It has become standard practice in philosophy to follow Richard Foley (1992) in distinguishing between an epistemology of belief and an epistemology of degrees of belief. The former distinguishes among three doxastic attitudes: belief, disbelief (i.e., believing something not to be the case), and suspension of belief. Key to the latter epistemology is the idea that beliefs come in various strengths, where the strength of a belief can be measured on a continuous scale (typically, the unit interval but that is a matter of convention). Many have hoped that because the two epistemologies appear to differ only in assuming a different granularity while looking at the same attitude – that of belief – there must be some way to connect them. Finding one or more bridge principles between these epistemologies seems a worthy endeavor indeed, for given such principles, we might be able to derive truths about the notion of categorical belief from insights about the notion of graded belief, or vice versa.

Rational agents represent their beliefs about the world in two rather different ways. One way is by means of qualitative beliefs, or yes-or-no beliefs: Here a proposition p is either believed, or disbelieved (i.e., ¬p is believed), or belief about p is suspended. Thereby, with “rational belief that p” it is meant that the proposition p is believed to be true, or accepted as being true, whereby the system of beliefs satisfies certain rationality conditions.

The lottery paradox (Kyburg 1961) calls into question some of our most basic assumptions about rational belief: Some of them concern rational all-or-nothing belief (binary belief, categorical belief, belief simpliciter); some concern rational numerical degrees of belief (credences, numerical strengths of belief, quantitative belief); and some pertain to ways of relating the two kinds of belief rationally.

It is widely accepted that our belief-attitudes come in two kinds: so-called outright beliefs and degrees of belief. Outright beliefs are coarse-grained: I can believe something, disbelieve it, or suspend judgment about it. Degrees of belief, by contrast, are fine-grained. I can be completely certain that something is false, or completely certain that it is true, or have any degree of confidence in between those extremes. We can model those fine-grained degrees of belief using real numbers, with 0 representing the lowest possible degree of belief, and 1 the highest.1

In this chapter, I review recent empirical findings on knowledge attributions in lottery cases and report a new experiment that advances our understanding of the topic. The main novel finding is that people deny knowledge in lottery cases because of an underlying qualitative difference in how they process probabilistic information. “Outside” information is generic and pertains to a base rate within a population. “Inside” information is specific and pertains to a particular item’s propensity. When an agent receives information that 99 percent of all lottery tickets lose (outside information), people judge that she does not know that her ticket will lose. By contrast, when an agent receives information that her specific ticket is 99 percent likely to lose (inside information), people judge that she knows that her ticket will lose. Despite this difference in knowledge judgments, people rate the likelihood of her ticket losing exactly the same in both cases (i.e., 99 percent). The results shed light on other factors affecting knowledge judgments in lottery cases, including formulaic expression and participants’ own estimation of whether it is true that the ticket will lose. The results also undermine previous hypotheses offered for knowledge denial in lottery cases, including the hypotheses that people deny knowledge because they either deny justification or acknowledge a chance for error.

In this chapter, we will revisit a recent solution to the lottery paradox by Igor Douven (2008b) that we believe, has been underappreciated. More specifically, we aim to show the following: First, Douven’s solution is best seen as epistemic rule consequentialist at heart and, once thus seen, it is more attractive than it might seem at first glance and indeed more than Douven himself would have us think. Second, Douven’s specific way of implementing epistemic rule consequentialism does not offer a fully satisfactory solution to the lottery paradox. Fortunately, however, a better alternative is available. Finally, third, we will work towards an epistemic rule consequentialist solution to the related preface paradox. Interestingly enough, while the lottery paradox does support the alternative form of rule consequentialism over Douven’s, in case of the preface paradox, it does not matter which version of the view one adopts. Both lead to the same result.

The lottery paradox exposes some tensions in our natural ways of thinking about probabilities, and in how we think about belief itself. This chapter explores the paradox from a psychological angle, arguing that it arises from the flexibility of our cognitive capacities to represent (and reason about) the empirical realm. A better understanding of these capacities can give us a clearer sense of our theoretical options. Ultimately, I take a broad view of the paradox: In my view, it can be triggered not only by discussion of games with stipulated odds but by topics of all sorts. However, it will be simplest to start with an example inspired by Kyburg’s (1961) classic discussion, in which you hold one ticket in a fair lottery, with odds of (let us say) a million to one, in which the draw has been held but the single winner not yet announced. It is very likely that your ticket has lost, but what is the significance of this high likelihood for the rationality of believing that your ticket has lost? If we insist that a threshold of .999999 is not high enough for rational belief, it may seem we are trapping ourselves in skepticism: Surely many of the ordinary things we rationally believe about the world are less certain than logical truths. On the other hand, if we do believe that this ticket has lost, by symmetry we should say the same for any of the other tickets in the lottery, and as long as conjunction of rational beliefs is a rational operation, it seems we would be rational to deduce that all the tickets have lost, in contradiction to our other beliefs about this fair lottery.