, Volume 176, Issue 2, pp 177–225 | Cite as

From the Knowability Paradox to the existence of proofs

  • W. DeanEmail author
  • H. Kurokawa


The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Löf using the sort of propositional modal operators which are employed in the original derivation of the Paradox. Instead we propose that the central tenet of verificationism is most accurately formulated as follows: if \({\varphi}\) is true, then there exists a proof of \({\varphi}\). Building on the work of Artemov (Bull Symb Log 7(1): 1–36, 2001), a system of explicit modal logic with proof quantifiers is introduced to reason about such statements. When the original reasoning of the Paradox is developed in this setting, we reach not a contradiction, but rather the conclusion that there must exist non-constructed proofs. This outcome is evaluated relative to the controversy between Dummett and Prawitz about proof existence and bivalence.


Knowability Paradox Fitch Verificationism Intuitionistic logic BHK interpretation Existence predicate Logic of proofs Potential proof Bivalence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Artemov S.N. (2001) Explicit provability and constructive semantics. The Bulletin of Symbolic Logic 7(1): 1–36CrossRefGoogle Scholar
  2. Artemov S., Iemhoff R. (2007) The basic intuitionistic logic of proofs. Journal of Symbolic Logic 72(2): 439–451CrossRefGoogle Scholar
  3. Beall J.C. (2000) Fitch’s proof, verificationism and the knower paradox. Australasian Journal of Philosophy 78(2): 241–247CrossRefGoogle Scholar
  4. Beeson M. (1985) Foundations of constructive mathematics: Metamathematical studies. Springer Verlag, BerlinGoogle Scholar
  5. Binmore, K., & Shin, H. (1993). Algorithmic knowledge and game theory. In C. Bicchieri & M. L. D. Chiara (Eds.), Knowledge, belief, and strategic interaction (Chap. 9, pp. 141–154). Cambridge University Press.Google Scholar
  6. Boolos G. (1992) The Logic of provability. Cambridge University Press, CambridgeGoogle Scholar
  7. Brogaard, B., & Salerno, J. (2002). Fitch’s Paradox of knowability. In E. Zalta (Ed.), The Stanford encyclopedia of philosophy. Stanford: Stanford University. Cited in
  8. Brouwer L. (1948) Essentieel negative eigennschappen [Essentially negative properties]. Indagationes Math 10: 322–323Google Scholar
  9. Brouwer, L. (1983). Consciousness, philosophy and mathemtics. In Philosophy of mathematics: Selected papers (2nd ed., pp. 90–96). New York: Cambridge University Press.Google Scholar
  10. Burgess, J. P. (Forthcoming). Can truth out? In J. Salerno (Ed.), New essays on the knowability paradox. Oxford: Oxford University Press.Google Scholar
  11. Cozzo C. (1994) What can we learn from the paradox of knowability? Topoi 13(2): 71–78CrossRefGoogle Scholar
  12. Dean, W., & Kurokawa, H. (2008a). On semantics for iQLP and related systems (Manuscript).Google Scholar
  13. Dean, W., & Kurokawa, H. (2008b). The knower paradox and the quantified logic of proofs. In A. Hieke (Ed.), Proceedings of the Austrian Ludwig Wittgenstein Society (Vol. 31).Google Scholar
  14. Dummett M. (1977) Elements of intuitionism. Clarendon Press, OxfordGoogle Scholar
  15. Dummett, M. (1978). The philosophical basis of intuitionistic logic. In Truth and other enigmas (pp. 215–247). New York: Oxford University Press.Google Scholar
  16. Dummett M. (1982) Realism. Synthese 52(1): 55–132CrossRefGoogle Scholar
  17. Dummett M. (1987) Reply to Prawitz. In: Taylor B.M. (eds) Michael Dummett: Contribution to philosophy. Martinus Nijhoff Publishers, Dortrecht, pp 117–165Google Scholar
  18. Dummett M. (1991) The logical basis of metaphysics. Duckworth, LondonGoogle Scholar
  19. Dummett, M. (1993). What is a theory of meaning? (II). In The seas of language (pp. 67–137). Oxford: Oxford University Press.Google Scholar
  20. Dummett M. (1998) Truth from the constructive standpoint. Theoria 64: 123–138Google Scholar
  21. Dummett M. (2001) Victor’s error. Analysis 61: 1–2CrossRefGoogle Scholar
  22. Dummett M. (2007) Reply to Künne. In: Auxier R., Hahn L. (eds) The philosophy of Michael Dummett. Open Court Press, La Salle, IL, pp 345–350Google Scholar
  23. Edgington D. (1985) The paradox of knowaiblity. Mind 94: 557–568CrossRefGoogle Scholar
