Simplified Kripke-Style Semantics for Some Normal Modal Logics

Pietruszczak, Andrzej; Klonowski, Mateusz; Petrukhin, Yaroslav

doi:10.1007/s11225-019-09849-2

Cited by 6 publications

(3 citation statements)

References 10 publications

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“…We have also considered the extension with the axiom D, the logic KD45(G), and have shown to be captured by normalised possibilistic Gödel Kripke models. In this way, we have obtained many-valued Gödel generalizations of the results reported by Pietruszczak in [15] about simplified semantics for several classical modal logics. We have also shown the decidability of those logics.…”

Section: Discussionmentioning

confidence: 72%

Simplified Kripke semantics for K45-like Godel modal logics and its axiomatic extensions

Rodríguez¹,

Tuyt²,

Godo³

et al. 2021

Preprint

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Section: Discussionmentioning

confidence: 72%

Simplified Kripke semantics for K45-like Godel modal logics and its axiomatic extensions

Rodríguez¹,

Tuyt²,

Godo³

et al. 2021

Preprint

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“…Remark 1 In Petruszczak (2009) it is indicated that simplified Kripke frames could indeed be used for the semantics of systems K45, KB5 and KD45, using subsets of propositional valuations in place of relations, as we proposed. He proves it by constructing specific accessibility relations equivalent to such subsets, as in (Banerjee & Dubois, 2009) for MEL, while the completeness proof in (Banerjee & Dubois, 2014) (and here) is direct.…”

Section: From Mel To Mel +mentioning

confidence: 88%

On the relation between possibilistic logic and modal logics of belief and knowledge

Banerjee

Dubois

Godo³

et al. 2017

Journal of Applied Non-Classical Logics

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Possibility theory and modal logic are two knowledge representation frameworks that share some common features, such as the duality between possibility and necessity, as well as some obvious differences since possibility theory is graded but is not primarily a logical setting. In the last thirty years there have been a series of attempts, reviewed in this paper, for bridging the two frameworks in one way or another. Possibility theory relies on possibility distributions and modal logic on accessibility relations, at the semantic level. Beyond the observation that many properties of possibility theory have qualitative counterparts in terms of axioms of well-known modal logic systems, the first works have looked for (graded) accessibility relations that can account for the behavior of possibility and necessity measures. More recently, another view has emerged from the study of logics of incomplete information, which is no longer based on general Kripke-like models. On the one hand, possibilistic logic, closely related to possibility theory, mainly handles beliefs having various strengths. On the other hand, in the so-called meta-epistemic logic (MEL) an agent can express both all-or-nothing beliefs and explicitly ignored facts, by only using modal formulas of depth 1, and no objective ones; its semantics is based on subsets of interpretations viewed as epistemic states. The system MEL + is a KD45-like extension of MEL with objective formulas. Generalized possibilistic logic (GPL) extends both possibilistic logic and MEL, and has a semantics in terms of sets of possibility distributions. After a survey of these different attempts, the paper presents GPL + , a graded counterpart of MEL + that extends both MEL and GPL by allowing objective (sub)formulas. The axioms of GPL + are graded counterparts of those of MEL + , the semantics being based on pairs made of an interpretation (representing the real state of facts) and a possibility distribution (representing an epistemic state). S5-like extensions of MEL + and GPL + , called MEL ++ and GPL ++ respectively, are also considered. Soundness and completeness of GPL + and and GPL ++ are established. The paper also discusses the difference between MEL and S5 used as a standard epistemic logic, or used as a logic for rough sets that accounts for indiscernibility rather than incomplete information. We highlight the square of opposition as a common structure underlying modal logic, possibility theory, and rough set theory.

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“…Semantics for extensions of K5 (see [26,28]). Everywhere not ρRρ for the root ρ, set C is a finite cluster, and ⊔ denotes disjoint union.…”

Section: Definition 2 (Kripke Semantics)mentioning

confidence: 99%

Uniform Interpolation via Nested Sequents

Giessen¹,

Jalali²,

Kuznets³

2021

Logic, Language, Information, and Computation

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We introduce a Gentzen-style framework, called layered sequent calculi, for modal logic K5 and its extensions KD5, K45, KD45, KB5, and S5 with the goal to investigate the uniform Lyndon interpolation property (ULIP), which implies both the uniform interpolation property and the Lyndon interpolation property. We obtain complexityoptimal decision procedures for all logics and present a constructive proof of the ULIP for K5, which to the best of our knowledge, is the first such syntactic proof. To prove that the interpolant is correct, we use modeltheoretic methods, especially bisimulation modulo literals.⋆ Supported by a UKRI Future Leaders Fellowship, 'Structure vs Invariants in Proofs', project reference MR/S035540/1. ⋆⋆ Acknowledges the support of the Netherlands Organization for Scientific Research under grant 639.073.807 and the Czech Science Foundation Grant No. 22-06414L. ⋆ ⋆ ⋆ Supported by the Austrian Science Fund (FWF) ByzDEL project (P33600).

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Simplified Kripke-Style Semantics for Some Normal Modal Logics

Cited by 6 publications

References 10 publications

Simplified Kripke semantics for K45-like Godel modal logics and its axiomatic extensions

Simplified Kripke semantics for K45-like Godel modal logics and its axiomatic extensions

On the relation between possibilistic logic and modal logics of belief and knowledge

Uniform Interpolation via Nested Sequents

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