A theory of belief is presented in which uncertainty has two dimensions. The two dimensions have a variety of interpretations. The article focusses on two of these interpretations.
The first is that one dimension corresponds to probability and the other to ''definiteness,'' which itself has a variety of interpretations. One interpretation of definiteness is as the ordinal inverse of an aspect of uncertainty called ''ambiguity'' that is often considered important in the decision theory literature. (Greater ambiguity produces less definiteness and vice versa.) Another interpretation of definiteness is as a factor that measures the distortion of an individual's probability judgments that is due to specific factors involved in the cognitive processing leading to judgments. This interpretation is used to provide a new foundation for support theories of probability judgments and a new formulation of the ''Unpacking Principle'' of Tversky and Koehler.
The second interpretation of the two dimensions of uncertainty is that one dimension of an event A corresponds to a function that measures the probabilistic strength of A as the focal event in conditional events of the form AjB; and the other dimension corresponds to a function that measures the probabilistic strength of A as the context or conditioning event in conditional events of the form CjA: The second interpretation is used to provide an account of experimental results in which for disjoint events A and B; the judge probabilities of AjรฐA,Bร and BjรฐA,Bร do not sum to 1.
The theory of belief is axiomatized qualitatively in terms of a primitive binary relation h on conditional events. (AjBhCjD is interpreted as ''the degree of belief of AjB is greater than the degree of belief of CjD:'') It is shown that the axiomatization is a generalization of conditional probability in which a principle of conditional probability that has been repeatedly criticized on normative grounds may fail.
Representation and uniqueness theorems for the axiomatization demonstrate that the resulting generalization is comparable in mathematical richness to finitely additive probability theory.