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A unified account of the conjunction fallacy by coherence

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Abstract

We propose a coherence account of the conjunction fallacy applicable to both of its two paradigms (the M–A paradigm and the A–B paradigm). We compare our account with a recent proposal by Tentori et al. (J Exp Psychol Gen 142(1): 235–255, 2013) that attempts to generalize earlier confirmation accounts. Their model works better than its predecessors in some respects, but it exhibits only a shallow form of generality and is unsatisfactory in other ways as well: it is strained, complex, and untestable as it stands. Our coherence account inherits the strength of the confirmation account, but in addition to being applicable to both paradigms, it is natural, simple, and readily testable. It thus constitutes the next natural step for Bayesian theorizing about the conjunction fallacy.

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Notes

  1. Many influential accounts of the conjunction fallacy in the literature have been Bayesian. The main exception is the representativeness account by Tversky and Kahneman (1983). We agree with its critics (e.g. Gigerenzer 1996) that the informal and fuzzy characterization of representativeness seriously limits its explanatory power. In this paper we only examine the Bayesian accounts of the conjunction fallacy.

  2. See, e.g. Tentori et al. (2013).

  3. The precise condition for \(\hbox {Con}(h_{1}, e)~<~\hbox {Con}(h_{1} \wedge h_{2}, e)\) depends on the way we measure confirmation. We will address this problem in the next section.

  4. We insert the qualification “appreciably” here because the conjunction fallacy is not expected if \(\hbox {Con}(h_{1 }\wedge h_{2}, e)\) is only negligibly greater than \(\hbox {Con}(h_{1}, e)\), so that the difference does not draw the participants’ attention. The same point applies, mutatis mutandis, to the qualification “appreciably” in other accounts of the conjunction fallacy. We assume, in other words, that there are thresholds of appreciable difference in probability, confirmation, coherence, etc. to be determined empirically, though we do not investigate the issue in this paper.

  5. See Bar-Hillel (1980) and Koehler (1996) for influential discussions of the base rate fallacy.

  6. In a recent paper Roche and Shogenji (2014) list thirteen Bayesian measures of confirmation.

  7. Two measures C(h, e) and \(C^*(h, e)\) are ordinally equivalent to each other if and only if for any two ordered pairs \({<}h_{1}, e_{1}{>}\) and \({<}hy_{2}, e_{2}{>}\), \(C(h_{1}, e_{1}) {<}/=/{>} C(h_{2}\), \(e_{2})\) if and only if \(C^*(h_{1}, e_{1}) {<}/=/{>} C^*(h_{2},e_{2})\).

  8. They prove that \(\hbox {Con}(h_{1}, e)<\hbox {Con}(h_{1} \wedge h_{2}, e)\) from the conditions (i) and (ii) on six popular measures of confirmation, where (i) and (ii) are understood formally as \(\hbox {Pr}(h_{1}{\vert }e) \le \hbox {Pr}(h_{1})\) and \(\hbox {Pr}(h_{2}{\vert }e \wedge h_{1})~>~\hbox {Pr}(h_{2}{\vert }h_{1})\), respectively.

  9. In his account of the conjunction fallacy Shogenji (2012) argues for a particular measure J(h, e). He takes J(h, e) to be the measure of “justification”, but it behaves much like a measure of confirmation.

  10. “M”, “B” and “A” in their paper correspond to “e”, “\(h_{1}\)” and “\(h_{2}\)” in this paper, respectively. The M–A paradigm is so called because of the strong correlation between M and A (e and \(h_{2})\); the A–B paradigm is so called because of the strong correlation between A and B (\(h_{1}\) and \(h_{2})\).

  11. Siebel (2002) proposes a coherence account of the Linda Scenario. It should be noted that Siebel’s account is not Bayesian since he adopts the constraint-satisfaction model of coherence (Thagard 1989, 1992, Ch. 4). Shogenji (2012) also mentions a coherence account of the Linda Scenario and his analysis of coherence is Bayesian, but he does not pursue the idea in part because it is not much different from the confirmation account with regard to the M–A paradigm, which is his focus. Schippers (2016) proposes an account of the Linda case in terms of “contrastive coherence”. The concept of contrastive coherence is of limited use in analyzing the conjunction fallacy since it only allows for qualitative evaluation—whether x coheres better with y than with z. We need quantitative evaluation for predicting relative frequencies of the conjunction fallacy. The concept of contrastive coherence is also limited to pairwise coherence, while a unified account of the conjunction fallacy we develop in this section applies a measure of coherence to tripletons.

  12. Assuming that \(C(x, y)=0\) for the neutral point at which there is neither confirmation nor disconfirmation. Replace 0 with k if the neutral point is set at \(k \ne \) 0.

  13. This is easy to prove by Bayes’ Theorem.

  14. It is understood here that incoherence is a probabilistic generalization of inconsistency, just as disconfirmation is a probabilistic generalization of refutation. Statements that are incoherent can still be true at the same time, though incoherence makes it less likely, just as a statement that is disconfirmed by evidence can still be true though disconfirmation makes it less likely. Some readers familiar with the literature may worry here about “the impossibility results” (Bovens and Hartmann 2003; Olsson 2005) that there is no probabilistic measure of coherence that is truth conducive, i.e. there is no probabilistic measure of coherence such that the more coherent the set is, the more probable the conjunction of its members is, ceteris paribus. However, this problem arises when the reliability of the evidence producer is a factor, which is not the case in the conjunction fallacy. It is easy to show (e.g. Shogenji 1999) that coherence is truth conducive, ceteris paribus, with regard to the conjunction of the members when the reliability of the evidence producer is not a factor.

