The assumptions underlying the generalized matching law

Allen (1981) derived the power-function generalization of the matching law from a functional equation involving relative response rates on three concurrently available schedules of reinforcement. This paper defines the conditions (relative homogeneity and independence) under which a more general class of behavioral laws reduces to the power law. The proof also removes two deficiencies of Allen's result (discussed by Houston, 1982), which are, first, that his derivation produces a power law without a bias coefficient, and second, that it holds only for experiments with three or more concurrent schedules.     

On the psychophysical law and estimation procedures in psychophysical scaling

A general formulation of the power law is presented which has two special features: (1) negative exponents are admissible; and (2) the log law is a special limiting case. Estimation procedures, which provide joint estimates of the exponent and the absolute threshold, are derived for the direct ratio scaling methods. A solution is provided for theaveraging problem for ratio production and bisection scaling, two methods generating observations on the physical scale, and Monte Carlo methods are used to evaluate the resulting estimators.     

To honor Fechner and obey Stevens: relationships between psychophysical and neural nonlinearities

G. T. Fechner (1860/1966) famously described two kinds of psychophysics: Outer psychophysics captures the black box relationship between sensory inputs and perceptual magnitudes, whereas inner psychophysics contains the neural transformations that Fechner's outer psychophysics elided. The relationship between the two has never been clear. Moreover, psychophysical power laws are found in almost every sensory system, yet the vast majority of neurons show sigmoid nonlinearities. Here, we selectively review the literatures on psychophysical and physiological nonlinearities and show how they can be placed within a framework for understanding the relationship between inner and outer psychophysics: a neural organization with a logical structure commensurate to outer psychophysical theory. In theoretical treatments of Stevens's law, the power law is a consequence of combining a Weber's law scaling of inputs with a Weber's law-like scaling of sensation magnitudes, yielding an exponent that is the ratio of the Weber constants. A neural derivation using physiological sigmoid nonlinearities should be commensurate to this internal logic. There is a class of models in which two nonlinear neural mechanisms (e.g., a sensory channel and the cortical numerosity mechanism tapped by magnitude estimation) are coupled through feedback, yielding power law behavior as an emergent property of the system, with an exponent that is a ratio of neural coupling strengths. Rather than a discrepancy between psychophysics and physiology, these models suggest complementarity between inner and outer psychophysics, because the Weber constants required for outer psychophysics modeling can be derived from the sigmoid nonlinearities of inner psychophysics.     

One hundred years of laws in psychology

Mainstream psychology in the 20th century has been conceived as a nomothetic science, but few psychological "laws" have been proposed. A PsycLit search of journal abstracts from 1900 to 1999 yielded a total of 3,093 "law" citations, or 22 per 10,000 entries, with two psychophysical laws (Weber's law and Stevens's power law) and two learning laws (Herrnstein's matching law and Thorndike's law of effect) as the most frequently cited. The number of law citations has been decreasing throughout the century, to 10 per 10,000 entries in the last decade, with few references to laws of recent origin. This could be the result of increasing doubts about the lawfulness of psychological processes coupled with a general preference for less ambitious terms (such as effects, principles, models, or functions).     

Variations in the relationship between radius of curvature and velocity as a function of joint motion

A power law describes the relationship between the geometric properties of a trajectory (radius of curvature) and movement kinematics (tangential velocity) in curved drawing movements. Although the power law is a general law of motion, there are conditions under which it degrades. In particular, the power law may be less explanatory of movements around certain joints. The present study considered how varying motion around different joints influenced the fit of the power law. Motions associated with finger and wrist, or elicited by an isometric force production task, were compared. The power law was most explanatory of finger motion and isometric production and least explanatory of wrist motion. The fit of the power law for finger and wrist motion suggested separate laws for each joint system. Since the fit of the power law was better for finger than for wrist motion, there is some suggestion that the power law better explains motion around fewer or simpler joint systems.     

Perceptual invariance and the psychophysical law

The psychophysical power law, demonstrated empirically by Stevens, is derivable from two postulates concerning (1) the tendency of perceptual organizations to model the environment and (2) the tendency for perceptual relations to remain constant. The two postulates appear to rule out logarithmic laws of the Fechner type. The theory is extendable to multidimensional perceptions such as those encountered in speech and color vision. The power transformations produced by the sense organs may be of such form that they produce a perceptual space in which subsequent transformations need be only linear.     

