Теория нормативных предлозений

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O regułach rachunku zdań

Streszczenie Zmienn istotn wyraenia rachunku zda nazywam zmienn, której warto przy pewnym ukadzie wartoci innych zmiennych wpywa na warto caego wyraenia.     

О правилах исчисления предложе ий

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Prototetyka St. Lesniewskiego

Streszczenie Prototetyka St. Len'ewskiego jest uogólnieniem dwuwartociowego rachunku zda. Wystpuj w niej obok terminów tego rachunku funktory (zarówno stae jak i zmienne) tych wszystkich kategorii semantycznych, jakie mog by zdefiniowane, gdy punktem wyjcia jest kategor a zda.W pracy zreferowane s trzy systemy prototetyki. Terminem pierwotnym dwu z nich jest implikacja, terminem pierwotnym systemu trzeciego jest równowano. Systemy o terminie pierwotnym implikacji róni s. reguami wnioskowania. W jednym z nich obowizuje regua weryfikacji, w drugim regua ekstensjonalnoci. Pierwsza z tych regu jest uogólnieniem nastpujcej reguy rachunku zda, wzbogaconego o terminy 0 i 1:Wyraenie jest tez systemu, gdy tezami systemu s oba wyraenia powstae z przez podstawienie za pewn jego zmienn symboli 0 i 1.W myl reguy ekstensjonalnoci tezami systemu s prawa ekstensjonalnoci, sformuowane dla funktorów dowolnych kategorii semantycznych wac-wych prototetyce.W pracy podaj dowody równowanoci trzech systemów prototetyki i dowód ich zupenoci.Znaczna cz wyników Leniewskiego, zreferowanych w pracy, nie bya dotd opublikowana. Opracowujc te wyniki opieraem si na notatkach z wykadów Leniewskiego, spisanych przez jego uczniów. Oryginalne notatki Leniewskiego ulegy zniszczeniu w czasie powstania warszawskiego.     

An efficient punched-card method of computing epsilonX,epsilonX2, epsilonXY,and higher moments

A method of computing X, X ², XY and higher moments on IBM equipment is described. The basic method is that of successively summary punching, collating a variable number of blank cards behind these summary cards, gang-punching the data from the summary cards into the blank cards, and totalling the entries on these summary cards. The method appears to have several advantages over those previously described, especially if coded data are used.     

Estimation of parameters in a truncated trivariate normal distribution

This paper gives maximum-likelihood estimators for certain parameters in a truncated trivariate normal distribution when the values of the other parameters are known. The estimators are functions of a random sample. Approximate variances and covariances of the estimators, when the sample size is large, are also given. The type of truncation considered is merely restriction of the range of one of the variates, whose true mean and variance are assumed to be known. Two cases of such restriction are treated: (a) ( x < + ); (b) (- <x '), where and ' are arbitrary cutoff points which are assumed to be known. A precise statement of the estimation problem is given in Section 1. Section 2 contains preliminary calculations. The estimators appear in Section 3. The asymptotic variances and covariances of the estimators are given in Section 4. The estimators and their asymptotic variances and covariances can be easily specialized to be suitable for the case of a certain truncated bivariate normal distribution (Sections 3 and 4).     

О геометрической интерпретации логических выражений

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Über die Messung der psychischen Ermüdung mit Hilfe des Flimmertests

Ohne ZusammenfassungErklärung der Abkürzungen Add.-Lstg. Additionsleistung - AZ Arbeitszeit - B/sec Anzahl der Blitze je Sekundes - KP Kurzpause - Lstg. Leistung - M Mittelwert (arithm. Mittel) - TZ Teilzeit - Vfr. Verschmelzungsfrequenz = Flimmergrenze - VL Versuchsleiter - Vp. Versuchsperson - Vpn. Versuchspersonen - Mittlere Abweichung vom Mittelwert - M Mittlerer Fehler des Mittelwertes     

The unit hierarchy and its properties

A correlation matrix may be expanded as the weighted sum of a series of unit hierarchies. The properties of the unit hierarchy are not only of theoretical interest for themselves, but lead to simpler modes of practical calculation. The analysis is analogous to a spectral set of projective operations in quantum-theory: and the analogy itself suggests many further problems and solutions.     

Synthesis of variance

The distribution of a linear combination of two statistics distributed as is Chi-square is studied. The degree of approximation involved in assuming a Chi-square distribution is illustrated for several representative cases. It is concluded that the approximation is sufficiently accurate to use in many practical applications. Illustrations are given of its use in extending the Chi-square, the Student t and the Fisher z tests to a wider range of problems.     

An I.B.M. technique for the computation of ΣX 2 and ΣXY

Given I.B.M. cards punched with scores (or any numbers)—but not their squares—a method is presented of tabulating them (on the No. 405 alphameric I.B.M. tabulator) so as to obtain the sum of squares. The technique is also adaptable to summation of cross-products. The principle is an extension of the Mendenhall-Warren-Hollerith technique of vertical progressive digiting, without the necessity of manual addition or summary-punching, and is designed for machines not equipped with the card cycle total transfer device or progressive total device. Use is made of counter rolling. Efficient use of machine capacity is made only when intercorrelations between no more than two variables are requiredin addition to sums of squares. A resumé of some techniques now commonly employed is included.The author is indebted to Dr. Paul Dwyer, Associate Professor of Mathematics, University of Michigan, for valuable criticism of the original draft; and to Mr. Alan Meacham, in charge of the University's Tabulating Station, for testing the method.     

