The rule in question is Bayes' Theorem, surely? Given that you know A with some probability, you compute P(B) as P(B|A)P(A) + P(B|~A)P(~A). The practical problem is that many deductions are in the form of implications (A=>B, itself true with some non-100% probability since you might be mistaken even in that), so while you might have a clear idea of P(B|A), you're entirely in the dark about P(B|~A).

Also, if you try to use this for inductive rather than deductive reasoning then it runs into the usual problems with picking your prior. Then there's the usual set of pathological cases (a red swivel chair is supporting evidence for the statement "all ravens are black" because it's a clear example of the logically equivalent "all non-black things are non-ravens"; before the year 2000 all supporting evidence for "all emeralds are green" was also supporting evidence for "all emeralds are grue"); I'm not sure whether those can be rephrased as problems with prior choice or whether they're a further layer of annoyance even once you've sorted out your prior.

Are they pathological when almost everything[1] is a non-black non-raven? :)

I thought Bayesian thinking was supposed to not suffer from that problem, but haven't worked out the details yet (eg. http://plato.stanford.edu/entries/epistemology-bayesian/ doesn't satisfy me). I think the flaws in applying the reasoning might involve the alternatives (we naturally assume *most* ravens are *certainly* black) and whether we know the number of objects and the number of ravens.

[1] Perhaps even measure-theoretically almost everything.