Bayesian reasoning with ifs and ands and ors

Abstract

The Bayesian approach to the psychology of reasoning generalizes binary logic, extending the binary concept of consistency to that of coherence, and allowing the study of deductive reasoning from uncertain premises. Studies in judgment and decision making have found that people’s probability judgments can fail to be coherent. We investigated people’s coherence further for judgments about conjunctions, disjunctions and conditionals, and asked whether their coherence would increase when they were given the explicit task of drawing inferences. Participants gave confidence judgments about a list of separate statements (the statements group) or the statements grouped as explicit inferences (the inferences group). Their responses were generally coherent at above chance levels for all the inferences investigated, regardless of the presence of an explicit inference task. An exception was that they were incoherent in the context known to cause the conjunction fallacy, and remained so even when they were given an explicit inference. The participants were coherent under the assumption that they interpreted the natural language conditional as it is represented in Bayesian accounts of conditional reasoning, but they were incoherent under the assumption that they interpreted the natural language conditional as the material conditional of elementary binary logic. Our results provide further support for the descriptive adequacy of Bayesian reasoning principles in the study of deduction under uncertainty.

Probabilistic inferences from conjoined to iterated conditionals

“…There is one-premise centering: inferring if A then B from the single premise AB. And two-premise centering: inferring if A then B from the two separate premises A and B. Centering is valid for quite a wide range of conditionals ( [19,20,66]). It is clearly valid for the material conditional, since not-A or B must be true when A and B is true.…”

Facts and Possibilities: A Model‐Based Theory of Sentential Reasoning

“…Conditionals of the sort, if A then B, for instance, do not correspond to the so-called material conditional of logic or to conditional probabilities (see also Byrne & Johnson-Laird, 2009;Schroyens, 2010). Probabilistic approaches to reasoning postulate that the probability calculus should replace logic, especially for conditionals, and that what matters is probabilistic validity (Adams, 1998)-p-validity, for short-which holds whenever a conclusion is no less probable than its premises given any consistent assignment of probabilities (e.g., Chater & Oaksford, 2009;Cruz, Baratgin, Oaksford, & Over, 2015;Evans, 2012;Oaksford & Chater, 2007;Over, 2009;Pfeifer, 2013;Pfeifer & Kleiter, 2009). A consequence of the model theory, however, is that probabilities are not part of the fundamental meanings of compound assertions.…”

Probabilities, causation, and logic programming in conditional reasoning: reply to Stenning and Van Lambalgen (2016)

“…Moreover, these coherent probability intervals for the conclusion are a deductive consequence of the premises and their probabilistic interpretation. It has been shown that peoples' assignments of conclusion probabilities respect probabilistic coherence (Cruz et al, 2015). This is not the same as…”

“…Yet there is good evidence that people are a sensitive to this "blurred" notion (Cruz, Baratgin, Oaksford, & Over, 2015). According to a probabilistic account based on what Edgington (1995) called the Equation, that is, Pr(if p then q) = Pr(q|p), the probabilities of the premises of an inference place coherence constraints on the range of values that the probability of the conclusion can take.…”