Counterfactuals Based on Real Possible Worlds

Abstract

Napoleon had won the battle of Waterloo, he would not have died on St. Helena. Under what conditions is this counterfactual true? The Lewis-Stalnaker semantics for conditionals, based upon possible states of affairs or possible worlds, give the following answer ([10], [18]). First, If A then B is vacuously true if there are no A-worlds, i.e. possible worlds in which A is true. Napoleon's winning the battle were somehow physically or logically impossible, the counterfactual would be vacuously true. For Stalnaker, If A then B is non-vacuously true if the A-world which differs least from the actual world is a B-world. For Lewis, since there may be no unique possible world which differs least, the conditional is nonvacuously true if some A-world is a B-world, together with every A-world at least as similar to the actual world. Thus if Napoleon dies elsewhere than on St. Helena in the closest or every closest possible world in which he wins at Waterloo, the counterfactual is true. Otherwise the counterfactual is false. The most obvious difficulty about these semantics lies in determining the degree of similarity a set of possible worlds bears to the actual world. Can possible worlds be inspected and compared? Is similarity not an inherently vague notion in any case, making it impossible to say whether two worlds which have X in common but not Y are more or less similar than two worlds which have Y in common but not X? Lewis accepts this inherent vagueness ([10]: 1). By contrast, what will be attempted in this paper is to present a theory of possible worlds which provides an unambiguous criterion of similarity in terms of a precisely defined concept of closeness. The possible worlds appealed to are real in a sense analogous to that in which transfinite cardinals are real. As Cantor showed, the study of transfinite sets is not the study of a capricious and arbitrary domain-the product of one man's imagination-but a realm with a structure as hard and objective as the crystalline structure of a diamond. Possible worlds are not mathematical entities of course