Whether you admit it or not, no matter what data you have, you implicitly have a prior probability for just about every proposition you can think of. If you say, “I have no idea whether that’s true or not,” you’re really just saying, “My prior is 50%.”
This is the kind of claim one not uncommonly sees from people hawking the Bayesian snake oil, but it deserves to be called out more, since it is at least problematic -- the 'whether you admit it or not' is effectively a 'regardless of your large amounts of evidence apparently to the contrary' -- and on closer investigation doesn't have all that much to recommend it. There are a lot of propositions of which we can think yet have no reason to think we have any prior probabilities for unless you assume Carroll's second sentence -- for instance, propositions we don't understand, or are only just now coming to understand.
But the second sentence is doubtful at least outside very specific contexts. For one thing, claims about having no idea are claims about inquiry -- "I have no idea whether that's true or not" at least implies that I do not regard myself as having done any inquiry that would be appropriate for coming to a conclusion -- whereas claims about prior probabilities are not.
For another, even if we take "I have no idea whether that's true or not" to mean just "I have no belief that that is true and no belief that it is not", this is not equivalent to "My belief in that being true is equal to my belief in that not being true", which is the usual Bayesian interpretation under degrees-of-belief assumptions. Not having started on leaning is not the same as leaning to each side equally; but this particular take on the principle of indifference cannot distinguish the two at all. Saying that the probabilities on each side are undefined is not the same as saying that the probabilities are equal. Bayesians like Carroll are really denying that the former is possible (on what grounds is always vague enough, since we always get this kind of 'whether you admit it or not' move), but the point is that "I have no idea" allows it as a possibility and "My prior is 50%" does not, and thus that they will not always mean the same.
In addition, because Carroll holds that "there is no objective, cut-and-dried procedure for setting your priors" in other cases, it's odd that we have a cut-and-dried procedure for setting our priors for that one case, out of infinitely many mathematically possible cases, at 50%. The problems with simply assuming a principle of indifference without further clarification are well known, but if there's no procedure for priors, there's actually no reason to think that people will always, or even usually, have only one prior. Have several, and get a range, or at least a diversity of answers that may be more scattershot or less scattershot. If there's no objective, cut-and-dried procedure for setting priors, and, as we certainly can to some extent, we can imaginatively or sympathetically participate more than one perspective, it seems entirely possible that we can take a stance on which A is 15% and also take a separate stance on which it is 60%, without knowing at present which is better or on which we'll settle. In such a case, we could say we have no idea whether A is true or not, and yet the probability assignment for is not at 50% at any point at all. Allowing that we can do this arguably explains some probability mistakes people often make -- we are confusing two different assessments for the reason that we are making both and not actively distinguishing them -- and, again, without an objective procedure for setting priors, it's difficult to say why one would think it impossible.
Of course, there are some independent reason to take probabilities, including priors, as things only having meaning within the context of a particular inquiry establishing them (a possible take-away, to give just one example of such a reason, from both Bertrand's paradox and proposed responses to it); and if one takes such a view, as I do, there is independent reason to reject Carroll's claim -- one would take seriously the first point above.