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

ISBN-10: 0124254012

ISBN-13: 9780124254015

All of the sciences — actual, organic, and social — have a necessity for quantitative size. This influential sequence, Foundations of Measurement, demonstrated the formal foundations for size, justifying the task of numbers to things by way of their structural correspondence.
Volume I introduces the designated 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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Extra resources for Additive and Polynomial Representations

Example text

Independence laws play a very prominent role in the discussion of additive and polynomial conjoint measurement (Chapters 6 and 7), in utility measurement (Chapter 8), and in multidimensional proximity measurement (Chapter 13). The final axiom, 5, is of a very different nature. It is hardly ever satisfied by accident. If Ax represents a finite set of levels of some factor and A2 represents a different factor, there is no reason whatsoever to suppose that when we move from {bx, b2) to the next higher level of Ax, say (ax, b2), the effect is exactly the same as when we move to the next higher level of A2, say φλ, α2).

Moreover, such systematic rejection tells us a good deal about what is wrong, and it may suggest either other measurement schemes that will work or a different choice of basic factors. Thus, one value of a satisfactory axiomatization is that it provides a set of relatively simple, conceptually distinct, empirically testable conditions to be tested. The problems of error and of selection of objects to be tested have no easy solutions, however ; they must be tackled with whatever experimental and statistical tools are available.

22 where the additive conjoint representation holds, but Axiom 5 is violated. As it happens, there are very few examples of what we consider satisfactory necessary and sufficient axiomatizations. Why is this? 4. 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.

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

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