Hi,
I am the author. It wasn’t a mistranslation. The logical equivalence was not translated into anything. It was merely intended to break down A according to its logical consequences shared with B. I never wrote “P(A v B) + P(A v ~B),” because that would be irrelevant.
DanielLC,
Hi, I am the author.
The =||= just means bientailment. It’s short for,
A |= (A v B) & (A v ~B) and (A v B) & (A v ~B) |= A
Where |= means entailment or logical consequence. =||= is analogous to a biconditional.
The point is that each side of a bientailment is logically equivalent, but the breakdown allows us to see how B alters the probability of different logical consequences of A.