Necessary and sufficient conditions for existence and uniqueness of weak qualitative probability structures
Michael A. Cohen · Journal of Mathematical Psychology · 1991
This paper provides necessary and sufficient conditions for the existence and uniqueness of a weak qualitative probability representation of an arbitrary qualitative probability space. Should a weak probability representation exist, necessary and sufficient conditions are obtained for each weak probability representation to also strongly represent the qualitative probability space. These are equivalent to the Archimedean conditions of P. Suppes and M. Zanotti (1976, Journal of Philosophical Logic, 5, 431–438). In addition, upper and lower bounds for the probability of a specific set consistent with a given qualitative probability relation are obtained. These results are obtained by embedding the qualitative probability space in an ordered vector space and studying the properties of positive linear functionals on this space. It is shown that if a weak qualitative probability representation of an arbitrary space is unique, then the set of weak probability representations of finite subalgebras weakly converge to the representation of the entire space. We briefly study uniqueness of finite qualitative probability spaces. For this case, uniqueness is entirely determined by the equivalences of the system.