Interval-valued Probabilities

It was John Maynard Keynes [13] who first forcibly argued that probabilities cannot be simply ordered. There are cases, he argued, in which the probability of the hypothesis h can be regarded neither as greater than that of hypothesis k, nor less than that of hypothesis k, nor yet equal to that of hypothesis k. Although he did not provide a mathematical structure for his probability values, he did give us some hints: Every probability is comparable to the 0 probability and to the 1 probability. In general, some probabilities (those that can be based on a correct application of the principle of indifference) have rational numerical values, and can serve to bound those that are not precisely comparable to rational valued probabilities. The structure looks something like this, where a/h represents the (not necessarily numerical) probability of the proposition a relative to the (total) evidence h: