Completeness of S4 for the Lebesgue Measure Algebra

Lando, Tamar

doi:10.1007/s10992-010-9161-3

Cited by 10 publications

(12 citation statements)

References 4 publications

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“…In Section 8 we prove our fifth general completeness result: a normal modal logic L is a logic above S4 iff it is the logic of a subalgebra of a homomorphic image of R + . This requires that L 2 is an interior image of R, a result first proved in [26] and [22]. We give an alternate proof of this result.…”

Section: §1 Introduction Topological Semantics For Modal Logic Was mentioning

confidence: 84%

“…We next build an interior map from any (non-trivial) real interval onto L 2 . Such maps have already appeared in the literature; see [26,22]. The sum of disjoint open intervals produces an open subspace X of R and a general space over X whose logic is L. Thus, general spaces over open subspaces of R give rise to all logics above S4.…”

Section: Completeness For General Spaces Over Open Subspaces Of Rmentioning

confidence: 99%

“…We next build an interior map from any (nontrivial) real interval onto L 2 . Such maps have already appeared in the literature; see [22,26] In realizing L 2 as an interior image of a nontrivial real interval I , our method differs from both Kremer's [22] and Lando's [26] methods. Kremer utilizes a decomposition of I (indeed of any complete dense-in-itself metrizable space) into equivalence classes indexed by L 2 , and maps each element in a class to its index.…”

Section: = Log(a) For Some a ∈ S(c + )mentioning

confidence: 99%

“…Since T 2 is not an interior image of C, in order to obtain our completeness results for general spaces over C, we work with the infinite binary tree with limits L 2 . This uncountable tree has been an object of recent interest [22,26]. In particular, Kremer [22] proved that if we equip L 2 with the Scott topology, and denote the result by L 2 , then T + 2 is isomorphic to a subalgebra of L + 2 as well as L 2 is an interior image of any complete dense-in-itself metric space.…”

Section: §1 Introduction Topological Semantics For Modal Logic Was mentioning

confidence: 99%

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TOPOLOGICAL COMPLETENESS OF LOGICS ABOVES4

Bezhanishvili¹,

Gabelaia²,

Lucero-Bryan³

2015

J. symb. log.

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It is a celebrated result of McKinsey and Tarski [28] that S4 is the logic of the closure algebra X + over any dense-in-itself separable metrizable space. In particular, S4 is the logic of the closure algebra over the reals R, the rationals Q, or the Cantor space C. By [5], each logic above S4 that has the finite model property is the logic of a subalgebra of Q + , as well as the logic of a subalgebra of C + . This is no longer true for R, and the main result of [5] states that each connected logic above S4 with the finite model property is the logic of a subalgebra of the closure algebra R + .In this paper we extend these results to all logics above S4. Namely, for a normal modal logic L, we prove that the following conditions are equivalent: (i) L is above S4, (ii) L is the logic of a subalgebra of Q + , (iii) L is the logic of a subalgebra of C + . We introduce the concept of a well-connected logic above S4 and prove that the following conditions are equivalent: (i) L is a well-connected logic, (ii) L is the logic of a subalgebra of the closure algebra T + 2 over the infinite binary tree, (iii) L is the logic of a subalgebra of the closure algebra L + 2 over the infinite binary tree with limits equipped with the Scott topology. Finally, we prove that a logic L above S4 is connected iff L is the logic of a subalgebra of R + , and transfer our results to the setting of intermediate logics.Proving these general completeness results requires new tools. We introduce the countable general frame property (CGFP) and prove that each normal modal logic has the CGFP. We introduce general topological semantics for S4, which generalizes topological semantics the same way general frame semantics generalizes Kripke semantics. We prove that the categories of descriptive frames for S4 and descriptive spaces are isomorphic. It follows that every logic above S4 is complete with respect to the corresponding class of descriptive spaces. We provide several ways of realizing the infinite binary tree with limits, and prove that when equipped with the Scott topology, it is an interior image of both C and R. Finally, we introduce gluing of general spaces and prove that the space obtained by appropriate gluing involving certain quotients of L 2 is an interior image of R. Contents2000 Mathematics Subject Classification. 03B45; 03B55.

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Section: §1 Introduction Topological Semantics For Modal Logic Was mentioning

confidence: 84%

Section: Completeness For General Spaces Over Open Subspaces Of Rmentioning

confidence: 99%

Section: = Log(a) For Some a ∈ S(c + )mentioning

confidence: 99%

Section: §1 Introduction Topological Semantics For Modal Logic Was mentioning

confidence: 99%

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TOPOLOGICAL COMPLETENESS OF LOGICS ABOVES4

Bezhanishvili¹,

Gabelaia²,

Lucero-Bryan³

2015

J. symb. log.

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“…We call the resulting space the infinite binary tree with limits and denote it by L 2 = (L 2 , ≤). This uncountable tree has been an object of recent interest [9,8]. In particular, [8] uses L 2 in a crucial way to obtain strong completeness of S4 for any dense-in-itself metric space.…”

Section: For Topological Spaces X Y We Recall That a Map F : X → Ymentioning

confidence: 99%

Topological completeness of extensions of S4

Bezhanishvili

Gabelaia²,

Lucero-Bryan

EPiC Series in Computing

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It was left as an open problem in (Bezhanishvili and Gabelaia, 2011) whether a connected normal extension of S4 without FMP is also the modal logic of some subalgebra of R+.Our purpose here is to solve this problem affirmatively by showing that each connected normal extension of S4 (with or without FMP) is in fact the modal logic of some subalgebra of R+. We also prove that each normal extension of S4 (with or without FMP) is the modal logic of a subalgebra of Q+, as well as the modal logic of a subalgebra of C+.

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Strong Completeness of S4 for the Real Line

Kremer

2021

Outstanding Contributions to Logic

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Completeness of S4 for the Lebesgue Measure Algebra

Cited by 10 publications

References 4 publications

TOPOLOGICAL COMPLETENESS OF LOGICS ABOVES4

TOPOLOGICAL COMPLETENESS OF LOGICS ABOVES4

Topological completeness of extensions of S4

Strong Completeness of S4 for the Real Line

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