An algebraic look at filtrations in modal logic

Conradie, Willem; Morton, Wilmari; Alten, C. J. Van

doi:10.1093/jigpal/jzt001

Cited by 5 publications

(3 citation statements)

References 11 publications

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“…The method can be developed either algebraically [28,29] or frame-theoretically [26,33], and the two are connected via duality [24,25]. For a recent account of filtrations we refer to [16,18]. In this section we give a slightly different account which is more suited for our purposes, and also discuss the connection with stable homomorphisms and the closed domain condition.…”

Section: Proof By Stone Duality It Is Sufficient To Show Thatmentioning

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: Proof By Stone Duality It Is Sufficient To Show Thatmentioning

confidence: 99%

Stable Canonical Rules

Bezhanishvili¹,

Bezhanishvili²,

Iemhoff³

2016

J. symb. log.

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“…We recall that an algebraic account of filtrations in modal logic was first given in [27,28] (see also [25,26]). For a more recent discussion of filtrations algebraically we refer to [3,15,20]. Here we follow the construction discussed in [3, sec.…”

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confidence: 99%

Stable Modal Logics

Bezhanishvili

Ilin

2018

The Review of Symbolic Logic

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Stable logics are modal logics characterized by a class of frames closed under relation preserving images. These logics admit all filtrations. Since many basic modal systems such as K4 and S4 are not stable, we introduce the more general concept of an M-stable logic, where M is an arbitrary normal modal logic that admits some filtration. Of course, M can be chosen to be K4 or S4. We give several characterizations of M-stable logics. We prove that there are continuum many S4-stable logics and continuum many K4-stable logics between K4 and S4. We axiomatize K4-stable and S4-stable logics by means of stable formulas and discuss the connection between S4-stable logics and stable superintuitionistic logics. We conclude the article with many examples (and nonexamples) of stable, K4-stable, and S4-stable logics and provide their axiomatization in terms of stable rules and formulas.

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“…It follows from results in [4] that 3 B is a normal operator extending 3 d , and hence that B |= ϕ ≈ .…”

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confidence: 97%

An analogue of Bull's theorem for Hybrid Logic

Robinson¹,

Conradie²

EPiC Series in Computing

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Hybrid logic extends modal logic with a special sort of variables, called nominals, which are evaluated to singletons in Kripke models by valuations, thus acting as names for states in models. Various syntactic mechanisms for exploiting and enhancing the expressive power gained through the addition of nominals can be included, most characteristically the satisfaction operator, @_ip, allowing one to express that p holds at the world named by a nominal i.R.A. Bull famously proved that each normal extension of S4.3 has the finite modelproperty. In the current paper, we prove a hybrid analogue of Bull's result. Like the proof of Bull's original result, ours is algebraic, and thus our secondary aim with this work is to illustrate the usefulness of algebraic methods within hybrid logic research, a field where such methods have been largely ignored.

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An algebraic look at filtrations in modal logic

Cited by 5 publications

References 11 publications

Stable Canonical Rules

Stable Canonical Rules

Stable Modal Logics

An analogue of Bull's theorem for Hybrid Logic

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