A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology

McKinsey, J. C. C.

doi:10.2307/2267105

Cited by 215 publications

(99 citation statements)

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“…The filtration method is the main tool for establishing the finite model property in modal logic. 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].…”

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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“…The following completeness theorem is taken from Rasiowa & Sikorski (1963, Theorem XI, 9.1, (vii)), which is in turn derived from McKinsey (1941) and McKinsey & Tarski (1944) Theorem 2.1 is well-known, especially when X = Q, R, or C. For X = Q, there is a new and more accessible proof in van Benthem et al (2006); and, for X = R, there are new and more accessible proofs in Aiello et al (2003), Bezhanishvili & Gehrke (2005), Mints & Zhang (2005) and Hodkinson (2012). For X = Q, Theorem 2.2 is easy to prove and seems to be well-known (though we have not found an explicit statement of it in the literature): we sketch the easy proof in Section §3 below.…”

Section: Strong Completeness For the Infinite Binary Tree With Limitsmentioning

confidence: 99%

“…A proof of the decomposition lemma. The proof here of Lemma 7.1 is adapted from the proof in Rasiowa & Sikorski (1963) of Theorem III, 7.1, itself derived from Tarski (1938), andMcKinsey &Tarski (1944). We had to add minor considerations in order to ensure the ε-clause.…”

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confidence: 99%

Strong Completeness of S4 for Any Dense-in-Itself Metric Space

Kremer

2013

The Review of Symbolic Logic

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Abstract. In the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4's completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the question of strong completeness for these spaces has remained open, together with the more general question of strong completeness for any dense-in-itself metric space. In this paper, we prove that S4 is strongly complete for any dense-in-itself metric space.

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“…Orlov and Gödel understood it as a logic of 'provability' (in order to provide a classical interpretation for the intuitionistic logic of Brouwer and Heyting) and Lewis as a logic of necessity and possibility, that is, as a modal logic. That it can be regarded as the logic of topological spaces was discovered by Stone (1937), Tarski (1938), Tsao-Chen (1938) and McKinsey (1941).…”

Section: Modal Logic Of Topological Spacesmentioning

confidence: 99%

Spatial Logic + Temporal Logic = ?

Kontchakov

Kurucz

Wolter

et al. 2007

Handbook of Spatial Logics

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IntroductionAs follows from the title of this chapter, our primary aim is to analyse possible solutions to the equationwhere the items on the left-hand side are some standard spatial and temporal logics, and + is some 'operator' combining these two logics into a single one. The question we are concerned with is how the computational complexity and the expressive power of the component logics are related to the complexity and expressiveness of the resulting spatio-temporal logic x under various combination operators +.

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A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology

Cited by 215 publications

References 3 publications

Stable Canonical Rules

Stable Canonical Rules

Strong Completeness of S4 for Any Dense-in-Itself Metric Space

Spatial Logic + Temporal Logic = ?

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