The Logic of Qualitative Probability

In this paper we present a theory of qualitative probability. Work in the area goes back at least to de Finetti. The usual approach is to specify a binary operator with φ ψ having the intended interpretation that φ is not more probable than ψ. We generalise these approaches by extending the domain of the operator from the set of events to the set of finite sequences of events. If Φ and Ψ are finite sequences of events, Φ Ψ has the intended interpretation that the summed probabilities of the elements of Φ is not greater than the sum of those of Ψ. We provide a sound and complete axiomatisation for this operator over finite outcome sets, and show that this theory is sufficiently powerful to capture the results of axiomatic probability theory. We argue that our approach is simpler and more perspicuous than previous accounts. As well, we prove that our approach generalises the two major accounts for finite outcome sets.