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., 2006, “Modal Logic in Mathematics,” in P. Blackburn, et al. (eds.), Handbook of Modal Logic, Amsterdam: Elsevier, pp. 927–970. (Scholar)
  • 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)
  • Boolos, G., 1979, The Unprovability of Consistency: An Essay in Modal Logic, Cambridge: Cambridge University Press. (Scholar)
  • –––, 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., 1985, 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)


  • 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 1, 1986, pp. 300–302. (Scholar)
  • –––, 1931, “Über Formal Unentscheidbare Sätze der Principia Mathematica und Verwandter Systeme I,” Monatshefte für Mathematik und Physik, 38: 173–198. (Scholar)
  • Halbach, V., and A. Visser, 2014, “The Henkin Sentence,” in M. Mazano, I. Sain, and E. Alonso (eds.), The Life and Work of Leon Henkin: Essays on His Contributions, Dordrecht: Springer International Publishing, pp. 249–263. (Scholar)
  • Henkin, L., 1952, “A Problem Concerning Provability,” Journal of Symbolic Logic, 17: 160. (Scholar)
  • –––., 1954, “Review of G. Kreisel: On a Problem of Leon Henkin’s,” Journal of Symbolic Logic, 19(3): 219–220. (Scholar)
  • Hilbert, D. and P. Bernays, 1939, Grundlagen der Mathematik, volume 2, Berlin/Heidelberg/New York: Springer-Verlag. (Scholar)
  • Kreisel, G., 1953, “On a Problem of Leon Henkin’s,” Indagationes Mathematicae, 15: 405–406. (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.
  • Magari, R., 1975a, “The Diagonalizable Algebras,” Bollettino della Unione Mathematica Italiana, 12: 117–125. (Scholar)
  • –––, 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)
  • Smoryński, C., 1991, “The Development of Self-reference: Löb’s Theorem,” in T. Drucker (ed.), Perspectives on the History of Mathematical Logic, Basel: Birkhäuser, pp. 110–133. (Scholar)

Cut-elimination for provability logic

The fixed point theorem

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. 2009, “Ordinal Completeness of Bimodal Provability Logic GLB,” In International Tbilisi Symposium on Logic, Language, and Computation, Berlin: Springer-Verlag, pp. 1–15. (Scholar)
  • Beklemishev, L.D., G. Bezhanishvili, and T. Icard, 2009, “On Topological Models of GLP,” in R. Schindler (ed.), Ways of Proof Theory (Ontos Mathematical Logic: Volume 2), Frankfurt: Ontos Verlag, pp. 133–153. (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)
  • –––, 2003, “Intuitionistic Logic and Modality via Topology,” Annals of Pure and Applied Logic, 127: 155–170. (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)
  • Goré, R. and J. Kelly, 2007, “Automated Proof Search in Gödel-Löb Provability Logic,”, British Logic Colloquium 2007, available at (Scholar)
  • Hakli, R. and S. Negri, 2012, “Does the Deduction Theorem Fail for Modal Logic?,” Synthese 187(3): 849–867. (Scholar)
  • Icard, T.F. III, 2011, “A Topological Study of the Closed Fragment of GLP,” Journal of Logic and Computation, 21(4): 683–696; first published online 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

The scope of provability logic: boundaries

  • Berarducci, A. and R. Verbrugge, 1993, “On the Provability Logic of Bounded Arithmetic,” Annals of Pure and Applied Logic, 61: 75–93. (Scholar)
  • Buss, S.R., 1986, Bounded Arithmetic, Naples: Bibliopolis. (Scholar)
  • de Jongh, D.H.J., M. Jumelet and F. Montagna, 1991, “On the Proof of Solovay’s Theorem,” Studia Logica, 50(1): 51–70. (Scholar)

