Philosophy The Lottery and Preface Paradoxes
Igor Douven
  • LAST REVIEWED: 08 October 2015
  • LAST MODIFIED: 19 March 2013
  • DOI: 10.1093/obo/9780195396577-0196


People speak about their own and other people’s beliefs both in categorical terms and in terms of degrees. Philosophers accordingly distinguished between an epistemology of belief and an epistemology of degrees of belief. A question that has intrigued many is how the two epistemologies are connected. A partial answer has been that the rationality of categorical beliefs supervenes on rationally held graded beliefs, in the sense that there cannot be a change in the former without there being some change in the latter, and where it is typically assumed that rationally held graded beliefs are representable by a probability function. Arguably, the simplest way to ensure this connection of supervenience is via a principle sometimes called the “Lockean thesis” (LT), which holds that it is rational to believe A if and only if it is rational to believe A to a degree above θ, where θ is some threshold value close to 1. While this thesis appears plausible on its face, it is beset by what are generally known as “the lottery paradox” and “the preface paradox.” These paradoxes rely on two principles, along with the Lockean thesis, namely, the “conjunction principle” (if it is rational to believe A and it is rational to believe B, then it is rational to believe A and B) and the “no contradictions principle” (it is never rational to believe an explicit contradiction). Many have remarked that, on their faces, the conjunction principle (CP) and the no contradictions principle (NCP) appear at least as plausible as the Lockean thesis. Which of the three principles is to be abandoned is a matter of continuing debate and controversy.

General Overviews

Wheeler 2007 provides a good overview of the literature on the lottery paradox. Sorensen 2011 discusses a number of epistemic paradoxes, including the lottery and preface paradoxes. There are several books dealing with paradoxes in general that also contain useful discussions of the lottery and preface paradoxes. Particularly valuable in this respect is Olin 2003. Other books that deserve mentioning are Clark 2002 and Rescher 2001. Haack 1978 does not specifically address the paradoxes at issue here, but it contains an excellent chapter on paradoxes generally. Salerno 2011 discusses the lottery paradox in the broader context of epistemic philosophy of logic.

  • Clark, Michael. Paradoxes from A to Z. London: Routledge, 2002.

    E-mail Citation »

    This is a collection of brief but useful discussions of a wide range of paradoxes, including the lottery and preface paradoxes.

  • Haack, Susan. Philosophy of Logics. Cambridge, UK: Cambridge University Press, 1978.

    DOI: 10.1017/CBO9780511812866E-mail Citation »

    Chapter 8 is devoted to paradoxes and includes a discussion of what counts as a good solution to a paradox; this discussion should be read by anyone interested in paradoxes.

  • Olin, Doris. Paradox. Chesham, UK: Acumen, 2003.

    E-mail Citation »

    An excellent general discussion of paradoxes precedes detailed expositions of the lottery and preface paradoxes (among other paradoxes discussed in the book).

  • Rescher, Nicholas. Paradoxes: Their Roots, Range, and Resolution. Chicago: Open Court, 2001.

    E-mail Citation »

    This is possibly the most comprehensive general book on paradoxes; it also contains brief discussions of the lottery and preface paradoxes.

  • Salerno, Joe. “Epistemic Philosophy of Logic.” Oxford Bibliographies Online. 2011.

    DOI: 10.1093/obo/9780195396577-0068E-mail Citation »

    This contains a discussion of various epistemic paradoxes, including the lottery paradox.

  • Sorensen, Roy. “Epistemic Paradoxes.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2011.

    E-mail Citation »

    This entry contains one section on each of the paradoxes at issue.

  • Wheeler, Gregory. “A Review of the Lottery Paradox.” In Probability and Inference: Essays in Honour of Henry E. Kyburg, Jr. Edited by William Harper and Gregory Wheeler, 1–31. London: King’s College Publications, 2007.

    E-mail Citation »

    This informed survey of much of the literature on the lottery paradox pays special attention to connections of the lottery paradox with some technical literature, such as the topic of nonmonotonic reasoning.

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