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

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

All of the sciences — actual, organic, and social — have a necessity for quantitative dimension. This influential sequence, Foundations of Measurement, demonstrated 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 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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INTRODUCTION and many different axiom systems lead to measurement, the procedure for obtaining the numbers always reduces to one of three basic methods. 1 do give internally consistent numerical answers. In Chapters 3-9 the representation and uniqueness theorems are reduced to applications of the theorems in Chapter 2. 1 carefully to gain a good intuitive idea of how numerical scales are constructed. 1 is applied to the situation at hand. To understand this, you may find it useful to scan the statements of the theorems in Chapter 2.

In other cases, where a o b is not defined for all a and b, this kind of solvability is less well concealed. Nevertheless, it should always be considered as a specific assumption, for it plays much the same role in extensive measurement as does the other kind of solvability axiom in other measurement systems. 24 1. INTRODUCTION Axiom 5 for equally spaced, additive conjoint structures falls in none of the three classes, but in effect it is of the solvability type. 2, by choosing al9a2,bl9 then selecting b2 with (ax, b2) ~ {bx, a2).

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).

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