Lottery paradox

Although the first published statement of the lottery paradox appears in Kyburg's 1961 Probability and the Logic of Rational Belief, the first formulation of the paradox appears in his "Probability and Randomness", a paper delivered at the 1959 meeting of the Association for Symbolic Logic, and the 1960 International Congress for the History and Philosophy of Science, but published in the journal Theoria in 1963. This paper is reprinted in Kyburg (1987).

The lottery paradox has become a central topic within epistemology, and the enormous literature surrounding this puzzle threatens to obscure its original purpose. Kyburg proposed the thought experiment to get across a feature of his innovative ideas on probability (Kyburg 1961, Kyburg and Teng 2001), which are built around taking the first two principles above seriously and rejecting the last. For Kyburg, the lottery paradox is not really a paradox: his solution is to restrict aggregation.

Even so, for orthodox probabilists the second and third principles are primary, so the first principle is rejected. Here too one will see claims that there is really no paradox but an error: the solution is to reject the first principle, and with it the idea of rational acceptance. For anyone with basic knowledge of probability, the first principle should be rejected: for a very likely event, the rational belief about that event is just that it is very likely, not that it is true.

Most of the literature in epistemology approaches the puzzle from the orthodox point of view and grapples with the particular consequences faced by doing so, which is why the lottery is associated with discussions of skepticism (e.g., Klein 1981), and conditions for asserting knowledge claims (e.g., J. P. Hawthorne 2004). It is common to also find proposed resolutions to the puzzle that turn on particular features of the lottery thought experiment (e.g., Pollock 1986), which then invites comparisons of the lottery to other epistemic paradoxes, such as David Makinson's preface paradox, and to "lotteries" having a different structure. This strategy is addressed in (Kyburg 1997) and also in (Wheeler 2007), which includes an extensive bibliography.

Philosophical logicians and AI researchers have tended to be interested in reconciling weakened versions of the three principles, and there are many ways to do this, including Jim Hawthorne and Luc Bovens's (1999) logic of belief, Gregory Wheeler's (2006) use of 1-monotone capacities, Bryson Brown's (1999) application of preservationist para-consistent logics, Igor Douven and Timothy Williamson's (2006) appeal to cumulative non-monotonic logics, Horacio Arlo-Costa's (2007) use of minimal model (classical) modal logics, and Joe Halpern's (2003) use of first-order probability.

Finally, philosophers of science, decision scientists, and statisticians are inclined to see the lottery paradox as an early example of the complications one faces in constructing principled methods for aggregating uncertain information, which is now a discipline of its own, with a dedicated journal, Information Fusion, in addition to continuous contributions to general area journals.

• List of paradoxes

• Arlo-Costa, H. (2005). "Non-Adjunctive Inference and Classical Modalities", The Journal of Philosophical Logic, 34, 581–605.
• Brown, B. (1999). "Adjunction and Aggregation", Nous, 33(2), 273–283.
• Douven and Williamson (2006). "Generalizing the Lottery Paradox", The British Journal for the Philosophy of Science, 57(4), pp. 755–779.
• Halpern, J. (2003). Reasoning about Uncertainty, Cambridge, MA: MIT Press.
• Hawthorne, J. and Bovens, L. (1999). "The Preface, the Lottery, and the Logic of Belief", Mind, 108: 241–264.
• Hawthorne, J.P. (2004). Knowledge and Lotteries, New York: Oxford University Press.
• Klein, P. (1981). Certainty: a Refutation of Scepticism, Minneapolis, MN: University of Minnesota Press.
• Kroedel, T. (2012). "The Lottery Paradox, Epistemic Justification and Permissibility", Analysis, 72(1), 57-60.
• Kyburg, H.E. (1961). Probability and the Logic of Rational Belief, Middletown, CT: Wesleyan University Press.
• Kyburg, H. E. (1983). Epistemology and Inference, Minneapolis, MN: University of Minnesota Press.
• Kyburg, H. E. (1997). "The Rule of Adjunction and Reasonable Inference", Journal of Philosophy, 94(3), 109–125.
• Kyburg, H. E., and Teng, C-M. (2001). Uncertain Inference, Cambridge: Cambridge University Press.
• Lewis, D. (1996). "Elusive Knowledge", Australasian Journal of Philosophy, 74, pp. 549–67.
• Makinson, D. (1965). "The Paradox of the Preface", Analysis, 25: 205–207.
• Pollock, J. (1986). "The Paradox of the Preface", Philosophy of Science, 53, pp. 246–258.
• Smullyan, Raymond (1978). What is the name of this book?. Prentice–Hall. p. 206. ISBN 0-13-955088-7.
• Wheeler, G. (2006). "Rational Acceptance and Conjunctive/Disjunctive Absorption", Journal of Logic, Language, and Information, 15(1-2): 49–53.
• Wheeler, G. (2007). "A Review of the Lottery Paradox", in William Harper and Gregory Wheeler (eds.) Probability and Inference: Essays in Honour of Henry E. Kyburg, Jr., King's College Publications, pp. 1–31.

• Links to James Hawthorne's papers on the logic of nonmonotonic conditionals (and Lottery Logic)