Is There a Monadic as well as a Dyadic Bayesian Logic? Two Logics Explaining Conjunction 'Fallacies'

Is there a Monadic as well as a Dyadic Bayesian Logic? Two Logics Explaining Conjunction ‘Fallacies’ Momme von Sydow (momme.von-sydow@psychologie.uni-heidelberg.de University of Heidelberg, Department of Psychology, Hauptstr. 47-51, D-69117 Heidelberg, Germany Abstract Formal logic and probability theory are often considered the most fundamental norms of rational thought, but their application to psychological tasks has raised serious doubts about human rationality. A central finding is that people sometimes judge the probability of a conjunction to be higher than that of its conjuncts (conjunction fallacies, CFs). Bayes- ian logic (BL, von Sydow, 2011) formalizes subjective proba- bilities of noisy-logical explanatory patterns (pattern pro- babilities) instead of extensional probabilities (relative fre- quencies), and predicts a system of rational inclusion fal- lacies. This paper distinguishes a monadic from a dyadic pat- tern explanation of CFs; it tests two corresponding formaliza- tions of BL (the former concerned with cells, the latter with marginals); and it models pattern probabilities in a novel way (based on acceptance thresholds). In an experiment we varied observed frequencies and formulations. The results deviate ra- dically from narrow norms but they corroborate the idea of monadic and dyadic pattern probabilities. Keywords: Probability judgment; bias; conjunction fallacy; inclusion fallacy; inductive Bayesian logics; predication. Standard propositional logic and standard probability theory are often considered to be the most fundamental norms of rational thought. Formalizing what is rational has been the subject of study over many centuries. Over 2000 years ago, Aristotle founded formal logic (Analytica Priora and Posteriora) and basic ideas of probabilistic reasoning (To- pica). Since logical positivism, the basic calculi of formal logics (Frege, Wittgenstein, Russell, Whitehead) and proba- bility theory (Kolmogorov) have been rigidly formalized in their present form. Yet the standards of rational thought are still being elaborated. Formal epistemology, logics, and mathematics have continued to develop non-standard systems of logic and probability, as either alternatives or refinements to these calculi (e.g., many-valued logics, non- monotonic logics, default logics, and belief functions). Psychology, however, has often been focused solely on these two basic calculi. This may have contributed to the view that human judgment is essentially biased. In the well known Linda task, people make probability judgments about a person X who is described to be a feminist, often P(X is bank teller & feminist) > P(X is feminist). This has been called a ‘conjunction fallacy,’ as the probability of a smaller set (the conjunction) cannot exceed that of the larger (one of the conjuncts, Kahneman & Tversky, 1982, cf. 1996). In my view, Gigerenzer (1996) correctly criticized Kahneman and Tversky’s bias-and-heuristic approach, not only because the suggested heuristics were underspecified but also because content-blind application of the “narrow norms” of logics and probability were misguided in the first place. However, Kahneman and Tversky’s (1996) warnings against norma- tive agnosticism seem not without foundation as well. It is proposed here that in different situations either a monadic or a dyadic Bayesian inductive logic can explain a class of conjunction fallacies (CFs). It is generally suggested, independent of details of the proposed Bayesian models, that there may be monadic CFs based on a probabilistic ‘monadic’ interpretation of two conjuncts. That is, a statement like “X’s are young and Xs are female” is presumably judged as true if X’s are typically young and X’s are typically female, even if individual X’s are usually not both young and female (monadic interpretation involving a conjunction of two separate monadic propositions). In contrast, “Xs are young women”, or “Xs are people who are young and female” seems to refer to the traditional interpretation of a conjunction as intersection—that is, Xs are individually usually both young and female (dyadic interpretation). Although deterministically a dyadic interpretation concerned with two marginal probabilities, and a dyadic interpretation concerned with logical joint probability distribution (logical patterns) coincide, they can differ in a probabilistic context. Generally, we pursue a third way between a simple application of narrow norms and normative agnosticism by advocating a domain-specific rational approach linked to Anderson’s rational analysis and inspired by the Bayesian renaissance in higher cognition (M. Oaksford, N. Chater, J. Tenenbaum) (cf. von Sydow, 2011). Truth Table Logics and Probability Theory – Two Narrow Norms for Predication? The Narrow Norm of Formal Logic If concerned with the truth of sentences involving logical connectives between predicates, such as AND, OR, EITHER OR or NEITHER NOR, standard logical truth-table definitions provide a truth criterion applicable even to contingent sentences: “people from the Linda school become bank tellers AND feminists” or “ravens are black AND they can fly. ” Yet interpreting such sentences as ∀(x) B(x) ∧ F(x) presents two problems. First, the problem of exceptions – predication often involves exceptions (e.g., white albino ravens). Almost none of the assertive predications uttered would be justified, based on a ‘deterministic’ logical truth criterion. A significant sugges- tion to solve this is instead to use a high probability criterion (cf. Schurz, 2005), where the probability of an empirically justified proposition surpasses a given threshold ϕ. Second, the problem of interpretation is that normal language con- nectives (e.g., ‘AND’) do not always correspond to analo- gous ideal language connectives (e.g., the logical conjunc- tion). Some conjunction fallacies may indeed be attributed to misinterpreted connectives (cf. Hilton, 1995, Hertwig,