In the very next equation after "A = (A v B) & (A v ~B)", you write:
p(A v B|B) – p(A v B) + p(A v ~B|B) – p(A v ~B) = .15
This is the equation where you put in the plus signs. Additionally, you can break things down like that inside the P() operator, but you can't just move that to outside the P() operator, because things might be correlated (and, in the case of B and ~B, certainly are).
Well, it wasn't actually an equation. That's why I used the =||= symbol. It was a bientailment. It asserts logical equivalence (in classical logic), and it means something slightly different than an equals symbol. The equation with the plus signs and the logical equivalence shouldn't be confused.
In 1983 Karl Popper and David Miller published an argument to the effect that probability theory could be used to disprove induction. Popper had long been an opponent of induction. Since probability theory in general, and Bayes in particular is often seen as rescuing induction from the standard objections, the argument is significant.
It is being discussed over at the Critical Rationalism site.