Conditional chance and rational credence
Two initially plausible claims:
- Sometimes, a possible chance function conditionalized on a proposition A yields another possible chance function.
- Any rational prior credence function Cr conditional on the hypothesis Ch=f that f is the (actual, present) chance function should coincide with f; i.e., Cr(A / Ch=f) = f(A) for all A (provided that Cr(Ch=f)>0).
Claim 1 is a supported by the popular idea that chances evolve by conditionalizing on history, so that the chance at time t2 equals the chance at t1 conditional on the history of events between t1 and t2. Claim 2 is a weak form of the Principal Principle and often taken to be a defining feature of chance.
But the two claims can't be true together. The reductio is simple. Let f and f' be candidate chance functions such that f' comes from f by conditionalizing on some proposition A (and thus f(A)>0). Let Cr and Cr' be rational credence functions such that Cr(Ch=f)>0 and Cr'(Ch=f')>0. Trivially, Cr(Ch=f / Ch=f) = 1. By claim 2, Cr(Ch=f / Ch=f) = f(Ch=f). So f(Ch=f)=1. By completely parallel reasoning, f'(Ch=f')=1. But conditionalizing on a proposition with positive probability leaves propositions with probability 1 untouched. Since f(Ch=f)=1, this means that f'(Ch=f)=1. But f' and f are distinct probability measures, which means that Ch=f and Ch=f' are incompatible. So f' can't assign probability 1 to both Ch=f and Ch=f'.
It seems to me that all current attempts to prove triviality results for counterfactual conditionals might suffer from a version of this problem.