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.     

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.     

Access,phenomenology and sorites

The non‐transitivity of the relation looks the same as has been used to argue that the relation has the same phenomenal character as is non‐transitive—a result that jeopardizes certain theories of consciousness. In this paper, I argue against this conclusion while granting the premise by dissociating lookings and phenomenology; an idea that some might find counter‐intuitive. However, such an intuition is left unsupported once phenomenology and cognitive access are distinguished from each other; a distinction that is conceptually and empirically grounded.     

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.     

Logic of Knowledge and Utterance and the Liar

We extend the ordinary logic of knowledge based on the operator K and the system of axioms S₅ by adding a new operator U, standing for the agent utters , and certain axioms and a rule for U, forming thus a new system KU. The main advantage of KU is that we can express in it intentions of the speaker concerning the truth or falsehood of the claims he utters and analyze them logically. Specifically we can express in the new language various notions of lying, as well as of telling the truth. Consequently, as long as lying or telling the truth about a fact is an intentional mode of the speaker, we can resolve the Liar paradox, or at least some of its variants, turning it into an ordinary (false or true) sentence. Also, using Kripke structures analogous to those employed by S. Kraus and D. Lehmann in [3] for modelling the logic of knowledge and belief, we offer a sound and complete semantics for KU.