ABSTRACT
What is missing is that the partial operation ED is not defined for sufficiently many pairs of events. This can be partially rectified by letting X = X / -.t ,
~= ~ / -.t , and defining E9 by: for each A, B, C in X, A E9 B = C if and only if for some x in A, y in B, z in C, x EB y = z, and considering the totally ordered structure ~ = <X, ~, E9). ~ is very close to an extensive structure. Its primary lack is that local definability and Archimedean may not hold. However, rather plausible axioms in terms of the primitives ~ and U can be given that guarantee that ~ is an extensive structure. Such an extensive structure ~ in this paper will be called a qualitative probability structure. The interested reader should consult Luce [1965], or Krantz et al. [1971, Chapter 5] or Fine [1971a], [1971b] for a detailed axiomatization. It should also be noted that it is inherent in the nature of probability, which has 0 as a maximal element, that e must be a partial, not a closed, operation.
