Representational Measurement Theory

Abstract

This chapter focuses on empirical structures, with an ordering attribute, leading to numerical or geometric representations. A historical sketch of physical, behavioral, and social science measurement until 1950 is followed by modern approaches to representational measurement in the behavioral sciences. Among these structures are: difference, additive conjoint, averaging, and non‐additive concatenation and conjoint structures. Derived measurement is described by distributive linkages between two structures with a common attribute and is exemplified by physical, utility, and psychophysical examples. Two mathematical results are needed in each case: the existence of structure‐preserving numerical representations and their uniqueness . The latter has been analyzed generically. Using these ideas, models of magnitude estimation and production are summarized. The nature of axiomatization is discussed in terms of types of logical languages, the role of certain second‐order axioms, and issues of consistency and independence of axioms. Important results about the impossibility of simple axiomatizations of classes of finite structures are described along with mention of some problems.

An Algebraic Approach to Unital Quantities and their Measurement

“…tions of measurement Perhaps the most elaborate alternative attempt at investigating the nature of quantities, including length, time, mass, velocity and so forth, and their measurement is found in the foundations of measurement, framed and discussed in [13], [15], and [14]. Among social scientists and philosophers of science this approach is widely thought to be the richest storehouse of information on the foundations of deterministic measurement.…”

“…By claiming that this construction is to be grounded in basic procedures for assigning numbers to objects or events on the basis of qualitative observations of attributes ( [13], pp. 1-2), RTM purports to explain and justify without contentious metaphysical commitments or circular theoretical assumptions our passage from simple qualitative observations to quantitative ones ( [14], p. 4).…”

“…14 So now we can make the claim about the flagpole's length mathematically explicit and rigorous as follows:…”

The concept of a scale in accounting measurement

“…However, that is not to say that Stevens' (1951) list of scales is exhaustive. Narens (1981aNarens ( , 1981b shows that there are scales between the ratio and interval scales, but none of them has yet played a role in actual scientific measurement (Luce & Suppes, 2001). For this reason, the discussion on scales in this study is limited to Stevens' (1951) list outlined above.…”

Fatores sensoriais visuais que influenciam o dimensionamento subjetivo na percepção de tamanho: um estudo de escalonamento psicofísico

“…Portanto, o problema não é como obter a experiência perceptual, mas como descrever e investigá-las de modo a serem expressas e compartilhadas para os outros (Ehrenstein & Ehrenstein, 1999) Com a disseminação dos primeiros métodos psicofísicos, a partir do início da década de 1870, alguns psicólogos tentaram importar ideias de medidas da Física, mas devido às dificuldades, eles deduziram que os modelos clássicos de medida das outras ciências eram, em sua maioria, inadequados para o fenômeno psicológico (Luce & Suppes, 2002).…”