Skip to main content
SpringerLink
Log in

Logics of (Formal and Informal) Provability

  • Chapter
  • First Online:
Introduction to Formal Philosophy

Part of the book series: Springer Undergraduate Texts in Philosophy ((SUTP))

  • 124k Accesses

Abstract

Provability logics are, roughly speaking, modal logics meant to capture the formal principles of various provability operators (which apply to sentences) or predicates (which apply to sentence names).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Notice however that different interpretations might make different principles plausible. For instance, (M) is not too convincing in the deontic reading, for unfortunately, not all that should be the case indeed is the case.

  2. 2.

    See [14] for an extensive introduction to issues related to definability of properties of frames.

  3. 3.

    A short historical remark: Gödel suggested the use of explicit proofs to interpret S4 in a public lecture in Vienna, without describing the logic. The content of the lecture has been published in 1995, and the logic of proofs was formulated independently before that publication.

References

  1. Anderson, A. R. (1956). The formal analysis of normative systems (Technical report), DTIC Document.

    Google Scholar 

  2. Artemov, S. (1985). Nonarithmeticity of truth predicate logics of provability. Doklady Akademii Nauk SSSR, 284(2), 270–271.

    Google Scholar 

  3. Artemov, S. (1986). On modal logics axiomatizing provability. Mathematics of the USSR-Izvestiya, 27(3), 401–429.

    Article  Google Scholar 

  4. Artemov, S. (1987). Arithmetically complete modal theories. American Mathematical Society Translations, 2(135), 39–54.

    Google Scholar 

  5. Artemov, S. (1994). Logic of proofs. Annals of Pure and Applied Logic, 67(1–3), 29–59.

    Article  Google Scholar 

  6. Artemov, S. (1995). Operational modal logic (Technical report), Cornell University, MSI, pp. 95–29.

    Google Scholar 

  7. Artemov, S. (1998). Logic of proofs: A unified semantics for modality and λ-terms (Technical report), Cornell University, CFIS, pp. 98–06.

    Google Scholar 

  8. Artemov, S. (2001). Explicit provability and constructive semantics. Bulletin of Symbolic logic, 7, 1–36.

    Article  Google Scholar 

  9. Artemov, S. (2006). Modal logic in mathematics. In P. Blackburn, J. van Benthem, & F. Wolter (Eds.), Handbook of modal logic (pp. 927–970). Burlington: Elsevier.

    Google Scholar 

  10. Artemov, S. (2007). On two models of provability. In D. M. Gabbay, S. S. Goncharov, & M. Zakharyaschev (Eds.), Mathematical problems from applied logic II (pp. 1–52). New York: Springer.

    Google Scholar 

  11. Artemov, S. N., & Beklemishev, L. D. (2005). Provability logic. In D. M. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (2nd ed., pp. 189–360). Dordrecht: Springer.

    Google Scholar 

  12. Beklemishev, L. D. (1990). On the classification of propositional provability logics. Mathematics of the USSR-Izvestiya, 35(2), 247–275.

    Article  Google Scholar 

  13. Beklemishev, L. D. (2003). Proof-theoretic analysis by iterated reflection. Archive for Mathematical Logic, 42(6), 515–552.

    Article  Google Scholar 

  14. Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal logic. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  15. Boolos, G. (1993). The logic of provability. Cambridge: Cambridge University Press.

    Google Scholar 

  16. Boolos, G., & Sambin, G. (1991). Provability: The emergence of a mathematical modality. Studia Logica, 50(1), 1–23.

    Article  Google Scholar 

  17. Carnap, R. (1934). Logische Syntax der Sprache. Berlin/Heidelberg: Springer.

    Book  Google Scholar 

  18. Cieśliński, C. (2017). The epistemic lightness of truth. Deflationism and its logic. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  19. Cieśliński, C. (2016). Minimalism and the generalisation problem: On Horwich’s second solution. Synthese, 1–25. https://doi.org/10.1007/s11229-016-1227-5.

    Article  Google Scholar 

  20. Feferman, S., Dawson, J. W., Jr., Kleene, S. C., Moore, G. H., Solovay, R. M., & van Heijenoort, J. (Eds.). (1986). Kurt Gödel: Collected works. Volume 1: Publications 1929–1936. Oxford/New York: Oxford University Press.

