 The non-propositional nature of the conditional is essential here. Suppose we just redescribe the four lines above as four possible worlds, four ways the world might be, in two of which the conditional is true—as Stalnaker did. And suppose we start off thinking each of the four is equally likely. Then we learn \({\sim}(A \amp B)\): the first line goes out. We learn nothing other than that. \(\mathbf{p}_A (B)\) is now 0. But \(\mathbf{p}\)(if \(A, B)\) is not 0: the third line remains a possibility, and we haven’t eliminated that. (Indeed, if probabilities change by conditionalization, the third line now has probability 1/3.) In short, no two contingent propositions are probabilistically independent in all probability distributions. But the conditional—not a proposition—is stipulated to be independent of \(A\). On Bradley’s theory, having learned that \({\sim}(A \amp B), \mathbf{p}_A (B) = \mathbf{p}\)(if \(A, B) = 0\); the third line gets 0; should \(A\) turn out to be false, it’s false that if \(A, B\).