Revising the concept of lawhood: special sciences and natural kinds

The Kripkean conception of natural kinds (kinds are defined by essences that are intrinsic to their members and that lie at the microphysical level) indirectly finds support in a certain conception of a law of nature, according to which generalizations must have unlimited scope and be exceptionless to count as laws of nature. On my view, the kinds that constitute the subject matter of special sciences such as biology may very well turn out to be natural despite the fact that their essences fail to be microphysical or micro-based. On the causal conception of natural kinds I privilege, the naturalness of a kind is a function of the fact that it figures prominently in at least one causal law. However, there is a strong tendency prevailing among contemporary philosophers to assume that, in order to count as proper laws generalizations must be expectionless. Since most generalizations tracked down by the special sciences turn out not to fulfill these criteria, what this conception of a law implies is that most of the generalizations the special sciences trade in are not proper laws. It follows that, on this view, most if not all of the kinds the special sciences dealing with turn out not to constitute natural kinds, understood as kinds to which bona fide laws apply. In order to establish that the non-microstructurally defined kinds that fall within the domain of enquiry of the special sciences are eligible for the status of natural kind, I must therefore establish that generalizations needn’t have unlimited scope and be exceptionless to count as laws of nature. This is precisely what I seek to do in this paper. I begin by arguing that the question “what is a law of nature?” is most naturally interpreted as the question “what features must generalizations exhibit in order to ground scientific explanations?” and by offering reasons to believe that generalizations needn’t be exceptionless and have unlimited scope to play the crucial role laws have been thought to play in scientific explanation. Drawing on Sandra Mitchell [Mitchell, S. (2000). Philosophy of Science, 67, 242–265] and James Woodward’s [Woodward, J. (1997). Philosophy of science, 64 (proceedings), 524–541; Woodward, J. (2000). British Journal for the philosophy of science, 51(2), 197–254; Woodward, J. (2001). Philosophy of science, 68, 1–20] work, I subsequently develop an alternative account of the criteria generalizations must satisfy in order to count as laws of nature, which at least some of the generalizations of the special sciences turn out to fulfill. I thus give credence to the idea that at least some of the kinds that fall within the domain of the special sciences figure in laws of nature, and I thereby restore the possibility that some special science kinds deserve to be deemed natural.     

Love,Power and Consistency: Scotus’ Doctrines of God’s Power,Contingent Creation,Induction and Natural Law

I first examine John Duns Scotus’ view of contingency, pure possibility, and created possibilities, and his version of the celebrated distinction between ordained and absolute power. Scotus’ views on ethical natural law and his account of induction are characterised, and their dependence on the preceding doctrines detailed. I argue that there is an inconsistency in his treatments of the problem of induction and ethical natural law. Both proceed with God’s contingently willed creation of a given order of laws, which can be revoked and replaced with a new order of laws. In the case of ethical natural law God promulgated the Decalogue, for example; in the case of nature, there are physical laws that can be known by induction. Scotus exalts the freedom of God and the mutability of ethical natural law in order to explain exceptions to it disclosed by revelation (for example, the Old Testament command to Abraham to kill Isaac). Yet he treats ethical natural laws as (mostly) not universal and immutable. In contrast, he holds that we can arrive at knowledge of the universal and immutable laws of nature, except for those regularities that result from free will. Finally, I present several ways of characterising this tension between Scotus’ doctrines.     

Some functional measurement procedures for determining the psychophysical law

Various statistical procedures are ieveloped for determining the psychophysical law within the context of a functional measurement approach to studying stimulus integration in perception. The specific results are limited to additive or multiplicative psychological laws, but the generalization to alternative cognitive algebras is evident. Estimation of parameters of the hypothesized psychophysical law and test of the hypothesis that the row psychophysical law is the same as the column psychophysical law in a two-factor stimulus design is considered for various possible psychophysicaldaws, including linear, polynomial, and power laws.     

The power law for the perception of rotation by airline pilots

The purpose of this study was to determine the power laws for the perception of rotation about the three major body axes. Eighteen airline pilots made magnitude estimates of 5-sec pulses of nine angular accelerations having a range of acceleration x time of 10–150 deg/sec. The results showed that (1) the power law with an exponent of 1.4 describes the subjective motion of these pilots for all three major body axes, (2) the power law also describes the perception of motion for individual pilots with a substantial range of exponents, (3) there were significant correlations among the exponents for the three body axes, and (4) the data suggest that the power law over the wide range used may be more complex than implied by a formula with a single exponent.     

An experimental comparison of stimulus and response translated power functions for brightness

A threshold parameter has been introduced into the power law in two different ways, one involving, translation on the stimulus axis (the Ф-Law) and the second involving translation on the response axis (the Ψ-Law). In this study the two laws were compared using the data from 6 Ss making magnitude estimation judgments of brightness. Although the prevalent view appears to favor the Ф -Lau, the data of this study clearly dictate a choice of the Ψ-Law over the $-Law for brightness.     

Are There Cross-Cultural Legal Principles? Modal Reasoning Uncovers Procedural Constraints on Law

Despite pervasive variation in the content of laws, legal theorists and anthropologists have argued that laws share certain abstract features and even speculated that law may be a human universal. In the present report, we evaluate this thesis through an experiment administered in 11 different countries. Are there cross-cultural principles of law? In a between-subjects design, participants (N = 3,054) were asked whether there could be laws that violate certain procedural principles (e.g., laws applied retrospectively or unintelligible laws), and also whether there are any such laws. Confirming our preregistered prediction, people reported that such laws cannot exist, but also (paradoxically) that there are such laws. These results document cross-culturally and –linguistically robust beliefs about the concept of law which defy people's grasp of how legal systems function in practice.     

