Approximate probability and expectation of gambles

Abstract

scientific procedures of measurement have such an approximate character and there is consequently an obvious need to capture in explicit axiomatic form the implicit assumptions underlying actual procedures. The second reason is of more general philosophical interest. The historical tendency since the work of Simpson, Lagrange, and Laplace in the eighteenth century is to treat the approximate character of measure ment procedures as due to errors of measurement arising either from 'systematic' or 'random' effects. This concept of error matches up with Laplacean determinism to give weight to a theory of exact results of mea surement being in principle, if not in practice, possible. The fairy-tale quality of this view of measurement in physics should be evident enough two hundred years after its first telling. Even if it is not, I shall not attempt to make it evident here, but restrict myself to the more special arena of personal beliefs and decisions and their measurement. It is surely surprising if not literally paradoxical that the view of ideal measurement characteristic of classical determinism has been carried over almost untouched to modern theories of partial belief and decision. It is, for example, a characteristic requirement of Savage's (1954) theory of decisions that the axioms on preferences between decisions be strong enough to lead to a numerical representation in terms of a unique proba bility distribution on beliefs and a utility function on consequences unique up to a linear transformation. As in the case of classical physics there is a mythology of ideal measure ment leading to unique numerical results which is widespread in con temporary theories of belief and decision. But the most casual common