Bayes Logic and Bayes Nets - Causal vs. Non-causal Induction and Inference with Logical Patterns of Correlations
Final Report Abstract
This DFG project has investigated issues at the intersection of logical judgments and probability judgments, which is central to the philosophical and psychological rationality debates. It is argued that the ‘narrow norm’ (Gigerenzer) of extensional probabilities (relative frequencies) cannot provide an adequacy criterion for logical predication. For example, since P(ravens are black AND they can fly) > P(ravens are black OR they can fly or both) is fallacious (A ∧ B is a subset of A v B), one would never be entitled to prefer a conjunction over an inclusive disjunction. In principle, a more specific predication could never be preferred over a more general one, based on standard extensional probability. In the project that was part of the SPP 1516 “New Models of Rationality,” we elaborated Bayesian models of predication and probability judgment that could address this fundamental problem. The advocated Bayesian logics (BL), was tested and corroborated during the funding-period with regard to several novel predictions. BL provides an intensional probability measure for describing the similarity of empirically found patterns to explanatory logical patterns. The pattern probabilities of BL supplement the more direct (extensional) application of standard probabilities. However, BL predicts a system of rational inclusion ‘fallacies.’ It does not deal with the question of how probable an element falls into a set (such as extensional probabilities), but rather how well a pattern of observed probabilities could have been produced by an ideal logical pattern (the explanans). I now outline the major topics and results of the project’s funding-period: We discussed central problems of the broader rationality debate. First, I addressed problems of a Darwinian metaphysics. This topic is related to my other work, since Bayesian accounts provide obstacles for such a metaphysics. Second, we discussed the enormous impact as well as crucial limitations of Tversky & Kahneman’s bias-and-heuristics research program. - In a longer paper I demonstrated that BL predicts a system of inclusion fallacies and that the results cannot be explained by prominent accounts of conjunction fallacies. - Furthermore, the idea of a generalised system of inclusion fallacies for (dyadic) logical predications was applied to sequential learning tasks. - Moreover, it was shown that a pattern account is applicable to specific (“this raven is…”) as well as to general predication (“ravens are…”). Additionally, we developed three more specific models of rational predication to account for nonstandard representations. BL relies on given representations, on the hypothesis space, and on partitioning. Standard BL only applies to standard dyadic and dichotomous interpretation of connectives. Although acknowledging deviations from this standard assumption at some point slowed down publications, it broadened the scope of BL and led to the exploration of interesting new phenomena: First, whereas standard BL would interpret conditionals as normal dyadic connectives (material implications), Bayesian Logic of Conditionals [BLC] builds on three alternative approaches and develops an intensional model of conditional based on these ideas that additionally accounts for the problem of inclusion. - Second, we addressed plausible non-standard meanings of connectives arising from probabilistic representations. It is known that the normal language-term ‘AND,’ for example, may not refer to the logical conjunction but rather to a disjunction. Misunderstanding AND may thus be a cause of CFs. The main point here, however, is that a probabilistic interpretation requires differentiating between a logical conjunction as intersection and a conjunction of monadic statements A and B, each based on marginal probabilities only. - A third model development accounts for the difference between a dichotomic and a polytomic dimension. The idea arose from empirical deviations, but the model actually follows from the same idea as standard (dyadic) BL when applied to a different representation. First results necessitate a pattern approach and, for instance, rule out a confirmation approach. Additionally, I have worked on causal Bayes nets that have gained increasing acceptance in recent decades but that do not represent logical proposition (i.e., noisy-or and noisy-and-not gates do not link to logical prediction). We started exploring these topics and published some first results on the violations of the Markov condition. Overall, this research project has corroborated and elaborated a rational intensional approach of predication and probability judgments that transcends a direct application of the narrow norm of extensional probabilities. We conducted several successful tests of BL involving new predictions and extensions to various representations. BL provides a unified normative and psychological account of noisy-logical predication and of an important but understudied class of probability judgments. BL seems to provide – at least approximately – a rational explanation for a corresponding central class of inclusion ‘fallacies’ and thus contributes substantially to central issues of the rationality debate.
Publications
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(2012). From Darwinian Metaphysics towards Understanding the Evolution of Evolutionary Mechanisms (revised ed., 481 pp). Göttingen: Universitätsverlag Göttingen. ISBN: 978-3-86395-006-4
von Sydow, Momme
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(2013). System implemented by a processor controlled machine for inductive determination of pattern probabilities of logical connectors. Patent No.: US 8,463,734 B2
von Sydow, M.
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(2014). Bayesian Mental Models of Conditionals. Cognitive Processing, 15 (Special Issue: Proceedings of the 12th Biannual Conference of the German Cognitive Science Society), 148-151
von Sydow, Momme
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(2015). Heuristics and Biases: Beyond Tversky and Kahneman's (1974) Judgment under Uncertainty. In Michael W. Eysenck & David Groome. Cognitive Psychology: Revisiting the Classical Studies, 146-161. Los Angeles and London: Sage. ISBN (Hardback): 978-1-4462-9447-5
Fiedler, Klaus & Momme von Sydow
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(2015). Pattern Probabilities for Non-Dichotomous Events: A New Rational Contribution to the Conjunction Fallacy Debate. In D. Noelle et al. (eds.), Proceedings of the Thirty-Seventh Annual Conference of the Cognitive Science Society, 2511-2516. Austin, TX: Cognitive Science Society. ISBN: 978-0-9911967-2-2
von Sydow, Momme
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(2016). Towards a Pattern-Based Logic of Probability Judgments and Logical Inclusion “Fallacies.” Thinking & Reasoning
von Sydow, Momme
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(2016). Transitive Reasoning Distorts Induction in Causal Chains. Memory & Cognition
von Sydow, Momme, Hagmayer, York, & Meder, Björn
