The Bayesian Logic of the Conjunction Fallacy: Effects of Probabilities and Frequencies in Contingency Tables

In this paper a Bayesian logic of the conjunction fallacy (CF) is advocated as a normative and descriptive proposal for testing hypotheses about dyadic logical connectors. According to traditional extensional probability a violation of additivity and, in particular, a violation of P(A) ≥ P(A ∧ B) or of P(Linda is a bank teller) ≥ P(Linda is a bank teller AND an active feminist) is a fallacy. The psychological literature has adopted this interpretation. In contrast, the proposed Bayesian model formulates qualitative as well as quantitative conditions under which such a judgment is reasonable. Qualitatively, the model is applicable to situations in which probabilities have to be taken (or are taken) not directly as extensional probabilities, but as (posterior) probabilities for alternative logical hypotheses about whole situations. If the preconditions of the model are given, Bayesian logic should be applicable to novel situations, like highly transparent tasks, even if frequency information is provided in a fully specified contingency table. Quantitatively, the model makes predictions about the effects of extensional probability patterns and resulting Bayesian probabilities and corresponding ‘CFs’. Additionally, while keeping the probability patterns constant, the model predicts effects of different underlying frequencies (sample sizes). In the experiment two quantitative factors were varied using highly transparent tasks with explicit frequency information in a contingency table. Despite using these strict conditions, the results supported the predicted occurrence of ‘double CFs’ and the differential effects of sample size.