For AI purposes we ask what kinds of counterfactuals are useful to humans, as they are also likely to be the kinds useful to computer programs. Concentrating on counterfactuals that are useful in that believing them usefully affects behavior may also have some philosophical benefits, because the reasoning leading to a useful counterfactual and the useful conclusions drawn from it provide some guidance about the best kind of theory.
We begin with some examples of the uses of particular counterfactuals.
Each of these counterfactuals tells us something about how the world works. We can use this advice in future, if we find ourselves in a similar situation. The notion that the counterfactual is applicable on similar occasions is important. If we are to use a counterfactual to predict in a new occasion s, there must be some kind of test, whether or not the new occasion s is sufficiently similar to the situation that gave rise to the counterfactual. This test is given by an approximate theory.
The kind of approximate theories that are relevant to counterfactuals are those covering only some aspects of the world. The theory used in the car example does not include facts that determine whether or not another car can come over the hill. Likewise the skier theory does not determine if a skier bends his knees. The theories allowing inferences about Caesar in Korea respectively relate to his character as a general and the weapons he had. ``If wishes were horses ... '', takes only advantage of the semantic parallel between having a wish and having a material object. Approximate theories are more thoroughly discussed in [McCarthy, 2000].