Probabilistic opinion pooling generalized Part two: Premise-based opinion pooling

We consider the classical problem of aggregating di¤erent individuals’probability assignments (opinions) over a -algebra of events. In practice, some events represent basic propositions, such as ‘it rains’or ‘CO2 emissions cause global warming’, while others represent combinations thereof, for instance disjunctions (unions) of basic events. It is plausible to treat the basic events as premises when forming collective opinions, i.e., to …rst form a collective opinion on each basic event based on the individuals’opinions on it, and then form collective opinions on other events by taking into account the collective opinions formed on basic events. We formally de…ne such premise-based opinion pooling (by a single axiom, independence on premises), and prove that it forces the collective opinion on any premise to be a linear combination of the individuals’opinions, as long as unanimities are respected and the premises are interconnected in a suitable sense. This result is obtained in di¤erent variants, some of which relax linearity to neutrality. The paper is the second of two self-contained, but technically related companion papers inspired from judgment aggregation theory.