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Title
Abstract
Introduction
Semantical Analysis of the Knowability Paradox
Conclusion
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Philosophy

Discovering knowability: a semantic analysis

Profile image of Sergei ArtemovSergei Artemov

2013, Synthese

https://doi.org/10.1007/S11229-012-0168-X
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Abstract

Abstract In this paper, we provide a semantic analysis of the well-known knowability paradox stemming from the Church–Fitch observation that the meaningful knowability principle all truths are knowable, when expressed as a bi-modal principle ♢, yields an unacceptable omniscience property all truths are known. We offer an alternative semantic proof of this fact independent of the Church–Fitch argument. This shows that the knowability paradox is not intrinsically related to the Church–Fitch proof, nor to the Moore sentence upon which it ...

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References (66)

  1. Artemov, S. N. (2001). Explicit provability and constructive semantics. Bulletin of Symbolic Logic, 7(1), 1-36.
  2. Artemov, S. N. (2008). The logic of justification. Review of Symbolic Logic, 1(4), 477-513.
  3. Artemov, S. N., & Protopopescu, T. (2010). Knowability from a logical point of view. Technical Report TR-2010008. CUNY PhD Program in Computer Science.
  4. Beall, J. C. (2009). Knowability and possible epistemic oddities. In J. Salerno (Ed.), New essays on the knowability paradox (pp. 105-127). Oxford: Oxford University Press.
  5. Brogaard, B., & Salerno, J. (forthcoming). Fitch's paradox of knowability. In E. N. Zalta (Ed.) The Stanford Encyclopedia of Philosophy (Fall 2012 Edn.). http://plato.stanford.edu/archives/fall/2012/ entries/fitchparadox/.
  6. Brogaard, B., & Salerno, J. (2002). Fitch's paradox of knowability. Analysis, 62(2), 143-151.
  7. Church, A. (2009). Referee reports on Fitch's "a definition of value." In J. Salerno (Ed.), New essays on the knowability paradox (pp. 13-20). Oxford: Oxford University Press.
  8. Costa-Leite, A. (2004). Combining possiblity and knowledge. In F. Carnielli, F. Dionisio, & P. Mateus (Eds.), Proceedings of the workshop on the combination of logics: Theory and applications. Lis- bon: IST-Department of Mathematics, Center for Logic and Computation.
  9. Costa-Leite, A. (2006). Fusions of Modal Logic in Fitch's Paradox. Croatian Journal of Philosophy, 17, 281-290.
  10. De Vidi, T., & Kenyon, D. (2006). Introduction. In T. De Vidi & D. Kenyon (Eds.), A Logical approach to philosophy: Essays in honour of Graham Solomon (pp. 1-21). Berlin: Springer.
  11. De Vidi, T., & Solomon, G. (1997). in intuitionistic modal logic. Australasian Journal of Philoso- phy, 75(2), 201-213.
  12. De Vidi, T., & Solomon, G. (2001). Knowability and intuitionistic logic. Philosophia, 28, 319-334.
  13. Dean, W., & Kurokawa, H. (2010). From the knowability paradox to the existence of proofs. Syn- these, 175(2), 177-225.
  14. Dummett, M. (2001). Victor's error. Analysis, 61, 1-2.
  15. Dummett, M. (2009). Fitch's paradox of knowability. In J. Salerno (Ed.), New essays on the knowability paradox, Chapter 4 (pp. 51-52). Oxford: Oxford University Press.
  16. Edgington, D. (1985). The paradox of knowability. Mind, 94(376), 557-568.
  17. Edgington, D. (2010). Possible knowledge of unknown truth. Synthese, 173(1), 41-52.
  18. Ewald, W. B. (1963). Intuitionistic tense and modal logic. Journal of Symbolic Logic, 51(1), 166-179.
  19. Fara, M. (2009). Knowability and the capacity to knowability. Synthese, 173(1), 53-73.
  20. Fischer, M. (2010). Some remarks on restricting the knowability principle. Synthese, 1-26. doi:10.1007/ s11229-010-9833-0.
  21. Fitch, F. (1963). A logical analysis of some value concepts. Journal of Symbolic Logic, 28(2), 135-142.
  22. Fitting, M. (2005). The logic of proofs, semantically. Annals of Pure and Applied Logic, 132, 1-25.
  23. Fitting, M. (2008). A quantified logic of evidence. Annals of Pure and Applied Logic, 152, 67-83.
  24. Fitting, M., & Mendelsohn, R. (1998). First-order modal logic. Dordrecht: Kluwer Academic Publishers.
  25. Gabbay, D. M., Kurucz, A., Wolter, F., & Zakharyaschev, M. (1998). Multi-dimensional modal logic: Theory and application. Studies in logic and the foundations of mathematics (Vol. 148). Amsterdam: Elsevier.
  26. Gödel, K. (1986). Eine Interpretation des intuitionistischen Aussagenkalkuls. Ergebnisse Math. Kolloq., 4, 39-40, 1933. English translation in: (S. Feferman et al., Eds.), Kurt Gödel collected works (Vol. 1, pp. 301-303). Oxford: Oxford University Press, 1986.
  27. Gödel, K. (1995). Vortrag bei Zilsel/Lecture at Zilsel's (1938a). In S. Feferman, J. W. Dawson, Jr, W. Goldfarb, C. Parsons, & R. M. Solovay (Eds.), Unpublished essays and lectures, volume III of Kurt Gödel collected works (pp. 86-113). Oxford: Oxford University Press.
  28. Hand, M. (2009). Performance and paradox. In J. Salerno (Ed.), New essays on the knowability paradox, Chapter 17 (pp. 283-301). Oxford: Oxford University Press.
