Timothy Williamson has claimed to prove that regularity must fail even in a nonstandard setting, with a counterexample based on tossing a fair coin infinitely many times. I argue that Williamson’s argument is mistaken, and that a corrected version shows that it is not regularity which fails in the non-standard setting but a fundamental property of shifts in Bernoulli processes.

Pruss uses an example of Lester Dubins to argue against the claim that appealing to hyperreal-valued probabilities saves probabilistic regularity from the objection that in continuum outcome-spaces and with standard probability functions all save countably many possibilities must be assigned probability 0. Dubins’s example seems to show that merely finitely additive standard probability functions allow reasoning to a foregone conclusion, and Pruss argues that hyperreal-valued probability functions are vulnerable to the same charge. However, Pruss’s argument relies on the rule of (...) conditionalisation, but I show that in examples like Dubins’s involving nonconglomerable probabilities, conditionalisation is self-defeating. (shrink)

Timothy Williamson claimed to prove with a coin-tossing example that hyperreal probabilities cannot save the principle of regularity. A premise of his argument is that two specified infinitary events must be assigned the same probability because, he claims, they are isomorphic. But as has been pointed out, they are not isomorphic. A way of framing Williamson’s argument that does not make it depend on the isomorphism claim is in terms of shifts in Bernoulli processes, the usual mathematical model of sequential (...) coin tossing. But even so framed, the argument still fails. (shrink)