, Volume 79, Issue 5, pp 971–999 | Cite as

Very Improbable Knowing

Original Article


Improbable knowing is knowing something even though it is almost certain on one’s evidence at the time that one does not know that thing. Once probabilities on the agent’s evidence are introduced into epistemic logic in a very natural way, it is easy to construct models of improbable knowing, some of which have realistic interpretations, for instance concerning agents like us with limited powers of perceptual discrimination. Improbable knowing is an extreme case of failure of the KK principle, that is, of a case of knowing something even though one does not know at the time that one knows that thing. A generalization of the argument yields cases of improbable rationality, in which it is rational for one to do something even though it is almost certain on one’s evidence at the time that it is not rational for one to do that thing. When the models are elaborated to represent appearances and beliefs as well as knowledge, they turn out to contain Gettier cases. Neglect of the possibility of improbable knowing may cause some sceptical claims and claims of the non-closure of knowledge under competent deduction to look more plausible than they deserve to. A formal appendix explores the closely related question of the conditions under which a reflection principle is violated. The principle says that the evidential probability of a proposition conditional on the evidential probability of that proposition’s being c is itself c.


  1. Conee, E., & Feldman, R. (2011). Response to Williamson. In Dougherty 2011.Google Scholar
  2. Dorr, C. (2008). How vagueness could cut out at any order. Unpublished MS.Google Scholar
  3. Dougherty, T. (Ed.). (2011). Evidentialism and its discontents. Oxford: Oxford University Press.Google Scholar
  4. Gettier, E. (1963). Is justified true belief knowledge? Analysis, 23, 121–123.CrossRefGoogle Scholar
  5. Gibbons, J. (2001). Knowledge in action. Philosophy and Phenomenological Research, 62, 579–600.CrossRefGoogle Scholar
  6. Goldman, A. (1976). Discrimination and perceptual knowledge. The Journal of Philosophy, 73, 771–791.CrossRefGoogle Scholar
  7. Greenough, P., & Pritchard, D. (Eds.). (2009). Williamson on knowledge. Oxford: Oxford University Press.Google Scholar
  8. Hawthorne, J. (2004). Knowledge and lotteries. Oxford: Clarendon Press.Google Scholar
  9. Hawthorne, J., & Lasonen-Aarnio, M. (2009). Knowledge and objective chance. In Greenough and Pritchard 2009.Google Scholar
  10. Hawthorne, J., & Stanley, J. (2008). Knowledge and action. The Journal of Philosophy, 105, 571–590.Google Scholar
  11. Hintikka, J. (1962). Knowledge and belief. Ithaca, NY: Cornell University Press.Google Scholar
  12. Hyman, J. (1999). How knowledge works. The Philosophical Quarterly, 49, 433–451.CrossRefGoogle Scholar
  13. Lasonen-Aarnio, M. (2008). Single premise deduction and risk. Philosophical Studies, 141, 157–173.CrossRefGoogle Scholar
  14. Lemmon, E. J. (1967). If I know, do I know that I know? In A. Stroll (Ed.), Epistemology. New York: Harper & Row.Google Scholar
  15. Mahtani, A. (2008). Can vagueness cut out at any order? Australasian Journal of Philosophy, 86, 499–508.CrossRefGoogle Scholar
  16. Radford, C. (1966). Knowledge—By examples. Analysis, 27, 1–11.Google Scholar
  17. Stalnaker, R. (1999). Context and content: Essays on intentionality in speech and thought. Oxford: Oxford University Press.CrossRefGoogle Scholar
  18. Stalnaker, R. (2006). On logics of knowledge and belief. Philosophical Studies, 128, 169–199.CrossRefGoogle Scholar
  19. Williamson, T. (1990). Identity and discrimination. Oxford: Blackwell.Google Scholar
  20. Williamson, T. (1999). On the structure of higher-order vagueness. Mind, 108, 127–143.CrossRefGoogle Scholar
  21. Williamson, T. (2000). Knowledge and its limits. Oxford: Oxford University Press.Google Scholar
  22. Williamson, T. (2005). Contextualism, subject-sensitive invariantism and knowledge of knowledge. The Philosophical Quarterly, 55, 213–235.CrossRefGoogle Scholar
  23. Williamson, T. (2007a). On being justified in one’s head. In M. Timmons, J. Greco, & A. R. Mele (Eds.), Rationality and the Good: Critical essays on the ethics and epistemology of Robert Audi. Oxford: Oxford University Press.Google Scholar
  24. Williamson, T. (2007b). How probable is an infinite sequence of heads? Analysis, 67, 173–180.CrossRefGoogle Scholar
  25. Williamson, T. (2008). Why epistemology can’t be operationalized. In Q. Smith (Ed.), Epistemology: New philosophical essays. Oxford: Oxford University Press.Google Scholar
  26. Williamson, T. (2009). Reply to John Hawthorne and Maria Lasonen-Aarnio. In Greenough and Pritchard 2009.Google Scholar
  27. Williamson, T. (2011). Improbable knowing. In Dougherty 2011.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.University of OxfordOxfordUK

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