On a General Bayesian Pattern Logic of Frequency-Based Logical Inclusion Fallacies

On a General Bayesian Pattern Logic of Frequency-Based Logical Inclusion Fallacies Momme von Sydow (momme.von-sydow@bio.uni-goettingen.de) Department of Psychology, Georg-August-Universitat Gottingen, Gosslerstr. 14, D-37073 Gottingen, Germany Abstract Bayesian logic provides a rational model of probability judgments deviating from the standard extensional norm of extensional probability. It formalizes the general idea of an inductive pattern logic that may resolve paradoxes of inclusion. Bayesian logic predicts that it should be possible to generalize the phenomenon of frequency-based logical con- junction fallacies to a system of logical inclusion fallacies. In Experiment 1 quantitative conditions for conjunction fallacies and the role of negations are investigated. Experiment 2 pro- vides a first test of the postulated more general system of lo- gical inclusion fallacies. The results of both experiments confirmed the proposed pattern logic and its formalization as Bayesian logic. Other theories of the conjunction fallacy cannot readily explain this class of frequency-based and pattern-based inclusion fallacies. Whether there are simpler heuristics that may perhaps explain these data as well should be investigated in the future. Keywords: Probability judgments; conjunction fallacy; logic; predication; paradoxes of inclusion. Predication: Paradoxes of Logic and Probability Theory Human predication about a type of entities (the subject class X) often deals with a logical relation (here using the general symbol: l ) between two predicates (predicate A and predicate B). Take the ubiquity of the predication of the relation ‘A and B’, understood as logical conjunction (A l B = A ∧ B): “Ravens are black and they can fly. ” “Professional basketball players are quick and tall. ” Formulated generally: “X are A and B”. Traditionally, a necessary condition for a valid logical predication is that the predicate is true, in the sense that all observed empirical cases confirm the classes of a con- nective’s truth table that are true (Frege, Russell, Whitehead, Wittgenstein, Popper). Using predicate logic, one would treat a predication as involving an implicit uni- versal quantifier, “all X are A and B”. (An alternative is to assume an implicit existential quantifier, cf. Brentano’s works.) Such an interpretation of predication entails two pro- blems: (1) The problem of exceptions: There are exceptions to most general predications that are concerned with contingent empirical matters (cf. the problem of mono- tonicity). Actually, there are for example white ravens (whole species of white ravens and cases of albinism) and there are ravens that cannot fly. (2) The problem of inclusions: The predication “ravens are black and they can fly” paradoxically entails that one may equally use the predicate of the inclusive disjunction “ravens are black OR they can fly or both” (∀x (B(x) ∧ F(x)) => (B(x) ∨ F(x))). Although this is, of course, true according to (extensional) logic, it is a psychologically uninformative conclusion, given that we are interested in an overall characterization of a situation. For any situation in which one would use an AND-predicate, one could equally use an OR-predicate. This problem may also lie at the heart of the paradoxes of implication in the narrow sense (cf. Evans & Over, 2004). From “ravens are black and they can fly” it follows that “If ravens are black then they can fly” (B ∧ F) => (B → F). There is a bunch of these paradoxes, discussed in the literature, e.g.: ¬B => (B → F); F => (B → F); (¬B ∧ F) => (B → F); (¬B ∧ ¬F) => (B → F). Still, here we will not directly concentrate on conditionals, because there may be specific problems of conditionals, due to a subjunctive inter- pretation, an incomplete representation or a reference to causal relations. However, we saw that analogous paradoxes arise for other connectives as well. I generally call them paradoxes of inclusion. Even if one takes a different view on the paradoxes of implication, this does not resolve the paradox that ∀x B(x) ∧ F(x) implies a justification for the predication ∀R B(x) ∨ F(x). The subset relation between an AND-set and an OR-set is generally undisputed. We here investigate this more general subset relation without investigating questions of implication directly. An apparent solution to the first problem, the problem of exceptions, is the use of high probabilities as a necessary condition for prediction (apart from existential prediction). The sentence “Ravens are black and they can fly” may be interpreted as a claim that the probability P(B & F|R) is high (cf. Adams, 1986). This would offer a rational basis for predication, but would not resolve the paradox of inclusion. Conjunction Fallacy Debate: Problems of Extensional Probability Judgments A related debate – but normally discussed separately – is concerned with the assignment of probabilities of conjunc- tive events. Here subjects’ intuition seems to deviate from the perhaps most simple norm of (extensional) probability theory (Kolmogorov, 1933): In the classical Linda task, first proposed by Tversky and Kahneman (1982, 1983; cf. Kahneman & Frederick, 2002) participants were presented with a story: “Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. ” As dependent variable subjects had to rank statements to their probability, including: “Linda is a bank teller” (B) and “Linda is a bank teller and she’s active in the feminist movement” (B and F). Most participants estimate P(B and F) > P(B). This is known as conjunction fallacy (CF). According to extensional probability theory, the con- junction ‘B ∧ F’ can never be more probable than one of its components, ‘B’ or ‘F’ respectively, since any second pro- perty delimits the set size of the first one. Likewise also non-standard accounts of probability, like the Dempster- Shafer belief functions, Cohen’s Baconian probabilities, or