Admissible Bases Via Stable Canonical Rules

Bezhanishvili, Nick; Gabelaia, David; Ghilardi, Silvio; Jibladze, Mamuka

doi:10.1007/s11225-015-9642-z

Cited by 8 publications

(8 citation statements)

References 22 publications

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“…In [10] the technique of stable canonical rules is utilized to give an alternative proof of the existence of explicit bases of admissible rules for the intuitionistic logic, S4, and K4. + (A, D)), and since this class is a variety, it is closed under homomorphic images, so C ∈ U(S K4 + (A, D)).…”

Section: Finite Si K4-algebra D I ⊆ a I And For Each Si K4-algmentioning

confidence: 99%

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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Section: Finite Si K4-algebra D I ⊆ a I And For Each Si K4-algmentioning

confidence: 99%

Stable Canonical Rules

Bezhanishvili¹,

Bezhanishvili²,

Iemhoff³

2016

J. symb. log.

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“…We prove that K is axiomatised by (8). Let A ∈ K and A i be an arbitraryminimal element of K ′ \ K. Since by (a) K is a down set, A i A.…”

Section: Stable K-potent Logicsmentioning

confidence: 93%

“…But then, by Definition 4.8 item 2, A |= ζ(A i ). As A i was arbitrary, A validates all formulas in (8). Vice versa, if A ∈ K then by (b), there exists a finite B ∈ K ′ \ K such that B A.…”

Section: Stable K-potent Logicsmentioning

confidence: 97%

“…(4) In order to remove the need for a unique second-last element one can work with canonical rules instead of canonical formulas; see [22] and [5] for similar results for modal logics. In [22] and [8] these canonical rules are used for obtaining bases for admissible rules for transitive modal logics and intermediate logics. Furthermore, these rules yield alternative proofs of the decidability of the admissibility problem for these logics.…”

Section: Further Directionsmentioning

confidence: 99%

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Canonical formulas for k-potent commutative, integral, residuated lattices

2017

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Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Indeed, they provide a uniform and semantic way of axiomatising all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective, canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper, we export the machinery of canonical formulas to substructural logics by introducing canonical formulas for k-potent, commutative, integral, residuated lattices (k-CIRL). We show that any subvariety of k-CIRL is axiomatised by canonical formulas. The paper ends with some applications and examples

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“…The situation has been changed with Jeřábek's paper [31] and his observation that multi-conclusion inference rules may be used for the canonical axiomatization of intermediate and modal logics. This topic was recently undertaken in many papers [2,4,3,5,6,13,14,15,16,24,30,29,32].…”

Section: Introductionmentioning

confidence: 99%

On the Blok-Esakia theorem for universal classes

Stronkowski

2018

Preprint

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The Blok-Esakia theorem states that there is an isomorphism from the lattice of intermediate logics onto the lattice of normal extensions of Grzegorczyk modal logic. The extension for multiconclusion consequence relations was obtained by Emil Jeřábek as an application of canonical rules. We show that Jeřábek's result follows already from Blok's algebraic proof.We also prove that the properties of strong structural completeness and strong universal completeness are preserved and reflects by the aforementioned isomorphism. These properties coincide with structural completeness and universal completeness respectively for singleconclusion consequence relations and, in particular, for logics.

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Admissible Bases Via Stable Canonical Rules

Abstract: Abstract.We establish the dichotomy property for stable canonical multi-conclusion rules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics.

Cited by 8 publications

References 22 publications

Stable Canonical Rules

Stable Canonical Rules

Canonical formulas for k-potent commutative, integral, residuated lattices

On the Blok-Esakia theorem for universal classes

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