“Those who find this confusing may find it helpful to study mathematical logic, which trains one to make very sharp distinctions between the proposition P, a proof of P, and a proof that P is provable”
This is a bit of a side question, but wouldn’t a proof that P is provable be a proof of P? In fact, it sounds like a particularly elegant form of proof.
No, because you can’t say anything about the relationship of P(A) in comparison to P(C|D)