An Impossibility Theorem on Beliefs in Games

A paradox of self-reference in beliefs in games is identified, which yields a game-theoretic impossibility theorem akin to Russell’s Paradox. An informal version of the paradox is that the following configuration of beliefs is impossible:Ann believes that Bob assumes thatAnn believes that Bob’s assumption is wrongThis is formalized to show that any belief model of a certain kind must have a ‘hole.’ An interpretation of the result is that if the analyst’s tools are available to the players in a game, then there are statements that the players can think about but cannot assume. Connections are made to some questions in the foundations of game theory.Special Issue Ways of Worlds II. On Possible Worlds and Related Notions Edited by Vincent F. Hendricks and Stig Andur Pedersen     

Another Paradox In Naive Set-Theory

Reasonning in naive set theory (with unlimited comprehension), we derive a paradox (a formal contradiction) which can be seen as a variant of the Burali-Forti paradox. Presented by Robert Goldblatt     

How to eliminate self-reference: a précis

We provide a systematic recipe for eliminating self-reference from a simple language in which semantic paradoxes (whether purely logical or empirical) can be expressed. We start from a non-quantificational language L which contains a truth predicate and sentence names, and we associate to each sentence F of L an infinite series of translations h ₀(F), h ₁(F), ..., stated in a quantificational language L ^*. Under certain conditions, we show that none of the translations is self-referential, but that any one of them perfectly mirrors the semantic behavior of the original. The result, which can be seen as a generalization of recent work by Yablo (1993, Analysis, 53, 251–252; 2004, Self-reference, CSLI) and Cook (2004, Journal of Symbolic Logic, 69(3), 767–774), shows that under certain conditions self-reference is not essential to any of the semantic phenomena that can be obtained in a simple language.     

The knowledge content of statistical data

An information-theoretic framework is used to analyze the knowledge content in multivariate cross classified data. Several related measures based directly on the information concept are proposed: the knowledge content (S) of a cross classification, its terseness (Zeta), and the separability (Gamma _X ) of one variable, given all others. Exemplary applications are presented which illustrate the solutions obtained where classical analysis is unsatisfactory, such as optimal grouping, the analysis of very skew tables, or the interpretation of well-known paradoxes. Further, the separability suggests a solution for the classic problem of inductive inference which is independent of sample size.     

Reasoning to what is true in fiction

The paper discusses the principle by which we reason to what is true in fiction. The focus is David Lewis's article Truth in Fiction (1978) which proposes an analysis in terms of counterfactuals and possible worlds. It is argued thatLewis's account is inadequate in detail and also in principle in that it conflicts radically with basic and familiar tenets of literary criticism. Literary critical reasoning about fiction concerns not the discovery of facts in possible worlds but the recovery of meanings in interpretative frameworks. The model theoretic approach fails to account for common literary or rhetorical devices like unreliable narration, connotation and point of view. And in explaining indeterminacy of content in terms of truth-value gaps it gives too simplistic an account of critical reasoning about character motivation and thematic development. A more adequate account of content-indeterminacy can be provided through a comparison of the interpretation of fiction with the interpretation of human action. A broader motif in the paper is the underlying tension between what is required for the logic of fiction and what is required for the aesthetics of fiction.     

Classical conditioning and retention of the infant's eyelid response: effects of age and interstimulus interval

Independent groups of 10-,20-, and 30-day-old infants were subjected to a classical eyelid conditioning procedure involving either a 500- or a 1500-msec interstimulus interval (ISI). Ten days later, all received a second conditioning session. A reliable increase in conditioned responding was observed at all ages but only by infants receiving the 1500-msec ISI. Although age was not a significant factor in any conditioning measure except final performance level, which was greater for the oldest than for the youngest group, it did influence long-term retention. A reliable memory component was observed in the Session 2 performance of infants initially trained at 20 and 30 days but not at 10 days. These data demonstrate the importance of temporal parameters in the formation of conditioned associations very early in infancy and provide evidence for the long-term behavioral consequences of those associations.     

Imitation and novelty in product development

A human stigmergy model of product development through a series of incremental novelties and imitations is obtained by observing the isomorphism between solution trees generated by groups of humans participating in online games like Foldit, self-organizing topics in online learning systems, product evolution through a combination of innovation and imitation in the cell phone industry, and social networks formed over the past 40 years in creation of the Boston biotech commons. The model incorporates two very simple rules: (1) preferential attachment, and (2) the combination of existing product designs into a single design. A computer model used to simulate the model produces product evolution networks isomorphic to observed solution trees and product evolution trees. Additionally, industries with a high degree of novelty produce a product evolution network with scale-free and betweenness centrality structure.     

Performed or observed keyboard actions affect pianists' judgements of relative pitch

Action can affect visual perception if the action's expected sensory effects resemble a concurrent unstable or deviant event. To determine whether action can also change auditory perception, participants were required to play pairs of octave-ambiguous tones by pressing successive keys on a piano or computer keyboard and to judge whether each pitch interval was rising or falling. Both pianists and nonpianist musicians gave significantly more “rising” responses when the order of key presses was left-to-right than when it was right-to-left, in accord with the pitch mapping of the piano. However, the effect was much larger in pianists. Pianists showed a similarly large effect when they passively observed the experimenter pressing keys on a piano keyboard, as long as the keyboard faced the participant. The results suggest that acquired action–effect associations can affect auditory perceptual judgement.