Aggregating credences into beliefs

Hybrid belief aggregation addresses aggregation of individual probabilistic beliefs into collective binary beliefs. In line with the development of judgment aggregation theory, our research delves into the identification of precise agenda conditions associated with some key impossibility theorems in the context of hybrid belief aggregation. We determine the necessary and sufficient level of logical interconnection between the propositions in an agenda for some key impossibilities to arise. Specifically, we prove three characterization theorems about hybrid belief aggregation: (i) Precisely the path-connected and pair-negatable agendas lead to the "oligarchy result" — only oligarchic rules satisfy universal domain, proposition-wise independence, respect for unanimity, and deductive closure of collective beliefs. (ii) Precisely the negation-connected agendas lead to the "triviality result" — only unanimity rules satisfy those conditions as well as anonymity. (iii) Precisely the blocked agendas lead to the "non-existence result" — no rules satisfy those conditions as well as completeness and consistency of collective beliefs. Furthermore, we compare these novel findings with existing agenda-theoretic characterization theorems in the domains of judgment aggregation and belief binarization.