From Degrees of Belief to Binary Beliefs: Lessons from Judgment-Aggregation Theory

Abstract

What is the relationship between degrees of belief and (all-or-nothing) beliefs? Can the latter be expressed as a function of the former, without running into paradoxes? We reassess this “belief-binarization” problem from the perspective of judgmentaggregation theory. Although some similarities between belief binarization and judgment aggregation have been noted before, the literature contains no general study of the implications of aggregation-theoretic impossibility and possibility results for belief binarization. We seek to fill this gap. At the centre of this paper is an impossibility theorem showing that, except in simple cases, there exists no belief-binarization rule satisfying four baseline desiderata (“universal domain”, “belief consistency and completeness”, “propositionwise independence”, “certainty preservation”). We show that this result is a corollary of the judgment-aggregation variant of Arrow’s impossibility theorem and explore several escape routes from it.