Set theory

Editor’s Note: This article was written by former ADV author and editor-in-chief Iva Kopraleva.

Everyone who has studies mathematics even at the most basic level has an understanding of Set Theory. A set is generally defined as a collection of objects. We can have a set of real numbers (R); natural numbers (N); integers (Z); etc. Although in mathematics Set Theory usually applies to abstract mathematical objects it is easy to think of examples from our everyday life to illustrate the idea. For example, chairs by definition belong to the set of furniture. But how can we use this line of reasoning to solve political science puzzles?

Charles Ragin comes up with the idea that scholars can take advantage of the existence of sets in the social world, in order to determine which are the necessary and/or sufficient conditions or a certain outcome to occur. To give a simple example, one might hypothesize that if a country is located in Western Europe this is a sufficient condition for said country to be a democracy today. Nevertheless, democratic countries exist outside of Western Europe as well, therefore, it is not a necessary condition for a country to be located in this region in order for it to be a democracy. The conclusion from this example is rather trivial but it illustrates well how one might think in terms of sets (the set of countries from Western Europe and the set of democracies in the world) and conditions.

In fact, the logical foundation of this set-theoretical methodology, named by Ragin Qualitative Comparative Analysis (QCA) can be found in Mill’s famous methods of agreement and difference. The method of agreement refers to a scenario where everything but one element is different in two cases and yet they result in the same outcome. According to the method of difference, on the other hand, everything but one element is similar in two cases and yet their outcome is different. These two logical constructions allow us to infer that the only similar/different element is in fact the cause for the similarity/difference in the outcome.

The QCA methodology uses this logic by dichotomizing both the variables which serve for conditions and the outcome we are interested in. This allows the researcher to track which conditions are present (have a value of 1) or absent (have a value of 0) when the outcome is present across multiple cases. On the basis of this analysis an inference can be made with regard to the necessary and/or sufficient conditions which need to be present, or alternatively, absent, in order for the outcome to occur. As the methodology progresses, the concept of the so called fuzzy sets appear which allows objects to be partial members of a certain set. Then the variables (or conditions) are ‘calibrated’ in order to reflect this partial membership and can be assigned values in the interval between 0 and 1.

It is important to note here that QCA has the potential to solve a lot of the inherent problems of other methods. On the one hand, QCA allows us to systematically analyse multiple cases, normally in the form of a medium-N study. This has certain advantages over the in-depth case study analyses because it allows us to draw more general conclusions. On the other hand, the QCA method can address a number of challenges posed by statistical methods. Firstly, the conditions which determine the presence/absence of the outcome are not assumed to be independent from one another. Secondly, the set-theoretical method does not assume linearity, that is to say the presence and the absence of the outcome do not necessarily result from the same conditions. Thirdly, QCA allows for the possibility of equifinality. This means that more than one configuration of conditions (or path) can lead to the presence or, respectively, the absence of the outcome. Finally, statistical models require large number of observations in order to produce robust results, something which is not always feasible in political science due to the limited number of empirical cases in existence for certain phenomena.

In conclusion, Set Theory can give a researcher (or a student) a different perspective on the cases they are studying. Instead of searching for linear relationships or sui generis explanations for the social phenomena around us, it might prove useful to think of the different cases as configurations of conditions and attempt to distill which of them are necessary or sufficient or both. QCA was already used to solve a number of important and extremely diverse political science puzzles including the emergence of the social security state; the consolidation of democracy; explaining US urban regimes; and many more to be found here. Would you use QCA for your own research project? Feel free to let us know in the comment section.

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