From Causal Models To Counterfactual Structures

Galles and Pearl [1998] claimed that ``for recursive models, the causal model framework does not add any restrictions to counterfactuals, beyond those imposed by Lewis's [possible-worlds] framework.''  This claim is shown to be false.  Indeed, the opposite claim is true: recursive models are shown to correspond precisely to a subclass of (possible-world) counterfactual structures.  On the other hand, a slight generalization of recursive models, models where all equations have unique solutions, is shown to be incomparable in expressive power to counterfactual structures, despite the fact that the Galles and Pearl arguments should apply to them as well.  The problem with the Galles and Pearl argument is identified: an axiom that they viewed as irrelevant, because it involved disjunction (which was not in their language), is not irrelevant at all.