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.
DanielLC,
Hi, I am the author.
The =||= just means bientailment. It's short for,
A |= (A v B) & (A v ~B) and (A v B) & (A v ~B) |= A
Where |= means entailment or logical consequence. =||= is analogous to a biconditional.
The point is that each side of a bientailment is logically equivalent, but the breakdown allows us to see how B alters the probability of different logical consequences of A.
Thank you, I think I understand most of it now. I don't see (at this hour) where the absolute values come from, but that doesn't seem to matter much. Let's focus on this line:
The conjunction of those two does contradict itself, and if you actually write out the probability of the contradiction -- using the standard product rule p(CD)=p(D)p(C|D) rather than multiplying their separate... (read more)