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Discovering knowability: a semantic analysis

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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 \({\diamondsuit}\), 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 relies, but rather to the knowability principle itself. Further, we show that, from a verifiability perspective, the knowability principle fails in the classical logic setting because it is missing the explicit incorporation of a hidden assumption of stability: ‘the proposition in question does not change from true to false in the process of discovery.’ Once stability is taken into account, the resulting stable knowability principle and its nuanced versions more accurately represent verification-based knowability and do not yield omniscience.

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References

  • Artemov S. N. (2001) Explicit provability and constructive semantics. Bulletin of Symbolic Logic 7(1): 1–36

    Article  Google Scholar 

  • Artemov S.N. (2008) The logic of justification. Review of Symbolic Logic 1(4): 477–513

    Article  Google Scholar 

  • Artemov, S. N., & Protopopescu, T. (2010). Knowability from a logical point of view. Technical Report TR-2010008. CUNY PhD Program in Computer Science.

  • Beall J. C. (2009) Knowability and possible epistemic oddities. In: Salerno J. (Ed.) New essays on the knowability paradox. Oxford University Press, Oxford, pp 105–127

    Chapter  Google Scholar 

  • 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/.

  • Brogaard B., Salerno J. (2002) Fitch’s paradox of knowability. Analysis 62(2): 143–151

    Article  Google Scholar 

  • 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.

  • Costa-Leite A. (2004) Combining possiblity and knowledge. In: Carnielli F., Dionisio F., Mateus P. (Eds.) Proceedings of the workshop on the combination of logics: Theory and applications. IST-Department of Mathematics, Center for Logic and Computation, Lisbon

    Google Scholar 

  • Costa-Leite A. (2006) Fusions of Modal Logic in Fitch’s Paradox. Croatian Journal of Philosophy 17: 281–290

    Google Scholar 

  • De Vidi T., Kenyon D. (2006) Introduction. In: De Vidi T., Kenyon D. (Eds.) A Logical approach to philosophy: Essays in honour of Graham Solomon. Springer, Berlin, pp 1–21

    Chapter  Google Scholar 

  • De Vidi T., Solomon G. (1997) \({\square}\) in intuitionistic modal logic. Australasian Journal of Philosophy 75(2): 201–213

    Article  Google Scholar 

  • De Vidi T., Solomon G. (2001) Knowability and intuitionistic logic. Philosophia 28: 319–334

    Article  Google Scholar 

  • Dean W., Kurokawa H. (2010) From the knowability paradox to the existence of proofs. Synthese 175(2): 177–225

    Article  Google Scholar 

  • Dummett M. (2001) Victor’s error. Analysis 61: 1–2

    Article  Google Scholar 

  • 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.

  • Edgington D. (1985) The paradox of knowability. Mind 94(376): 557–568

    Article  Google Scholar 

  • Edgington D. (2010) Possible knowledge of unknown truth. Synthese 173(1): 41–52

    Article  Google Scholar 

  • Ewald W. B. (1963) Intuitionistic tense and modal logic. Journal of Symbolic Logic 51(1): 166–179

    Article  Google Scholar 

  • Fara M. (2009) Knowability and the capacity to knowability. Synthese 173(1): 53–73

    Article  Google Scholar 

  • Fischer, M. (2010). Some remarks on restricting the knowability principle. Synthese, 1–26. doi:10.1007/s11229-010-9833-0.

  • Fitch F. (1963) A logical analysis of some value concepts. Journal of Symbolic Logic 28(2): 135–142

    Article  Google Scholar 

  • Fitting M. (2005) The logic of proofs, semantically. Annals of Pure and Applied Logic 132: 1–25

    Article  Google Scholar 

  • Fitting M. (2008) A quantified logic of evidence. Annals of Pure and Applied Logic 152: 67–83

    Article  Google Scholar 

  • Fitting M., Mendelsohn R. (1998) First-order modal logic. Kluwer Academic Publishers, Dordrecht

    Book  Google Scholar 

  • 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.

  • 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.

  • 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.

  • 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.

  • Hand M., Kvanvig J. L. (1999) Tennant on knowability. Australasian journal of philosophy 77(4): 422–428

    Article  Google Scholar 

  • Hart W. D., McGinn C. (1976) Knowledge and necessity. Journal of Philosophical Logic 5(2): 205–208

    Article  Google Scholar 

  • Jenkins C. (2009) The mystery of the disappearing diamond. In: Salerno J. (Ed.) New essays on the knowability paradox. Oxford University Press, New York, pp 205–222

    Google Scholar 

  • Kleene S. (1952) Introduction to metamathematics. Ishi Press, New York

    Google Scholar 

  • Kripke S. (1980) Naming and necessity. Basil Blackwell, Oxford

    Google Scholar 

  • Kvanvig J. (2006) The knowability paradox. Oxford University Press, Oxford

    Book  Google Scholar 

  • Kvanvig J. (2009) Restriction strategies for knowability. In: Salerno J. (Ed.) New essays on the knowability paradox. Oxford University Press, New York, pp 205–222

    Chapter  Google Scholar 

  • Melia J. (1991) Anti-realism untouched. Mind 100(3): 341–342

    Article  Google Scholar 

  • 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.

