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.
