Journal of Symbolic Logic 58 (1):249-290 (1993)
PA is Peano Arithmetic. Pr(x) is the usual Σ1-formula representing provability in PA. A strong provability predicate is a formula which has the same properties as Pr(·) but is not Σ1. An example: Q is ω-provable if PA + ¬ Q is ω-inconsistent (Boolos ). In  Dzhaparidze introduced a joint provability logic for iterated ω-provability and obtained its arithmetical completeness. In this paper we prove some further modal properties of Dzhaparidze's logic, e.g., the fixed point property and the Craig interpolation lemma. We also consider other examples of the strong provability predicates and their applications
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Provability Algebras and Proof-Theoretic Ordinals, I.Lev D. Beklemishev - 2004 - Annals of Pure and Applied Logic 128 (1-3):103-123.
Topological Completeness of the Provability Logic GLP.Lev Beklemishev & David Gabelaia - 2013 - Annals of Pure and Applied Logic 164 (12):1201-1223.
On Provability Logics with Linearly Ordered Modalities.Lev D. Beklemishev, David Fernández-Duque & Joost J. Joosten - 2014 - Studia Logica 102 (3):541-566.
Kripke Semantics for Provability Logic GLP.Lev D. Beklemishev - 2010 - Annals of Pure and Applied Logic 161 (6):756-774.
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