(II. Teil) Individuum,Gesellschaft und Staat

Ohne ZusammenfassungI. Teil: Synthese, Vo.l VII (1948–49) S. 536–549     

Efficiency of multiple-choice tests as a function of spread of item difficulties

The validity of a univocal multiple-choice test is determined for varying distributions of item difficulty and varying degrees of item precision. Validity is a function of _d ² + _v ² , where _d measures item unreliability and _v measures the spread of item difficulties. When this variance is very small, validity is high for one optimum cutting score, but the test gives relatively little valid information for other cutting scores. As this variance increases, eta increases up to a certain point, and then begins to decrease. Screening validity at the optimum cutting score declines as this variance increases, but the test becomes much more flexible, maintaining the same validity for a wide range of cutting scores. For items of the type ordinarily used in psychological tests, the test with uniform item difficulty gives greater over-all validity, and superior validity for most cutting scores, compared to a test with a range of item difficulties. When a multiple-choice test is intended to reject the poorestF per cent of the men tested, items should on the average be located at or above the threshold for men whose true ability is at theFth percentile.This research was performed under contract Nop 536 with the Bureau of Naval Personnel, and received additional support from the Bureau of Research and Service, College of Education, University of Illinois.     

A simplified method for approximating multiple regression coefficients

A method of exhaustion has been described for calculating regression coefficients. This method dispenses with the solution of simultaneous equations but utilizes a process of successive extraction in obtaining's, where each successive is maximized. This procedure permits the worker to discard as he goes along those weights which are deemed unsatisfactory for purposes of prediction. The coefficients and theR in a problem involving a criterion and six independent variables were calculated in sixty minutes. TheR's obtained by this method are smaller than those yielded by the Doolittle technique, but in problems which have been considered this discrepancy has not exceeded .05.     

The selective efficiency of a test battery

In industrial acceptance sampling one frequently makes use of operating characteristic curves to describe the discriminating power of a particular sampling plan. Similarly, it is possible to demonstrate the selective efficiency of a test battery in terms of (a) the Applicant's Operating Characteristic (A.O.C.); (b) the Selector's Operating Characteristic (S.O.C.). The A.O.C. determines the chance of selection by means of a test for any given level of true ability. The S.O.C. connects functionally probability of success on the criterion with the predictor scores of a battery. For the case of a normal bivariate distribution the exact mathematical expressions of the OC curves are derived in terms of the correlation coefficient, the cut-off points and, and the predictor and criterion scoresX andY (in standard measures). The Efficiency IndexH is defined as the percentage of successful subjects gained by the use of a test battery, taking chance selection as a yardstick for comparison. Its optimum, for fixed and , is derived. The distribution law of the criterion scores of selectees is deduced and its first four moments are shown to depart little from normality for cases usually encountered in practice. A Quality-Gain diagram graphically illustrates the improvements secured. Another simple device, the Cost-Utility diagram, explains to management the full implications of selecting personnel by means of a test battery. Neither of the diagrams requires an understanding of the correlation coefficient. The confidence belt of the OC curves, the standard error of the mean criterion score of selectees and the standard error of the predicted number of successful applicants are determined. Finally, the full theory is applied in detail to a real test battery.This paper is the result of an investigation undertaken by the South African National Institute for Personnel Research, Johannesburg, and was completed at the Educational Testing Service, Princeton. The author wishes to thank these organizations for permission to publish this paper.     

The contribution of an orthogonal multiple factor solution to multiple correlation

A method is indicated by which multiple factor analysis may be used in determining a number,r, and then in selectingr predicting variables out ofn variables so that each of the remainingn-r variables may be predicted almost as well from ther variables as it could be predicted from all then—1 variables.     

A note on correcting for chance success in objective tests

The conventional scoring formula to correct for guessing is derived and is compared with a regression method for scoring which has been recently proposed by Hamilton. It is shown that the usual formula,S=R–W/(n–1), yields a close approximation (correct within one point) to the maximum-likelihood estimate of an individual's true score on the test, if we assume that the individual knows or does not know the answer to each item, that guessing at unknown items is random, and that success at guessing is governed by the binomial law. It is also shown that the usual scoring formula yields an unbiased estimate of the individual's true score, when the true score is defined as the mean score over an indefinitely large number of independent attempts at the test or at equivalent (parallel) tests.     

A technique for criterion-keying and selecting test items

For multiple-choice tests where noa priori key exists, the initial selection of a key for maximum validity may be made on the basis of the number of persons choosing each alternative and their mean criterion score. The keying formula is derived. Once the initial keying has been done, further precision in keying and item selection may use, in addition, the mean total test score for persons choosing each alternative. Item-selection formulas suggested by Horst and by Gulliksen for maximizing test validity are both in the form of a ratio, an item-validity index divided by an item-reliability index. The formula derived here is shown to be equivalent to the numerators of these formulas. The expression in the denominators uses the total test score. Although a radical appears in the denominator of Horst's formula and not in the denominator of Gulliksen's formula, both of them select the same items in practice.The author gratefully acknowledges the suggestions and criticisms of Dr. Harold Gulliksen, Research Adviser at the Educational Testing Service.