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

This paper continues Dietrich and List’s [2011] work on propositionalattitude aggregation theory, which is a generalised 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 C.C. Chang’s [1958] 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] characterisation of linear opinion pools. Conceptually, this characterisation 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.

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