  24. Fagin R., Halpern J. (1988) Belief, awareness, and limited reasoning. Artificial Intelligence 34: 39–76CrossRefGoogle Scholar
  25. Fagin R., Halpern J.Y., Moses Y., Vardi M.Y. (1995) Reasoning about knowledge. MIT Press, CambridgeGoogle Scholar
  26. Fitch F. (1963) A logical analysis of some value concepts. Journal of Symbolic Logic 28(2): 135–142CrossRefGoogle Scholar
  27. Fitting, M. (2004) Quantified LP. Technical Report, CUNY PhD. Program in Computer Science Technical Report TR-2004019.Google Scholar
  28. Fitting M. (2005) The logic of proofs, semantically. Annals of Pure and Applied Logic 132(1): 1–25CrossRefGoogle Scholar
  29. Fitting M. (2006) A quantified logic of evidence. Electronic Notes in Theoretic Computer Science 143: 59–71CrossRefGoogle Scholar
  30. Gödel, K. (1986). An interpertation of intuitionistic propositional calculus. In S. Feferman, et al. (Eds.), Kurt Gödel. Collected works (Vol. I, pp. 144–195). Oxford: Oxford University Press.Google Scholar
  31. Goodman, N. D. (1970). A theory of constructions equivalent to arithmetic. In A. Kino, J. Myhill, & R. E. Vesley (Eds.), Proceedings Summer Conference on Intuitionism and Proof Theory, Buffalo, NY, USA, Aug 1968, Studies in logic and the foundations of mathematics (pp. 101–120). Amsterdam: North-Holland.Google Scholar
  32. Hand M. (2003) Knowablity and epistemic truth. Australasian Journal of Philosophy 81(2): 216–228CrossRefGoogle Scholar
  33. Hand, M. (2008). Performance and paradox. In New essay on the knowability paradox. Oxford: Oxford University Press.Google Scholar
  34. Hart W. (1979) The epistemology of abstract objects. Aristotelian Society Supplementary Volume 53: 163–165Google Scholar
  35. Hart W., McGinn C. (1976) Knowledge and necessity. Journal of Philosophical Logic 5(2): 205–208CrossRefGoogle Scholar
  36. Heyting A. (1974) Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie. Springer, BerlinGoogle Scholar
  37. Kahle R., Schroeder-Heister P. (2006) Introduction: Proof-theoretic semantics. Synthese 148(3): 503–506CrossRefGoogle Scholar
  38. Kreisel, G. (1962). Foundations of intuitionistic logic. In E. Nagel, P. Suppes, & A. Tarski (Eds.), LMPS (pp. 198–210).Google Scholar
  39. Kreisel G. (1967) Informal rigour and completeness proofs. In: Lakatos I. (eds) Problems in philosophy of mathematics. North Holland, Amsterdam, pp 138–171CrossRefGoogle Scholar
  40. Künne W. (2007) Two Principles concerning truth. In: Auxier R., Hahn L. (eds) The philosophy of Michael Dummett. Open Court Press, La Salle, IL, pp 315–344Google Scholar
  41. Kvanvig J. (1995) The knowability paradox and the prospects for anti-realism. Noûs 29(4): 481–500CrossRefGoogle Scholar
  42. Kvanvig J. (2006) The knowability paradox. Clarendon Press, OxfordCrossRefGoogle Scholar
  43. Martin-Löf P. (1984) Intuitionistic type theory. Bibliopolis, NapoliGoogle Scholar
  44. Martin-Löf, P. (1991). A path from logic to metaphysics. In Atti del Congresso Nuovi problemi della logica e della filosopfia della scienza, Viareggio, 8–13 gennaio, 1990 (Vol. II, pp. 141–149). Bologna.Google Scholar
  45. Martin-Löf P. et al (1995) Verificationism then and now. In: DePauli-Schimanovich W. et al. (eds) The foundational debate. Kluwer, The Netherlands, pp 187–196Google Scholar
  46. Martin-Löf, P. (1996). On the meanings of the logical constants and the justifications of the logical laws. Nordic Journal of Philosophical Logic, 1(1), 11–60. Text of lectures originally given in 1983 and distributed in 1985.Google Scholar
  47. Martin-Löf P. (1998) Truth and knowability: On the principles C and K of Michael Dummett. In: Dales G., Oliveri G. (eds) Truth in mathematics. Oxford University Press, New York, pp 105–114Google Scholar
  48. Martino E., Usberti G. (1994) Temporal and atemporal truth in intuitionistic mathematics. Topoi 13: 83–92CrossRefGoogle Scholar
  49. McCarty D. (2006) The coherence of antirealism. Mind 115: 947–956CrossRefGoogle Scholar
  50. McKinsey J.C.C., Tarski A. (1948) Some theorems about the sentential calculi of Lewis and Heyting. Journal of Symbolic Logic 13: 1–15CrossRefGoogle Scholar
  51. Mkrtychev, A. (1997). Models for the logic of proofs. In S. Adian & A. Nerode (Eds.), Proceeding of LFCS (pp. 266–275). Heidelberg: Springer.Google Scholar
