Comparative probability for conditional events : a new look through coherence

We study, from the standpoint of coherence, comparative probabilities on an arbitrary family E of conditional events. Given a binary relation less than or equal to, coherence conditions on less than or equal to are related to de Finetti's coherent betting system: we consider their connections to the usual properties of comparative probability and to the possibility of numerical representations of less than or equal to. In this context, the numerical reference frame is that of de Finetti's coherent subjective conditional probability, which is not introduced (as in Kolmogoroffs approach) through a ratio between probability measures. Another relevant feature of our approach is that the family E need not have any particular algebraic structure, so that the ordering can be initially given for a few conditional events of interest and then possibly extended by a step-by-step procedure, preserving coherence.