"Previously, the most popular philosophy of science was probably Karl Popper's falsificationism - this is the old philosophy that the Bayesian revolution is currently dethroning. Karl Popper's idea that theories can be definitely falsified, but never definitely confirmed, is yet another special case of the Bayesian rules; if p(X|A) ~ 1 - if the theory makes a definite prediction - then observing ~X very strongly falsifies A. On the other hand, if p(X|A) ~ 1, and we observe X, this doesn't definitely confirm the theory; there might be some other condition B such that p(X|B) ~ 1, in which case observing X doesn't favor A over B. For observing X to definitely confirm A, we would have to know, not that p(X|A) ~ 1, but that p(X|~A) ~ 0, which is something that we can't know because we can't range over all possible alternative explanations."
Of course, this raises the possibility that we can reformulate logical positivism in Bayesian terms:
For every meaningful proposition Q there is some finitely executable experimental test E for which:
That is, for a proposition to be meaningful, there must be some repeatable experience that is more likely to occur if the proposition is true than if it is false.
Definitely food for thought.
I'm just beginning to read about something called postpositivism. At first glance, postpositivism this looks like another sad attempt to revive metaphysics by focusing on the weaknesses of prototypical logical positivism all-the-while ignoring the central principle of positivism. In other words, postpositivism seems to score highly on the waffle-o-meter... Stay tuned!