Using Eight-Variable Karnaugh Maps to Unravel Hidden Technicalities in Qualitative Comparative Analysis

We use a regular and modular eight-variable Karnaugh map to reveal some technical details of Boolean minimization usually employed in solving problems of Qualitative Comparative Analysis (QCA). We utilize as a large running example a prominent eight-variable political-science problem of sparse diversity (involving a partially-defined Boolean function (PDBF), that is dominantly unspecified). We recover the published solution of this problem, showing that it is merely one candidate solution among a set of many equally-likely competitive solutions. We immediately obtain one of these rival solutions, that looks better than the published solution in two aspects, namely: (a) it is based on a smaller minimal set of supporting variables, and (b) it provides a more compact Boolean formula. However, we refrain from labelling our solution as a better one, but instead we stress that it is simply un-comparable with the published solution. The comparison between any two rival solutions should be context-specific and not tool-specific. In fact, the Boolean minimization technique, borrowed from the area of digital design, cannot be used as is in QCA context. A more suitable paradigm for QCA problems is to identify all minimal sets of supporting variables (possibly via integer programming), and then obtain all irredundant disjunctive forms (IDFs) for each of these sets. Such a paradigm stresses inherent ambiguity, and does not seem appealing as the QCA one, which usually provides a decisive answer (irrespective of whether it is justified or not).The problem studied herein is shown to have at least four distinct minimal sets of supporting variables with various cardinalities. Each of the corresponding functions does not have any non-essential prime implicants, and hence each enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. Moreover, each of them is unate as it is expressible in terms of un-complemented literals only. Political scientists are invited to investigate the meanings of the (so far) abstract formulas we obtained, and to devise some context-specific tool to assess and compare them.

Boolean Algebra

This chapter addresses Boolean algebra, which is based on Boolean logic. In the social sciences, Boolean algebra comes under different labels. It is often used in set-theoretic and qualitative comparative analysis to assess complex causation that leads to particular outcomes involving different combinations of conditions. The basic features of Boolean algebra are the use of binary data, combinatorial logic, and Boolean minimization to reduce the expressions of causal complexity. By calculating the intersection between the final Boolean equation and the hypotheses formulated in Boolean terms, three subsets of causal combinations emerge: hypothesized and empirically confirmed; hypothesized, but not detected within the empirical evidence; and causal configurations found empirically, but not hypothesized. This approach is both holistic and analytic because it examines cases as a whole and in parts.

Aggregation Bias and Ambivalent Cases: A New Parameter of Consistency to Understand the Significance of Set-theoretic Sufficiency in fsQCA

The Boolean minimization, used in fuzzy-set qualitative comparative analysis (fsQCA) to establish sufficient relationships between conditions and outcome, automatically produces false positive subset relationships in the presence of random data. However, because this type of aggregation bias mainly produces ambivalent subset relationships between the condition(s) and the outcome, such false positives do not pose a problem for the fsQCA results per se. The aggregation bias has a negative impact on fsQCA analysis only because the consistency score is not able to detect set-theoretic subset relationships. Indeed, the existent parameter of consistency does not distinguish whether the subset relationship between conditions and outcome is the result of the mere Boolean minimization or whether it has set-theoretic significance. This article proposes a new consistency formula that provides information about subset relationships between conditions and outcome and detects the difference between randomly-generated subsets and meaningful subset relationships. The new parameter of consistency proposed here can be considered as an additional tool to test the significance of a meaningful sufficient relationship without being subject to the aggregation bias.

On Minimum Representations of Matched Formulas (Extended Abstract)

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. We present here two results for matched CNFs: The first result is a shorter and simpler proof of the fact that Boolean minimization remains complete for the second level of polynomial hierarchy even if the input is restricted to matched CNFs. The second result is structural --- we show that if a Boolean function f admits a representation by a matched CNF then every clause minimum CNF representation of f is matched.

Standards of Good Practice and the Methodology of Necessary Conditions in Qualitative Comparative Analysis

The analysis of necessary conditions for some outcome of interest has long been one of the main preoccupations of scholars in all disciplines of the social sciences. In this connection, the introduction of Qualitative Comparative Analysis (QCA) in the late 1980s has revolutionized the way research on necessary conditions has been carried out. Standards of good practice for QCA have long demanded that the results of preceding tests for necessity constrain QCA's core process of Boolean minimization so as to enhance the quality of parsimonious and intermediate solutions. Schneider and Wagemann's Theory-Guided/Enhanced Standard Analysis (T/ESA) is currently being adopted by applied researchers as the new state-of-the-art procedure in this respect. In drawing on Schneider and Wagemann's own illustrative data example and a meta-analysis of thirty-six truth tables across twenty-one published studies that have adhered to current standards of good practice in QCA, I demonstrate that, once bias against compound conditions in necessity tests is accounted for, T/ESA will produce conservative solutions, and not enhanced parsimonious or intermediate ones.

On Minimum Representations of Matched Formulas

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. Each matched CNF is trivially satisfiable (each clause can be satisfied by its representative variable). Another property which is easy to see, is that the class of matched CNFs is not closed under partial assignment of truth values to variables. This latter property leads to a fact (proved here) that given two matched CNFs it is co-NP complete to decide whether they are logically equivalent. The construction in this proof leads to another result: a much shorter and simpler proof of the fact that the Boolean minimization problem for matched CNFs is a complete problem for the second level of the polynomial hierarchy. The main result of this paper deals with the structure of clause minimum CNFs. We prove here that if a Boolean function f admits a representation by a matched CNF then every clause minimum CNF representation of f is matched.