L. J. Savage [The foundations of statistics (1972; Zbl 0276.62006)] has studied qualitative probability relations in order to justify the axioms of numerical probability from purely ordinal arguments, for the purpose of modelling subjective uncertainty judgements. Such an attempt is made here for possibility measures, in a finite setting. Axioms are suggested which a binary relation “at least as possible as” must satisfy so that the only numerical uncertainty measures compatible with such a relation are possibility measures.
Moreover, a general notion of belief structure is proposed in order to encompass both probability and possibility theories within the same qualitative setting. This is done by relaxing the qualitative “additivity” axiom. The set functions agreeing with such belief structures satisfy a general decomposability property previously suggested by H. Prade [IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 260-283 (1985; Zbl 0565.68089)] and the author.