Towards a pattern-based logic of probability judgements and logical inclusion “fallacies”

ABSTRACT

Probability judgements entail a conjunction fallacy (CF) if a conjunction is estimated to be more probable than one of its conjuncts. In the context of predication of alternative logical hypothesis, Bayesian logic (BL) provides a formalisation of pattern probabilities that renders a class of pattern-based CFs rational. BL predicts a complete system of other logical inclusion fallacies (IFs). A first test of this prediction is investigated here, using transparent tasks with clear set inclusions, varying in observed frequencies only. Experiment 1 uses data where BL makes dominant predictions; Experiment 2's predictions were less clear, and we additionally investigated judgements about the second-most probable hypotheses. The results corroborated a pattern-probability account and cannot be easily explained by other theories of CFs (e.g. inverse probability, confirmation). IFs were not limited to conjunctions, but rather occurred systematically for several logical connectives. Thus, pattern-based probability judgements about logical relations may constitute a basic class of intuitive but potentially rational probability judgements.

KEYWORDS:

• Probability judgements
• conjunction fallacy
• inclusion fallacies
• Bayesian models
• rationality

Acknowledgments

Portions of the work were presented at the CogSci (Amsterdam, 2009). Further developments were presented at subsequent CogSci conferences (Sapporo, 2012; Berlin, 2013; Quebec, 2014; Passadena, 2015). I would like to thank David Over, Klaus Oberauer, Valerie Thompson, and anonymous reviewers for valuable comments on earlier versions of the manuscript. I thank Anna Gast, Christina Botros, Johanna Frisch and Urszula Mihułowicz for conducting the studies and Martha Cunningham, Larry Fiddick, Johanna Frisch, York Hagmayer, Björn Meder, Ralf Mayrhofer, and Jonathan Nelson for helpful comments in an early phase of the manuscript.

Disclosure statement

No conflict of interest was reported by the author. However, the author mentioned a related patent in the submission process.

Notes

¹ The inclusion rule likewise holds for most non-standard accounts of probability (cf. Hájek, 2001; Fitelson, 2006; Foley, 2009; Kern-Isberner, 2001; Schurz, 2005; cf. von Sydow, 2011).

² Cf. the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Although they do not provide probabilities, they deal with the trade-off between goodness of fit and complexity of models (here, number of free parameters) (Burnham & Anderson, 2004).

³ The belief update for the subjective cell probabilities uses a Dirichlet distribution for independent events, corresponding to the multinomial distribution in Step 3 of the belief-update of ideal pattern-hypotheses. This flexible distribution, however, does not resolve the problem of inclusion, but only of the use of pattern-hypotheses.

⁴ Formally, the ideal cell probability of a logically false cell is .25*r (with a maximum of .25) and that of a true cell is t – r (t – .25), with t as defined above.

⁵ Von Sydow (2006) proposed generalising the WST model, traditionally limited to conditionals, to other connectives, but that account differs from the present one as well.

⁶ Apart from more specific conditions, they still advocate a confirmation approach, predicting CFs if and only if the confirmation c(B & F|e) is larger than c(B |e) (or than c(F|e). It is not clear to me whether a newly proposed condition, c(B, F|e) > 0, actually involves a deviation from this basic position (Tentori et al., 2013, p. 249).

⁷ A further explanatory factor for the findings of Tentori et al. (2013, Experiment 2) may be relevant. After a Linda story, confirmation predicts P(bankteller ∧ feminist) > P(bank teller ∧ owns dark shoes), since the Linda story confirms the former and not the latter conjunction. However, one may have to differentiate relevance from confirmation: Imagine keeping the Linda story without changing the confirmation relations but changing the relevance of “black shoes”, e.g. by inviting Linda to a party where one must wear dark shoes. In such cases, confirmation may invariably continue to predict no selection of “feminist & dark shoes” over “feminist & bank teller” and even over “feminist”; but, I would expect such selections would become more common.

⁸ Polytomous representations may play a role for all sorts of interpretations of ordinary language ‘AND,’ whether it is a dyadic, a marginal or a sum interpretation (cf. von Sydow, 2014, 2015).

Additional information

Funding

The study was funded by the Deutsche Forschungsgemeinschaft (DFG, Sy 111/2-1; part of the priority program “New Models of Rationality” [SPP 1516]).