 Conditional Proof fails for Stalnaker’s semantics. “\(A\) or \(B\); \({\sim}A\); so \(B\)” is of course valid. But (*) “\(A\) or \(B\), therefore \({\sim}A\gt B\)” is not: it can be true that Ann or Mary cooked the dinner (for Ann cooked it); yet false that in the nearest world to the actual world in which Ann did not cook it, Mary cooked it.