The pseudodiagnosticity trap: Should participants consider alternative hypotheses?

The pseudodiagnosticity task has been used as an example of the tendency on the part of participants to incorrectly assess Bayesian constraints in assessing data, and as a failure to consider alternative hypotheses in a probabilistic inference task. In the task, participants are given one value, the anchor value, corresponding to P(D1|H) and may choose one other value, either P(D1|¬!H), P(D2|H), or P(D2|not;!H). Most participants select P(D2|H), or P(D2|¬!H) which have been considered inappropriate (and called pseudodiagnostic) because only P(D1|¬!H) allows use of Bayes' theorem. We present a new analysis based on probability intervals and show that selection of either P(D2|H), or P(D2|¬!H) is in fact pseudodiagnostic, whereas choice of P(D1|¬!H) is diagnostic. Our analysis shows that choice of the pseudodiagnostic values actually increases uncertainty regarding the posterior probability of H, supporting the original interpretation of the experimental findings on the pseudodiagnosticity task. The argument illuminates the general proposition that evolutionarily adaptive heuristics for Bayesian inference can be misled in some task situations.     

Semantic Coherence and Inconsistency in Estimating Conditional Probabilities

Two studies examined semantic coherence and internal inconsistency fallacies in conditional probability estimation. Problems reflected five distinct relationships between two sets: identical sets, mutually exclusive sets, subsets, overlapping sets, and independent sets (a special case of overlapping sets). Participants estimated P(A), P(B), P(A|B), and P(B|A). Inconsistency occurs when this constellation of estimates does not conform to Bayes' theorem. Semantic coherence occurs when this constellation of estimates is consistent with the depicted relationship among sets. Fuzzy‐trace theory predicts that people have difficulty with overlapping sets and subsets because they require class‐inclusion reasoning. On these problems, people are vulnerable to denominator neglect, the tendency to ignore relevant denominators, making the gist more difficult to discern. Independent sets are simplified by the gist understanding that P(A) provides no information about P(B), and thus, P(A|B) = P(A). The gist for identical sets is that P(A|B) = 1.0, and the gist of mutually exclusive sets is that P(A|B) = 0. In Study 1, identical, mutually exclusive, and independent sets yielded superior performance (in internal inconsistency and semantic coherence) than subsets and overlapping sets. For subsets and overlapping sets, interventions clarifying appropriate denominators generally improved semantic coherence and inconsistency, including teaching people to use Euler diagrams, 2 × 2 tables, or relative frequencies. In Study 2, with problems about breast cancer and BRCA mutations, there was a strong correlation between inconsistency in conditional probability estimation and conjunction fallacies of joint probability estimation, suggesting that similar fallacious reasoning processes produce these errors. Copyright © 2012 John Wiley & Sons, Ltd.     

EDGAR MORIN'S PARADIGM OF COMPLEXITY AND GÖDEL'S INCOMPLETENESS THEOREM

This article shows that in two respects, Gödel's incompleteness theorem strongly supports the arguments of Edgar Morin's complexity paradigm. First, from the viewpoint of the content of Gödel's theorem, the latter justifies the basic view of complexity paradigm according to which knowledge is a dynamic, unfinished process, and develops by way of self-criticism and self-transcendence. Second, from the viewpoint of the proof procedure of Gödel's theorem, the latter confirms the complexity paradigm's circular line of inference through which is formed the all-round knowledge of a concrete object.     

Rule Separation and Embedding Theorems for Logics Without Weakening

A full separation theorem for the derivable rules of intuitionistic linear logic without bounds, 0 and exponentials is proved. Several structural consequences of this theorem for subreducts of (commutative) residuated lattices are obtained. The theorem is then extended to the logic LR ⁺ and its proof is extended to obtain the finite embeddability property for the class of square increasing residuated lattices.     

Vagueness-Adaptive Logic: A Pragmatical Approach to Sorites Paradoxes

Kripke-completeness of every classical modal logic with Sahlqvist formulas is one of the basic general results on completeness of classical modal logics. This paper shows a Sahlqvist theorem for modal logic over the relevant logic Bin terms of Routley-Meyer semantics. It is shown that usual Sahlqvist theorem for classical modal logics can be obtained as a special case of our theorem.     

Age and probabilistic reasoning: Biases in conjunctive,disjunctive and Bayesian judgements in early and late adulthood

Probabilistic reasoning plays an essential part in many aspects of our daily routine and it has been argued that as we grow older, the need to make judgements under uncertainty becomes increasingly important. Two studies were conducted to establish whether the propensity to commit probabilistic reasoning errors increased with age. Young (aged 16–24), middle aged (25–54), and older persons (55 years and above) were included. Study 1 revealed systematic biases and errors across a range of judgement tasks. However, no evidence of any age effect in Bayesian inference, the incidence of the conjunction fallacy, or in the number of disjunction errors was found. The results obtained in Study 1 were replicated in Study 2, where the potential mediating role of working memory processes and intellectual capacity were explicitly assessed. While some aspects of probabilistic reasoning performance were correlated with measures of intelligence and working memory functioning among young adults, this was much less evident in older persons. The present findings are discussed in relation to the evolution of the dualistic heuristic–analytical system over the adult lifespan. Copyright © 2005 John Wiley & Sons, Ltd.     

Gleason's Theorem Has a Constructive Proof

Gleason's theorem for R ³ says that if f is a nonnegative function on the unit sphere with the property that f(x) + f(y) + f(z) is a fixed constant for each triple x,y,z of mutually orthogonal unit vectors, then f is a quadratic form. We examine the issues raised by discussions in this journal regarding the possibility of a constructive proof of Gleason"s theorem in light of the recent publication of such a proof.     

RELIGION AND THE THEORIES OF SCIENCE: A RESPONSE TO BARBOUR

Abstract. This paper offers a detailed response to “Religion and the Theories of Science” in Barbour's Gifford Lectures I. Topics include: complementarity, indeterminacy, parts and wholes, and Bell's theorem in quantum theory; metaphysical issues raised by relativity theory and thermodynamics, principally the problem of temporality and “top-down” versus “bottom—up” causality; design arguments and the origins of the universe in astronomy and creation; and God's action in the context of evolution and continuing creation. Areas of agreement and disagreement between Barbour and myself over philosophical and theological implications are presented, and endnotes indicate further areas of conversation.