On Provability Logics with Linearly Ordered Modalities

Studia Logica 102 (3):541-566 (2014)
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We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment of GLP Λ



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The Logic of Provability.Philip Scowcroft - 1995 - Philosophical Review 104 (4):627.
Topological completeness of the provability logic GLP.Lev Beklemishev & David Gabelaia - 2013 - Annals of Pure and Applied Logic 164 (12):1201-1223.
Provability algebras and proof-theoretic ordinals, I.Lev D. Beklemishev - 2004 - Annals of Pure and Applied Logic 128 (1-3):103-123.
Kripke semantics for provability logic GLP.Lev D. Beklemishev - 2010 - Annals of Pure and Applied Logic 161 (6):756-774.

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