Linked bibliography for the SEP article "Provability Logic" by Rineke (L.C.) Verbrugge

Linked bibliography for the SEP article "Provability Logic" by Rineke (L.C.) Verbrugge

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General references on provability logic

Artemov, S.N. and L.D. Beklemishev, 2004, “Provability Logic,” in Handbook of Philosophical Logic, Second Edition, D. Gabbay and F. Guenthner, eds., Volume 13, Dordrecht: Kluwer, pp. 229–403. (Scholar)
Artemov, S.N., 2006, “Modal Logic in Mathematics,” in P. Blackburn, et al. (eds.), Handbook of Modal Logic, Amsterdam: Elsevier, pp. 927–970. (Scholar)
Boolos, G., 1979, The Unprovability of Consistency: An Essay in Modal Logic, Cambridge: Cambridge University Press. (Scholar)
Boolos, G., 1993, The Logic of Provability, New York and Cambridge: Cambridge University Press. (Scholar)
de Jongh, D.H.J. and G. Japaridze, 1998, “The Logic of Provability,” in Handbook of Proof Theory, Buss, S.R. (ed.), Amsterdam: North-Holland, pp. 475-546. (Scholar)
Lindström, P., 1996, “Provability Logic—A Short Introduction,” Theoria, 52(1–2): 19–61. (Scholar)
Segerberg, K., 1971, An Essay in Classical Modal Logic, Uppsala: Filosofiska Föreningen och Filosofiska Institutionen vid Uppsala Universitet. (Scholar)
Švejdar, V., 2000, “On Provability Logic,” Nordic Journal of Philosophy, 4: 95–116. (Scholar)
Smoryński, C., 1995, Self-Reference and Modal Logic, New York: Springer-Verlag. (Scholar)
Verbrugge, R. 1996, “Provability” in The Encyclopedia of Philosophy (Supplement), D.M. Borchert (ed.), New York: Simon and Schuster MacMillan, pp. 476–478. (Scholar)
Visser, A., 1998, “Provability Logic,” in Routledge Encyclopedia of Philosophy, W. Craig (ed.), London: Routledge, pp. 793–797. (Scholar)

History

van Benthem, J.F.A.K., 1978, “Four Paradoxes,” Journal of Philosophical Logic, 7(1): 49–72. (Scholar)
Boolos, G. and G. Sambin, 1991, “Provability: The Emergence of a Mathematical Modality,” Studia Logica, 50(1): 1–23. (Scholar)
Gödel, K., 1933, “Eine Interpretation des Intuitionistischen Aussagenkalküls,” Ergebnisse eines Mathematischen Kolloquiums, 4: 39–40; translation “An Interpretation of the Intuitionistic Propositional Calculus,” in K. Gödel, Collected Works, S. Feferman et al. (eds.), Oxford and New York: Oxford University Press, Volume 3, 1995, pp. 296–302. (Scholar)
Gödel, K., 1931, “Über Formal Unentscheidbare Sätze der Principia Mathematica und Verwandter Systeme I,” Monatshefte für Mathematik und Physik, 38: 173–198. (Scholar)
Henkin, L., 1952, “A Problem Concerning Provability,” Journal of Symbolic Logic, 17: 160. (Scholar)
Hilbert, D. and P. Bernays, 1939, Grundlagen der Mathematik, volume 2, Berlin/Heidelberg/New York: Springer-Verlag. (Scholar)
Lewis, C.I., 1912, “Implication and the Algebra of Logic,” Mind, 21: 522–531. (Scholar)
Löb, M.H., 1955, “Solution of a Problem of Leon Henkin,” Journal of Symbolic Logic, 20: 115–118. (Scholar)
Macintyre, A.J. and H. Simmons, 1973, “Gödel's Diagonalization Technique and Related Properties of Theories,” Colloquium Mathematicum, 28: 165–180. (Scholar)
Magari, R., 1975a, “The Diagonalizable Algebras,” Bollettino della Unione Mathematica Italiana, 12: 117–125. (Scholar)
Magari, R., 1975b, “Representation and Duality Theory for Diagonalizable Algebras,” Studia Logica, 34(4): 305–313. (Scholar)
Smiley, T.J., 1963, “The Logical Basis of Ethics,” Acta Philosophica Fennica, 16: 237–246. (Scholar)

