Judgment aggregation: a short introduction

s from the speci…c decision problem and aggregation rule in question. My exposition of this model follows the formalism introduced in List and Pettit (2002) and extended beyond standard propositional logic by Dietrich (2007a). There is a …nite set of (two or more) individuals, who have to make judgments on some propositions.3 Propositions are represented by sentences from propositional logic or a more expressive logical language, and they are generally denoted p, q, r and so on. Propositional logic can express ‘atomic propositions’, which do not contain any logical connectives, such as the proposition that we can a¤ord a budget de…cit or the proposition that spending on education should be increased, as well as ‘compound propositions’, with the logical connectives not, and, or, if-then, if and only if, such as the proposition that if we can a¤ord a budget de…cit, then spending on education should be increased. Instead of propositional logic, any logic with some minimal properties can be used, including expressively richer logics such as predicate, modal, deontic and conditional logics (Dietrich 2007a). Crucially, the logic allows us to distinguish between ‘consistent’and ‘inconsistent’sets of propositions. For example, the set fp, q, p and qg is consistent while the sets fp, if p then q, not qg or fp, not pg are not.4 The set of propositions on which judgments are to be made in a particular decision problem is called the ‘agenda’. Formally, the ‘agenda’is de…ned as a non-empty subset of the logical language, which is closed under negation, i.e., if p is on the agenda, then so is not p.5 In the government example, the agenda contains the propositions p, if p then q, q and their negations. In the court example, it contains p, q, r and their negations, but here there is an additional 3The agenda characterization results discussed further below require three or more individuals. 4 In propositional logic, a set of propositions is ‘consistent’if all its members can be simultaneously true, and ‘inconsistent’otherwise. More generally, consistency is de…nable in terms of a more basic notion of ‘logical entailment’(Dietrich 2007a). 5For some formal results, it is necessary to exclude tautological or contradictory propositions from the agenda. Further, some results simplify when the agenda is assumed to be a …nite set of propositions. In order to avoid such technicalities, I make these simplifying assumptions (i.e., no tautologies or contradictions, and a …nite agenda) throughout this paper. To render …niteness compatible with negation-closure, I assume that double negations cancel each other out; more elaborate constructions can be given.