Perhaps the most prominent current definition of (actual) causality is due to Halpern and Pearl. It is defined using causal models (also known as structural equations models). We abstract the definition, extracting its key features, so that it can be applied to any other model where counterfactuals are defined. By abstracting the definition, we gain a number of benefits. Not only can we apply the definition in a wider range of models, including ones that allow, for example, backtracking, but we can apply the definition to determine if A is a cause of B even if A and B are formulas involving disjunctions, negations, beliefs, and nested counterfactuals (none of which can be handled by the Halpern-Pearl definition). Moreover, we can extend the ideas to getting an abstract definition of explanation that can be applied beyond causal models. Finally, we gain a deeper understanding of features of the definition even in causal models.

Galles and Pearl 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 examined carefully, with the goal of clarifying the exact relationship between causal models and Lewis's framework. 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.

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.