Counterfactuals and comparative possibility

In the last dozen years or so, our understanding of modality has been much improved by means of possible-world semantics: the project of analyzing modal language by systematically specifying the conditions under which a modal sentence is true at a possible world. I hope to do the same for counterfactual conditionals. I write A [5]-~ C for the counter-factual conditional with antecedent A and consequent C. It may be read as ' I f it were the case that A, then it would be the case that C' or some more idiomatic paraphrase thereof. 1. ANALYSES I shall lead up by steps to an analysis I believe to be satisfactory. ANALYSIS O. A D-~ C is true at worm i iff C holds at every A-world such that ~. "A-world', of course, means 'world where A holds'. The blank is to be filled in with some sort of condition restricting the A-worlds to be considered. The condition may depend on i but not on A. For instance, we might consider only those A-worlds that agree with i in certain specified respects. On this analysis, the counterfactual is some fixed strict conditional. N o matter what condition we put into the blank, Analysis 0 cannot be correct. For it says that if A [:]-+/~ is true at i, B holds at every A-world such that-. In other words, there are no AB-worlds such that-. Then A B []-* C and A B D-* C are alike vacuously true, and-(AB Fq-* C) and-(ABE]-*C) are alike false, for any C whatever. On the contrary: it can perfectly well happen that A[~-~B is true, yet AB[-q-,(A is non-vacuous, and ABI~-~C is false. In fact, we can have an arbitrarily long sequence like this of non-vacuously true counterfactuals and true denials of their opposites: