I think that the contribution that Bayesian methodology makes toward good criticism of a scientific hypothesis is that to "do the math", you need to be able to compute P(E|H). If H is a bad explanation, you will notice this when you try to determine (before you see E) how you would go about computing P(E|H). Alternately, you discover it when you try to imagine some E such that P(E|H) is different from P(E|not H).

No, you don't assign probabilities to criticisms, as such. But I do think that every atomic criticism of a hypothesis H contains at its heart a conditional proposition of the form (E|H) or else a likelihood odds ratio P(E|H)/P(E|not H) together with a challenge, "So how would you go about calculating that?"

Incidentally, you also ought to look at some of the earlier postings where EY was, in effect, using naive Bayes classifiers to classify (i.e. create ontologies), rather than using Bayes's theorem to evaluate hypotheses that predict. Also take a look at Pearl's book to get a modern Bayesian view of what explanation is all about.