Formal Models and Axiomatic Measurement

Abstract

The paper addresses questions related to the interpretation of numerical representations of structures satisfying certain axiomatic properties. It is argued that a statement like ‘situation A is two times as risky as situation B’, Is not meaningful as It depends on the particular representation chosen out of many ones which are axiomatically equivalent, even though each of them entails a ratio scale. Instead, it is meaningful that ‘the risk of A is equivalent to the risk of (B and B)’. Furthermore, it is demonstrated that a different equivalent representation may imply a property which was not axiomized, but which is empirically meaningful and testable. Finally it is shown that a numerical representation which satisfies the axioms of an empirical structure, may not be meaningful as it may entail empirically contradictory implications, due to, e.g., violation of dimensional invariance.

This is a modified and extended version of a paper earlier published in Dutch as: Formele modellen en axiomatische meetmethoden, of: wat betekent een kwantitatieve formulering?, In H.F.M. Crombag, L.J.Th. van der Kamp, & C.A.J. Vlek (Red.) (1987): De Psychologie Voorbij, (Bundel aangeboden aan prof. dr.J. P. van der Geer bij zijn afscheid als hoogleraar methodenleer). Lisse: Swets & Zeitlinger. I am grateful to R.Duncan Luce for pointing out a weakness in my exposition of Pollatsek & Tversky’s Theory of Risk in an earlier version of this paper.

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