Frequency of occurrence as a psychophysical continuum: Weber’s fraction,Ekman’s fraction,range effects,and the phi-gamma hypothesis

Using the continuum of frequency of occurrence of words in English, it was found that: (1) errors in judgment are distributed lognormally rather than normally, and therefore the standard method of calculating Weber’s fraction underestimates its definition, (2) Weber’s fraction has an extremely large value of 3.3, (3)Ekman’s fraction equals 1.81, not .03 as with sensory continua, and (4)the logarithm of the dynamic range times Stevens’ law exponent equals 3.83, not 1.53 as with sensory continua. The last two results favor Teghtsoonian’s underlying sensory scale interpretation over Poulton’s range effects interpretation of the range-exponent relation.     

Perceptual cues used in reproduction of inspired volume

Man is capable of reproducing an inspired lung volume (IV) with relative precision (±3 %–8 %). To investigate the importance of cues which may be used to determine IV, we performed two experiments. In each, the subject inspired a “standard” breath (from resting end~xpired lung volume) corresponding to 25%, 50%, or 75% of inspiratory capacity (IC) and then expired nearly completely. From this expired volume, the subject either tried to reproduce the tidal volume (i.e., volume of air inspired) of the first breath (volume reproduction) or to reproduce the previous end-inspired lung volume (position reproduction). The constant error indicated that subjects overestimated tidal volume by about 200 ml. Despite this overestimation, the variability of reproduction was less for volume reproduction than for position reproduction when IV was small. The JND for volume reproduction increased linearly with increasing IV, whereas the JND for position reproduction was similar at all IVs so that the Weber ratio decreased. These observations suggest that, in the perception of inspired volume, the sensation of end-inspired position is important when the inspired volume is large, but that subjects apparently rely on some other cue to reproduce small lung volumes accurately.     

When are semantic targets detected faster than visual or acoustic ones?

Duration-discrimination data from an experiment using empty auditory intervals in a two-alternative forced-choice paradigm are presented. The observed functional relationship between standard deviation of the psychometric density function and stimulus duration is shown to be fit significantly better by a Weber’s law model of duration discrimination than by Creelman’s counter model. Both models fail to predict the rapid rise in the Weber fraction observed for durations longer than about 2 sec. However, the Weber’s law model, based on a generalization of Weber’s law, accurately predicts the initial drop in the Weber fraction for very short durations and the observed constancy of the Weber fraction for durations up to 2 sec.     

Difference threshold for stimulus length under simultaneous and nonsimultaneous viewing conditions

The present study was concerned with Weber’s Law as it is related to the discriminability of the lengths of lines. Experiments were conducted to investigate three questions: (I) Is Weber’s Law equally applicable to simultaneous and nonsimultaneous viewing conditions? (2) Is the relationship between the stimulus sizes and the values ofDLs described more adequately by the function proposed by Weber or a generalized Weber’s Law stated by Miller? and (3) Is Weber’s Law better approximated by proximal or distal size? It was demonstrated that the discriminability of the lengths of lines follows Weber’s Law under the nonsimultaneous viewing condition, but not under the simultaneous viewing condition. Under the nonsimultaneous viewing condition, it was noted that the generalized Weber’s Law as stated by Miller described the relationship between the DL and stimulus size significantly better than the function proposed by Weber. From the results pertaining to the third question, it was not possible to determine whether the proximal or the distal size follows Weber’s Law more closely.     

Counting processes in human timing

A subject reproducing a long duration, t, may time out either a single interval of duration t or a succession of n intervals, each of duration t/n. It is shown that a class of timing models obeying Weber’s law predicts the variance of reproductions of t to be a decreasing function of the number of subdivisions, n. In contrast, a second class of proportional variance models, which includes Creelman’s pulse counter model (1962), predicts no change in the variance as a function of n. Data are presented from a duration reproduction experiment in which subjects counted silently at a specified rate up to a given number and then responded. Several statistics involving the variance of the reproduced durations are shown to be predicted significantly better by the Weber’s law class of models than by the proportional variance class of models.     

Magnitude estimation of apparent sums and differences

Os first scaled two continua by magnitude estimation: apparent area of circles and loudness of 1,000-Hz tones. They then gave magnitude estimations of apparent sums and apparent differences for IS pairs of stimuli on each of the two continua. The scales for sums and differences were in some cases nearly linearly related to the power function obtained when the same as scaled the underlying continuum. However, systematic departure from linearity was the usual result. The power law exponents obtained were generally smaller than those usually reported for the two sensory continua.     