  29. Hand, M., & Kvanvig, J. L. (1999). Tennant on knowability. Australasian Journal of Philosophy, 77(4), 422-428.
  30. Hart, W. D., & McGinn, C. (1976). Knowledge and necessity. Journal of Philosophical Logic, 5(2), 205-208.
  31. Jenkins, C. (2009). The mystery of the disappearing diamond. In J. Salerno (Ed.), New essays on the knowability paradox (pp. 205-222). New York: Oxford University Press.
  32. Kleene, S. (1952). Introduction to metamathematics. New York: Ishi Press.
  33. Kripke, S. (1980). Naming and necessity. Oxford: Basil Blackwell.
  34. Kvanvig, J. (2006). The knowability paradox. Oxford: Oxford University Press.
  35. Kvanvig, J. (2009). Restriction strategies for knowability. In J. Salerno (Ed.), New essays on the knowability paradox (pp. 205-222). New York: Oxford University Press.
  36. Melia, J. (1991). Anti-realism untouched. Mind, 100(3), 341-342.
  37. Negri, S., Maffezioli, P., & Naibo, A. (2012). The Church-Fitch knowability paradox in the light of structural proof theory. Synthese, 1-40. doi:10.1007/s11229-012-0061-7.
  38. Peacocke, P., & Boghossian, C. (2000). Introduction. In New essays on the a priori (pp. 1-10). Oxford: Oxford University Press.
  39. Percival, P. (1990). Fitch and intuitionistic knowability. Analysis, 50(3), 182-187.
  40. Priest, G. (2009). Beyond the limits of knowledge. In J. Salerno (Ed.), New essays on the knowability paradox (pp. 93-104). Oxford: Oxford University Press.
  41. Prior, A. N. (1957). Time and Modality. Oxford: Oxford University Press.
  42. Proietti, C. (2001). Intuitionistic epistemic logic, Kripke models and Fitch's paradox. Journal of Philosophical Logic, 1-24. doi:10.1007/s10992-011-9207-1.
  43. Rabinowicz, W., & Segerberg, K. (1994). Actual truth, possible knowledge. Topoi, 13(2), 101-116.
  44. Rasmussen, S. A. (2009). The paradox of knowability and the mapping objection. In J. Salerno (Ed.), New essays on the knowability paradox (pp. 53-75). Oxford: Oxford University Press.
  45. Restall, G. (2009). Not every truth is knowable: At least not all at once. In J. Salerno (Ed.), New essays on the knowability paradox (pp. 339-354). Oxford: Oxford University Press.
  46. Rückert, H. (2004). A solution to Fitch's paradox of knowability. In S. Rahman & J. Symons (Eds.), Logic, epistemology and the unity of science (pp. 351-380). Dordrecht: Kluwer Academic Publishers.
  47. Salerno, J. (1999). Revising the logic of logical revision. Philosophical Studies, 99, 2211-2227.
  48. Shapiro, S. (2006). Externalism, anti-realism, and the kk-thesis. In T. De Vidi & D. Kenyon (Eds.), A logical approach to philosophy (pp. 22-35). Berlin: Springer.
  49. Tennant, N. (1997). The taming of the true. Oxford: Oxford University Press.
  50. Tennant, N. (2001). Is every truth knowable: Reply to Williamson. Ratio, 14, 263-280.
  51. Tennant, N. (2009). Revamping the restriction strategy. In J Salerno (Ed.), New essays on the knowability paradox (pp. 223-238). New York: Oxford University Press.
  52. Tennant, N. (2010). Williamson's woes. Synthese, 173(1), 9-23.
  53. van Benthem, J. (2009). Actions that make us know. In J. Salerno (Ed.), New essays on the knowability paradox (pp. 129-146). Oxford: Oxford University Press.
  54. Wansing, H. (2002). Diamonds are a philosopher's best friend: The knowability paradox and modal epistemic relevance logic. Journal of Philosophical Logic, 36(6), 591-612.
  55. Williamson, T. (1982). Intuitionism disproved?. Analysis, 42(4), 203-207.
  56. Williamson, T. (1988). Knowability and constructivism. Philosophical Quarterly, 38(153), 422-432.
  57. Williamson, T. (1992). On intuitionistic epistemic modal logic. Journal of Philosophical Logic, 21, 63-89.
  58. Williamson, T. (1994). Never say never. Topoi, 13, 135-145.
  59. Williamson, T. (2000a). Knowledge and its limits. Oxford: Oxford University Press.
  60. Williamson, T. (2000b). Tennant on knowable truth. Ratio, XIII, 99-114.
  61. Williamson, T. (2009). Tennant's troubles. In J. Salerno (Ed.), New essays on the knowability para- dox (pp. 183-205). New York: Oxford University Press.
  62. Wolter, F., & Zakharyaschev, M. (1997). On the relation between intuitionistic and classical modal logics. Algebra and Logic, 36, 121-155.
  63. Wolter, F., & Zakharyaschev, M. (1999a). Intuitionistic modal logics as fragments of classical bimodal logics. In E. Orlowska (Ed.), Logic at work (pp. 168-186). Berlin: Springer. Synthese (2013) 190:3349-3376
  64. Wolter, F., & Zakharyaschev, M. (1999b). Intuitionistic modal logic. In A. Cantini, E. Casari, & P. Minari (Eds.), Logic and foundations of mathematics (pp. 227-238). Dordrecht: Kluwer Aca- demic Publishers.
  65. Wright, C. (1983). Can a Davidsonian meaning-theory be construed in terms of warranted assertibility? In C. Wright (Ed.), Realism, meaning and truth (pp. 286-315). Oxford: Blackwell.
  66. Wright, C. (2001). On being in a quandry: Relativism, vagueness and logical revisionism. Mind, 110(437), 45-98.

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