First-Order Intuitionistic Epistemic Logic

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, to the best of our knowledge, no analysis of these results in a constructive meta-logic has been conducted.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. We expect that our methods used for mechanising cut-elimination and decidability also extend to other modal logics (and have verified this observation for the classical modal logic K).

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“…Sequent calculus representations for IEL have been proposed by Krupski [18], Su and Sano [28], and more recently Fiorino [9].…”

Section: Cut-free Sequent Calculusmentioning

confidence: 99%

“…Proof theory of IEL has been studied by Krupski [18], Su and Sano [27,28], and more recently Fiorino [9]. Su and Sano propose a cut-free sequent calculus for IEL (and an extension of IEL with quantifiers).…”

Section: Related Workmentioning

confidence: 99%

Constructive and Mechanised Meta-Theory of Intuitionistic Epistemic Logic

Hagemeier

Kirst

2021

Logical Foundations of Computer Science

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“…In [4] and also in its journal version [22] the logics IEL − and IEL, which are weaker systems close to the logic IEL + , were investigated. In the work [33] by Su and Sano Kripke-style semantics for the logics QIEL − and QIEL, the predicate versions of IEL − and IEL, were considered. The completeness theorem for both QIEL − and QIEL with respect to Kripke semantics was proved there.…”

Section: § 1 Introductionmentioning

confidence: 99%

“…The completeness theorem for both QIEL − and QIEL with respect to Kripke semantics was proved there. The method in [33] was based on constructing sequent calculus for these logics and considering the set of saturated sequents as a Kripke model. Below we give a different proof of the completeness theorem for the logic QIEL + (or QH4), namely, we consider intuitionistic saturated theories.…”

Section: § 1 Introductionmentioning

confidence: 99%

The predicate version of the joint logic of problems and propositions

Оноприенко

2022

Sb. Math.

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We consider the joint logic of problems and propositions $\mathrm{QHC}$ introduced by Melikhov. We construct Kripke models with audit worlds for this logic and prove the soundness and completeness of $\mathrm{QHC}$ with respect to this type of model. The conservativity of the logic $\mathrm{QHC}$ over the intuitionistic modal logic $\mathrm{QH4}$, which coincides with the ‘lax logic’ $\mathrm{QLL}^+$, is established. We construct Kripke models with audit worlds for the logic $\mathrm{QH4}$ and prove the corresponding soundness and completeness theorems. We also prove that the logics $\mathrm{QHC}$ and $\mathrm{QH4}$ have the disjunction and existence properties. Bibliography: 33 titles.

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“…Moreover, paper [7] extends Gentzen calculus LJ with two rules for the connective K that do not fulfil the subformula property. We also quote [9], that presents a calculus for First Order IEL that extends LJ with new rules for the connective K and the result is a logical apparatus that fulfil the subformula property. Apart the aspect related to the subformula property and computational complexity, in the quoted papers there is no investigation about efficiency in proof search.…”

Section: Introductionmentioning

confidence: 99%

Linear Depth Deduction with Subformula Property for Intuitionistic Epistemic Logic

Fiorino¹

2021

Preprint

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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 to 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 the invalidity.

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First-Order Intuitionistic Epistemic Logic

Cited by 8 publications

References 9 publications

Constructive and Mechanised Meta-Theory of Intuitionistic Epistemic Logic

Constructive and Mechanised Meta-Theory of Intuitionistic Epistemic Logic

The predicate version of the joint logic of problems and propositions

Linear Depth Deduction with Subformula Property for Intuitionistic Epistemic Logic

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