Fuzzy-Set Analysis

Fuzzy-set analysis (FSA) is one branch of qualitative comparative analysis (QCA). The idea of fuzzy sets goes back to earlier insights from computer science and formal logic, which had led to the development of a fuzzy-set approach already in the 1960s. As George Klir, Ute St. Clair, and Bo Yuan (1997) have shown, the difference between fuzzy sets and conventional (“crisp”) sets is that set memberships are unequivocal in crisp sets. For example, the element “Sunday” is a member in the set of all days of the week (membership value of 1); whereas the element “January” is not a member of that same set (membership value of 0). By contrast, fuzzy sets have the characteristic that nonperfect set membership can exist. This is especially relevant considering that social science concepts very often have fuzzy boundaries. Democracies are usually not perfect democracies but just democracies to a certain degree. Following this reasoning, most countries have both a partial membership in the set of “democracies” as well as in the set of “nondemocracies.” This is also a fundamental difference to crisp sets: In fuzzy sets, one and the same case can have a (partial) membership both in the original set and in the complementary set. A dichotomy is implicitly maintained, but membership degrees render it possible to differentiate for the degree. QCA as a technique is based [Page 945]on the notion of a set. Since crisp-set qualitative comparative analysis (csQCA) can be criticized for requiring dichotomous conditions and outcomes, and since many phenomena in the social sciences are not simply dichotomous, it is understandable why fuzzy sets are attractive for an inclusion in QCA. Below, this entry discusses some of its major features and current developments.

Both the dichotomous version csQCA and fuzzy-set qualitative comparative analysis (fsQCA) are based on an algebra that is different from mainstream linear algebra, as it is applied in conventional statistical methods. csQCA is based on Boolean algebra and fsQCA on fuzzy algebra. However, it is wrong to assume that fuzzy algebra would be an extension of Boolean algebra; rather, Boolean algebra should be seen as a special case of fuzzy algebra. Indeed, all rules for fuzzy algebra are also valid for Boolean algebra but not vice versa. Dichotomies (as used in csQCA and for Boolean algebra) are nothing else than very restricted versions of fuzzy scales, containing just the values of 0 and 1. One of the most difficult (and also most contested) points with regard to FSA is the decision about the attribution of fuzzy values to individual cases. More concretely, if the level of democracy of a given country has to be coded, the question is how the fuzzy values (between 0 and 1) are attributed to single countries, or, in other words, which indicators have to be chosen to attribute a lower fuzzy value to one country and a higher value to another. One possibility is to use already existing quantitative indicators and to standardize them on a range between 0 and 1. However, as Charles Ragin (2000) notes, such a strategy would overlook the central aspect of the coding, namely, that fuzzy values represent the degree of presence of a concept and that they are based on theoretical knowledge. Furthermore, most social science concepts suffer from not having easily definable quantitative equivalents. Therefore, an important part of FSA deals with the translation of qualitative degrees of the presence/absence of a concept into quantitative measures. Indeed, “[i]n the hands of a social scientist […], a fuzzy set can be seen as a fine-grained, continuous measure that has been carefully calibrated using substantive and theoretical knowledge” (p. 7). This also means that in-depth case knowledge is the undeniable prerequisite for a meaningful FSA. Therefore, the coding process usually comes more toward the end of the research process. Recently, Ragin has proposed two more methods to calibrate fuzzy values: a so-called direct and an indirect method that both combine the (theory driven) setting of qualitative anchors with quantitative standardizations in between these anchors (for details, see Ragin, 2008, 71ff).

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