But not both: the exclusive disjunction in qualitative comparative analysis (QCA)

Abstract

The application of Boolean logic using Qualitative Comparative Analysis (QCA) is becoming more frequent in political science but is still in its relative infancy. Boolean 'AND' and 'OR' are used to express and simplify combinations of necessary and sufficient conditions. This paper draws out a distinction overlooked by the QCA literature: the difference between inclusive-and exclusive-or (OR and XOR). It demonstrates that many scholars who have used the Boolean OR in fact mean XOR, discusses the implications of this confusion and explains the applications of XOR to QCA. Although XOR can be expressed in terms of OR and AND, explicit use of XOR has several advantages: it mirrors natural language closely, extends our understanding of equifinality and deals with mutually exclusive clusters of sufficiency conditions. XOR deserves explicit treatment within QCA because it emphasizes precisely the values that make QCA attractive to political scientists: contextualization, confounding variables, and multiple and conjunctural causation.The primary function of the phrase 'either…or' is 'to emphasize the indifference of the two (or more) things…but a secondary function is to emphasize the mutual exclusiveness', that is, either of the two, but not both (Simpson 2004). The word 'or' has two meanings: Inclusive 'or' describes the relation 'A or B or both', and can also be written 'A and/or B'.Exclusive 'or' describes the relation 'A or B, but not both'. 1 Qualitative Comparative Analysis (QCA) currently deploys only Boolean 'AND' and the inclusive disjunction 'OR' to express and simplify combinations of necessary and sufficient conditions. But in this paper I argue that QCA should also use the exclusive disjunction 'XOR', because the XOR mirrors natural language closely, deals with mutually exclusive clusters of sufficiency conditions, and extends our understanding of equifinality -the notion that different causal paths can lead to the same outcome. The exclusive disjunction deserves explicit treatment because it emphasizes precisely the values that make QCA attractive to political scientists: contextualization and complexity, confounding variables, and multiple, conjunctural causation (Schneider and Wagemann 2006).The exclusive, not the inclusive, disjunction should be used wherever there is logical or practical incompatibility between two or more competing paths to an outcomethey cannot both be taken at the same time. Regrettably, the distinction between inclusive and exclusive 'or' has been treated with casual disregard across the discipline of political science, not just those parts of it that use the QCA method. 'A or B' is often implicitly taken to mean: 'A or B (but not both)' rather than 'A or B (or both)', with resultant confusion. The exclusive disjunction is already widely discussed by linguists, philosophers and logicians, and used extensively in the electrical engineering literature (Hurford 1974;Akers 1959;Jennings 1994;Fleisher, Tavel, and Yeager 1983). It is also commonly, 1 In H.P. Grice's terms, the word '...

A configurational perspective on design elements and user governance engagement in blockchain platforms

“…The identified solutions can also be presented in Boolean expression form, as follows: where “+” indicates logical OR, or non‐exclusive disjunction (Hackett, 2015), suggesting that either one of the configurations or the combination of the configurations can achieve high level of governance engagement. “*” is logical AND, and the arrow refers to the logical implication sign.…”

“…where "+" indicates logical OR, or non-exclusive disjunction (Hackett, 2015), suggesting that either one of the con- (absence). Analysis minimum thresholds for the High Generative Governance Engagement and Low Generative Governance Engagement configurations, respectively: raw consistency = 0.80, 0.80; PRI consistency = 0.75, 0.50; frequency = 1 cases/configurations.…”

The Goldilocks principle: applying the exclusive disjunction to fuzzy sets

“…Scholars using fsQCA often describe relations between exclusive equifinal explanatory conditions because QCA assumes multiple conjunctural causation (Rihoux 2006). Hackett (2015) employs examples from crisp-set QCA to demonstrate the uses of the exclusive disjunction, but users of fuzzy-set QCA also describe relations between clusters of equifinal causal conditions that are exclusive, rather than inclusive, in nature. For example, italicization of the words 'either...or' in Arvind and Stirton's article on the adoption of the Napoleonic code suggests that the authors intend the resulting formula to express an exclusive relationship between INUS conditions:…”

“…QCA currently uses only Boolean conjunction AND (*) and inclusive disjunction OR (+) to express combinations of necessary and sufficient conditions for outcomes. A recent argument put forward in Hackett (2015) is that QCA should also adopt the Boolean XOR (⊕), the exclusive disjunction, for three reasons: (i) confusion between inclusive and exclusive 'or' leads to a loss of logical clarity, (ii) incorporating the exclusive disjunction mirrors natural language closely and extends our understanding of equifinality: the notion that different causal paths can lead to the same outcome, and (iii) exclusivity is a defining element of political life, in which trade-offs must often be made and priorities decided upon under conditions of competition between societal groups for scarce resources (Hackett 2015). This paper expands upon Hackett's work by explaining how the Boolean XOR can be applied in the case of fuzzy set Qualitative Comparative Analysis (fsQCA), and not just crisp set Qualitative Comparative Analysis (csQCA).…”

“…The final section offers two methods for applying the exclusive disjunction to fuzzy sets: mechanical and calibrated. The mechanical approach, familiar from Hackett (2015), simply breaks the XOR down into its components to calculate the value of the set membership of the whole. The new calibrated approach provides the tools for researchers to incorporate crucial case-specific knowledge in determining how similar and how large the set memberships need to be in order to violate the rule 'A or B but not both'.…”