Might Counterfactuals and Conditional Excluded Middle

Lewis vs. Stalnaker

According to Lewis' semantics for counterfactuals, A>C is true if and only if (1) there is no A-worlds, or (2) there is some A&C-world which is more similar to our actual world than any other A&~C-worlds. This formulation allows that a counterfactual, A>C, can be true without there being the closest A-world; it might be the case that for any A-world, there is another A-world, which is closer to the actual world than that A-world.

This is, according to Lewis, as it should be. Lewis rejects the Limit Assumption according to which when we compare overall similarities between the antecedent-worlds, we can eventually reach a limit: some set of antecedent-worlds which are closest to our actual worlds. More precisely, the assumption says that for every world W and antecedent A, if A is not logically impossible or unaccessible from the actual world, there are always worlds (or at least one world) where A is true which are closest to our actual world. Lewis rejects that idea because the assumption rules out the possibility that worlds might get infinitely closer and closer to the actual worlds without limit. (For Lewis' argument and examples against the Limit Assumption, see his Counterfactuals (1973), pp. 19-21). However, invoking the Limit Assumption is harmless in our context, for it will not be playing any role in what follows. So, for simplicity, we can introduce the Limit Assumption.

Given this assumption, Lewis' analysis amounts to this: A>C is true (non-vacuously) if and only if C is true at all the closest A-worlds. This analysis allows that there might be a tie for closest, which Stalnaker denies. According to Stalnaker's analysis, A>C is true (non-vacuously) if and only if C is true at the closest A-world. Here Stalnaker assumes that there is always a unique closest A-world, so-called the Uniqueness Assumption. Lewis called it the Stalnaker Assumption. It can be easily seen that the Uniqueness Assumption implies but not implied by the Limit Assumption.

Might-Counterfactuals

Might-counterfactuals have the form "if it were the case that A, it might be the case that C," more simply, A>mC. Lewis defines might-counterfactuals in terms of (would)-counterfactuals: A>mC =df. ~(A>~C). On the face of it, this seems reasonable; when one says that, for example, "If I had looked in my pocket, I might have found a penny," you can contradict this by saying, "(Even) if you had looked in your pocket, you would have not found a penny."

So, under the Limit Assumption, the truth condition of might-counterfactuals amounts to this: A>mC is true if and only if C is true at some of the closest A-worlds.

Clearly, this sort of definition is not available in Stalnaker ?s framework, given the Uniqueness Assumption. It can be seen that under the definition A>mC =df. ~(A>~C), the difference between would- and might-counterfactuals collapses in Stalnaker's framework: given the Uniqueness Assumption, A>C implies A>mC (except in the vacuous case) and vice versa; they turn out to be equivalent.

So Stalnaker proposes a different view of might-counterfactuals, what may be called an epistemic view. According to this view, might-counterfactuals involve another sort of modality, epistemic possibility. Stalnaker says that A>mC usually means something like "Nothing I know rules out its being the case that A>C." More precisely, according to this view, A>mC is true if and only if it (epistemically) might have been the case that if it had been that A, it would have been that C. (See Stalnaker (1981), "A defense of Conditional Excluded Middle," in William Harper, Robert Stalnaker and Glenn Pearce (eds.) Ifs, pp. 87-104. Also see Keith DeRose (1999), "Can it be that it would have been even though it might not have been?", Philosophical Perspective, vol 13: pp. 385-413.)

This looks right at least for some cases. But it has been observed that in many other cases the meaning of 'might' is not epistemic. Consider the following case due to Bennett (2003, p. 190): At a particular time John knows nothing ruling out the possibility of the truth of A>C, and so by Stalnaker's theory he can truly assert A>mC at that time; yet it is downright impossible that A and C should both hold. For example, John says "If I had looked in the haystack, I might have found the needle there," and this it true by Stalnaker's theory; but really it is false, because the needle was not in the haystack. Thus Stalnaker's account of 'might' assigns a wrong truth value in such cases, which involve a non-epistemic 'might'.

Conditional Excluded Middle

The law of excluded middle states that a proposition is either true or false; in other words, it must be the case that one of P and ~P is true. Given this law, if there is always a unique closest A-world, then at that world one of C and ~C must be true. That is, under the Uniqueness Assumption, the following formula is valid:

(A>C) or (A>~C)

which is Conditional Excluded Middle (CEM).

On the face of it, CEM seems reasonable. It can explain why we often do not distinguish between the external negation of a whole conditional ~(A>C) and the internal negation of the consequent A>~C. The latter implies the former both on Lewis' and Stalnaker's theories and, given CEM (or given the Uniqueness Assumption), the former also implies the latter.

So Stalnaker's theory may be thought as having one virtue that it makes valid CEM. Also, consider the following famous Bizet-Verdi example:

(1) If Bizet and Verdi were compatriots, Bizet would be Italian. (2) If Bizet and Verdi were compatriots, Bizet would be French.

Lewis' analysis verdicts that both (1) and (2) are false, for the worlds where Bizet is Italian and the worlds where Bizet is French are tied. After all, Lewis is committed to say such things as this:

If Bizet and Verdi were compatriots, Bizet either would or would not be Italian; nevertheless, it is not the case that if Bizet and Verdi were compatriots, Bizet would be Italian; and it is not the case that if Bizet and Verdi were compatriots, Bizet would not be Italian.

That is, (A > (C or ~C)) & ~(A>C) & ~(A>~C)

This sounds like a contradiction; and, given CEM, we cannot truly say such a thing.