and it contains a copy of IPC via the embedding $\varphi\mapsto\square\varphi$. In this article, we consider $L3$, i.e. $L$ augmented with S3 modal axioms, define basic epistemic extensions and prove completeness w.r.t. algebraic semantics. The resulting logics combine classical knowledge and belief with intuitionistic truth. Some epistemic laws of Intuitionistic Epistemic Logic studied by Artemov and Protopopescu [1] are reflected by classical modal principles. In particular, the implications "intuitionistic truth $\Rightarrow$ knowledge $\Rightarrow$ classical truth" are represented by the theorems $\square\varphi\rightarrow K\varphi$ and $K\varphi\rightarrow\varphi$ of our logic $EL3$, where we are dealing with classical instead of intuitionistic knowledge. Finally, we show that a modification of our semantics yields algebraic models for the systems of Intuitionistic Epistemic Logic introduced in [1].

Intuitionistic Epistemic Logic, Kripke Models and Fitch’s Paradox

The present work is motivated by two questions. (1) What should an intuitionistic epistemic logic look like? (2) How should one interpret the knowledge operator in a Kripke-model for it? In what follows we outline an answer to (2) and give a model-theoretic definition of the operator K. This will shed some light also on (1), since it turns out that K, defined as we do, fulfills the properties of a necessity operator for a normal modal logic. The interest of our construction also lies in a better insight into the intuitionistic solution to Fitch’s paradox, which is discussed in the third section. In particular we examine, in the light of our definition, DeVidi and Solomon’s proposal of formulating the verification thesis as \(\phi \rightarrow \neg \neg K\phi\). We show, as our main result, that this definition excapes the paradox, though it is validated only under restrictive conditions on the models.