Abstract
It is a scientific platitude that there can be neither precise control nor prediction of phenomena without measurement. Disciplines as diverse as cosmology and social psychology provide evidence that it is nearly useless to have an exactly formulated quantitative theory, if empirically feasible methods of measurement cannot be developed for a substantial portion of the quantitative concepts of the theory. Given a physical concept like that of mass or a psychological concept like that of habit strength, the point of a theory of measurement is to lay bare the structure of a collection of empirical relations which may be used to measure the characteristic of empirical phenomena-corresponding to the concept. Why a collection of relations? From an abstract standpoint, a set of empirical data consists of a collection of relations between specified objects. For example, data on the relative weights of a set of physical objects are easily represented by an ordering relation on the set; additional data, and a fortiori an additional relation, are needed to yield a satisfactory quantitative measurement of the masses of the objects.
Reprinted from The Journal of Symbolic Logic 23 (1958), 113–128. Written jointly with Dana Scott.
We would like to record here our indebtedness to Alfred Tarski, whose clear and precise formulation of the mathematical theory of models has greatly influenced our presentation (Tarski, 1954, 1955). Although our theories of measurement do not constitute special cases of the arithmetical classes of Tarski, the notions are closely related, and we have made use of results and methods from the theory of models.
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Notes
Although in most mathematical contexts imbeddability is defined in terms of isomorphism rather than homomorphism, for theories of measurement this is too restrictive. However, the notion of homomorphism used here is actually closely connected with isomorphic imbeddability and the facts are explained in detail in Section II.
In some contexts we shall say that the class K is a theory of measurement of type s relative to R. Notice that a consequence of this definition is that if K is a theory of measurement, then so is every subclass of K closed under isomorphism. Moreover, the class of all systems imbeddable in members of K is also a theory of measurement.
In this connection see Sierpinski (1934, Section 7, pp. 141ff.) in particular Proposition C75, where of course different terminology is used.
It is sufficient here to consider a relational system isomorphic to the ordering of the ordinals of the second number class or to the lexicographical ordering of all pairs of real numbers.
A simple ordering is imbeddable in Re, if and only if it contains a countable dense subset. For the exact formulation and a sketch of a proof, see Birkhoff (1948, pp. 31–32, Theorem 2).
The word ‘countable’ means at most denumerable, and it refers to the cardinality of the domains of the relational systems.
See Luce ( 1956, Section 2, p. 181). The axioms given here are actually a simplification of those given by Luce.
The authors are indebted to the referee for pointing out the work by Hailperin (1954), which suggested this general definition.
Te proofs of both these facts about H are very similar to the corresponding proofs in Suppes and Winet (1955; Article 8 in this volume).
Article 8 in this volume.
Essentially this example was first given in another context by Herman Rubin to show that a particular set of axioms is defective.
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© 1969 Springer Science+Business Media Dordrecht
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Suppes, P. (1969). Foundational Aspects of Theories of Measurement. In: Studies in the Methodology and Foundations of Science. Synthese Library, vol 22. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3173-7_4
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DOI: https://doi.org/10.1007/978-94-017-3173-7_4
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