Qualitative probability, developed by Bruno de Finetti (1937) and Leonard Savage (1954), asks when a qualitative ordering of events by "more likely than" can be represented by a numerical probability function. This connects measurement theory to the foundations of probabilistic reasoning and decision under uncertainty.
Axioms for Qualitative Probability
2. A ≽ ∅ for all events A (non-negativity)
3. Ω ≻ ∅ (non-triviality)
4. If A ∩ C = B ∩ C = ∅, then A ≽ B ⟺ A ∪ C ≽ B ∪ C
The fourth axiom (de Finetti's additivity condition) is the qualitative analog of finite additivity: it states that the relative likelihood of two disjoint events is unaffected by adding a third disjoint event to both. This is the simplest condition for representation, but it is not sufficient for countably additive probability on infinite spaces.
Savage's Framework
Savage (1954) embedded qualitative probability within a richer framework of decision under uncertainty, deriving both probability and utility simultaneously from axioms on preferences over acts. His sure-thing principle (P2) plays a role analogous to the independence axiom in expected utility theory. Savage's framework remains the most complete axiomatic foundation for Bayesian decision theory.