Cut-elimination for provability logic

Goré, R. and R. Ramanayake, 2008, “Valentini's Cut-Elimination for Provability Logic Resolved,” in Advances in Modal Logic Volume 7, C. Areces and R. Goldblatt (eds.), London: College Publications, pp. 67-86. (Scholar)
Negri, S., 2005, “Proof Analysis in Modal Logic,” Journal of Philosophical Logic, 50: 507–544. (Scholar)
Valentini, S., 1983, “The Modal Logic of Provability: Cut-Elimination,” Journal of Philosophical Logic, 12: 471–476. (Scholar)

Possible worlds semantics and topological semantics

Abashidze, M., 1985, “Ordinal Completeness of the Gödel-Löb Modal System,” (in Russian) in Intensional Logics and the Logical Structure of Theories, Tbilisi: Metsniereba, pp. 49–73. (Scholar)
Aiello, M., I. Pratt-Hartmann and J. van Benthem (eds.), 2007, Handbook of Spatial Logics, Berlin: Springer-Verlag. (Scholar)
Beklemishev, L.D., G. Bezhanishvili and T. Icard, 2009, “On Topological Models of GLP,” Logic Group Preprint Series 278, Utrecht: University of Utrecht. Available at http://preprints.phil.uu.nl/lgps/. (Scholar)
Beklemishev, L.D. 2010, “Ordinal completeness of bimodal provability logic GLB,” Logic Group Preprint Series 282, Utrecht: University of Utrecht. Available at http://preprints.phil.uu.nl/lgps/. (Scholar)
Blass, A., 1990, “Infinitary Combinatorics and Modal Logic,” Journal of Symbolic Logic, 55(2): 761–778. (Scholar)
Esakia, L., 1981, “Diagonal Constructions, Löb's Formula and Cantor's Scattered Spaces,” (in Russian), in Studies in Logic and Semantics, Z. Mikeladze (ed.), Tbilisi: Metsniereba, pp. 128–143. (Scholar)
Esakia, L., 2003, “Intuitionistic Logic and Modality via Topology,” Annals of Pure and Applied Logic, 127: 155–170. (Scholar)
Goré, R. and J. Kelly, 2007, “Automated Proof Search in Gödel-Löb Provability Logic,”, British Logic Colloquium 2007, available at http://www.dcs.bbk.ac.uk/~roman/blc/. (Scholar)
Goré, R., 2009, “Machine Checking Proof Theory: An Application of Logic to Logic,” In ICLA '09: Proceedings of the 3rd Indian Conference on Logic and Its Applications, Berlin: Springer-Verlag, pp. 23-35. (Scholar)
Hakli, R. and S. Negri, 2010 (to appear), “Does the Deduction Theorem Fail for Modal Logic?,” Synthese. (Scholar)
Icard, T.F. III, 2009, “A Topological Study of the Closed Fragment of GLP,” Journal of Logic and Computation, Advance Access published online on August 12, 2009, doi:10.1093/logcom/exp043. (Scholar)
Japaridze, G.K., 1986, The Modal Logical Means of Investigation of Provability, Thesis in Philosophy (in Russian), Moscow. (Scholar)
McKinsey, J.C.C. and A. Tarski, 1944, “The Algebra of Topology,” Annals of Mathematics, 45: 141–191. (Scholar)

Provability and Peano Arithmetic

Davis, M., 1958, Computability and Unsolvability, New York, McGraw-Hill; reprinted with an additional appendix, New York, Dover Publications 1983. (Scholar)
Hájek, P. and P. Pudlák, 1993, Metamathematics of First-Order Arithmetic, Berlin: Springer-Verlag. (Scholar)
Solovay, R.M., 1976, “Provability Interpretations of Modal Logic,” Israel Journal of Mathematics, 25: 287–304. (Scholar)

