The logic of qualitative probability

Abstract

In this paper we present a theory of qualitative probability. The usual approach of earlier work was to specify a binary operator ⪯ on formulas with ϕ⪯ψ having the intended interpretation that the event expressed by ϕ is no more probable than that expressed by 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 combined probabilities of the elements of Φ are no greater than those of Ψ. A sound and complete axiomatisation for this operator over finite outcome sets is given. We argue that our approach is more perspicuous and intuitive than previous accounts. As well, we show that the approach is sufficiently expressive to capture the results of axiomatic probability theory and to encode rational linear inequalities. We also prove that our approach generalises the two major accounts for finite outcome sets.