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By David H. Krantz, R. Duncan Luce, Patrick Suppes, Amos Tversky

All of the sciences — actual, organic, and social — have a necessity for quantitative size. This influential sequence, Foundations of Measurement, validated the formal foundations for dimension, justifying the project of numbers to things by way of their structural correspondence.
Volume I introduces the particular mathematical effects that serve to formulate numerical representations of qualitative constructions. quantity II extends the topic towards geometrical, threshold, and probabilistic representations, and quantity III examines illustration as expressed in axiomatization and invariance.

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With only the ordinal uniqueness of Theorem 3, however, there is no suitable notion of approximation, and so the definition of φ is not very satisfactory. Finally, we note that Debreu (1954, Lemma II) has shown that the orderpreserving function φ of Theorem 2 can always be constructed so as to be continuous in every natural topology on the simple order <Λ, >>. This is easy to prove once all the required definitions are given, but we shall not do so since the ideas involved are not used elsewhere in this book.

Of course, we can select a new set of objects that exhibits a refined version of the same phenomenon on the new balance. Nevertheless, the pattern of improved approximations is such that we elect to retain the assumption of an underlying weak ordering of weight and to say that, in any particular set of observations, there are systematic errors due to imperfections in the observational situation. The existence of error also has implications for the construction of measurement scales. 2, such constructions most often involve a counting-of-units procedure based on some 28 1.

5) and otherwise we describe in nontechnical terms some of the types of axioms typically found in measurement systems. 1 Necessary Axioms The previous discussion suggests that much of the effort in analyzing measurement goes into finding a good axiom system. It should be clear by now that at least one axiom is required to construct a representation. Specifically, if > is an arbitrary binary relation on A, then there need not be any homomorphism of {A, >> into is transitive (for if a > b and b > c, then φ(α) > φφ) > φ(ς)\ hence φ(α) ^ φ(ϋ), which 22 1.

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