Additive and Polynomial Representations by David H. Krantz, R. Duncan Luce, Patrick Suppes, Amos

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, verified the formal foundations for dimension, justifying the task of numbers to things by way of their structural correspondence.
Volume I introduces the unique mathematical effects that serve to formulate numerical representations of qualitative buildings. 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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One demand is for the axioms to have a direct and easily understood meaning in terms of empirical operations, so simple that either they are evidently empirically true on intuitive grounds or it is evident how systematically to test them. In part, simplicity and clarity of meaning lie in the eye of the beholder. By the time you finish this book, some axioms may be clear which now might leave you aghast. Axiomatization is partly a search for simplicity and partly a restructuring of the axiomatizer's cognitive processes so that more things seem simple.

CHOOSING AN AXIOM SYSTEM 25 into a particular numerical structure is very heterogeneous and may include rather unusual and difficult-to-describe or pathological instances as well as more regular ones. Thus, the conditions which completely characterize such a set of structures are probably too complicated to be useful; in any event, they are not known. More systematic results, clarifying the above informal statement, are found in Chapter 9, Measurement Inequalities, and Chapter 18, Axiomatizability.

Little needs to be added here except to note that the two factor aspect of the objects is more likely to lead to violations of transitivity. These will occur in any situation where attention is sometimes exclusively focused on one factor, sometimes on the other. Axioms 3 and 4 are empirical laws of a very interesting type. We call them independence laws. Axiom 3 asserts that the ordering of ^-effects is independent of the choice of a fixed level in A2, which we abbreviate by saying that Ax is independent of A2.

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