On Provability Logics with Linearly Ordered Modalities

Beklemishev, Lev D.; Fernández–Duque, David; Joosten, Joost J.

doi:10.1007/s11225-013-9490-7

Cited by 29 publications

(51 citation statements)

References 19 publications

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“…However, the formula M + (ϕ) is, in a sense, inconvenient since its size does not depend on the size of ϕ and, additionally, M + (ϕ) is not necessarily in the language of ϕ. We borrow a result from [10] to improve upon that. Let m 1 ϕ 1 , m 2 ϕ 2 , .…”

Section: Further Resultsmentioning

confidence: 99%

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A many-sorted variant of Japaridze’s polymodal provability logic

Berger

Beklemishev

Tompits

2018

Logic Journal of the IGPL

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We consider a many-sorted variant of Japaridze's polymodal provability logic GLP. In this variant, propositional variables are assigned sorts n < ω, where variables of sort n are arithmetically interpreted as Πn+1-sentences of the arithmetical hierarchy. We prove that the many-sorted variant is arithmetically complete with respect to this interpretation.The Logics GLP, GLP * , and J * . The polymodal provability logic GLP is formulated in the language of the propositional calculus enriched by unary connectives 0 , 1 , 2 , . . ., called

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Section: Further Resultsmentioning

confidence: 99%

“…For the other direction, suppose GLP R + Θ (ϕ) → ϕ. It follows from results of[7] together with a result of[10] that this implies…”

mentioning

confidence: 92%

A many-sorted variant of Japaridze’s polymodal provability logic

Berger

Beklemishev

Tompits

2018

Logic Journal of the IGPL

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“…We note that replacing propositional constants by propositional variables in a formula does not change its derivability in GLP. Therefore, using Lemma 3.5 (which was essentially proved in [21]), we easily see that the new formulations of Lemmas 3.6 and 6.3 (to be given below) are equivalent to the original ones.…”

Section: Consider the Setmentioning

confidence: 90%

Linear $ \mathrm{GLP}$-algebras and their elementary theories

Pakhomov¹

2016

Izv. Math.

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The polymodal provability logic GLP was introduced by Japaridze in 1986. It is the provability logic of certain chains of provability predicates of increasing strength. Every polymodal logic corresponds to a variety of polymodal algebras. Beklemishev and Visser asked whether the elementary theory of the free GLP-algebra generated by the constants 0, 1 is decidable [1]. For every positive integer n we solve the corresponding question for the logics GLPn that are the fragments of GLP with n modalities. We prove that the elementary theory of the free GLPn-algebra generated by the constants 0, 1 is decidable for all n. We introduce the notion of a linear GLPn-algebra and prove that all free GLPn-algebras generated by the constants 0, 1 are linear. We also consider the more general case of the logics GLPα whose modalities are indexed by the elements of a linearly ordered set α: we define the notion of a linear algebra and prove the latter result in this case.

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“…Remark 4.10. The given proof also works for the more general analogs of RC, e.g., for logics with linearly ordered sets of modalities (see [1]).…”

Section: We Say That a Sequentmentioning

confidence: 96%

Positive provability logic for uniform reflection principles

Beklemishev

2014

Annals of Pure and Applied Logic

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We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant 'true' by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform reflection schemata in arithmetic, possibly of unrestricted logical complexity. We formulate an arithmetically complete calculus with modalities labeled by natural numbers and ω, where ω corresponds to the full uniform reflection schema, whereas n < ω corresponds to its restriction to arithmetical Π n+1 -formulas. This calculus is shown to be complete w.r.t. a suitable class of finite Kripke models and to be decidable in polynomial time.

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On Provability Logics with Linearly Ordered Modalities

Cited by 29 publications

References 19 publications

A many-sorted variant of Japaridze’s polymodal provability logic

A many-sorted variant of Japaridze’s polymodal provability logic

Linear $ \mathrm{GLP}$-algebras and their elementary theories

Positive provability logic for uniform reflection principles

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