Abstract
This paper raises a principled objection against the idea that Bayesian confirmation theory can be used to explain the conjunction fallacy. The paper demonstrates that confirmation-based explanations are limited in scope and can only be applied to cases of the fallacy of a certain restricted kind. In particular; confirmation-based explanations cannot account for the inverse conjunction fallacy, a more recently discovered form of the conjunction fallacy. Once the problem has been set out, the paper explores four different ways for the confirmation theorist to come to terms with the problem, and argues that none of them are successful.
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Notes
This idea seems to have originated with Keynes (1921).
All these measures except for \(J\), \(N\) and \(K\) are taken from a list in Crupi et al. (2008, p. 184, table 1.)
Another descriptive use of these measures is due to Tentori et al. (2007) who tests which measure best captures our actual judgments of confirmation. This use of the confirmation measures is distinct from the use mentioned in the main text and we will not discuss it in this paper.
If we stipulate that \(l_3\) = ‘Linda is active in the feminist movement’ (and that \(l_{2}\) is thus equivalent to \(l_{1} \wedge l_3\)), then these assumptions amount to (1) \(\mathop {\text {P}}\nolimits (l_3 \mid d \wedge l_{1}) > \mathop {\text {P}}\nolimits (l_3 \mid l_{1}\)) and, (2) \(\mathop {\text {P}}\nolimits (l_{1} \mid d) < \mathop {\text {P}}\nolimits (l_{1})\).
There are some differences between confirmation theoretic accounts regarding the exact details of how to explain the conjunction fallacies. According to the most worked out account due to Tentori et al. (2013), the probability of a conjunction fallacy is a decreasing function of Con(\(h_1, d\)) and an increasing function of Con(\(h_2, d|h_1\)) and Con(\(h_2, h_1| d\)).
This is controversial, but at the very least, if the confirmation theoretic account is correct there might not be any probabilistic fallacy involved in the conjunction fallacy. It might still be considered a mistake of some kind to interpret ‘\(x\) is more likely than \(y\) given \(d\)’ in terms of ‘\(d\) better confirms \(x\) than it does \(y\)’ though.
In their discussion of the conjunction fallacy Tentori et al. (2013) claim that their account actually has a certain degree of generality since it can be applied to both the ‘M–A paradigm’—cases like the Linda-case, and to the ‘A–B paradigm’ where it is not the case that there is an initial description that confirms the added conjunct, but rather a degree of confirmation between the two conjuncts. However, it seems to us that this form of generality is only apparent. On Tentori et al’s account the probability of a conjunction fallacy depends on three factors Con(\(h_1, d\)), Con(\(h_2, d|h_1\)) and Con(\(h_2, h_1| d\). The third is used to explain the M–A cases and the second is used to explain the A–B cases, no single factor is used to explain both.
An ‘interpretation’ is just an assignment of entities to the variables that occur in the definition of the confirmation measures. In three of the four interpretations, it is assumed that there are two different hypotheses which are identified with \(s_{1}\) and \(s_{2}\) respectively. All four interpretations assign different entities to \(e\).
This interpretation was suggested by Tomoji Shogenji at one point in personal correspondence about his particular measure. Since it is applicable to all measures it is reused here.
If the probability that an arbitrary member of a certain finite category \(c\) has a certain property \(p\) is \(x\), then the probability that all members of \(c\) has \(p\) is \(x^{|c|}\) (given that the probabilities are independent of each other).
This account was proposed by Shogenji (in personal communication) in response to the criticism offered above.
See footnote 6.
This is what Shogenji (2012, p. 35) calls equi-maximality. According to him a measure is equi-maximal if its value is maximal when \(\mathop {\text {P}}\nolimits (h \mid e) = 1\) regardless of the value of \(\mathop {\text {P}}\nolimits (h)\).
Since \(s_{1}\) entails \(s_{2}\) we know that \(\mathop {\text {P}}\nolimits (s_{1}) \leqslant \mathop {\text {P}}\nolimits (s_{2})\). By Bayes theorem, we know that if \(\mathop {\text {P}}\nolimits (s_{1}) = \mathop {\text {P}}\nolimits (s_{2})\), then \(\mathop {\text {P}}\nolimits (s_{1} \mid s_{2}) = \mathop {\text {P}}\nolimits (s_{2} \mid s_{1})\), and so, since we know that \(\mathop {\text {P}}\nolimits (s_{1} \mid s_{2}) \ne \mathop {\text {P}}\nolimits (s_{2} \mid s_{1})\) we know by modus tollens that \(\mathop {\text {P}}\nolimits (s_{1}) \ne \mathop {\text {P}}\nolimits (s_{2})\).
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Acknowledgments
While working on the paper we benefitted from discussing it with James Hampton, Erik Olsson, Stefan Schubert, Aron Vallinder, and in particular Tomoji Shogenji. Sadly, while the paper was still under review, Elias Assarsson unexpectedly passed away. The paper is dedicated to his loving memory.
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Elias Assarsson: Deceased.
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Jönsson, M.L., Assarsson, E. A problem for confirmation theoretic accounts of the conjunction fallacy. Philos Stud 173, 437–449 (2016). https://doi.org/10.1007/s11098-015-0500-7
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DOI: https://doi.org/10.1007/s11098-015-0500-7