[Correction: In the original version of this comment, I claimed that the linked post was mistaken on a point of mathematics: specifically, I said that it is not the case that P(A|B) – P(A)= P(A v B|B) – P(A v B) + P(A v ~B|B) – P(A v ~B). However, Guy Srinivasan pointed out that my supposed disproof contained a mistake. Redoing my calculations, I find that the equation in question is in fact an identity. I regret the error.]
P(Av~B|B) does not equal P(A). P((Av~B) & B) equals P(A).
Edit: it doesn't, of course. P(Av~B|B) = P(A|B) and the other thing I said is just silly.
In 1983 Karl Popper and David Miller published an argument to the effect that probability theory could be used to disprove induction. Popper had long been an opponent of induction. Since probability theory in general, and Bayes in particular is often seen as rescuing induction from the standard objections, the argument is significant.
It is being discussed over at the Critical Rationalism site.