Consequence Relations and Admissible Rules

We generalize the (∧, ∨)-canonical formulas to (∧, ∨)-canonical rules, and prove that each intuitionistic multi-conclusion consequence relation is axiomatizable by (∧, ∨)-canonical rules. This yields a convenient characterization of stable superintuitionistic logics. The (∧, ∨)-canonical formulas are analogues of the (∧, →)-canonical formulas, which are the algebraic counterpart of Zakharyaschev's canonical formulas for superintuitionistic logics (si-logics for short). Consequently, stable si-logics are analogues of subframe si-logics. We introduce cofinal stable intuitionistic multi-conclusion consequence relations and cofinal stable si-logics, thus answering the question of what the analogues of cofinal subframe logics should be. This is done by utilizing the (∧, ∨, ¬)-reduct of Heyting algebras. We prove that every cofinal stable si-logic has the finite model property, and that there are continuum many cofinal stable si-logics that are not stable. We conclude with several examples showing the similarities and differences between the classes of stable, cofinal stable, subframe, and cofinal subframe si-logics.formulas that provide a uniform axiomatization for large classes of si-logics and transitive modal logics. Zakharyaschev [26,27] generalized these results by introducing canonical formulas, which axiomatize all si-logics and all transitive modal logics. Jeřábek [21] further generalized Zakharyaschev's approach by defining canonical multi-conclusion rules that axiomatize all intuitionistic multi-conclusion consequence relations and all transitive modal multi-conclusion consequence relations.The algebraic counterparts of Zakharyaschev's canonical formulas for silogics are the (∧, →)-canonical formulas of [4]. The (∧, →)-canonical formula of a finite subdirectly irreducible (s.i.) Heyting algebra A fully describes the ∨-free reduct of A, but describes the behavior of the missing connective ∨ only partially. In fact, the algebraic content of Zakharyaschev's closed domain condition (CDC) is encoded by D ⊆ A 2 , where the behavior of ∨ is described fully. The (∧, →)-canonical formulas, though syntactically quite different, serve the same purpose as Zakharyaschev's canonical formulas in providing a uniform axiomatization of all si-logics.One of the main technical tools in developing the theory of (∧, →)-canonical formulas is Diego's theorem [11] that the variety of implicative meet-semilattices is locally finite. Another locally finite variety closely related to the variety of Heyting algebras is that of bounded distributive lattices. This suggests a different approach to canonical formulas, which was developed in [5], where (∧, ∨)-canonical formulas were introduced. The (∧, ∨)-canonical formula of a finite s.i. Heyting algebra A fully describes the bounded lattice reduct of A, and only partially the behavior of the missing connective →. In this case the CDC is encoded by D ⊆ A 2 , where the behavior of → is described fully. As in the (∧, →)-case, the (∧, ∨)-canonical formulas provide a uniform axiomatization of a...

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“…For more details see [18,21]. A multi-conclusion rule is an expression of the form Γ/Δ, where Γ and Δ are finite sets of formulas.…”

Section: Intuitionistic Multi-conclusion Consequence Relationsmentioning

confidence: 99%

Cofinal Stable Logics

Bezhanishvili

Ilin

2016

Stud Logica

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“…We use [12,15,22, 36] as our main references for the basic theory of normal modal logics, including their algebraic and relational semantics, and the dual equivalence between modal algebras and modal spaces (descriptive Kripke frames). We also use [14] for universal algebra, [23,31] for modal rules, and [20,21] for multi-conclusion modal rules. …”

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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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“…On a more abstract level, there are two definitions of admissibility in the literature that in most instances amount to the same notion. Although intuitively clear, the proper connection between the two is not completely straightforward [14,17]. And it is mostly considered only for admissibility with respect to the set of all substitutions.…”

Section: Discussionmentioning

confidence: 99%

Structural completeness in propositional logics of dependence

Iemhoff

Yang

2016

Arch. Math. Logic

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In this paper we prove that three of the main propositional logics of dependence (including propositional dependence logic and inquisitive logic), none of which is structural, are structurally complete with respect to a class of substitutions under which the logics are closed. We obtain an analogues result with respect to stable substitutions, for the negative variants of some well-known intermediate logics, which are intermediate theories that are closely related to inquisitive logic.

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Consequence Relations and Admissible Rules

Cited by 28 publications

References 12 publications

Cofinal Stable Logics

Cofinal Stable Logics

Stable Canonical Rules

Structural completeness in propositional logics of dependence

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