There appear to be two sorts of doxastic attitudes, propositional beliefs in the traditional sense (belief that your shoes are tied) and Bayesian credences (degree of belief .998 that your shoes are tied). Deductive consistency and closure traditionally define coherence for belief.Probability theory traditionally defines coherence for degrees of belief. That leaves open the question how propositional and probabilistic beliefs should cohere with one another. We refer to that as the problem of doxastic coherence. We explicate doxastic coherence in terms of rationality constraints on doxastic correspondences that map propositional belief states to Bayesian credal states. Of particular interest is the principle that doxastic correspondences should preserve diachronic coherence. By way of application, we propose a concrete family of coherent doxastic correspondences. Furthermore, we show that the familiar Lockean proposal that one should believe the propositions that pass a credal threshold is incoherent in a number of important respects.

Suppose I hold a ticket – ticket #5472, say – in a fair 10,000-ticket lottery with a single winner. Suppose the lottery has been drawn, but I’m yet to hear the result. Suppose I already believe, however, that ticket #5472 has lost, based on the fact that there is only one winning ticket and 9,999 losers. Surely, given these odds, I am justified in believing that ticket #5472 has lost. There’s nothing special, however, about the ticket that I happen to be holding; it has as good a chance of being the winner as any other ticket. As a result, if I’m justified in believing that ticket #5472 has lost, then I should be justified in believing the same thing about ticket #1, about ticket #2, about ticket #3 … right up to ticket #10,000.

It is rational to believe all sorts of things, even things about which we are not absolutely certain, and about which we admit that there is a chance that we are wrong. And we know that it is possible to have excellent evidence for propositions that are false. For example, I believe – rationally – that I will be in Munich next month, based on my intention to be there, having taken all requisite steps to enact my plan, such as making a flight reservation, and yet I admit that there is a small chance that I will not in fact be in Munich. A small percentage of transatlantic flights are cancelled, there is a small chance of a natural disaster such as a volcano eruption that grounds all flights across Europe, and so on. Yet when it comes to some propositions, we can have apparently very strong evidence that makes their truth extremely likely – even more likely than that I will be in Munich next month – without its being rational to believe them. For example, there is a good argument that it is not rational for me to believe that my single ticket in a million-ticket lottery will lose, based on the evidence that 999,999 tickets will lose.1 Is this correct? If so, how can we explain it?

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: The set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this problem generalizes: There exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and nontriviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any nontrivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms that this “holistic” relation could take.

We talk and think about our beliefs both in qualitative terms – as when we say that we believe A, or disbelieve A, or are agnostic about A – and in quantitative terms, as when we say that we believe A to a certain degree, or are more strongly convinced of A than of B. Traditionally, analytic philosophers, especially epistemologists, have focused on categorical (all-or-nothing) beliefs, to the almost complete neglect of graded beliefs. On the other hand, the Bayesian boom that started in the late 1980s has led many philosophers to concentrate fully on graded beliefs; these philosophers have sometimes rejected talk about categorical beliefs as being unscientific and as therefore having no place in a serious epistemology.