The Bayesian logic of frequency-based conjunction fallacies - Sydow, Momme von (Journal of mathematical psychology, 2011)
An inductive, pattern-sensitive Bayesian logic (BL) is proposed as a normative and descriptive model for probability judgments about hypotheses involving probabilistic logical connectives. The model explains a specific class of frequency-based conjunction fallacies (CFs). It is suggested that the pattern probabilities calculated by BL may serve as a criterion of noisy-logical predication, resolving some paradoxes of predication. The model is developed for frequency information in 2×2 contingency tables. According to standard probability theory, a violation of the conjunction rule, P(A)≥P(A∧B) (e.g., P(ravens are black) ≥ P(ravens are black AND they can fly)), is always a fallacy. A frequentist interpretation of probability has exculpated participants from committing CFs when one is concerned with single events. Here a pattern-based Bayesian interpretation of probabilities of (noisy) dyadic logical predications is elaborated, predicting frequency-based but rational ‘CFs’. BL formalizes the probabilities of logical patterns, integrating over noise levels. BL, for instance, predicts double CFs, differential sample-size effects, and pattern sensitivity. Three experiments provide a first corroboration that BL is also an adequate empirical model to predict logical probability judgments based on 2×2 contingency tables. BL may shed light on the more general rationality debate.
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