Similarity, feature discovery, and the size principle

In this paper we consider the “size principle” for featural similarity, which states that rare features should be weighted more heavily than common features in people’s evaluations of the similarity between two entities. Specifically, it predicts that if a feature is possessed by n objects, the expected weight scales according to a 1/n law. One justification of the size principle emerges from a Bayesian analysis of simple induction problems ( [Tenenbaum and Griffiths, 2001a] and [Tenenbaum and Griffiths, 2001b]), and is closely related to work by Shepard (1987) proposing universal laws for inductive generalization. In this article, we (1) show that the size principle can be more generally derived as an expression of a form of representational optimality, and (2) present analyses suggesting that across 11 different data sets in the domains of animals and artifacts, human judgments are in agreement with this law. A number of implications are discussed.     

Pure timing in temporal differentiation

Temporal control appears to depend on whether the critical durations are those of stimuli or those of responses. Stimulus timing (temporal discrimination) supports Weber's law, whereas response timing (temporal differentiation) indicates decreasing relative sensitivity with longer time intervals. The two types of procedure also yield different conclusions in scaling experiments designed to study the functional midpoint of two or more durations (temporal bisection procedures). In addition, the fractional-exponent power relation between emitted and required duration usually found with animals in differentiation experiments conflicts with deductions from formal analyses. The experiment reported here derived from considering differentiation arrangements as schedules of reinforcement. When analyzed from this perspective, the procedures are tandem schedules involving a required pause followed by a response, and it is the pause alone that involves temporal control. A choice procedure separated timing from responding, and enabled observations of pause timing in isolation. Pure temporal control in differentiation consisted of linear overestimation of the standard duration, and Weber's law described sensitivity. These results indicate that the two problems, the fractional-exponent power relations and the apparently different nature of sensitivity in differentiation and discrimination, disappear when temporal control is observed alone in differentiation.     

Drawing Movements as an Outcome of the Principle of Least Action

According to the two-thirds power law the cube of the speed of a drawing movement is proportional to the radius of curvature of the trajectory, and the coefficient of proportionality has the meaning of mechanical power. We derive this empirical law from the variational principle known in physics as the principle of least action. It states that if a movement between two points of a given path obeys the two-thirds law, then the amount of work required to execute a trajectory in a fixed time is minimal. In this strict sense one may say that among infinitely many ways to execute a given path, the central nervous system chooses the most economical. We show that the kinematic equations for all drawing movements are solutions of a certain differential equation with a single (time-variable) coefficient. We consider several special cases of drawing movements corresponding to simplest forms of this coefficient. Copyright 2001 Academic Press.     

NOTE ON THE VALIDITY OF THE POWER LAW

On the basis of previous experimental findings, it is argued that the power law is not based on a true (subjective) ratio scale, but on a quasi-ratio scale. Hence, and in spite of the results of the cross-modality matching, the validity of the power law, as well as the validity of models and equations in which a subjective scale is involved, is doubted. It is further indicated that the validity of Brown's law can, under special conditions, be tested by the relation between subjective variables of velocity, length and time. Finally, the relation between the power law and Luce's theorem is discussed briefly.     

Is There an Intrinsic Criterion for Causal Lawlike Statements?

A scientific mathematical law is causal if and only if it is a process law that contains a time derivative. This is the intrinsic criterion for causal laws we propose. A process is a space-time line along which some properties are conserved or vary. A process law contains a time variable, but only process laws that contain a time derivative are causal laws. An effect is identified with what corresponds to a time derivative of some property or magnitude in a process law, whereas the other terms correspond to the cause(s). According to our criterion, causes are simultaneous with their effects and causality has no temporal direction. Several examples from natural and social disciplines support the applicability of our criterion to all scientific laws. Various objections to our proposal are presented and refuted. The merits our intrinsic theory of causality vis-à-vis the Salmon–Dowe conserved quantity theory are discussed.     

Reduction Invariance and Prelec's Weighting Functions

Within the framework of separable utility theory, a condition, called reduction invariance, is shown to be equivalent to the 2-parameter family of weighting functions that Prelec (1998) derived from the condition called compound invariance. Reduction invariance, which is a variant on the reduction of compound gambles, is appreciably simpler and more easily testable than compound invariance, and a simpler proof is provided. Both conditions are generalized leading to more general weighting functions that include, as special cases, the families of functions that Prelec called exponential-power and hyperbolic logarithm and that he derived from two other invariance principles. However, of these various families, only Prelec's compound-invariance family includes, as a special case, the power function, which arises from the simplest probabilistic assumption of reduction of compound gambles. Copyright 2001 Academic Press.