It’s not that we would believe the statement and its negation are both true; rather, we would believe that the statement is true with probability x and false with probability 1-x, as usual.
Then I don’t understand why you said this earlier:
we expect that if ZFC were inconsistent, we would have found a contradiction by now
The consistency of ZFC is an arithmetical statement. You say we haven’t yet found a disproof for it, so we should believe it more; but we haven’t found a disproof of its negation either, so we should believe it less! Isn’t this incoherent by Bayesian lights? Or am I misunderstanding something about your idea?
Then I don’t understand why you said this earlier:
The consistency of ZFC is an arithmetical statement. You say we haven’t yet found a disproof for it, so we should believe it more; but we haven’t found a disproof of its negation either, so we should believe it less! Isn’t this incoherent by Bayesian lights? Or am I misunderstanding something about your idea?