  52. Montague R., Kaplan D. (1960) A paradox regained. Notre Dame Journal of Formal Logic 1: 79–90CrossRefGoogle Scholar
  53. Pagin P. (1998) Bivalence: Meaning theory vs metaphysics. Theoria 64: 37–66Google Scholar
  54. Parikh R. (1987) Knowledge and the problem of logical omniscience. In: Ras Z., Zemankova M. (eds) Methodologies for intelligent systems: Proceedings of the sixth international symposium. North Holland, Amsterdam, pp 432–439Google Scholar
  55. Percival P. (1990) Fitch and intuitionistic knowability. Analysis 50: 182–187CrossRefGoogle Scholar
  56. Prawitz D. (1977) Meanings and proofs: On the conflict between classical and intuitionistic logic. Theoria 43(1): 2–40CrossRefGoogle Scholar
  57. Prawitz D. (1980a) Ideas and results in proof theory. In: Fenstad J.E. (eds) Proceedings of the second Scandinavian logic symposium. North Holland, Amsterdam, pp 235–307Google Scholar
  58. Prawitz D. (1980b) Intuitionistic logic: A philosophical challenge. In: Right G.H. (eds) Logic and philosophy. Martinus Nijhoff Publishers, Hague, pp 1–10Google Scholar
  59. Prawitz D. (1987) Dummett on a theory of meaning and its impact on logic. In: Taylor B.M. (eds) Michael Dummett: Contribution to philosophy. Martinus Nijhoff Publishers, Dortrecht, pp 117–165Google Scholar
  60. Prawitz D. (1998a) Comments on Göran Sundholm’s paper “Proofs as acts and proofs as objects”. Theoria 64: 318–329Google Scholar
  61. Prawitz D. (1998b) Comments on Lars Bergström’s paper “Prawitz’s Version of Verificationism”. Theoria 64: 293–303Google Scholar
  62. Prawitz D. (1998c) Problems for a generalization of a verificationist theory of meaning. Topoi: An International Review of Philosophy 64: 87–92Google Scholar
  63. Prawitz D. (1998d) Truth and Objectivity from a verificationist point of view. In: Dales G., Oliveri G. (eds) Truth in mathematics. Oxford University Press, New York, pp 41–51Google Scholar
  64. Raatikainen P. (2004) Conceptions of truth in intuitionism. History and Philosophy of Logic 25(2): 131–145CrossRefGoogle Scholar
  65. Salerno, J. (2008). Knowablity noir. In New essays on the knowability paradox. Oxford: Oxford University Press.Google Scholar
  66. Scott, D. (1970). Constructive validity. In M. Laudet, D. Lacombe, L. Nolin, & M. Schützenberger (Eds.), Proceedings symposium on automatic demonstration, Versailles, France, December 1968, Vol. 125 of Lecture Notes in Mathematics (pp. 237–275). Berlin: Springer-Verlag.Google Scholar
  67. Smorynski C. (1985) Self-reference and modal logic. Springer-Verlag, BerlinGoogle Scholar
  68. Sundholm G. (1983) Constructions, proofs and the meaning of logical constants. Journal of Philosophical Logic 12: 151–172CrossRefGoogle Scholar
  69. Tennant N. (1997) The taming of the true. Oxford University Press, OxfordGoogle Scholar
  70. Troelstra, A., & van Dalen, D. (1988). Constructivism in mathematics (Vols. I, II). Amsterdam: North Holland.Google Scholar
  71. Usberti G. (1995) Significato e Conoscenza: Per una Critica del Neoverificazionismo. Guerini Scientifica, MilanoGoogle Scholar
  72. Wansing H. (2002) Diamonds are a philosopher’s best friends. Journal of Philosophical Logic 31: 591–612CrossRefGoogle Scholar
  73. Weinstein S. (1983) The intended interpretation of intuitionistic logic. Journal of Philosophical Logic 12: 261–270CrossRefGoogle Scholar
  74. Williamson T. (1982) Intuitionism disproved? Analysis 42: 203–207CrossRefGoogle Scholar
  75. Williamson T. (1987) On the paradox of knowability. Mind 96: 256–261CrossRefGoogle Scholar
  76. Williamson T. (1990) Two incomplete anti-realist modal epistemic logics. The Journal of Symbolic Logic 55(1): 297–314CrossRefGoogle Scholar
  77. Williamson T. (1992) On intuitionistic modal epistemic logic. Journal of Philosophical Logic 21: 63–89Google Scholar
  78. Wittgenstein, L. (1967). Remarks on the foundations of mathematics. In G. H. von Wright, R. Rees, & G. E. M. Anscombe (Eds.). MIT Press.Google Scholar
  79. Yavorsky R.E. (2001) Provability logics with quantifiers on proofs. Annals of Pure and Applied Logic 113(1–3): 373–387CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Graduate Center, City University of New YorkNew YorkUSA
  2. 2.Department of PhilosophyUniversity of WarwickCoventryUK

Personalised recommendations