Counterfactuals and Counterexamples
It is controversial whether indicative conditionals with false antecedents are generally true. As far as I know, which really is not very far at all, it is equally controversial whether counterfactual conditionals with necessarily false antecedents are generelly true. What's interesting is the different kinds of counterexamples that are brought forward against these views. For indicatives, the counterexamples are indicative conditionals with false antecedents that nevertheless appear to be false, e.g. "if I put diesel in my coffee, the coffee tastes fine." For counterfactuals however, the alleged counterexamples (brought forward e.g. by Field in §7.2 of Realism, Mathematics & Modality, Katz in §5 of "What mathematical knowledge could be", and Rosen in §1 of "Modal fictionalism fixed") are counterfactual conditionals with necessarily false antecedents that appear to be true, e.g. "if the axiom of choice were false, the cardinals wouldn't be linearly ordered". Isn't this quite puzzling? How can the fact that some instances are true be a problem for a theory that claims that all instances are true?
(The problem is supposed to be that these instances are not only true, but also interesting and informative. But this would only be a problem if the attacked theory claimed that all subjunctive conditionals with necessarily false antecedent were equally uninteresting and uninformative. Hence Field and Katz quote an unfortunate passage from Lewis where he says that "nothing sensible can be said about how things would be different if there were no number 17" (Plurality, p.111). But it is clear from Lewis' work on counterfactuals that he doesn't think that the same holds for all counterfactuals with necessarily false antecedents (see e.g. Counterfactuals, pp.24ff.). And even if he thought so: This view about what is interesting and sensible to say would be independent of his semantics for counterfactuals.)