The scope of provability logic

Artemov, S.N., 1985a, “Nonarithmeticity of Truth Predicate Logics of Provability,” Doklady Akademii Nauk SSSR, 284: 270–271 (in Russian); English translation in Soviet Mathematics Dokl, 32: 403–405. (Scholar)
Artemov, S.N., 1985b,“On Modal Logics Axiomatizing Provability,” Izvestiya Akad. Nauk SSSR, ser. mat., 49(6): 1123–1154 (in Russian); English translation in Math. USSR Izvestiya, 27(3). (Scholar)
Artemov, S.N. and L.D. Beklemishev, 1993, “On Propositional Quantifiers in Provability Logic,” Notre Dame Journal of Formal Logic, 34: 401–419. (Scholar)
Artemov, S.N., 1994, “Logic of Proofs,” Annals of Pure and Applied Logic, 67(2): 29–59. (Scholar)
Artemov, S.N. and F. Montagna, 1994, “On First-order Theories with Provability Operator,” Journal of Symbolic Logic, 59(4): 1139–1153. (Scholar)
Artemov, S.N., 2001, “Explicit Provability and Constructive Semantics,” Bulletin of Symbolic Logic, 7: 1–36. (Scholar)
Artemov, S.N. and R. Iemhoff, 2007, “The Basic Intuitionistic Logic of Proofs,” Journal of Symbolic Logic, 72(2): 439–451. (Scholar)
Beklemishev, L.D., 1989, “On the Classification of Propositional Provability Logics,” Izvestiya Akademii Nauk SSSR, ser. mat., 53(5): 915–943 (in Russian); English translation in Math. USSR Izvestiya, 35 (1990) 247–275. (Scholar)
Beklemishev, L.D., 1994, “On Bimodal Logics of Provability,” Annals of Pure and Applied Logic, 68: 115–160. (Scholar)
Beklemishev, L.D., 1996, “Bimodal Logics for Extensions of Arithmetical Theories,” Journal of Symbolic Logic, 61: 91–124. (Scholar)
Beklemishev, L.D., 1999, “Parameter-Free Induction and Provably Total Computable Functions,” Theoretical Computer Science, 224: 13–33. (Scholar)
Beklemishev, L.D., M. Pentus and N. Vereshchagin, 1999, Provability, Complexity, Grammars, American Mathematical Society Translations (Series 2, Volume 192). (Scholar)
Beklemishev, L.D., 2004, “Provability Algebras and Proof-Theoretic Ordinals, I,” Annals of Pure and Applied Logic, 128: 103–123. (Scholar)
Beklemishev, L.D., 2005, “Reflection Principles and Provability Algebras in Formal Arithmetic,” Uspekhi Matematicheskikh Nauk, 60(2): 3–78. (in Russian); English translation in: Russian Mathematical Surveys, 60(2) (2005): 197–268. (Scholar)
Beklemishev, L.D., 2006, “The Worm Principle,” in Lecture Notes in Logic 27. Logic Colloquium '02, Z. Chatzidakis, P. Koepke, and W. Pohlers (eds.), Natick (MA): AK Peters, pp. 75–95. (Scholar)
Beklemishev, L.D., J. Joosten and M. Vervoort, 2005, “A Finitary Treatment of the Closed Fragment of Japaridze's Provability Logic,” Journal of Logic and Computation, 15(4): 447–463. (Scholar)
Beklemishev, L.D. and A. Visser, 2006, “Problems in the Logic of Provability,” in Mathematical Problems from Applied Logic I: Logics for the XXIst Century, D.M. Gabbay, S.S. Goncharov and M. Zakharyashev (eds.), (International Mathematical Series, Volume 4), New York: Springer, pp. 77–136. (Scholar)
Berarducci, A., 1990, “The Interpretability Logic of Peano Arithmetic,” Journal of Symbolic Logic, 55: 1059–1089. (Scholar)
Berarducci, A. and R. Verbrugge, 1993, “On the Provability Logic of Bounded Arithmetic,” Annals of Pure and Applied Logic, 61: 75–93. (Scholar)
de Jongh, D.H.J. and F. Veltman, 1990, “Provability Logics for Relative Interpretability,” in P.P. Petkov (ed.), Mathematical Logic: Proceedings of the Heyting 1988 Summer School in Varna, Bulgaria, Boston: Plenum Press, pp. 31–42. (Scholar)
Guaspari, D. and R.M. Solovay, 1979, “Rosser Sentences,” Annals of Mathematical Logic, 16: 81–99. (Scholar)
Iemhoff, R., 2000, “A Modal Analysis of some Principles of the Provability Logic of Heyting Arithmetic,” in Advances in Modal Logic (Volume 2), M. Zakharyashev et al. (eds.), Stanford: CSLI Publications, pp. 319–354. (Scholar)
Iemhoff, R., 2001, “On the Admissible Rules of Intuitionistic Propositional Logic,” Journal of Symbolic Logic, 66: 281–294. (Scholar)
Iemhoff, R., 2003, “Preservativity Logic: An Analogue of Interpretability Logic for Constructive Theories,” Mathematical Logic Quarterly, 49(3): 1–21. (Scholar)
Ignatiev, K.N., 1993, “On Strong Provability Predicates and the Associated Modal Logics,” Journal of Symbolic Logic, 58: 249–290. (Scholar)
Jongh, D.H.J. de, M. Jumelet and F. Montagna, 1991, “On the Proof of Solovay's Theorem,” Studia Logica, 50: 51–70. (Scholar)
McGee, V. and G. Boolos, 1987, “The Degree of the Set of Sentences of Predicate Provability Logic that are True under Every Interpretation,” Journal of Symbolic Logic, 52: 165–171. (Scholar)
Montagna, F., 1992, “Polynomially and Superexponentially Shorter Proofs in Fragments of Arithmetic,” Journal of Symbolic Logic, 57: 844–863. (Scholar)
Shavrukov, V.Yu., 1988, “The Logic of Relative Interpretability over Peano Arithmetic,” Technical Report No. 5, Moscow: Steklov Mathematical Institute (in Russian). (Scholar)
Shavrukov, V.Yu., 1993a, “A Note on the Diagonalizable Algebras of PA and ZF,” Annals of Pure and Applied Logic, 61: 161–173. (Scholar)
Shavrukov, V.Yu., 1993b, “Subalgebras of Diagonalizable Algebras of Theories Containing Arithmetic,” Dissertationes Mathematicae, 323. (Scholar)
Shavrukov, V.Yu., 1997, “Undecidability in Diagonalizable Algebras,” Journal of Symbolic Logic, 62: 79–116. (Scholar)
Švejdar, V., 1983, “Modal Analysis of Generalized Rosser Sentences,” Journal of Symbolic Logic, 48: 986–999. (Scholar)
Troelstra, A.S., 1973, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Berlin: Springer-Verlag. (Scholar)
Vardanyan, V.A., 1986, “Arithmetic Complexiy of Predicate Logics of Provability and their Fragments,” Doklady Akademii Nauk SSSR, 288: 11–14 (in Russian); English translation in Soviet Math. Dokl., 33: 569–572. (Scholar)
Visser, A., 1980, Aspects of Diagonalization and Provability, Ph.D. Thesis, Utrecht: University of Utrecht. (Scholar)
Visser, A., 1990, “Interpretability Logic,” in Mathematical Logic: Proceedings of the Heyting 1988 Summer School in Varna, Bulgaria, P.P. Petkov (ed.), Boston: Plenum Press, pp. 175–209. (Scholar)
Visser, A., 1998, “An Overview of Interpretability Logic,” in Advances in Modal Logic (Volume 1), M. Kracht et al. (eds.), Stanford: CSLI Publications, pp. 307–359. (Scholar)
Visser, A., 1999, “Rules and Arithmetics,” Notre Dame Journal of Formal Logic, 40(1): 116–140. (Scholar)
Visser, A., 2002, “Substitutions of Σ₁ Sentences: Explorations between Intuitionistic Propositional Logic and Intuitionistic Arithmetic,” Annals of Pure and Applied Logic, 114: 227–271. (Scholar)
Visser, A. and M. de Jonge, 2006, “No Escape from Vardanyan's Theorem”, Archive of Mathematical Logic, 45(5): 539–554. (Scholar)
Visser, A., 2008, “Closed Fragments of Provability Logics of Constructive Theories,” Journal of Symbolic Logic, 73: 1081–1096. (Scholar)
Zambella, D., 1994, “Shavrukov's Theorem on the Subalgebras of Diagonalizable Algebras for Theories Containing IΔ₀+exp,” Notre Dame Journal of Formal Logic, 35: 147–157. (Scholar)