  15. The precise condition for \(\hbox {Coh}(h_{1}, e)~<~\hbox {Coh}(h_{1} \wedge h_{2}, e)\) depends on the way we measure coherence. We will discuss measures of coherence in Sect. 5 and demonstrate the precise behavior of our preferred measure in Sect. 6.

  16. As we will discuss in the next section, simpler probabilistic models also have an advantage in their predictive accuracy.

  17. One reader has questioned this judgment for the reason that being a feminist may explain Linda’s choice (surprising given e) to work as a bank teller. There may be some cultural and generational differences about the idea of being a feminist. If someone agrees with this reader (we don’t), then consider some other case of the M–A paradigm. Our general point is that \(h_{1}\) in the M–A paradigm is at odds with e while e and \(h_{2}\) are positively correlated, so that \(h_{1}\) is usually at odds with \(h_{2}\). As a result, \(h_{1}\) in the M–A paradigm usually lowers the probability of \(h_{2}\) against the background of e.

  18. There is a strong emphasis on condition (1) in the literature often at the expense of conditions (3) and (4). See, for example, Koscholke (2016) on how different Bayesian measures of coherence perform in a variety of “test cases”. We are skeptical that a single measure of coherence can accommodate all our intuitive judgments about coherence in different contexts. There may be such a measure, but what we seek here is a measure that is not just similar to the explicandum but illuminating, especially in analyzing the conjunction fallacy. If some people find our measure of coherence to be insufficiently similar to the prescientific concept of coherence, we have no objection to changing the term. Lewis (1946) called a set of mutually supporting beliefs “congruent”, and Chisholm (1966) called a set of mutually confirming propositions “concurrent”.

  19. Other complex measures of coherence with many components to weigh and tally include those proposed by Fitelson (2003), Meijs (2006), Roche (2013) and Schupbach (2011).

  20. This is commonly known as the problem of overfitting. See Burnham and Anderson (2002) on how the complexity of a probabilistic model affects the accuracy of predictions. Forster and Sober (1994) and Sober 2015, Ch. 2) provide an accessible account of the problem.

  21. See Koscholke and Schippers (2016) for this line of objection to “relative overlap measures” of coherence. We can think of even simpler (non-relative) overlap measures, such as \(X(x_{1}, \ldots , x_{n})=\hbox {Pr}(x_{1} \wedge \cdots \wedge x_{n})\) and \(Y(x_{1}, \ldots , x_{n})=1/\hbox {Pr}(x_{1} \wedge \cdots \wedge x_{n})\), but they are not appropriately sensitive to changes in the degree of coherence due to the addition of a member. For example, \(X(e, h_{1}, h_{2})\) is never greater than \(X(e, h_{1})\), while \(Y(e, h_{1}, h_{2})\) is never smaller than \(Y(e, h_{1})\).

  22. Schupbach (2011) also cites “the depth problem” as reason for modifying Shogenji’s measure, viz. \(S(x_{1}, x_{2}, x_{3})\) ignores the degrees of coherence among members of the subsets. For example, \(S(x_{1}, x_{2}, x_{3})\) is solely dependent on \(\hbox {Pr}(x_{1})\), \(\hbox {Pr}(x_{2})\), \(\hbox {Pr}(x_{3})\) and \(\hbox {Pr}(x_{1} \wedge x_{2} \wedge x_{3})\), and is insensitive to \(S(x_{1}, x_{2})\), \(S(x_{2}, x_{3})\) and \(S(x_{3}, x_{1})\). (See also Fitelson (2003) for this line of objection to Shogenji’s measure.) We set aside the depth problem here since there is no indication in cases of the conjunction fallacy that the degrees of coherence among members of the subsets are additional factors that influence the occurrence of the conjunction fallacy.

  23. If we want to make the neutral point zero instead of one, we can normalize \(S^{*}( x_{1}, {\ldots }, x_{n})\) by logarithm to obtain \(\log S(x_1 ,\ldots ,x_n )^{\frac{1}{n-1}}=\frac{1}{n-1}\log S(x_1 ,\ldots ,x_n ).\)

  24. See, for instance, the Inverse Conjunction Fallacy due to Jönsson and Hampton (2006). See Jönsson and Assarsson (2016) for some problems that this fallacy gives rise to for confirmation theoretic accounts of the conjunction fallacy.

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Acknowledgements

We would like to thank Stefan Schubert for feedback on an early version of this manuscript and anonymous referees of this journal for detailed comments and helpful suggestions.

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Authors and Affiliations

  1. Department of Philosophy, Lund University, Lund, Sweden

    Martin L. Jönsson

  2. Department of Philosophy, Rhode Island College, Providence, RI, USA

    Tomoji Shogenji

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  1. Martin L. Jönsson
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  2. Tomoji Shogenji
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Correspondence to Tomoji Shogenji.

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Jönsson, M.L., Shogenji, T. A unified account of the conjunction fallacy by coherence. Synthese 196, 221–237 (2019). https://doi.org/10.1007/s11229-017-1467-z

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