Bayesian epistemologists seem to think that Bayes' theorem (BT) has some special epistemological significance. Let's assume that BT provides us with a synchronic constraint on the coherence of one's degrees of belief (it tells us that whatever our degrees of beliefs in H, E, H given E, and E given H are at time t they have to be related so that Pr(H|E)t=(Pr(H)tPr(E|H)t)/Pr(E)t) and that synchronc coherence is a necessary but not sufficient condition for epistemic rationality. So far nothing epistemologically special about BT--every other theorem or axiom of probability theory also provides us with such a synchronic constraint.
Supposedly, however, BT does more than just that--it also tells us how to "conditionalize on new evidence". What I don't understand is how it is supposed to do so. As far as I can see, the theorem only tells us that the conditional probability of H given E, (Pr(H|E)t) is equal to (Pr(H)tPr(E|H)t)/Pr(E)t but this is only the old synchronic constraint again. It is only if we assume that, in observing that E, our degree of belief in H (Pr(H)t+1) becomes identical to our previous degree of belief in H given E (Pr(H|E)t) (i.e. if we assume that Pr(E)t+1=1 and Pr(H|E)t+1=Pr(H|E)t) that we can use BT to find out what that degree of belief was equal to. But then if this is the case BT in and of itself does not tell us what our degree of belief in H should be after the evidence is in. It only tell us what our degree of belief in Pr(H|E)t had to be before the new evidence was in.
Can someone please show me the error of my ways? Why do Bayesian epistemologists assume that BT plays any different role from that of other axioms of probability? In what sense it is providing us with anything other than a synchornic constraint on our degrees of belief?