Philosophical significance

Davis, M., 1990, “Is Mathematical Insight Algorithmic?,” Commentary on Roger Penrose, The Emperor's New Mind, Behavioral and Brain Sciences, 13: 659–660. (Scholar)
Davis, M., 1993, “How Subtle is Gödel's Theorem?” (Commentary on Roger Penrose, The Emperor's New Mind), Behavioral and Brain Sciences, 16: 611–612. (Scholar)
Egré, P., 2005, “The Knower Paradox in the Light of Provability Interpretations of Modal Logics,” Journal of Logic, Language, and Information, 14(1): 13–48. (Scholar)
Kaplan, D. and R. Montague, 1960, , “A Paradox Regained,” Notre Dame Journal of Formal Logic, 1(3): 79–90. (Scholar)
Montague, R., 1963, “Syntactical Treatments of Modality, With Corollaries on Reflection Principles and Finite Axiomatizability,” Acta Philosophica Fennica, 16: 153–67. (Scholar)
Quine, W.V., 1966, “Necessary Truth,” in Quine, W.V., The Ways of Paradox and Other Essays, New York: Random House, pp. 48–56. (Scholar)
Quine, W.V., 1953, “Three Grades of Modal Involvement,” in Proceedings of the 11th International Congress of Philosophy, Amsterdam: North-Holland, pp. 65-81; reprinted in Quine, W.V., 1966, The Ways of Paradox and Other Essays, New York: Random House, pp. 156–174. (Scholar)

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