To elaborate, A->B is an operation with a truth table:
A B A->B
T T T
T F F
F T T
F F T
The only thing that falsifies A->B is if A is true but B is false. This is different from how we usually think about implication, because it’s not like there’s any requirement that you can deduce B from A. It’s just a truth table.
But it is relevant to probability, because if A->B, then you’re not allowed to assign high probability to A but low probability to B.
EDIT: Anyhow I think that paragraph is a really quick and dirty way of phrasing the incompatibility of logical uncertainty with normal probability. The issue is that in normal probability, logical steps are things that are allowed to happen inside the parentheses of the P() function. No matter how complicated the proof of φ, as long as the proof follows logically from premises, you can’t doubt φ more than you doubt the premises, because the P() function thinks that P(premises) and P(logical equivalent of premises according to Boolean algebra) are “the same thing.”