This raises a question of the meaningfuless of second-order Bayesian reasoning. Suppose I had a prior for the probability of some event C of, say, 0.469. Could one object to that, on the grounds that I have assigned a probability of zero to the probability of C being some other value? A prior of independence of A and B seems to me of a like nature to an assignment of a probability to C.

In order to have a probability distribution rather than just a probability, you need to ask a question that isn't boolean, ie one with more than two possible answers. If you ask "Will this coin come up heads on the next flip?", you get a probability, because there are only two possible answers. If you ask "How many times will this coin come up heads out of the next hundred flips?", then you get back a probability for each number from 0 to 100 - that is, a probability distribution. And if you ask "what kind of coin do I have in my pocket?", then you get a function that takes any possible description (from "copper" to "slightly worn 1980 American quarter") and returns a probability of matching that description.