Journal of Logic, Language and Information

, Volume 14, Issue 1, pp 13–48 | Cite as

The Knower Paradox in the Light of Provability Interpretations of Modal Logic

  • Paul Égré


This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework of first-order arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368–387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338–355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287–304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy.


Believer Paradox epistemic logic hierarchy solutions to the semantic paradoxes Knower Paradox provability logic self-reference syntactical treatments of modalities 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, C.A., 1983, “The Paradox of the Knower,” The Journal of Philosophy 80, 338–355.Google Scholar
  2. Asher, N. and Kamp., H., 1989, “Self-reference, attitudes and paradox,” in Chierchia et al. (1989): 85–158.Google Scholar
  3. Bicchieri, C. and Dalla Chiara, M.L., eds., 1992, Knowledge, Belief and Strategic Interaction, Cambridge: Cambridge University Press.Google Scholar
  4. Binmore, K. and Shin, H.S., 1992, “Algorithmic knowledge and game theory,” in Bicchieri and Dalla Chiara, pp. 141–154.Google Scholar
  5. Boolos, G., 1993, The Logic of Provability, New York, Cambridge Unversity Press.Google Scholar
  6. Boolos, G. and Sambin, G., 1991, “Provability: The emergence of a mathematical modality,” Studia Logica 50, 1–23.Google Scholar
  7. Chellas, B.F., 1980, Modal Logic: An Introduction, Cambridge: Cambridge University Press.Google Scholar
  8. Chierchia, G., Partee, B. and Turner, R., eds., 1989, Properties, Types and Meaning, Vol. I, Foundational Issues, Dordrecht: Kluwer Academic Publisher, Studies in Linguistics and Philosophy.Google Scholar
  9. Cross, C.B., 2001a, “The Paradox of the Knower without epistemic closure,” Mind 110, 319–333.Google Scholar
  10. Cross, C.B., 2001b, “A theorem concerning syntactical treatments of non-idealized belief,” Synthese 129, 335–341.Google Scholar
  11. Cross, C.B., 2004, “More on the Paradox of the Knower without epistemic closure,” Mind 113, 109–114.Google Scholar
  12. Enderton, H.B., 1972, A Mathematical Introduction to Logic, San Diego, CA: Academic Press.Google Scholar
  13. Feferman, S., 1962, “Transfinite recursive progressions of axiomatic theories,” Journal of Symbolic Logic 27, 259–316.Google Scholar
  14. Field, H., 2002, “Saving the truth schema from paradox,” Journal of Philosophical Logic 31, 1–27.Google Scholar
  15. Friedman, H. and Sheard, M., 1987, “An axiomatic approach to self-referential truth,” Annals of Pure and Applied Logic 33, 1–21.Google Scholar
  16. Gödel, K., 1933, “Eine Interpretation des intuitionistischen Aussagenkalküls,” translated in, Collected Works, Vol. 1, K. Gödel, S. Feferman et al., eds., New York: Oxford University Press.Google Scholar
  17. Henkin, L., 1952, A Problem concerning provability,” Journal of Symbolic Logic 17, 160.Google Scholar
  18. Kaplan, D. and Montague, R., 1960, “A paradox regained,” Notre Dame Journal of Formal Logic 1, 79–90, repr. in Montague (1974), 271–85.Google Scholar
  19. Koons, R., 1992, Paradoxes of Belief and Strategic Rationality, New York: Cambridge University Press.Google Scholar
  20. Löb, M.H., 1955, “ Solution of a problem of Leon Henkin,” Journal of Symbolic Logic 20, 115–118.Google Scholar
  21. McGee, V., 1991, Truth, Vagueness and Paradox: An Essay on the Logic of Truth, Indianapolis: Hackett Publishing Company.Google Scholar
  22. Montague, R., 1963, “Syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability,” Acta Philosophica Fennica 16, 153–67, repr. in Montague (1974), 286–302.Google Scholar
  23. Montague, R., 1974, Formal Philosophy Selected Papers of Richard Montague, edited and with an introduction by R.H. Thomason, New Haven, Yale University Press.Google Scholar
  24. Perlis, D. and Subrahmanian, V., 1994, “Meta-languages, reflection principles and self-reference,” in Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. II: Deduction Methodologies, D. Gabbay, C.J. Hogger and J.A. Robinson, eds., Oxford University Press, pp. 323–358.Google Scholar
  25. Quine, W.V.O., 1940, Mathematical Logic, Revised Edn., Cambridge, MA: Harvard University Press, 1981.Google Scholar
  26. Quine, W.V.O., 1953, “Three grades of modal involvement,” in The Ways of Paradox and Other Essays, Cambridge, MA: Harvard University Press, pp. 158–176.Google Scholar
  27. Reinhardt, W.N., 1980, “Necessity predicates and operators,” Journal of Philosophical Logic 9, 437–450.Google Scholar
  28. Reinhardt, W.N., 1986, “Epistemic theories and the interpretation of Gödel’s incompleteness theorems,” Journal of Philosophical Logic 15, 427–474.Google Scholar
  29. Richard, M., 1990, Propositional Attitudes, An Essay on Thoughts and How We Ascribe Them, New York, Cambridge.Google Scholar
  30. Skyrms, B., 1978, “An immaculate conception of modality, or how to confuse use and mention,” The Journal of Philosophy 75, 368–387.Google Scholar
  31. Smoryński, C., 1985, Self-Reference and Modal Logic, New York: Springer Verlag.Google Scholar
  32. Smoryński, C., 1991, “The development of self-reference: Löb’s theorem,” in Perspectives on the History of Mathematical Logic, T. Drucker, ed., Boston, MA: Birkhäuser, pp. 110–133.Google Scholar
  33. Smullyan, R.M., 1986, “Logicians who reason about themselves,” in Reasoning About Knowledge, Proceedings of the TARK Conference, San Mateo, CA: Morgan Kaufman, pp. 341–352.Google Scholar
  34. Smullyan, R.M., 1992, Gödel’s Incompleteness Theorems, Oxford Logic Guides 19, New York: Oxford University Press.Google Scholar
  35. Solovay, R., 1976, “Provability interpretations of modal logic,” Israel Journal of Mathematics 25, 287–304.Google Scholar
  36. Thomason, R., 1977, “Indirect discourse is not quotational,” The Monist 60, 340–354.Google Scholar
  37. Thomason, R., 1980, “A note on syntactical treatments of modality,” Synthese 44, 391–395.Google Scholar
  38. Turner, R., 1990, Truth and Modality for Knowledge Representation, Cambridge, MA: MIT Press.Google Scholar
  39. Tymoczko, T., 1984, “An unsolved puzzle about knowledge,” Philosophical Quarterly 34, 437–458.Google Scholar
  40. Uzquiano, G., 2004, “The Paradox of the Knower without epistemic closure?,” Mind 113, 95–107.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.IHPSTParisFrance
  2. 2.Institut Jean-NicodParisFrance

Personalised recommendations