    Google Scholar 

  21. Fitting, M. (2003). A semantics for the logic of proofs (Technical report, TR – 2003012). CUNY Ph.D. Program in Computer Science.

    Google Scholar 

  22. Flagg, R. C., & Friedman, H. (1986). Epistemic and intuitionistic formal systems. Annals of Pure and Applied Logic, 32(1), 53–60.

    Article  Google Scholar 

  23. Gaifman, H. (2006). Naming and diagonalization, from cantor to Gödel to Kleene. Logic Journal of the IGPL, 14(5), 709–728.

    Article  Google Scholar 

  24. Gödel, K. (1933). Eine Interpretation des intuitionistischen Aussagenkalkuls. [Reprinted in [20]].

    Google Scholar 

  25. Goodman, N. D. (1984). Epistemic arithmetic is a conservative extension of intuitionistic arithmetic. Journal of Symbolic Logic, 49(1), 192–203.

    Article  Google Scholar 

  26. Grzegorczyk, A. (1967). Some relational systems and the associated topological spaces. Fundamenta Mathematicae, 60(3), 223–231.

    Article  Google Scholar 

  27. Hájek, P. & Pudlak, P. (1993). Metamathematics of first-order arithmetic (Vol. 2, pp. 295–297). Berlin: Springer.

    Book  Google Scholar 

  28. Halbach, V. (2011). Axiomatic theories of truth. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  29. Hedman, S. (2004). A first course in logic: An introduction to model theory, proof theory, computability, and complexity. Oxford/New York: Oxford University Press.

    Google Scholar 

  30. Henkin, L. (1952). A problem concerning provability. Journal of Symbolic Logic, 17(2), 160.

    Article  Google Scholar 

  31. Heyting, A. (1930). Die formalen Regeln der intuitionistischen Mathematik. Verlag der Akademie der Wissenschaften.

    Google Scholar 

  32. Heyting, A. (1931). Die intuitionistische Grundlegung der Mathematik. Erkenntnis, 2(1), 106–115.

    Article  Google Scholar 

  33. Heyting, A. (1934). Mathematische Grundlagenforschung Intuitionismus, Beweistheorie. Springer.

    Google Scholar 

  34. Hilbert, D., & Bernays, P. (1939). Grundlagen der Mathematik II. Springer.

    Google Scholar 

  35. Horsten, L. (1994). Modal-epistemic variants of Shapiro’s system of epistemic arithmetic. Notre Dame Journal of Formal Logic, 35(2), 284–291.

    Article  Google Scholar 

  36. Horsten, L. (1997). Provability in principle and controversial constructivistic principles. Journal of Philosophical Logic, 26(6), 635–660.

    Article  Google Scholar 

  37. Horsten, L. (1998). In defence of epistemic arithmetic. Synthese, 116, 1–25.

    Article  Google Scholar 

  38. Horsten, L. (2011). The Tarskian turn. Deflationism and axiomatic truth. Cambridge: MIT Press.

    Book  Google Scholar 

  39. Ignatiev, K. N. (1993). On strong provability predicates and the associated modal logics. The Journal of Symbolic Logic, 58(01), 249–290.

    Article  Google Scholar 

  40. Japaridze, G. K. (1985). The polymodal logic of provability. In Intensional Logics and Logical Structure of Theories: Material from the Fourth Soviet–Finnish Symposium on Logic, Telavi (pp. 16–48).

    Google Scholar 

  41. Japaridze, G. K. (1986). The modal logical means of investigation of provability. Ph.D. thesis, Moscow State University.

    Google Scholar 

  42. Japaridze, G., & de Jongh, D. (1998). The logic of provability. In S. R. Buss (Ed.), Handbook of proof theory (Vol. 137, pp. 475–550). Burlington: Elsevier.

