Incoherence without Exploitability

If you believe that your sister is in Tashkent and that Tashkent is in Uzbekistan, but you also believe that your sister is not in Uzbekistan, then your beliefs are not merely odd, but irrational. Moreover, it seems your beliefs are irrational in virtue of their having a certain structure, in particular their being logically inconsistent. It is a requirement of rationality that your beliefs be logically consistent. But belief is not an all or nothing affair. It comes in degrees. We believe some things more strongly than others. Your levels of confidence, or degrees of belief, are known as credences. Are there any rational constraints on the structure of your credences, just as there are rational constraints on the structure of your binary beliefs? Suppose you are certain that your sister is in Tashkent, and you regard it as rather likely that Tashkent is in Uzbekistan, but you also regard it as unlikely that your sister is in Uzbekistan. Your credences are in some sense incoherent and resemble inconsistent beliefs. Plausibly, you are irrational if you have these incoherent credences, just as you would be irrational in virtue of having inconsistent beliefs. But can we give any argument that this sort of incoherence is irrational? The Dutch Book Argument (DBA) is the most prominent argument that purports to show that it is indeed a requirement of rationality that your credences have a certain structure. It is a requirement of rationality that your credences be coherent, in a sense to be made precise in the next section.