# How disjunction works with the conditional excluded middle

I'm studying the semantics for counterfactuals, and I'm slightly confused about how certain inferences supposedly make the Conditional Excluded Middle (CXM) fail.

Formally, we can write the principle as: (φ □→ ψ) v (φ □→ ~ψ)

Enter the Bizet Verdi sentences:

(1) If Bizet and Verdi were compatriots, Verdi would french.

(2) If Bizet and Verdi were compatriots, Bizet would be Italian (that is, Verdi would not be french: ¬(1)).

I see how this fails intuitively on Lewis' semantics. There isn't a set of worlds around the world i from which the statements are made on, where the worlds in the set are as close as possible to i assuming the antecedent's truth, such that the following holds: on all worlds make (1) and not (2) true, or vice versa. At least on Lewis' account of similarity. We have a tie, but Lewis wants every world in the set which the counterfactual ranges over to "agree" on whether a "would" counterfactual true.

But it seems to me that Lewis' solution; they're both false, doesn't negate the CXM. Disjunction is inclusive. So the CXM seems formally satisfied if either (that is, at least one of) (1) or ¬(1) is true. But clearly, if we're saying ¬(1), as Lewis does, that satisfies the disjunction? The same could be said on a semantics that makes both true.

I'm probably missing something obvious here, and this will look like a silly error, but oh well.

• The disjunction is (if Bizet and Verdi were compatriots, Verdi would be French) v (if Bizet and Verdi were compatriots, it is not the case that Verdi would be French). On David Lewis' account, both counterfactuals are false, so the disjunction is false. For comparison, on Stalnaker's account, CXM holds, so Stalnaker would say that one of your sentences (1) and (2) is true, we just do not know which. Aug 30, 2019 at 20:41
• Oh snap. I misunderstood the scope of the negation I think. I've got myself in a bit of a muddle with this one haha Aug 30, 2019 at 21:11
• @Bumble That should be an answer. Sep 3, 2019 at 15:42