  • Peacocke, P., & Boghossian, C. (2000). Introduction. In New essays on the a priori (pp. 1–10). Oxford: Oxford University Press.

  • Percival P. (1990) Fitch and intuitionistic knowability. Analysis 50(3): 182–187

    Article  Google Scholar 

  • Priest G. (2009) Beyond the limits of knowledge. In: Salerno J. (Ed.) New essays on the knowability paradox. Oxford University Press, Oxford, pp 93–104

    Chapter  Google Scholar 

  • Prior A. N. (1957) Time and Modality. Oxford University Press, Oxford

    Google Scholar 

  • 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.

  • Rabinowicz W., Segerberg K. (1994) Actual truth, possible knowledge. Topoi 13(2): 101–116

    Article  Google Scholar 

  • Rasmussen S. A. (2009) The paradox of knowability and the mapping objection. In: Salerno J. (Ed.) New essays on the knowability paradox. Oxford University Press, Oxford, pp 53–75

    Chapter  Google Scholar 

  • Restall G. (2009) Not every truth is knowable: At least not all at once. In: Salerno J. (Ed.) New essays on the knowability paradox. Oxford University Press, Oxford, pp 339–354

    Chapter  Google Scholar 

  • Rückert H. (2004) A solution to Fitch’s paradox of knowability. In: Rahman S., Symons J. (Ed.) Logic, epistemology and the unity of science. Kluwer Academic Publishers, Dordrecht, pp 351–380

    Chapter  Google Scholar 

  • Salerno J. (1999) Revising the logic of logical revision. Philosophical Studies 99: 2211–2227

    Google Scholar 

  • Shapiro S. (2006) Externalism, anti-realism, and the kk-thesis. In: De Vidi T., Kenyon D. (Eds.) A logical approach to philosophy. Springer, Berlin, pp 22–35

    Chapter  Google Scholar 

  • Tennant N. (1997) The taming of the true. Oxford University Press, Oxford

    Google Scholar 

  • Tennant N. (2001) Is every truth knowable: Reply to Williamson. Ratio 14: 263–280

    Article  Google Scholar 

  • Tennant N. (2009) Revamping the restriction strategy. In: Salerno J (Ed.) New essays on the knowability paradox. Oxford University Press, New York, pp 223–238

    Chapter  Google Scholar 

  • Tennant N. (2010) Williamson’s woes. Synthese 173(1): 9–23

    Article  Google Scholar 

  • van Benthem J. (2009) Actions that make us know. In: Salerno J. (Ed.) New essays on the knowability paradox. Oxford University Press, Oxford, pp 129–146

    Chapter  Google Scholar 

  • 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

    Article  Google Scholar 

  • Williamson T. (1982) Intuitionism disproved?. Analysis 42(4): 203–207

    Article  Google Scholar 

  • Williamson T. (1988) Knowability and constructivism. Philosophical Quarterly 38(153): 422–432

    Article  Google Scholar 

  • Williamson T. (1992) On intuitionistic epistemic modal logic. Journal of Philosophical Logic 21: 63–89

    Google Scholar 

  • Williamson T. (1994) Never say never. Topoi 13: 135–145

    Article  Google Scholar 

  • Williamson T. (2000a) Knowledge and its limits. Oxford University Press, Oxford

    Google Scholar 

  • Williamson T. (2000b) Tennant on knowable truth. Ratio XIII: 99–114

    Article  Google Scholar 

  • Williamson T. (2009) Tennant’s troubles. In: Salerno J. (Ed.) New essays on the knowability paradox. Oxford University Press, New York, pp 183–205

    Chapter  Google Scholar 

  • Wolter F., Zakharyaschev M. (1997) On the relation between intuitionistic and classical modal logics. Algebra and Logic 36: 121–155

    Article  Google Scholar 

  • Wolter F., Zakharyaschev M. (1999a) Intuitionistic modal logics as fragments of classical bimodal logics. In: Orlowska E. (Ed.) Logic at work. Springer, Berlin, pp 168–186

    Google Scholar 

  • Wolter F., Zakharyaschev M. (1999b) Intuitionistic modal logic. In: Cantini A., Casari E., Minari P. (Eds.) Logic and foundations of mathematics. Kluwer Academic Publishers, Dordrecht, pp 227–238

    Chapter  Google Scholar 

  • 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.

  • Wright C. (2001) On being in a quandry: Relativism, vagueness and logical revisionism. Mind 110(437): 45–98

    Article  Google Scholar 

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  1. The CUNY Graduate Center, 365 Fifth Avenue, rm. 4329, New York City, NY, 10016, USA

    Sergei Artemov & Tudor Protopopescu

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  1. Sergei Artemov
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  2. Tudor Protopopescu
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Correspondence to Sergei Artemov.

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Artemov, S., Protopopescu, T. Discovering knowability: a semantic analysis. Synthese 190, 3349–3376 (2013). https://doi.org/10.1007/s11229-012-0168-x

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