A note on the ω-incompleteness formalization

The paper studies two formal schemes related to -completeness.LetS be a suitable formal theory containing primitive recursive arithmetic and letT be a formal extension ofS. Denoted by (a), (b) and (c), respectively, are the following three propositions (where (x) is a formula with the only free variable x): (a) (for anyn) ( _T (n)), (b) _{T x} Pr _T (^–(x)^–) and (c) _T x(x) (the notational conventions are those of Smoryski [3]). The aim of this paper is to examine the meaning of the schemes which result from the formalizations, over the base theoryS, of the implications (b) (c) and (a) (b), where ranges over all formulae. The analysis yields two results overS : 1. the schema corresponding to (b) (c) is equivalent to ¬Cons _T and 2. the schema corresponding to (a) (b) is not consistent with 1-CON _T. The former result follows from a simple adaptation of the -incompleteness proof; the second is new and is based on a particular application of the diagonalization lemma.Presented byMelvin Fitting     

A note on the proof theory the λII-calculus

The II-calculus, a theory of first-order dependent function types in Curry-Howard-de Bruijn correspondence with a fragment of minimal first-order logic, is defined as a system of (linearized) natural deduction. In this paper, we present a Gentzen-style sequent calculus for the II-calculus and prove the cut-elimination theorem.The cut-elimination result builds upon the existence of normal forms for the natural deduction system and can be considered to be analogous to a proof provided by Prawitz for first-order logic. The type-theoretic setting considered here elegantly illustrates the distinction between the processes of normalization in a natural deduction system and cut-elimination in a Gentzen-style sequent calculus.We consider an application of the cut-free calculus, via the subformula property, to proof-search in the II-calculus. For this application, the normalization result for the natural deduction calculus alone is inadequate, a (cut-free) calculus with the subformula property being required.This paper was written whilst the author was affiliated to the University of Edinburgh, Scotland, U.K. and revised for publication whilst he was affiliated to the University of Birmingham, England, U.K.Presented byDaniele Mundici     

A note on “pattern variants on a square field”

It is shown how results of Prokhovnik on the number of pattern variants that may be formed byk markers on a square network ofm ² positions may be derived more simply by means of a combinatorial theorem of Pólya's, which may also be used to solve systematically many other problems of this type.     

Motion and rest in the Pensées – A note on Pascal's modernism

International Journal for Philosophy of Religion -     

A Note on the “Halo Effect” in Ratings

Abstract

Sheldon, W. H. Atlas of Men. New York: Harper, 1954. Pp.—. Reviewed by Louise Bates Ames     

A note on the relation between coefficient alpha and guttman's “split-half” lower bounds

Use of the same term split-half for division of ann-item test into two subtests containing equal [Cronbach], and possibly unequal [Guttman], numbers of items sometimes leads to a misunderstanding about the relation between Guttman's maximum split-half bound and Cronbach's coefficient alpha.Coefficient alpha is the average of split-half bounds in the Cronbach sense and so is not larger than the maximum split-half bound in either sense whenn is even. Whenn is odd, however, splithalf bounds exist only in the Guttman sense and the largest of these may be smaller than coefficient alpha.     

What wealth-happiness paradox? A short note on the American case

Happiness scholars have tried to resolve the seeming paradox that as Americans’ wealth increased substantially over the last few decades, their happiness did not. This article questions whether the paradox is real. Demonstrations of the paradox almost always rely on GDP per capita as the measure of wealth, but that is a poor measure of a people’s well-being. It is heavily and increasingly skewed; it does not account for effort. Using instead measures of household income, male income, and average wages eliminates the paradox; these indicators of affluence have grown only slowly or declined in the same period, paralleling the changes in happiness scores. Moreover, using these indicators reveals a modest but real correlation between material well-being and national happiness.     

A note on Dasgupta’s Generalism

Philosophical Studies - Dasgupta (Philos Stud Int J Philos Anal Tradit 145(1):35–67, 2009) has argued that material individuals, such as particles and laptops, are metaphysically...     

The “ABCs” of REBT II: A Preliminary Study of Errors and Confusions Made by REBT Therapists

In this study, 19 REBT therapists from the Albert Ellis Institute??s referral list from a sample of 83 provided a specific ??ABC?? assessment of a problem and its solution that they would teach a class of counselling graduate students. This material was analysed for evidence of errors and confusions with respect to the ??ABCs?? of REBT. A total of 81 of such errors/confusions were found with most being made about beliefs at ??B??, particularly about rational beliefs. A variety of errors and confusions were also made about ??A?? and emotional ??Cs??. It was suggested that one way of addressing this state of affairs would be for the Albert Ellis Institute to introduce a theory test which applicants for Fellowship and Associate Fellowship status would have to pass before being awarded such status. Weaknesses of the current study were noted and suggestions for future research made.     

What really is turquoise? A note on the evolution of color terms

Summary In order to distinguish between the relativistic and the universalistic sematics in color terms, formal models in the framework of fuzzy-set theory are developed. These models can be used to generate empirically testable hypotheses about response latencies and the distribution of color terms in the visual spectrum.In Experiment I subjects had to name 20 colors in the blue-green area of the spectrum and 20 in the yellow-red area. Although the relative frequency data did not favor either model, the decision time data favored a specific universalistic model.Experiment II was intended to clarify the behavioral effects of basicality by investigating the differences in color naming of users and non-users of derived color terms as turquoise and orange. For users frequency data as well as response latencies from the unrestricted color-naming task conformed well with the predictions derived from the specific universalistic model, whereas the data for the non-users fell in between this model and the MIN-rule model. These results can be accounted for by a continuous model for basicality with a basicality parameter r.I want to thank J. Freyd/Stanford, P. Kay/Berkeley, and C. Freksa/Munich who have read earlier versions of this paper and have made many valuable suggestions concerning the theoretical framework, the readability and the style in general. The discussion about the experimental results in L. Zadeh's seminar in Berkeley has stimulated me to work further on a unified theory of color naming. The motivating critique of two anonymous reviewers has helped me to express the results more clearly and to rethink the arguments more thoroughly     

A note on the deontic system DL of Jones and Pörn

I would like to thank Ingmar Pörn for valuable comments on an earlier version of this paper.