Vagueness, Uncertainty and Degrees of Belief: Two Kinds of Indeterminacy—One Kind of Credence

If we think, as Ramsey did, that a degree of belief that P is a stronger or weaker tendency to act as if P , then it is clear that not only uncertainty, but also vagueness, gives rise to degrees of belief. If I like hot coffee and do not know whether the coffee is hot or cold, I will have some tendency to reach for a cup; if I like hot coffee and know that the coffee is borderline hot, I will have some tendency to reach for a cup. Suppose that we take degrees of belief arising from uncertainty to obey the laws of probability and that we model vagueness using degrees of truth. We then encounter a problem: it does not look as though degrees of belief arising from vagueness should obey the laws of probability. One response would be to countenance two different sorts of degrees of belief: degrees of belief arising from uncertainty, which obey the laws of probability; and degrees of belief arising from vagueness, which obey a different set of laws. I argue, however, that if a degree of belief that P is a stronger or weaker tendency to act as if P , then this option is not open. Instead, I propose an account of the behaviour of degrees of belief which integrates subjective probabilities and degrees of truth. On this account, degrees of belief are expectations of degrees of truth. The account explains why degrees of belief behave in accordance with the laws of probability in cases involving only uncertainty, while also allowing degrees of belief to behave differently in cases involving only vagueness, and in mixed cases involving both uncertainty and vagueness. Justifications of the account in terms of Dutch books, and in terms of epistemic accuracy, are presented. 1 Probabilities and Degrees of Truth When the idea of degrees of truth—and the formalism of fuzzy set theory and logic—were new, there was apparently some confusion about the difference between probabilities and degrees of truth. Those days are behind us. Let S be the proposition that it is raining in Sydney at midday on 1 January 2012. If I tell you (say on the evening of 31 December 2011) that the probability of S is 0.7, then whether the probability involved is supposed to be objective or subjective, the idea is that at midday on 1 January 2012, either it will (clearly, definitely) be raining in Sydney, or it will (clearly, definitely) not be, but at the time at which I speak, either I simply do not know which it will be (if the probability is subjective) or it is objectively chancy which it will be (if the probability is objective). If, on the other hand, I tell you that the degree of truth of S is 0.7, then the idea is that at midday on 1 January 2012, it is neither clearly, definitely raining in Sydney, nor clearly, definitely not raining in Sydney—rather, the weather is in a state which is, say, borderline between a very heavy fog and a light rain. I take it that this distinction between probabilities and degrees of truth is now well understood. Of course some say that we do not need degrees of truth—that we need only probabilities, perhaps only subjective probabilities, in order to give a good account of our world and of our place in it—but this claim is based on an understanding of the distinction between probabilities and degrees of truth. Nevertheless, deep problems—both conceptual and technical—remain, concerning the relationships between probabilities and degrees of truth. For it is not as though the two notions can live separately: they come into contact and begin to interact, in puzzling ways, when we consider the notion of degree of belief. The aim of this paper is to resolve the puzzle and sort out the relationship.