    Chapter  Google Scholar 

  43. Kaye, R. (1991). Models of Peano Arithmetic. Oxford: Oxford University Press.

    Google Scholar 

  44. Ketland, J. (2005). Deflationism and the Gödel phenomena: Reply to tennant. Mind, 114(453), 75–88.

    Article  Google Scholar 

  45. Kolmogorov, A. (1932). Zur Deutung der Intuitionistischen Logik. Mathematische Zeitschrift, 35(1), 58–65.

    Article  Google Scholar 

  46. Kossak, R. (2006). The structure of models of Peano Arithmetic. Oxford: Clarendon Press.

    Book  Google Scholar 

  47. Leitgeb, H. (2009). On formal and informal provability. In O. Bueno & Ø. Linnebo (Eds.), New waves in philosophy of mathematics (pp. 263–299). New York: Palgrave Macmillan.

    Chapter  Google Scholar 

  48. Löb, M. H. (1955). Solution of a problem of Leon Henkin. The Journal of Symbolic Logic, 20(02), 115–118.

    Article  Google Scholar 

  49. Magari, R. (1975). The diagonalizable algebras (the algebraization of the theories which express Theor.:II). Bollettino della Unione Matematica Italiana, 4(12), 117–125.

    Google Scholar 

  50. Magari, R. (1975). Representation and duality theory for diagonalizable algebras. Studia Logica, 34(4), 305–313.

    Article  Google Scholar 

  51. Marfori, M. A. (2010). Informal proofs and mathematical rigour. Studia Logica, 96, 261–272.

    Article  Google Scholar 

  52. McKinsey, J. C., & Tarski, A. (1948). Some theorems about the sentential calculi of Lewis and Heyting. The Journal of Symbolic Logic, 13(01), 1–15.

    Article  Google Scholar 

  53. Mkrtychev, A. (1997). Models for the logic of proofs. In S. Adian & A. Nerode (Eds.), Logical foundations of computer science ’97 (pp. 266–275). Berlin/Heidelberg: Springer.

    Chapter  Google Scholar 

  54. Montagna, F. (1978). On the algebraization of a Feferman’s predicate. Studia Logica, 37(3), 221–236.

    Article  Google Scholar 

  55. Montagna, F. (1979). On the diagonalizable algebra of Peano Arithmetic. Bollettino della Unione Matematica Italiana, 16(5), 795–812.

    Google Scholar 

  56. Montagna, F., et al. (1984). The predicate modal logic of provability. Notre Dame Journal of Formal Logic, 25(2), 179–189.

    Article  Google Scholar 

  57. Myhill, J. (1960). Some remarks on the notion of proof. Journal of Philosophy, 57(14), 461–471.

    Article  Google Scholar 

  58. Nogina, E. (1994). Logic of proofs with the strong provability operator (Technical report). Institute for Logic, Language and Computation, University of Amsterdam, ILLC Prepublication Series ML-94-10.

    Google Scholar 

  59. Nogina, E. (1996). Grzegorczyk logic with arithmetical proof operators. Fundamentalnaya i Prikladnaya Matematika, 2(2), 483–499.

    Google Scholar 

  60. Nowell-Smith, P., & Lemmon, E. (1960). Escapism: The logical basis of ethics. Mind, 69(275), 289–300.

    Article  Google Scholar 

  61. Orlov, I. E. (1928). The calculus of compatibility of propositions. Mathematics of the USSR, Sbornik, 35, 263–286.

    Google Scholar 

  62. Pawlowski, P., & Urbaniak, R. (2018). Many-valued logic of informal provability: A non-deterministic strategy. The Review of Symbolic Logic, 11(2), 207–223.

    Article  Google Scholar 

  63. Segerberg, K. K. (1971). An essay in classical modal logic (The Philosophical Society in Uppsala). Uppsala: Uppsala University.

    Google Scholar 

  64. Shapiro, S. (1985). Epistemic and intuitionistic arithemtic. In Intensional mathematics. New York: North Holland.

    Chapter  Google Scholar 

  65. Shavrukov, V. Y. (1997). Undecidability in diagonalizable algebras. The Journal of Symbolic Logic, 62(01), 79–116.

    Article  Google Scholar 

  66. Simpson, S. G. (2009). Subsystems of second order arithmetic (Vol. 1). Cambridge: Cambridge University Press.

    Book  Google Scholar 

  67. Smiley, T. J. (1963). The logical basis of ethics. Acta Philosophica Fennica, 16, 237–246.

    Google Scholar 

  68. Smith, P. (2007). An introduction to Gödel’s theorems. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  69. Smoryński, C. (1985). Self-reference and modal logic (Universitext). New York: Springer.

    Book  Google Scholar 

  70. Smoryński, C. (2004). Modal logic and self-reference. In D. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 11, pp. 1–53). Dordrecht: Springer.

    Google Scholar 

  71. Solovay, R. M. (1976). Provability interpretations of modal logic. Israel Journal of Mathematics, 25(3–4), 287–304.

    Article  Google Scholar 

  72. Sundholm, G. (1998). Proofs as acts and proofs as objects: Some questions for Dag Prawitz. Theoria, 64(2–3), 187–216.

    Google Scholar 

  73. Švejdar, V. (1983). Modal analysis of generalized Rosser sentences. The Journal of Symbolic Logic, 48(04), 986–999.

    Article  Google Scholar 

  74. Švejdar, V. (2000). On provability logic. Nordic Journal of Philosophical Logic, 4(2), 95–116.

    Google Scholar 

  75. Tarski, A., Mostowski, A., & Robinson, R. M. (1953). Undecidable theories. Amsterdam: Elsevier.

    Google Scholar 

  76. Troelstra, A. & van Dalen, D. (1988). Constructivism in mathematics (Vols. 1 and 2). Amsterdam: Elsevier.

    Google Scholar 

  77. Vardanyan, V. A. (1986). Arithmetic complexity of predicate logics of provability and their fragments. Soviet Mathematics Doklady, 33(3), 569–572.

    Google Scholar 

  78. Verbrugge, R. L. (2017). Provability logic. In The Stanford encyclopedia of philosophy (Fall 2017 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/fall2017/entries/logic-provability/.

  79. Visser, A. (1981). Aspects of diagonalization and probability. Ph.D. thesis, University of Utrecht.

    Google Scholar 

  80. Visser, A. (1984). The provability logics of recursively enumerable theories extending Peano Arithmetic at arbitrary theories extending Peano Arithmetic. Journal of Philosophical Logic, 13(1), 97–113.

    Article  Google Scholar 

  81. Visser, A. (1990). Interpretability logic. In P. Petkov (Ed.), Mathematical logic (pp. 175–209). Boston: Springer.

    Chapter  Google Scholar 

  82. Visser, A. (1997). A course on bimodal provability logic. Annals of Pure and Applied Logic, 73, 109–142 (1995); The Journal of Symbolic Logic, 62(02), 686–687.

    Article  Google Scholar 

  83. Visser, A. (1998). An overview of interpretability logic. In M. Kracht, M. de Rijke, H. Wansing, & M. Zakharyaschev (Eds.), Advances in modal logic (Vol. 1, pp. 307–359). Stanford: CSLI Publications.

    Google Scholar 

  84. Visser, A. (2008). Propositional combinations of σ 1 sentences in Heyting’s arithmetic. Logic Group Preprint Series, 117, 1–43.

    Google Scholar 

  85. Yavorskaya, T. (2001). Logic of proofs and provability. Annals of pure and applied logic, 113(1), 345–372.

    Article  Google Scholar 

  86. Yavorsky, R. E. (2001). Provability logics with quantifiers on proofs. Annals of Pure and Applied Logic, 113(1), 373–387.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Logic and Philosophy of Science, Ghent University, Ghent, Belgium

    Rafal Urbaniak & Pawel Pawlowski

  2. Institute of Philosophy, Sociology and Journalism, University of Gdansk, Gdansk, Poland

    Rafal Urbaniak

Authors
  1. Rafal Urbaniak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pawel Pawlowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rafal Urbaniak .

Editor information

Editors and Affiliations

  1. Division of Philosophy, Royal Institute of Technology (KTH), Stockholm, Sweden

    Sven Ove Hansson

  2. Center for Information and Bubble Studies, University of Copenhagen, Copenhagen, Denmark

    Vincent F. Hendricks

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Urbaniak, R., Pawlowski, P. (2018). Logics of (Formal and Informal) Provability. In: Hansson, S., Hendricks, V. (eds) Introduction to Formal Philosophy. Springer Undergraduate Texts in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-319-77434-3_9

Download citation

Publish with us

Policies and ethics

Search

Navigation