## Abstract

This paper takes issue with the current tendency in the literature on Qualitative Comparative Analysis (QCA) to settle for so-called *intermediate* solution formulas, in which parsimony is not maximized. I show that there is a tight conceptual connection between parsimony and causality: only maximally parsimonious solution formulas reflect causal structures. However, in order to maximize parsimony, QCA—due to its reliance on Quine-McCluskey optimization (Q-M)—is often forced to introduce untenable simplifying assumptions. The paper ends by demonstrating that there is an alternative Boolean method for causal data analysis, *viz.* Coincidence Analysis (CNA), that replaces Q-M by a different optimization algorithm and, thereby, succeeds in consistently maximizing parsimony without reliance on untenable assumptions.

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## Notes

- 1.
In this paper, the notion of a

*model*is used in terms of a full specification of the functional dependencies among endogenous and exogenous factors. In the QCA literature, a model is sometimes (e.g. Wright and Boudet 2012) simply taken to be a selection of endogenous and exogenous factors (without a specification of functional dependencies). - 2.
Some studies even settle for so-called

*conservative*solution formulas with all redundancies remaining (cf. Grofman and Schneider 2009). - 3.
There also exist non-causal applications of QCA. For instance, Mendel and Korjani (2012) use QCA as a method for

*linguistic summarization*, which is a data mining approach to extract patterns from databases. The goal of this approach is not to generate causal knowledge, causal explanations, or to test causal hypotheses; rather, it seeks to better understand and communicate about data. For such a purpose, solution formulas that are not maximally parsimonious may be very useful. Moreover, intermediate solutions may be informative with respect to correlations and associations among investigated factors; cf. e.g. (Grant, Morales, and Sallaz (2009), 344–345) who very explicitly abstain from causally interpreting intermediate solutions (though for different reasons than the ones advanced in this paper). - 4.
- 5.
Usually, in the prose around solution formulas in QCA studies only necessary conditions that consist of single factors are explicitly labeled “necessary conditions”. (A commendable exception is Bol and Luppi (2013), who systematize the search for complex necessary conditions within the QCA framework.) In fact, however, every QCA solution formula that identifies complex sufficient conditions for the absence of an effect is tantamount to a solution formula that identifies a complex necessary condition for the presence of the effect (by contraposition), and vice versa. For instance, a formula that identifies \(AC\) and \(\overline{D}\) as two alternative sufficient conditions for \(\overline{E}\) is logically equivalent to a formula that identifies \(\overline{A}D \vee \overline{C}D\) as necessary condition for \(E\). Disjunctive necessary conditions can be interpreted as imposing restrictions on the space of alternative causes of an effect: in the structure represented by (1), \(E\) has exactly three alternative causes.

- 6.
In the QCA literature, two rows \(c_i\) and \(c_h\) in a truth-table such that one and the same configuration of conditions is combined with the presence of the outcome in \(c_i\) and with its absence in \(c_h\) are often very misleadingly called “contradictory” (cf. e.g. Rihoux and De Meur 2009; Schneider and Wagemann 2012 §5.1; Rubinson 2013). In fact, however, such a pair of rows is far from being contradictory, rather, it merely entails that the relevant configuration of conditions is neither sufficient for the presence nor for the absence of the outcome.

- 7.
- 8.
Without providing any reasons, (Rubinson (2013), 2866) rightly deplores the fact that “far too many researchers automatically pick the intermediate solution, assuming that it must be best”. The most straightforward reason why a general preference of intermediate solutions is deplorable is simply that these solutions are not (causally) explanatory.

- 9.
A causal dependence among investigated conditions is but one reason why certain remainders turn out to be impossible—logical, conceptual, or mereological dependencies being further reasons. Thus, the arrow from \(A\) to \(C\) in Fig. 1 could also be interpreted in terms of such a non-causal form of dependence.

- 10.
In fact, the disjunction of solution terms of the conservative formula (4) is logically equivalent to the disjunction of configurations in rows \(c_1\) to \(c_5\) and \(c_7\) in Table 1, which are the configurations that are sufficient for \(E\) in that table. Neither the intermediate nor the parsimonious solution—(6) and (5)—preserve this logical equivalence.

- 11.
(Eliason and Stryker (2009), 26) implement a very analogous minimization idea in the context of a goodness-of-fit test for fuzzy-set solutions.

- 12.
Note that CNA does not preclude the addition of counterfactual configurations; it just does not require it. If there are good theoretical grounds for counterfactually supplementing the data, CNA does not prevent the researcher from doing so. Whereas in the case of QCA it is the algorithmic machinery of the method that calls for the introduction of counterfactual configurations in order to eliminate all redundancies from solution formulas, CNA leaves counterfactual considerations entirely up to the researcher’s background theory.

- 13.
The fact that QCA’s \(\mathcal {S}_3\) and CNA infer the same causal model for outcome \(E\) from Table 1 does not indicate that corresponding inferences are based on the same or even related assumptions. One and the same conclusion can be inferred from very different assumptions. For instance, “Socrates is mortal” can be inferred from the assumptions “Socrates is a man” and “All men are mortal”, or from “All immortal things are angels” and “Socrates is not an angel”, or from any contradiction, e.g. from “It rains and it does not rain”. A conclusion is established if it is not only validly inferred from a set of assumptions, but if the latter are moreover cogently justifiable.

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## Acknowledgments

I am indebted to Alrik Thiem and Carsten Schneider for very valuable discussions about the arguments presented in this paper as well as for comments on previous drafts. Moreover, I thank an anonymous reviewer for a helpful report on an earlier version of the paper. Finally, I am grateful to the Swiss National Science Foundation for generous support of this work (grant PP00P1_144736/1).

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Baumgartner, M. Parsimony and Causality.
*Qual Quant* **49, **839–856 (2015). https://doi.org/10.1007/s11135-014-0026-7

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### Keywords

- Boolean method
- Set-theoretic method
- Qualitative Comparative Analysis (QCA)
- Coincidence Analysis (CNA)
- INUS causation
- Quine-McCluskey optimization