  Binarizing belief aggregation addresses how to rationally aggregate
individual probabilistic beliefs into collective binary beliefs. Similar to the
development of judgment aggregation theory, formulating axiomatic requirements,
proving impossibility theorems, and identifying exact agenda conditions of
impossibility theorems are natural and important research topics in binarizing
belief aggregation. Building on our previous research on impossibility
theorems, we use an agenda-theoretic approach to generalize the results and to
determine the necessary and sufficient level of logical interconnection between
the issues in an agenda for the impossibility theorems to arise. We demonstrate
that (1) path-connectedness and even-negatability constitute the exact agenda
condition for the oligarchy result stating that binarizing belief aggregation
satisfying proposition-wise independence and deductive closure of collective
beliefs yields the oligarchies under minor conditions; (2)
negation-connectedness is the condition for the triviality result obtained by
adding anonymity to the oligarchy result; and (3) blockedness is the condition
for the impossibility result, which follows by adding completeness and
consistency of collective beliefs. Moreover, we compare these novel findings
with existing agenda-theoretic characterization theorems in judgment
aggregation and belief binarization.
