Axiomatizing the logic of comparative probability

Often where an axiomatization of an intensional logic using only finitely many axioms schemes and rulesof the simplest kind is unknown, one has a choice between an axiomatization involving an infinite family of axiom schemes and one involving nonstandard “Gabbay-style” rules. The present note adds another example of this phenomenon, pertaining to the logic comparative probability (“p is no more likely than q”). Peter Gärdenfors has produced an axiomatization involving an infinite family of schemes, and here analternative using a “Gabbay-style” rule is offered. Both axiomatizations depend on the Kraft-Pratt-Seidenberg theorem from measurement theory.