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


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.

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  1. In H.P. Grice’s terms, the word ‘or’ has a conversational implicature, something that can be worked out from how something was said, rather than simply what was said. Usually there are contextual clues that indicate whether the ‘or’ used is exclusive or inclusive. For example, the ‘or’ in ‘you may have tea or coffee’ sounds inclusive when stated in a neutral tone, but exclusive if the ‘or’ is given heavy emphasis. See Grice’s (1989) work Studies in the Way of Words for discussion of both conversational and conventional implicature.

  2. In Boolean logic, upper-case letters represent the presence of a condition and lower-case letters represent its absence.

  3. The pathway to intermediate policies would, like the paths to restrictive policies, also require the XOR as follows: \(\hbox {gIar} \oplus \hbox {gImAR} \rightarrow \hbox {Intermediate policies}\).

  4. Context also matters. A disjunct that seems obviously exclusive and therefore in no need of XOR formalization may not be so in other contexts. The ‘or’ in ‘male OR pregnant’, for example, is clearly exclusive amongst humans but inclusive amongst members of the Syngnathidae family, such as seahorses.

  5. Latter two examples taken from (Goertz 2003).


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Correspondence to Ursula Hackett.

Glossary of Terms

Glossary of Terms



Exclusive disjunction

A or B, but not both

Inclusive disjunction

A or B, or both

Logical incompatibility

Where two conditions cannot logically occur simultaneously

Practical incompatibility

Where two conditions cannot occur simultaneously because they would together violate a prior commitment or overshoot the outcome of interest

Target category

The outcome of interest

Intermediate outcome/Mid-range values

A distinctive outcome condition (dependent variable) that lies between two other conditions along a particular dimension, so that its values on that dimension are between the two extremes.

Overshoot/ Overkill

Where the presence of just one of two causal conditions (independent variables) results in the target intermediate outcome, but the presence of both causal conditions results in a value along a particular dimension that is higher than the target outcome. The target outcome is not reached.


Where the presence of precisely one of two causal conditions (independent variables) results in the target intermediate outcome, but the presence of neither causal condition results in a value along a particular dimension that is lower than the target outcome. The target outcome is not reached.

Fuzzy Set XOR application

Split the exclusive disjunction into its component parts to find the fuzzy set value of each case according to the formula: min(max(A,B),max(a,b))

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Hackett, U. But not both: the exclusive disjunction in qualitative comparative analysis (QCA). Qual Quant 49, 75–92 (2015).

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