Stalnaker's and Lewis's recent analyses of counterfactuals (see [1], 1-2], and [-3]) are based on the idea that given a possible world IV, other worlds can in principle be judged as to how similar they are to W. The authors assume, in other words, that possible worlds can be looked upon as approximations to W, and that of two such approximations one may be better than the other.
It will be convenient to adopt the following terminological convention: instead of saying that a sentence X is true in an approximation W' to W, we shall simply say that W' is an X-approximation to IV.
The authors tell us little about what exactly it takes for one approximation to W to be better than another. They take this to be a virtue of their theory ; since on their view, the vagueness of the notion of relative similarity among worlds affords an explanation of the (alleged) vagueness of counterfactual judgements. But, as David Lewis says, not anything goes.
One of the more obvious requirements which must be satisfied if the notion of world similarity is to make sense at all seems to be this: (R) Let a non-causal sentence B be logically and causally independent from A in IV. Moreover, let B be true in W. Then B is true in the best A-approximations to IV.
For surely good A-approximations to W will be worlds where the causal dependences among propositions are the same as in W. But B is clearly true in some worlds of this sort; and since B is true in W, such worlds approximate W better than the others. By way of a simple illustration of (R), consider a man -call him Jones -who is possessed of the following dispositions as regards wearing his hat. Bad weather invariably induces him to wear his hat. Fine weather, on the other hand, affects him neither way: on fine days he puts his hat on or leaves it on the peg, completely at random. Suppose, moreover, that actually the weather is bad, so Jones is wearing his hat. Writing A for