Causal Minimality in the Boolean Approach to Causal Inference

In J.L. Mackie’s (1974) influential account of causal regularities, a causal regularity for an effect factor E is a statement expressing that condition C is sufficient and necessary for (the presence or instantiation of) E (relative to a background or causal field), where C is in general a complex Boolean formula involving a number of factors. Without loss of generality, we can put C in disjunctive normal form, a disjunction of conjunctions whose conjuncts express presence or absence of factors. Since C is supposed to be sufficient and necessary for E, each conjunction therein expresses a sufficient condition. Mackie’s requirement is that such a sufficient condition should be minimal, in the sense that no conjunction of a proper subset of the conjuncts is sufficient for E. If this requirement is met, then every (positive or negative) factor that appears in the formula is (at least) an INUS condition: an Insufficient but Non-redundant part of an Unnecessary but Sufficient condition for E.

Mackie’s minimality or non-redundancy requirement has been criticized as too weak (Baumgartner 2008), and a stronger criterion is adopted in some Boolean methods for causal inference, which have found interesting applications in social science (e.g., Ragin and Alexandrovna Sedziaka 2013; Baumgartner and Epple 2014). In addition to minimization of sufficient conditions, the stronger criterion requires that the disjunctive normal form that expresses a necessary condition should be minimally necessary, in the sense that no disjunction of a proper subset of the disjuncts is necessary for the effect.

In this talk we identify another criterion of non-redundancy in this setting, which is a counterpart to the causal minimality condition in the framework of causal Bayes nets (Spirtes et al. 1993; Pearl 2000). We show that this criterion is in general even stronger than the two mentioned above. Moreover, we argue that (1) the reasons for strengthening Mackie’s criterion of non-redundancy support moving all the way to the criterion we identified, and (2) an argument in the literature against the causal minimality condition for causal systems with determinism also challenges Mackie’s criterion of non-redundancy, and an uncompromising response to the argument requires embracing the stronger criterion we identified. Taken together, (1) and (2) suggest that the Boolean approach to causal inference should either abandon its minimality constraint on causal regularities or embrace a stronger one.