Completeness and decidability of three logics of counterfactual conditionals1

Abstract

A sentence Q] 0-t y is intended to mean, roughly, that y holds in certain of the possible worlds in which Q] holds: those of them that are most closely similar to our actual world. We could capture this intention most straightforwardly by positing a function f which selects, for any sentence 9 and world i, a set f(v, i) of worlds regarded as the set of worlds most closely similar to i out of the worlds in which Q] holds. But this approach is open to objection. Just as no real number greater than 1 is closest to 0, so it may be that none of the worlds in which Q] holds is most closely similar to i. It may be that for each of them, there is another still closer. To meet this difficulty, we could introduce the notion of degrees of similarity between worlds, and take Q] 0-t y to mean that, unless no worlds in which v holds are similar to any degree to our actual world, there is some degree of similarity to our actual world within which there are