Krull Dimension in Modal Logic

Bezhanishvili, Guram; Bezhanishvili, Nick; Lucero-Bryan, Joel; Mill, Jan van

doi:10.1017/jsl.2017.14

Cited by 9 publications

(14 citation statements)

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“…We write ¬ to denote co-Heyting negation, and ← to denote co-Heyting implication. 7 6 For recent related literature see [7], where a Krull dimension is defined for any topological space and is used in obtaining fine-grained topological completeness results for modal and intermediate logics. If X is any topological space, we write O (X) for its collection of opens sets.…”

Section: Finite Esakia Duality Lemma 21 Can Be Lifted To a Contravamentioning

confidence: 99%

Tarski's theorem on intuitionistic logic, for polyhedra

Bezhanishvili

Marra

McNeill

et al. 2018

Annals of Pure and Applied Logic

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In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space R n with n 1 suffices, as does e.g. the Cantor space. In particular, intuitionistic logic cannot detect topological dimension in the frame of all open sets of a Euclidean space. By contrast, we consider the lattice of open subpolyhedra of a given compact polyhedron P ⊆ R n , prove that it is a locally finite Heyting subalgebra of the (non-locally-finite) algebra of all open sets of R n , and show that intuitionistic logic is able to capture the topological dimension of P through the bounded-depth axiom schemata. Further, we show that intuitionistic logic is precisely the logic of formulae valid in all Heyting algebras arising from polyhedra in this manner. Thus, our main theorem reconciles through polyhedral geometry two classical results: topological completeness in the style of Tarski, and Jaśkowski's theorem that intuitionistic logic enjoys the finite model property. Several questions of interest remain open. E.g., what is the intermediate logic of all closed triangulable manifolds? (16)].

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Section: Finite Esakia Duality Lemma 21 Can Be Lifted To a Contravamentioning

confidence: 99%

Tarski's theorem on intuitionistic logic, for polyhedra

Bezhanishvili

Marra

McNeill

et al. 2018

Annals of Pure and Applied Logic

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“…The formula χ F encodes the structure of the frame F in such a way that for any S4-frame G we have G ¬χ F iff F is not a p-morphic image of a generated subframe of G [11, Section 2, Lemma I]. The following generalizes Fine's result to the topological setting (see [3,Lemma 3.5]):…”

Section: A New Logic Arising From a Scattered Stone Spacementioning

confidence: 94%

“…In fact, we prove a stronger result that if X is a scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zerodimensional space, then L(X) is either S4.Grz or S4.Grz n for some n ≥ 1 depending on the Cantor-Bendixson rank of X. Our results are proved within ZFC, with key technical tool being the notion of modal Krull dimension introduced in [3].…”

Section: Introductionmentioning

confidence: 94%

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On modal logics arising from scattered locally compact Hausdorff spaces

Bezhanishvili

Lucero-Bryan

et al. 2019

Annals of Pure and Applied Logic

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On modal logics arising from scattered locally compact Hausdorff spaces Bezhanishvili, G.; Bezhanishvili, N.; Lucero-Bryan, J.; van Mill, J.Abstract. For a topological space X, let L(X) be the modal logic of X where is interpreted as interior (and hence ♦ as closure) in X. It was shown in [6] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, S4.Grz n (n ≥ 1), and their intersections arise as L(X) for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to [6, Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either S4.Grz or S4.Grz n for some n ≥ 1. In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space.2010 Mathematics Subject Classification. 03B45; 54G12.

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“…On the other hand, a logical approach gives rise to an entirely new set of problems in topology. For example, recent work on topological modal logic led to a novel concept of modal dimension for topological spaces [4] and also established some new connections between topological completeness of modal logics and large cardinal axioms of set theory [8].…”

Section: Introductionmentioning

confidence: 99%