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

Kremer, Philip

doi:10.1017/s1755020313000087

Cited by 27 publications

(44 citation statements)

References 17 publications

(31 reference statements)

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“…(He proved further results, in stronger languages, that imply our theorem 4.7 below for this particular space.) The field opened out when [13] proved that S4 is strongly complete over every dense-in-itself metric space, thereby strengthening McKinsey and Tarski's theorem.…”

Section: Introductionmentioning

confidence: 93%

Strong Completeness of Modal Logics Over 0-Dimensional Metric Spaces

Goldblatt

Hodkinson

2019

The Review of Symbolic Logic

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We prove strong completeness results for some modal logics with the universal modality, with respect to their topological semantics over 0-dimensional dense-inthemselves metric spaces. We also use failure of compactness to show that, for some languages and spaces, no standard modal deductive system is strongly complete.2. If X is the Cantor set, then in the language comprising [ =] and ✷, the system S4DT 1 S is strongly complete over X (corollary 5.12).3. If X is not homeomorphic to the Cantor set, then in the language comprising ∀ and [d], the system KD4U is strongly complete over X (corollary 4.8).The definitions of S4U, KD4U, and S4DT 1 S are not needed here. They can be found in, e.g., [11, §8.1] and [12], and [14, §2] for S4DT 1 S. We will use the completeness results from [10] and [14]. We lift them to strong completeness by methods similar to those of [13] for non-compact spaces, and first-order compactness for the Cantor set. Limitative results will also be given:4. Let X be a dense-in-itself metric space. In any language able to express ∀ and the tangle operators (or the mu-calculus), no modal deductive system is sound and strongly complete over X (corollary 3.2).5. Let X be an infinite compact T1 topological space. In any language able to express ∀ and [d], no modal deductive system is sound and strongly complete over X (theorem 5.1).So for the language comprising ∀ and [d], KD4U is sound and complete over every 0dimensional dense-in-itself metric space X (by the discussion following [10, theorem 8.4]), but by (3) and (5), it is strongly complete only when X is not compact. Over the Cantor set, no modal deductive system for this language is strongly complete.

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Section: Introductionmentioning

confidence: 93%

Strong Completeness of Modal Logics Over 0-Dimensional Metric Spaces

Goldblatt

Hodkinson

2019

The Review of Symbolic Logic

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“…We say that QH is strongly complete for X iff every consistent pair of nonempty sets of sentences is satisfiable in X. If X [X] is a topological 9 We note a counterintuitive but harmless consequence of the fact that…”

Section: Topological Semanticsmentioning

confidence: 99%

“…In Section 6, below, we define a topological space 2 ≤ω , the infinite binary tree with limits, and we show that QH is strongly complete for 2 ≤ω . For every dense-in-itself space X, [9] defines a continuous function f X : X → 2 ≤ω , and we would like to make use of f X to transfer strong completeness from 2 ≤ω back to X -even though f X sometimes fails to be an interior map. We will define a new notion of a quasi-interior map between topological spaces, which will induce a notion of a quasi-p-morphism between predicate topological spaces and predicate topological models.…”

Section: Satisfiability Transferring Mapsmentioning

confidence: 99%

Quantified Intuitionistic Logic Over Metrizable Spaces

Kremer

2019

The Review of Symbolic Logic

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In the topological semantics, quantified intuitionistic logic, QH, is known to be strongly complete not only for the class of all topological spaces but also for some particular topological spaces — for example, for the irrational line, ${\Bbb P}$, and for the rational line, ${\Bbb Q}$, in each case with a constant countable domain for the quantifiers. Each of ${\Bbb P}$ and ${\Bbb Q}$ is a separable zero-dimensional dense-in-itself metrizable space. The main result of the current article generalizes these known results: QH is strongly complete for any zero-dimensional dense-in-itself metrizable space with a constant domain of cardinality ≤ the space’s weight; consequently, QH is strongly complete for any separable zero-dimensional dense-in-itself metrizable space with a constant countable domain. We also prove a result that follows from earlier work of Moerdijk: if we allow varying domains for the quantifiers, then QH is strongly complete for any dense-in-itself metrizable space with countable domains.

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“…Here we only give a sketch of the basic idea underlying most of these proofs. Strong completeness of S4 with respect to any dense-in-itself metric separable space was recently shown in [76] and [77]. A full axiomatization of the space of rational numbers Q in the language with topological and temporal modalities F and P was given in [95].…”

Section: Topological Completenessmentioning

confidence: 99%