From Causal Models To Counterfactual Structures

TL;DR

This paper clarifies the relationship between causal models and counterfactual structures, showing recursive models correspond to a subclass of possible-worlds, while slight generalizations are incomparable in expressive power.

Contribution

It precisely characterizes the correspondence between recursive causal models and counterfactual structures, and identifies limitations of previous claims about their equivalence.

Findings

Recursive models correspond to a subclass of counterfactual structures.

Generalized models with unique solutions are incomparable to counterfactual structures.

An overlooked disjunctive axiom affects the interpretation of causal models.

Abstract

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.