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9On Topological Models of GLPIn Ralf Schindler (ed.), Ways of Proof Theory, De Gruyter. pp. 135-156. 2010.
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13On the limit existence principles in elementary arithmetic and Σ n 0 -consequences of theoriesAnnals of Pure and Applied Logic 136 (1-2): 56-74. 2005.We study the arithmetical schema asserting that every eventually decreasing elementary recursive function has a limit. Some other related principles are also formulated. We establish their relationship with restricted parameter-free induction schemata. We also prove that the same principle, formulated as an inference rule, provides an axiomatization of the Σ2-consequences of IΣ1.Using these results we show that ILM is the logic of Π1-conservativity of any reasonable extension of parameter-free Π…Read more
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40Franco Montagna’s Work on Provability Logic and Many-valued LogicStudia Logica 104 (1): 1-46. 2016.Franco Montagna, a prominent logician and one of the leaders of the Italian school on Mathematical Logic, passed away on February 18, 2015. We survey some of his results and ideas in the two disciplines he greatly contributed along his career: provability logic and many-valued logic
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20Reflection algebras and conservation results for theories of iterated truthAnnals of Pure and Applied Logic 173 (5): 103093. 2022.
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5Calibrating Provability Logic: From Modal Logic to Reflection CalculusIn Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic, Csli Publications. pp. 89-94. 1998.
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14Dick de Jongh and Franco Montagna. Provable fixed points. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 34 , pp. 229–250 (review)Journal of Symbolic Logic 58 (2): 715-717. 1993.
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11Smullyan Raymond M.. Diagonalization and self-reference. Oxford logic guides, no. 27. Clarendon Press, Oxford University Press, Oxford and New York1994, xv + 396 pp (review)Journal of Symbolic Logic 61 (3): 1052-1055. 1996.
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20Wolfgang Burr. Fragments of Heyting arithmetic. The journal of symbolic logic, vol. 65 , pp. 1223–1240 (review)Bulletin of Symbolic Logic 8 (4): 533-534. 2002.
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4A Note on Strictly Positive Logics and Word Rewriting SystemsIn Sergei Odintsov (ed.), Larisa Maksimova on Implication, Interpolation, and Definability, Springer Verlag. 2018.We establish a natural translation from word rewriting systems to strictly positive polymodal logics. Thereby, the latter can be considered as a generalization of the former. As a corollary we obtain examples of undecidable finitely axiomatizable strictly positive normal modal logics. The translation has its counterpart on the level of proofs: we formulate a natural deep inference proof system for strictly positive logics generalizing derivations in word rewriting systems. We also make some obse…Read more
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15A many-sorted variant of Japaridze’s polymodal provability logicLogic Journal of the IGPL 26 (5): 505-538. 2018.
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Leloup, G., Rings of monoids elementarily equivalent to polynomial rings Miller, C., Expansions of the real field with power functions Ozawa, M., Forcing in nonstandard analysis Rathjen, M., Proof theory of reflection (review)Annals of Pure and Applied Logic 68 343. 1994.
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8Calibrating Provability Logic: From Modal Logic to Reflection CalculusIn Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic, Csli Publications. pp. 89-94. 1998.
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18Provability, complexity, grammarsAmerican Mathematical Society. 1999.(2) Vol., Classification of Propositional Provability Logics LD Beklemishev Introduction Overview. The idea of an axiomatic approach to the study of ...
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49Provability logics for natural Turing progressions of arithmetical theoriesStudia Logica 50 (1). 1991.Provability logics with many modal operators for progressions of theories obtained by iterating their consistency statements are introduced. The corresponding arithmetical completeness theorem is proved
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37Topological completeness of the provability logic GLPAnnals of Pure and Applied Logic 164 (12): 1201-1223. 2013.Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces
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31Provability algebras and proof-theoretic ordinals, IAnnals of Pure and Applied Logic 128 (1-3): 103-123. 2004.We suggest an algebraic approach to proof-theoretic analysis based on the notion of graded provability algebra, that is, Lindenbaum boolean algebra of a theory enriched by additional operators which allow for the structure to capture proof-theoretic information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to 0 can be recovered from the corresponding algebra in a canonical way. This method also establishes links between proof-theoretic ordinal analysi…Read more
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Gödel’s theorem: an incomplete guide to its use and abuse (review)Bulletin of Symbolic Logic 13 (2): 241-242. 2007.
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29Induction Rules, Reflection Principles, and Provably Recursive FunctionsBulletin of Symbolic Logic 8 (2): 302. 2002.
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29A proof-theoretic analysis of collectionArchive for Mathematical Logic 37 (5-6): 275-296. 1998.By a result of Paris and Friedman, the collection axiom schema for $\Sigma_{n+1}$ formulas, $B\Sigma_{n+1}$ , is $\Pi_{n+2}$ conservative over $I\Sigma_n$ . We give a new proof-theoretic proof of this theorem, which is based on a reduction of $B\Sigma_n$ to a version of collection rule and a subsequent analysis of this rule via Herbrand's theorem. A generalization of this method allows us to improve known results on reflection principles for $B\Sigma_n$ and to answer some technical questions lef…Read more
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11Review: Per Lindstrom, Aspects of Incompleteness (review)Journal of Symbolic Logic 63 (4): 1606-1608. 1998.
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Omnibus Review (review)Journal of Symbolic Logic 58 (2): 715-717. 1993.Reviewed Works:Dick de Jongh, Franco Montagna, Provable Fixed Points.Dick de Jongh, Franco Montagna, Much Shorter Proofs.Alessandra Carbone, Franco Montagna, Rosser Orderings in Bimodal Logics.Alessandra Carbone, Franco Montagna, Much Shorter Proofs: A Bimodal Investigation.
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43Kripke semantics for provability logic GLPAnnals of Pure and Applied Logic 161 (6): 756-774. 2010.A well-known polymodal provability logic inlMMLBox due to Japaridze is complete w.r.t. the arithmetical semantics where modalities correspond to reflection principles of restricted logical complexity in arithmetic. This system plays an important role in some recent applications of provability algebras in proof theory. However, an obstacle in the study of inlMMLBox is that it is incomplete w.r.t. any class of Kripke frames. In this paper we provide a complete Kripke semantics for inlMMLBox . Firs…Read more
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152002 european summer meeting of the association for symbolic logic logic colloquium'02Bulletin of Symbolic Logic 9 (1): 71. 2003.
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1318th workshop on logic, language, information and computation (wollic 2011)Bulletin of Symbolic Logic 18 (1): 152-153. 2012.