(a) P is an explanation why Q and not S,
(b) P could not be an explanation why S, and
(c) Q and S are incompatible.
Then one can give contrastive explanations of libertarian-free actions. For instance, suppose that we are trying to explain why x did A rather than B. We can say:
- A and B are incompatible and x's reasons favored A with strength at least SA and favored B with strength at most SB.
- A and B are incompatible and x's reasons favored A with strength at most SA and favored B with strength at least SB.
We can also give contrastive explanations for quantum events. Suppose the electron in state |up>+|down> collapses to |up>. Explanation:
- |up> and |down> are incompatible and the electron's state contained at least proportion 2-1/2 of |up> and at most 2-1/2 of |down>.
There is a way in which my contrastivity requirement is a fairly natural weakening of the more common conditions:
(a) P is an explanation why Q and not S, and
(b') P&(Q or S) entails Q&~S.
For my (b) and (c) entail:
(b'') P&(P explains Q or P explains S) entails Q&~S,
which does look like a fairly natural weakening of (b').
In other words, we switch from the requirement that the explanation entail which of the alternatives should happen to the requirement that the explanation could only explain one of the two alternatives.