Download e-book for iPad: 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

ISBN-10: 0124254012

ISBN-13: 9780124254015

All of the sciences — actual, organic, and social — have a necessity for quantitative dimension. This influential sequence, Foundations of Measurement, tested the formal foundations for dimension, justifying the project of numbers to things by way of their structural correspondence.
Volume I introduces the detailed 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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3. , c2), then (ax, d2) > φλ, d2). 4. If(cx, i/2) > (Ci, ό2), /Ae« (ί/χ, α2) > {dx, b2). Note that Axioms 1 and 2 can be combined into the statement that (A, > > is a weak order. DEFINITION 4. e 0« independent conjoint structure. Define relations >x on Ax and >2on A2 by: α ι ^ ι ^ι iff there exists c2 in A2 with {αλ, c2) > (bx, c2); a2 > 2 b2 iff there exists ct in Ax with {cx , a2) > (cx , b2). It is easy to show (using Axioms 1-4) that {A{, > t > is a weak order, / = 1,2 (Exercise 11). Axiom 3 is precisely what is needed in order to prove that > !

3 Necessary and Sufficient Axiom Systems The ubiquity of these nonnecessary restrictions may well seem puzzlingIt seems far more desirable to find axiom systems composed entirely of necessary axioms that are also sufficient to prove the desired representation and uniqueness theorems. Such an axiom system is said to be necessary and sufficient for the representation. The advantage lies in the exclusion of examples such as on p. 22 where the additive conjoint representation holds, but Axiom 5 is violated.

But any definition of standard sequence involves finding exact replicas of a given object or interval. In the presence of error, we are confronted with two choices. The first, which often is the solution adopted, is to equate the replicas by a method that is much more expensive, but much more precise, than the method that is used to make comparisons in the field. For example, a good meter stick, calibrated in millimeters, should have the property that, if a comparison object falls clearly between the marks x and x + 1 millimeters, then with high probability its true length actually lies between x and x + 1 millimeters.

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

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