More Trouble for Regular Probabilities
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In standard probability theory, probability zero is not the same as impossibility. However, many have suggested that it should be—that only impossible events should have probability zero. In cases where infinitely many outcomes have equal probability, regularity requires that some probabilities are infinitesimal, but merely introducing infinitesimals does not solve all of the problems with regularity. We will see that regular probabilities are not invariant over rigid transformations, even for simple, bounded, countable, constructive, and disjoint sets of possible outcomes. Hence, regular chances cannot be determined by space-time invariant physical laws, and regular credences cannot satisfy seemingly reasonable symmetry principles. Moreover, the examples here are immune to the objections against Williamson’s infinite coin flips.