From Causal Models To Counterfactual Structures

📝 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.

💡 Analysis

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.

📄 Content

arXiv:1106.2647v2 [cs.AI] 17 Aug 2013 From Causal Models To Counterfactual Structures∗ Joseph Y. Halpern† Computer Science Department Cornell University halpern@cs.cornell.edu October 16, 2018 Abstract 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] frame- work.” 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 ax- iom that they viewed as irrelevant, because it involved disjunction (which was not in their language), is not irrelevant at all. 1 Introduction Counterfactual reasoning arises in broad array of fields, from statistics to economics to law. Not sur- prisingly, there has been a great deal of work on giving semantics to counterfactuals. Perhaps the best-known approach is due to Lewis [1973] and Stalnaker [1968], and involves possible worlds. The idea is that a counterfactual of the form “if A were the case then B would be the case”, typically written A ⪰B, is true at a world w if B is true at all the worlds closest to w where A is true. Of course, making this precise requires having some notion of “closeness” among worlds. More recently, Pearl [2000] proposed the use of causal models based on structural equations for reasoning about causality. In causal models, we can examine the effect of interventions, and answer questions of the form “if random variable X were set to x, what would the value of random variable Y be”. This suggests that causal models can also provide semantics for (at least some) counterfactuals. The relationship between the semantics of counterfactuals in causal models and in counterfactual structures (i.e., possible-worlds structures where the semantics of counterfactuals is given in terms of ∗A preliminary version of this paper appears in the Proceedings of the Twelfth International Conference on Principles of Knowledge Representation and Reasoning (KR 2010), 2010. †Supported in part by NSF grants IIS-0534064, IIS-0812045, and IIS-0911036, and by AFOSR grants FA9550-08-1-0438 and FA9550-09-1-0266, and ARO grant W911NF-09-1-0281. 1 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 intro- duced 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 arti- cle 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 restric- tions 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 (specifi- cally, 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

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