 Here again we run up against one of the deepest and oldest divides in formal epistemology, that between subjectivists and objectivists. The subjectivists hold that any assignment of probabilities is a legitimate, reasonable way to start one’s inquiry. One need only conform to the three probability axioms to be reasonable. They take this view largely because they despair of clarifying the PoI. They see no reason, for example, that we should follow Carnap in first dividing according to the number of \(\mathsf{T}\)s, and only then subdividing according to where in sequence those \(\mathsf{T}\)s appear. Closely related to this skepticism is a skepticism about the prospects for justifying the PoI, even once clarified, in a way that would put it on a par with the three axioms of probability. We haven’t yet touched on how the three axioms are supposed to be justified. But the classic story is this: a family of  theorems—Dutch book theorems (see entry)  and  representation theorems (see entry)—are  taken to show that any deviation from the three axioms of probability leads to irrational decision-making. For example, if you deviate from the axioms, you will accept a set of bets that is bound to lose money, even though you can see that losing money is inevitable a priori. These theorems don’t extend to violations of the PoI though, however it’s clarified. So subjectivists conclude that violating the PoI is not irrational.