Probability theory has been a remarkably fruitful theory, with applications in almost every branch of science. In philosophy, some important applications of probability theory go by the name Bayesianism; this has been an extremely successful program (see for example Howson and Urbach 2006; Bovens and Hartmann 2003; Talbott 2008). But probability theory seems to impute much richer and more determinate attitudes than seems warranted. What should your rational degree of belief be that global mean surface temperature will have risen by more than four degrees by 2080? Perhaps it should be 0.75? Why not 0.75001? Why not 0.7497? Is that event more or less likely than getting at least one head on two tosses of a fair coin? It seems there are many events about which we can (or perhaps should) take less precise attitudes than orthodox probability requires. Among the reasons to question the orthodoxy, it seems that the insistence that states of belief be represented by a single real-valued probability function is quite an unrealistic idealisation, and one that brings with it some rather awkward consequences that we shall discuss later. Indeed, it has long been recognised that probability theory offers only a rather idealised model of belief. As far back as the mid-nineteenth century, we find George Boole saying: