A might counterfactual is a statement of the form 'if so-and-so were the case then such-and-such might be the case'. I used to think that there are different kinds of might counterfactuals: that sometimes the 'might' takes scope over the entire conditional, and other times it does not.
For example, suppose we have an indeterministic coin that we don't toss. In this context, I'd say (1) is true and (2) is false.
(1) If I had tossed the coin it might have landed heads.
(2) If I had tossed the coin it would have landed heads.
These intuitions are controversial. But if they are correct, then the might counterfactual (1) can't express that the corresponding would counterfactual is epistemically possible. For we know that the would counterfactual is false. That is, the 'might' here doesn't scope over the conditional. Rather, the might counterfactual (1) seems to express the dual of the would counterfactual (2), as Lewis suggested in Counterfactuals: 'if A then might B' seems to be equivalent to 'not: if A then would not-B'.
On the other hand, consider the following situation. We know that the laws of nature entail either that whenever A happens then B happens or that whenever A happens then C happens; we don't know which. In the first case, if the laws of nature entail that whenever A happens then B happens (for ordinary A and B), it seems to me that (3) is true.
(3) If A had happened then B would have happened.
Similarly, in the second case, if the laws of nature entail that whenever A happens then C happens, then (4) is true.
(4) If A had happened then C would have happened.
So we know that one of (3) or (4) is true. Now consider the corresponding might counterfactuals.
(5) If A had happened then B might have happened.
(6) If A had happened then C might have happened.
Intuitively, these are both true as well. But if might counterfactuals are the dual of would counterfactuals, then (5) entails the negation of (4) and (6) entails the negation of (3), assuming B and C are logically incompatible. (By duality, (5) is equivalent to `it is not the case that if A had happened then not-B had happened'. This contradicts (4).) So the might counterfactuals (5) and (6) can't be the duals of the corresponding would counterfactuals.
Instead, the 'might' here does seem to scope over the corresponding would conditional: we don't know which of the would counterfactuals (3) and (4) is true, and that seems to be expressed by (5) and (6).
Are there also cases where 'might' takes narrow scope in the consequent of a would counterfactual? I remember that I used to think so, but sadly I can't remember any relevant example.
Over time, I changed my mind. Nowadays, I'd like to say that 'would' and 'might' are epistemic modals that are evaluated relative to a subjunctive supposition. That is, a subjunctive 'if' clause updates the information state of the utterance context by "imaging" on the antecedent; 'would' then expresses that the updated information state supports the consequent, while 'might' expresses that the updated information state is compatible with the consequent.
What does this view predict for the above two kinds of scenarios?
In the case of the indeterministic coin, the intuition that (1) is true and (2) false is vindicated. Supposing (subjunctively) that the coin is tossed, it is uncertain how it lands. So we can say that it might land heads, but not that it would land heads. In general, the new view essentially vindicates the duality of might and would counterfactuals: 'might B' is true relative to a certain subjunctive supposition A iff 'would not-B' is false relative to that supposition.
But now we run into trouble with the unknown laws scenario, where duality seems to fail.
To be sure, the intuitions here don't say that either (3) or (4) is actually assertable. Subjunctively supposing A, we can't say that B would have happened, nor that C would have happened. The subjunctive supposition A supports 'would B' only if it is evaluated under the indicative supposition that the laws of nature say 'if A then B'. But are we right when we judge that either (3) or (4) is true? Is the disjunction of (3) and (4) assertable even though neither of the disjuncts is assertable?
There are different ways to go here. One possibility is to revise the account I have sketched and argue that (would and might) counterfactuals are evaluated not relative to our subjective information state imaged on the antecedent, but relative to some more objective information state -- the (actual) objective chance function, for example. But it's not clear how that helps. One of (3) or (4) will come out as clearly true. But (5) and (6) come out false, unless 'might' scopes over the conditional. And I don't think it does. Consider (5').
(5') What if A had happened? It might be that B had happened.
This seems to me to say the same thing as (5). But it's not plausible that the 'might' in the second sentence somehow scopes over the 'if' in the first.
I'd rather stick with the idea that counterfactuals are evaluated relative to subjective information states. The problem raised by the laws case is then related to a more general problem: to explain why counterfactuals intuitively seem to describe objective and possibly unknown facts about the world.
Let's have a closer look at the "imaging" function that defines subjunctive supposition. Roughly speaking, when we subjunctively suppose A, we shift the (subjective) probability of any world w to the A-world closest to w. Which A-world is closest to w is determined by intrinsic facts about w: the laws, the past, or whatever. Some worlds are such that the closest A-worlds are B-worlds, others are not. On the view I sketched, 'if A then would B' is assertable only if the worlds in our subjective information state are all of the first kind. That's how counterfactuals appear to describe an objective feature of the world (and that's how the Lewis-Stalnaker account comes out almost right on the new account).
In the scenario with the unknown laws, we know that the world is one of two ways: the laws either say 'if A then B' or 'if A then C'. On the supposition that it is the first way, (3) is assertable; on the supposition that it is the second way, (4) is assertable. If we read 'S is true at w' as 'S is assertable on the supposition w', then (3) is true at some worlds in our information state and (4) is true at the remaining worlds. If 'A or B' is true at w iff one (or more) of A and B is true at w, the disjunction of (3) and (4) comes out true at all worlds in our information state. The disjunction is true even though neither disjunct is assertable.
That looks promising to me. But it all needs to be spelled out more carefully.