A generalised model of judgment aggregation

Abstract

Abstract. The new …eld of judgment aggregation aims to merge many individual sets of judgments on logically interconnected propositions into a single collective set of judgments on these propositions. Judgment aggregation has commonly been studied using classical propositional logic, with a limited expressive power and a problematic representation of conditional statements ("if P then Q") as material conditionals. In this methodological paper, I present a simple uni…ed model of judgment aggregation in general logics. I show how many realistic decision problems can be represented in it. This includes decision problems expressed in languages of classical propositional logic, predicate logic (e.g. preference aggregation problems), modal or conditional logics, and some multi-valued or fuzzy logics. I provide a list of simple tools for working with general logics, and I prove impossibility results that generalise earlier theorems.

A measure of distance between judgment sets

“…List and Pettit (2002) arrived at the first systematic result in this area by showing that no aggregation rule applied to an agenda containing certain propositions can satisfy four conditions. More general results have been obtained by Dietrich (2006Dietrich ( , 2007, Dietrich andList (2009, 2010), Dokow and Holzman (2010) and Nehring and Puppe (2010) amongst others. 3 A condition that has been central in the literature, particularly when deriving impossibility theorems, is the 'independence' condition.…”

Probabilistic Opinion Pooling

“…In recent years, there has been much work on the aggregation of binary opinions, where a group seeks to assign the values 'true'/'false'or 'yes'/'no'to a set of propositions, based on the individuals'assignments -a problem now known as judgment aggregation (e.g., List and Pettit 2002;Dietrich 2007;Dietrich and List 2007;Nehring and Puppe 2010;Dokow and Holzman 2010; for a recent review, see List 2012). Truth-value assignments, especially in classical propositional logic, can be viewed as degenerate probability assignments (restricted to the values 0 and 1).…”

Majority voting on restricted domains

“…We consider a group of individuals N = f1; 2; : : : ; ng (n 2) making judgments on some propositions represented in logic (Dietrich [4], generalizing List and Pettit [24], [25]). …”

“…3 As a notational convention, we cancel double-negations in front of propositions in X. 4 Further, for any Y X, we write Y = fp; :p : p 2 Y g to denote the (single-)negation closure of Y . Judgment sets.…”

“…But so far the only domain-restriction condition known to guarantee consistent majority judgments is List's unidimensional alignment ( [21], [22]), a global non-product domain condition based on orders of individuals. Here we use Dietrich's generalized model [4], which allows propositions to be expressed in rich logical languages. We include some proofs in the main text, others in the appendix.…”