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• W2400412945 abstract "Bayesian Logic and Trial-by-Trial Learning Momme von Sydow (momme.von-sydow@psychologie.uni-heidelberg.de) Klaus Fiedler (klaus.fiedler@psychologie.uni-heidelberg.de) University of Heidelberg, Department of Psychology, Hauptstr. 47-51, D-69117 Heidelberg, Germany Abstract Standard logic and probability theory are both beset with fundamental problems if used as adequacy criteria for relating logical propositions to learning data. We discuss the problems of exception, of sample size, and of inclusion. Bayesian pat- tern logic (‘Bayesian logic’ or BL for short) has been pro- posed as a possible rational resolution of these problems. BL can also be taken as psychological theory suggesting fre- quency-based conjunction fallacies (CFs) and a generalization of CFs to other logical inclusion fallacies. In this paper, this generalization is elaborated using trial-by-trial learning scena- rios without memory load. In each trial participants have to provide a probability judgment. Apart from investigating lo- gical probability judgments in this trial-by-trial context, it is explored whether under no memory load the propositional as- sessment of previous evidence has an influence on further probability judgments. The results generally support BL and cannot easily be explained by other theories of CFs. Keywords: Conjunction fallacy, probability judgments, trial- by-trial learning, Bayesian induction, logical predication. Standard Logic and Probability Theory as Criteria for True Logical Propositions The relationship between general logical propositions (or sentences) and evidence is fundamental to both epistemo- logy and psychology. We here investigate general predica- tion of logical relationships between two dichotomous attri- butes (or predicates), like “ravens are black and they can fly” (with the conjunction ‘and’). What would be an adequate justification for such a type of sentences? Arising from an old tradition going back to Aristotle, modern formal logic uses truth table definitions for all 16 logical connectives. The truth table definition may be used as a deterministic criterion of truth for empirical relation- ships. With regard to a conjunctive predication, like “ravens (R) are black (A) and they can fly (B)” (A ∧ B| R), the whole sentence is true (or, more correctly, ‘not false’) as long as one has observed only exemplars corresponding to true cells of a truth table (for the conjunction this is the ‘a-cell’, ‘A & B’). In contrast, the proposition would be falsified, if one observed a single case defined to be false (here: b-cell: ‘A & ¬B’; c-cell: ‘¬A & B’, or d-cell: ‘¬A & ¬B’). Problem of Exceptions Exceptions may not prove the rule, but in ordinary language exceptions are indeed regularly tolerated. This may reflect the deeper epistemolo- gical point that in the empirical world deterministic relationships are rather the exception than the rule. Actually, in philosophy of science it has been argued that strict falsifi- cationism would absurdly imply that all important theories would be falsified. Even more so in normal language, as evident from our deterministic example, there exist exceptions: white (albino) ravens as well as ravens that cannot fly. If exceptions are the rule for contingent, empirical relationships, it seems reasonable to replace the strict deterministic truth criteria of logic by a high- probability criterion (see Schurz, 2005): P(black & can fly | ravens) > Ψ, with Ψ > .5. However, the following two pro- blems beset a simple extensional probability criterion of truth as well as one based on standard formal logics. Problem of Sample Size If we had to access the truth of “ravens are black and they can fly” without previous know- ledge about ravens, either one confirmatory raven (A & B case) or many cases both equally yielded the same exten- sional probability of 1 (the number of confirmative cases di- vided by all cases). In the latter case, however, a higher sub- jective probability of this sentence seems justified. There- fore, a kind of second order probability, a probability con- cerning probabilities, is needed, as introduced in the model. Problem of Inclusion The extension (all cases falling into a set) of a subset can never be larger than that of a superset. Comparing conjunctions and inclusive disjunc- tions, it follows that P(ravens are black AND they can fly) ≤ P(ravens are black OR they can fly or both) [formally: P(A ∧ B|R) ≤ P(A ∨ B|R)]. If we use extensional probabilities as truth criterion, the second sentence can therefore never be ‘less true’ than the first one. If one assumes at least some exceptions, the latter is even ‘truer’ in principle. Going one step further, the logical tautology, allowing for all values (“Ravens are black or not, and they can fly or not”), is a priori the extensionally most probable sentence [P(A ∨ B|R) ≤ P(A T B|R) or even P(A ∨ B|R) < P(A T B|R)]. Using standard (extensional) probabilities as truth criterion, one would therefore always have to choose tautologies as the most suitable hypothesis, regardless of the evidence and of the properties in question. In conclusion, if a truth criterion should be informative about the observable world, simple extensional probabilities in principle cannot provide a rea- sonable truth criterion. Bayesian Logic Bayesian pattern logic (or ‘Bayesian logic’, BL, for short) formulates a second order probability that given data may have been generated by noisy-logical patterns of proba- bilities. The model provides a technical, rational solution to the three mentioned problems and – in approximation – a potential psychological model of human induction of noisy- logical relationships as well. The model is part of a renaissance of Bayesian approaches in cognitive science (e.g., Chater, Tenenbaum, Yuille, 2006; Oaksford & Chater, 2007, Kruschke, 2008). The following sketch is meant to clarify the main idea of Bayesian logic (for more detail, see von Sydow, 2011)." @default.
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