1 From Fuzzy Sets to Crisp Truth Tables

1. Overview One limitation of the truth table approach is that it is designed for causal conditions are simple presence/absence dichotomies (i.e., Boolean or "crisp" sets). Many of the causal conditions that interest social scientists, however, vary by level or degree. For example, while it is clear that some countries are democracies and some are not, there are many in-between cases. These countries are not fully in the set of democracies, nor are they fully excluded from this set. Fortunately, there is a well-developed mathematical system for addressing partial membership in sets, fuzzy-set theory. Section 2 of this paper provides a brief introduction to the fuzzy-set approach, building on Ragin (2000). Fuzzy sets are especially powerful because they allow researchers to calibrate partial membership in sets using values in the interval between 0 (nonmembership) and 1 (full membership) without abandoning core set theoretic principles, for example, the subset relation. Ragin (2000) demonstrates that the subset relation is central to the analysis of multiple conjunctural causation, where several different combinations of conditions are sufficient for the same outcome. While fuzzy sets solve the problem of trying to force-fit cases into one of two categories (membership versus nonmembership in a set), they are not well suited for conventional truth table analysis. With fuzzy sets, there is no simple way to sort cases according to the combinations of causal conditions they display because each case’s array of membership scores may be unique. Ragin (2000) circumvents this limitation by developing an algorithm for analyzing configurations of fuzzy-set memberships that bypasses truth tables altogether. While this algorithm remains true to fuzzy-set theory through its use of the containment (or inclusion) rule, it forfeits many of the analytic strengths and virtues that follow from analyzing evidence in terms of truth tables. For example, truth tables are very useful for investigating "limited diversity" and the consequences of different "simplifying assumptions" that follow from the use of different subsets of "remainders" to reduce complexity (see Ragin 1987; Ragin and Sonnett 2004). Analyses of this type are difficult without using a truth table as the starting point. Ragin and Sonnett (2004), for example, show how to use QCA to aid counterfactual analysis and link the analysis of counterfactual conditions to core practices in case-oriented research. Truth tables are central to the analysis of counterfactuals, and the techniques described in Ragin and Sonnett (2004) cannot be implemented without the aid of truth tables.