Universal algebra for general aggregation theory: Many-valued propositional-attitude aggregators as MV-homomorphisms

Abstract

This article continues Dietrich and List's (2010) work on propositional-attitude aggregation theory, which is a generalized unification of the judgement aggregation and probabilistic opinion-pooling literatures. We first propose an algebraic framework for an analysis of (many-valued) propositional-attitude aggregation problems. Then we shall show that systematic propositional-attitude aggregators can be viewed as homomorphisms—algebraically structure-preserving maps—in the category of Chang's (1958a, Trans. Am. Math. Soc., 88, 467–490) MV-algebras. (Proof idea: Systematic aggregators are induced by maps satisfying certain functional equations, which in turn can be verified to entail homomorphy identities.) Since the 2-element Boolean algebra as well as the real unit interval can be endowed with an MV-algebra structure, we obtain as natural corollaries two famous theorems: Arrow's theorem for judgement aggregation as well as McConway's (1981, J. Am. Stat. Assoc., 76, 410–414) characterization of linear opinion pools. Conceptually, this characterization of aggregators can be seen as justifying a certain structuralist interpretation of social choice. Technically and perhaps more importantly, it opens up a new methodology to social-choice theorists: the analysis of general aggregation problems by means of universal algebra.