closest worlds) was studied by Galles and Pearl [1998]. They argue that the relationship between the two approaches depends in part on whether we consider *recursive* (i.e., *acyclic*) models (those without feedback—see Section 2 for details). They reach the following conclusion [Pearl 2000, p. 242].1 In sum, for recursive models, the causal model framework does not add any restrictions to counterfactuals beyond those imposed by Lewis's framework; the very general concept of closest worlds is sufficient. Put another way, the assumption of recursiveness is so strong that it already embodies all other restrictions imposed by structural semantics. When we consider nonrecursive systems, however, we see that reversibility [a particular axiom introduced by Galles and Pearl] is not enforced by Lewis's framework.

This conclusion seems to have been accepted by the community. For example, in the Wikipedia article on "Counterfactual conditional" (en.wikipedia.org/wiki/Counterfactual conditional; Sept., 2009), it says "[The definition of counterfactuals in causal models] has been shown to be compatible with the axioms of possible world semantics."
In this paper I examine these claims and the proofs given for them more carefully, and try to settle once and for all the relationship between causal models and counterfactual structures. The first sentence in the statement above says "for recursive models, the causal model framework does not add any restrictions to counterfactuals beyond those imposed by Lewis's framework". It is not clear (at least to me)
exactly what this means. Recursive models are a well-defined subclass of causal models. Galles and Pearl themselves show that there are additional axioms that hold in recursive models over and above those that hold in counterfactual structures. Indeed, they show that the reversibility axiom mentioned above is valid in recursive models and is not valid in possible-worlds models. They also show that all the axioms of causal reasoning in the possible-worlds framework that they view as relevant (specifically, axioms that do not mention disjunction, since it is not in their language) hold in recursive causal models. Thus, the only conclusion that can be drawn from their argument is just the opposite to what they claim, namely, that the possible-worlds approach does not add any restrictions to causal reasoning beyond those imposed by causal models, since causal models satisfy all the axioms that counterfactual structures do, and perhaps more.2 Pearl [private communication, 2010] intended the clause "for recursive models" to apply to counterfactual structures as well as to structural models. However, the notion of a recursive counterfactual structure is not defined either by Galles and Pearl [1998] or Pearl [2000]. In fact, in general, the notion of recursive model as defined, for example, in [Pearl 2000, Definition 7.3.4], does not even make sense for counterfactual structures. I show that, if we focus on the language considered by Galles and Pearl and counterfactual structures appropriate for this language, there is a well-defined subclass of counterfactual structures that can justifiably be viewed as recursive (counterfactual) structures. I then show that precisely the same formulas in this language are valid in recursive causal models and recursive counterfactual structures. Put another way, at least as far as Galles and Pearl's language goes, recursive causal models and recursive counterfactual structures are equally expressive. Thus, by restricting to recursive counterfactual structures (as Pearl intended), the Galles-Pearl claim is, in a sense, correct (although the