Causal models, conditional logic, and cycles of counterfactual dependence

An important component in the interventionist account of causal explanation is an interpretation of counterfactual conditionals as statements about consequences of hypothetical interventions. The interpretation receives a formal treatment in the framework of functional causal models (aka structural equation models). In Pearl (2000)’s influential formulation, functional causal models are assumed to satisfy a “unique-solution” property; this class of Pearlian causal models includes the ones called recursive. Halpern (2013) showed that every recursive causal model is Stalnakerian --- in the sense that there is a Stalnakerian possible-world model that validates the exact same formulas as the causal model does (in a language that does not allow nested counterfactuals or counterfactuals with disjunctive antecedents) --- but some Pearlian models are not Stalnakerian. In this paper, we show that not every Stalnakerian causal model is recursive, and we provide a characterization of the class of Stalnakerian causal models and a complete axiomatization with respect to this class. Our characterization is philosophically interesting, in that the class of Stalnakerian causal models is shown to be precisely the class of Pearlian models that do not contain any cycle of counterfactual dependence (in a sense of counterfactual dependence akin to Lewis’s famous relation between distinct events). Finally, we go beyond the class of Pearlian causal models by also considering models that admit multiple solutions, and generalize the previous results.

Halpern, J. Y. (2013). From causal models to counterfactual structures. Review of Symbolic Logic 6(2): 305-22.

Pearl, J. (2009). Causality: Models, Reasoning, and Inference (2nd ed.). Cambridge, UK: Cambridge University Press.