面纱·身体·伊斯兰教——由法国“罩袍禁令”引发的思考

法国"罩袍禁令"一经颁布,世界舆论哗然。一块小小的面纱为何能引发世界如此广泛的争论?它在冲突或争论之中为何被赋予重要意义?面纱所遮掩的女性身体在其中又扮演何种角色?面纱、身体、伊斯兰教三者之间又存在怎样的内在关系?本文试图对这些问题进行分析,并提出女性身体除了表达一个族群对自己文化、宗教特性的坚守之外,也成为其反抗外来力量的工具和媒介。     

Intensity discrimination for pulsed sinusoids of various frequencies

Studies of pure-tone intensity discrimination have shown that Weber’s law fails for tones in the region of 1 kHz. In this experiment, intensity discrimination of pulsed sinusoids ranging in frequency from 0.15 to 12 kHz is investigated. For each tone in this region, Weber’s law is found to fail. Some theoretical implications of these results are discussed.     

Why is Law a Normative Discipline? on Hans Kelsen’s ‘Normology’

What does it mean to claim of law that it is a normative discipline? Can the answer be so simple that one need merely refer to law’s normative object of study and the conclusions that the legal participant must allegedly draw from this? What, in any case, is a ‘normative discipline’? The essay attempts to address these questions by analysing Hans Kelsen’s ‘normological’ theory of law through his work on sovereignty and especially by focusing on the normative character of Kelsen’s epistemological claims regarding law. A theoretical critique of Kelsen is offered through Edmund Husserl’s phenomenological account of logic as a normative discipline.     

On the ability to directly evaluate sensory ratios

The study explored whether participants could directly evaluate sensory ratios. The results indicate that they did not exhibit this ability for the extensive continuum of length and the intensive continua of brightness and brightness difference. Functional measurement analysis, combined with a chronometric analysis of responses, indicated that participants used mental counting for length and used difference evaluation for brightness and brightness difference. The finding that participants evaluated brightness differences under instructions to evaluate brightness ratios suggests that difference judgments may be a general human capability. It resolves questions raised by prior tests of Torgerson’s conjecture and by prior consistency tests of ratio evaluation.     

Incongruous and Creative Grace: Reading Paul and the Gift with Martin Luther

Paul and the Gift by John Barclay advances an interpretation of Paul’s theology of grace that resonates with Martin Luther’s reading: God’s gift is God’s Son, Jesus Christ, given for and to the unworthy. To imagine Luther reading Paul and the Gift is thus to conjure images of deep and fundamental consensus. But questions remain. Is the law a cultural canon of worth that God’s gift of Christ ignores, or is it, as God’s law, a fixed judgement that God’s grace contravenes? Does God give only ‘without regard to worth’ and thus with a kind of divine indifference to cultural indices of value, or does the gift of Christ contradict the conditions of its receipts and thus come in a way that is actually incongruous? With these questions, Luther might push back against Barclay. With others he would ask Barclay to go further. Is not God’s incongruous grace also and characteristically creative? How is the gift of Christ that God gave present to and for recipients as the gift God now gives? In all these ways, Luther’s theology of the word poses questions to or invites expansions of Barclay’s theology of grace.     

三数值加工系统假说：数值加工机制新探

数值加工的机制是数量认知领域的核心科学问题之一。三数值加工系统假说从新的角度阐释了各种数值加工机制的关系, 它认为认知系统通过三种不同的机制来快速分析非符号刺激的数值：感数(subitizing)机制精确地分析1~4个刺激的数值; 数量(numerosity)机制分析密度适中的刺激点阵的数量, 加工误差正比于被分析的数量, 符合韦伯定律; 当刺激的密度超出一定范围时, 密度(density)机制通过分析刺激密度来推断刺激的数量关系, 加工误差正比于被加工数值的平方根。一系列研究证实, 这三种数值加工机制具有不同的行为规律和脑电特征。未来研究需要探讨数量机制是否分别与感数机制、密度机制存在平行激活, 从而阐明三种数值加工机制的作用关系。     

Response differentiation: A psychophysical method for response produced stimuli

Twenty-four rats were reinforced for releasing a lever between t and t + .20 sec after pressing it. Response duration differentiation performance decreased monotonically as t was increased in length from .10 sec to 1.60 sec. A response analogue of Weber’s ratio was constant throughout the middle range of duration bandwidth requirements and increased at the ex tremely short bandwidth—a response-differentiation finding analogous to Weber’s law of stimulus discrimination. This finding was interpreted as support for the conceptualization that response differentiation represents a special instance of response discrimination or discrimination based on response-produced stimuli.     

