Cofinal Stable Logics

Bezhanishvili, Guram; Bezhanishvili, Nick; Ilin, Julia

doi:10.1007/s11225-016-9677-9

Cited by 10 publications

(29 citation statements)

References 19 publications

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“…These rules were recently introduced in [1], where it was shown that each normal modal multi-conclusion consequence relation is axiomatizable by stable multi-conclusion canonical rules. The same result for intuitionistic multi-conclusion consequence relations was established in [2].…”

Section: Introductionsupporting

confidence: 73%

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Admissible Bases Via Stable Canonical Rules

Bezhanishvili

Gabelaia²,

Ghilardi

et al. 2016

Stud Logica

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Section: Introductionsupporting

confidence: 73%

“…Now assume that a transitive modal space (W, R) does not validate (2). We show that then it does not validate (1).…”

Section: Proof Let ρ(A D) Be the Rulementioning

confidence: 90%

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Admissible Bases Via Stable Canonical Rules

Bezhanishvili

Gabelaia²,

Ghilardi

et al. 2016

Stud Logica

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“…For more in-depth development of the theory of stable modal systems see [5]. The theory of stable superintuitionistic logics and stable intuitionistic multi-conclusion consequence relations is developed in [3,4].Stable canonical rules have several applications. For example, they are utilized in [10] to obtain an alternative proof of the existence of explicit bases of admissible rules for the intuitionistic logic, K4, and S4.…”

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confidence: 99%

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Stable Canonical Rules

Bezhanishvili¹,

Bezhanishvili²,

Iemhoff³

2016

J. symb. log.

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Abstract. We introduce stable canonical rules and prove that each normal modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. We apply these results to construct finite refutation patterns for modal formulas, and prove that each normal modal logic is axiomatizable by stable canonical rules. We also define stable multi-conclusion consequence relations and stable logics and prove that these systems have the finite model property. We conclude the paper with a number of examples of stable and nonstable systems, and show how to axiomatize them. §1. Introduction. It is a well-known result of Zakharyaschev [38] that each normal extension of K4 is axiomatizable by canonical formulas. This result was generalized in two directions. In [2] it was generalized to all normal extensions of wK4, and in [21] Zakharyaschev's canonical formulas were generalized to multi-conclusion canonical rules and it was proved that each normal modal multi-conclusion consequence relation over K4 is axiomatizable by canonical rules.The key ingredients of Zakharyaschev's technique include the concepts of subreduction, closed domain condition, and selective filtration. While selective filtration is very effective in the transitive case [15], and also generalizes to the weakly transitive case [2,6], it is less effective for K. This is one of the reasons why canonical formulas and rules do not work well for K [15,21]. In [3] a different approach to canonical formulas for intuitionistic logic was developed that uses the technique of filtration instead of selective filtration. The new canonical formulas were called stable canonical formulas, and it was shown that each superintuitionistic logic is axiomatizable by stable canonical formulas.In this paper we generalize the technique of [3] to the modal setting. Since the technique of filtration works well for K, we show that this new technique is effective in the nontransitive case as well. We give an algebraic account of filtration, introduce stable canonical rules, and prove that each normal modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. This allows us to construct finite refutation patterns for modal formulas, and to show that each normal modal logic is axiomatizable by stable canonical rules. For normal extensions of K4 we prove that stable canonical rules can be replaced by stable canonical formulas, thus providing an alternative to Zakharyaschev's axiomatization [38]. This approach also yields a new class of multi-conclusion consequence relations and logics. Following [3], we call these systems stable. We show that every stable multi-conclusion consequence relation and every stable logic has the finite model property. We also give a number of examples of stable and nonstable logics, and show how to axiomatize them. For more in-depth development of the theory of stable modal systems see [5]. The theory of stable superintuitionistic logics and stable intuitionistic multi-conclusion consequence relations is developed in [3,4]...

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Jankov Formulas and Axiomatization Techniques for Intermediate Logics

Bezhanishvili

2022

Outstanding Contributions to Logic

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Cofinal Stable Logics

Cited by 10 publications

References 19 publications

Admissible Bases Via Stable Canonical Rules

Admissible Bases Via Stable Canonical Rules

Stable Canonical Rules

Jankov Formulas and Axiomatization Techniques for Intermediate Logics

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