Interpretability logic

  • Berarducci, A., 1990, “The Interpretability Logic of Peano Arithmetic,” Journal of Symbolic Logic, 55: 1059–1089. (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)
  • de Jongh, D.H.J., and A. Visser, 1991, “Explicit Fixed Points in Interpretability Logic,” Studia Logica, 50(1): 39–49. (Scholar)
  • Joosten, J.J., and Visser, A., 2000, “The Interpretability Logic of All Reasonable Arithmetical Theories,” Erkenntnis, 53(1-2): 3–26. (Scholar)
  • Montagna, F., 1987, “Provability in Finite Subtheories of PA,” Journal of Symbolic Logic, 52(2): 494–511. (Scholar)
  • Shavrukov, V.Yu., 1988, “The Logic of Relative Interpretability over Peano Arithmetic,” Technical Report No. 5, Moscow: Steklov Mathematical Institute (in Russian). (Scholar)
  • Visser, A., 1990, “Interpretability Logic,” in P.P. Petkov (ed.), Mathematical Logic: Proceedings of the Heyting 1988 Summer School in Varna, Bulgaria, Boston: Plenum Press, pp. 175–209. (Scholar)
  • –––, 1998, “An Overview of Interpretability Logic,” in M. Kracht et al. (eds.), Advances in Modal Logic (Volume 1), Stanford: CSLI Publications, pp. 307–359. (Scholar)

Propositional quantifiers

Japaridze’s bimodal and polymodal provability logics

  • Beklemishev, L.D., 2004, “Provability Algebras and Proof-Theoretic Ordinals, I,” Annals of Pure and Applied Logic, 128: 103–123. (Scholar)
  • –––, 2010a, “Kripke Semantics for Provability Logic GLP,” Annals of Pure and Applied Logic, 161(6): 756–774. (Scholar)
  • –––, 2010b, “On the Craig Interpolation and the Fixed Point Properties of GLP,” in S. Feferman et al. (eds.), Proofs, Categories and Computations (Tributes, 13), London: College Publications, pp. 49–60. (Scholar)
  • –––, 2011a, “A Simplified Proof of Arithmetical Completeness Theorem for Provability Logic GLP,” Proceedings Steklov Institute of Mathematics, 274(1): 25–33. (Scholar)
  • –––, 2011b, “Ordinal Completeness of Bimodal Provability Logic GLB,” in N. Bezhanishvili et al. (eds.), Logic, Language, and Computation, 8th International Tbilisi Symposium TbiLLC 2009 (Lecture Notes in Computer Science: Volume 6618), Heidelberg: Springer, pp. 1–15. (Scholar)
  • Beklemishev, L.D., and D. Gabelaia, 2013, “Topological Completeness of the Provability Logic GLP,” Annals of Pure and Applied Logic, 164(12): 1201–1223. (Scholar)
  • –––, 2014, “Topological Interpretations of Provability Logic,“ in G. Bezhanishvili (ed.), Leo Esakia on Duality in Modal and Intuitionistic Logics (Outstanding Contributions to Logic: Volume 4), Heidelberg: Springer, pp. 257–290. (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)
  • Fernández-Duque, D. and J.J. Joosten, 2014, “Well-orders on the Transfinite Japaridze Algebra,” Logic Journal of the IGPL, 22 (6): 933–963. (Scholar)
  • Ignatiev, K.N., 1993, “On Strong Provability Predicates and the Associated Modal Logics,” Journal of Symbolic Logic, 58: 249–290. (Scholar)
  • Japaridze, G., 1988, “The Polymodal Provability Logic,” In Intensional Logics and the Logical Structure of Theories: Material from the Fourth Soviet-Finnish Symposium on Logic, Telavi, pp. 16–48. (Scholar)
  • Pakhomov, F.N., 2014, “On the Complexity of the Closed Fragment of Japaridze’s Provability Logic,” Archive for Mathematical Logic, 53(7-8): 949–967. (Scholar)

Predicate 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 Doklady, 32: 403–405. (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)
  • 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 Mathematics Doklady, 33: 569–572. (Scholar)
  • Visser, A. and M. de Jonge, 2006, “No Escape from Vardanyan’s Theorem”, Archive of Mathematical Logic, 45(5): 539–554. (Scholar)

Other generalizations

Philosophical significance

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