Range of acceptable stimulus intensities: An estimator of dynamic range for intensive perceptual continua

The dynamic range (DR) of a sensory system is the span (usually given in log units) from the lowest to highest intensities over which a continuously graded response is evoked, and may be a distinctive feature of each such system. Teghtsoonian (1971) proposed that, although DR varies widely over sensory systems, itssubjective size (SDR) is invariant. Assuming the psychophysical power law, the exponent for any continuum is given by the ratio of subjective span to DR, both quantities expressed logarithmically. Thus, exponents are inversely related to DR and may be interpreted as indexes of it. Because DR can be difficult or even dangerous to measure directly, we sought to define a smaller range representing some fixed proportion of DR that could be used in its place to test the hypothesis of an invariant subjective range. Observers manipulated the intensities of five target continua to produce the broadest range they found acceptable and reasonably comfortable, a range of acceptable stimulus intensities (RASIN). Combined with an assumed constant SDR (derived from previous research), RASINs accurately predicted exponents obtained by magnitude production from the same observers on the five continua, as well as exponents reported in the literature.     

Mental body representations retain homuncular shape distortions: Evidence from Weber’s illusion

Mental body representations underlying tactile perception do not accurately reflect the body’s true morphology. For example, perceived tactile distance is dependent on both the body part being touched and the stimulus orientation, a phenomenon called Weber’s illusion. These findings suggest the presence of size and shape distortions, respectively. However, whereas each morphological feature is typically measured in isolation, a complete morphological characterization requires the concurrent measurement of both size and shape. We did so in three experiments, manipulating both the stimulated body parts (hand; forearm) and stimulus orientation while requiring participants to make tactile distance judgments. We found that the forearm was significantly more distorted than the hand lengthwise but not widthwise. Effects of stimulus orientation are thought to reflect receptive field anisotropies in primary somatosensory cortex. The results of the present study therefore suggest that mental body representations retain homuncular shape distortions that characterize early stages of somatosensory processing.     

Tests of the psychological meaning of the power law.

The purpose of this study was to establish a theoretical framework for Stevens' empirically derived power law. Three models were proposed to explain the power law. They respectively outline how sensory, stimulus, and response variables determine the judgmental behavior in a psychophysical task. A correlational study on individual differences in exponents was carried out to test the predictions derived from each model. The use of four different sensory continua and four scaling procedures provided the experimental means of manipulating the sensory, stimulus, and response variables in the scaling situation. The results showed that response variables are important determinants of judgmental behavior in psychophysical scaling. These findings suggest that subjects' responses to stimulus intensities in a scaling task are largely cognitive.     

Discriminating the volume of motion-defined solids

We investigated the ability of human observers to discriminate an important global 3-D structural property, namely volume, of motion-defined objects. We used convex transparent wire-frame objects consisting of about 12 planar triangular facets. Two objects, vertically separated by 7°, were shown simultaneously on a computer display. Both revolved at 67°/sec around a common vertical axis through their centers of mass. Observers watched the objects monocularly for an average of three full rotations before they responded. We measured volume discrimination as a function of absolute volume (3–48 cm³; 1 m viewing distance) and shape (cubes, rods, and slabs of different regularity). We found that (1) volume discrimination performance can be described by Weber’s law, (2) Weber fractions depend strongly on the particular combination of shapes used (regular shapes, especially cubes, are easiest to compare, and similar shapes are easier to compare than different shapes), and (3) humans use a representation of volume that is more veridical and stable in the sense of repeatability than a strategy based on the average visible (2-D) area would yield.     

Duration discrimination in a series of rhythmic events

Duration discrimination of the last of a series of four clicks was investigated. Examination of psychophysical functions from eight subjects revealed evidence for a Weber’s law model relating discrimination to base interclick interval. Also, the point of subjective equality was seen to change reliably as a function of base rate.     

The connection between magnitude and discrimination scales and direct and indirect scaling methods

(1) An analysis of the method of paired comparisons shows that Case V and Case VI, the latter characterized by log-normal distributions and Weber's law for subjective continua, are fundamentally indistinguishable. Case VI produces a log-arithmetic interval scale of subjective magnitude. (2) It is demonstrated that the difference between discrimination scales according to Case V and from category rating is due to the difference between intra- and interindividual variability yielding different Weber functions.This research was supported by the Humanistic Faculty of the University of Stockholm and by the Swedish Social Science Research Council. I am indebted to Mr. U. Forsberg for computational assistance.