Crisp-set QCA with Three-Valued Modal Logic

This paper deals with the methodological problem on how to compare social/political settings with regard to their effects on a qualitative dependent variable with binary values 0 and 1. In his classical crisp-set QCA-methodology (csQCA) Ch. Ragin proposes to translate the original data into Boolean expressions, which are subsequently simplified by means of the Quine-McCluskey procedure for further analyses and interpretation. Crisp-set QCA has three major problems with regard to its dependent variables: i) For some social/political settings the effects on the dependent variable may be unknown: due to limited diversity there are no real cases matching these settings. ii) For other social/political settings, the consequences may be insofar contradictory, as they entail different binary values for the same dependent variable. This situation e.g. occurs, if two actors are not totally constrained by the same social/political setting and use their decision latitude just in the opposite way. iii) Finally, the value of the dependent variable may be uncertain because of experts, who give fuzzy judgments about the value of the dependent variable. This is typical for situations, where concept building is not yet finished. Ragin has attempted to solve these problems by declaring some values as missing and by switching from Boolean to fuzzy logic. Both solutions are unsatisfactory: in qualitative analysis every piece of counter-evidence matters and thus must not be discarded as a missing value. Moreover, fuzzy logic quantifies propositions about society, which are often qualitative by their very nature. Consequently, this article proposes to use for QCA not the binary logic of G. Boole, but rather the three-valued logic of J. Lukasiewicz, which has a third indeterminate truth-state "i". It imposes itself to assign this third state "i" to those consequences of social/political settings, which are contradictory, uncertain, or missing, as described before. Like in conventional csQCA, the resulting three-valued data-tables can be translated into logical expressions. However, in order to be able to simplify these expressions by means of the mentioned Quine-McCluskey procedure and the corresponding software of Ch. Ragin, the dependent variable has to be treated with the standard operators from modal logic. The result is a split of the originally three-valued dependent variable into three binary variables, representing (i) necessary, (ii) possible, and (iii) impossible outcomes of the respective social/political settings. After returning to classical logic, these new binary variables can of course be treated with Ragin‘s crisp-set methodology. The output, however, is much richer than with the classical two-valued crisp-set approach of Ch. Ragin: with three-valued modal logic, it is e.g. much easier to study the additional conditions, which turn a possible outcome into a necessary or even an impossible one. This advantage will be demonstrated by the exemplary use of datasets about the political success of conflict movements, which Ragin originally studied in his book „The Comparative Method (chap. 8)“. This way it becomes possible to compare the results of the proposed new methodology with Ragin‘s original analysis, based on two-valued Boolean logic.