Judgment Aggregation. Gabriella Pigozzi

if ψ”) belong to L, and nothing else belongs to L.² We say that a formula is positive if its outermost connective is not a negation (e.g., p → q, ¬p ∨ q).³

      The meaning of a formula φ ∈ At is its truth value as specified by a valuation function V : L → {0, 1} where 0 stands for “false” and 1 for “true.” Each valuation V is an extension of some valuation V : At → {0, 1} of truth values to atoms, which obeys the following constraints:

. These constraints define the semantics of the logical connectives introduced above. When V(φ) = 1 (respectively, V(φ) = 0) we will often write V ⊨ φ (respectively, V ⊭ φ). If Φ is a set of formulae, we write V ⊨ Φ to express that for all φ ∈ Φ, V ⊨ φ, i.e., all formulae in φ are made true by V.

      We conclude with some auxiliary terminology concerning special classes of propositional formulae. A formula φ is a tautology if, for any valuation V, V ⊨ φ; it is a contradiction if, for any valuation V, V ⊭ φ; it is contingent if it is neither a tautology nor a contradiction. A set of formulae Φ is consistent if it has a model, that is, if there exists a valuation V, such that V ⊨ φ for each φ ∈ Φ; a formula φ is a logical consequence of a set of formulae Φ (in symbols, Φ ⊨ φ) if for every valuation V such that V ⊨ Φ, it is the case that V ⊨ φ

       Agendas

      With the machinery of propositional logic in place, we can frame the problem of the aggregation of judgments simply as a set of individuals or agents that are called to decide upon a given set of issues:

      Definition 2.1 Judgment aggregation problem. Let L be a propositional language on a given set of atoms At. A judgment aggregation problem for L is a tuple J = 〈N, A〉 where:

      • N is a finite non-empty set;

      • A ⊆ L such that

for some finite I ⊆ L which contains only positive contingent formulae.

      Set N is the set of individuals (or agents or voters). A is called the agenda and I is called the set of issues or the pre-agenda of A. An agenda based on a set of issues I will often be denoted ±I. Given an agenda A, we denote its pre-agenda by [A].⁴

      Intuitively, one can view a judgment aggregation problem as what specifies the space of possible situations in which the individuals in N have to reach some collective decision about the issues in I. An agenda A = ±I represents then all possible attitudes that can be assumed toward the issues in I. In the framework we are going to work with, such attitudes are of only two types: acceptance and rejection. The agenda is therefore a set of formulae which is closed under negation, i.e., ∀φ: φ ∈ A iff ¬φ ∈ A, and where double negations are eliminated. To make an example, the doctrinal paradox agenda

expresses all the acceptance/rejection attitudes that one individual can assume over the set of issues {p, q, p ∧ q}.

2.1.2 JUDGMENT SETS AND PROFILES

      Given a judgment aggregation problem, individuals are asked to express their opinions on the formulae of the agenda by accepting some and rejecting others. These opinions are called judgment sets and are defined as follows:

Definition 2.2 Judgment sets and profiles. Let J = 〈N, A〉 be a judgment aggregation problem. A judgment set for J is a set of formulae J ⊆ A such that:

      • J is consistent;

      • J is complete, i.e., ∀φ ∈ A, either φ ∈ J or ¬φ ∈ J.

      Instead of φ ∈ J we will often use the notation J ⊨ φ to indicate that φ belongs to judgment set J.⁵ The set of all judgment sets is denoted

where

denotes the power-set function. A judgment profile

is an |N|-tuple of judgment sets. With Pi we denote the i^th entry of P, i.e., the judgment set of agent i in P. For φ ∈ A, we use Pφ to denote the set of individuals accepting φ in

. Finally, we denote with P the set of all judgment profiles. Abusing notation, we will sometimes indicate that a judgment set Ji belongs to a profile P by writing Ji ∈ P.

      So individuals express their opinions through sets of formulae of the agenda: the formulae contained in the set are the ones that are accepted by the individual, the ones belonging to the complement of the set are the ones that are rejected by the individual. The consistency and completeness criteria formalize a notion of ‘rationality’ for the views that might be held by individuals. Such views cannot be internally contradictory (consistency) and cannot abstain from accepting or rejecting any of the issues posed by the agenda (completeness).⁶

      Remark 2.3 Deductive closure A set of formulae Φ is deductively closed (w.r.t. agenda A) if any φ ∈ A that follows logically from Φ is also contained in it: if Φ ⊨ φ, then φ ∈ Φ. Since judgment sets are sets of formulae that are consistent and complete, they are also deductively closed. However, a set of formulae that is consistent and deductively closed is not necessarily complete. When working with judgment sets the two notations φ ∈ J (membership) and J ⊨ φ (consequence) can be seen as notational variants. However, when working with sets of formulae that are not judgment sets by the letter of Definition 2.2—in our context these will typically be sets of formulae accepted by a group of individuals—we will keep the two notations distinct.

2.1.3 AGGREGATION FUNCTIONS

      The judgment aggregation problem consists in the aggregation of the individuals’ judgment sets into one collective judgment set. The aggregation of individual judgments is viewed as a function:

Definition 2.4 Aggregation function. Let J = 〈N, A〉 be a judgment aggregation problem. An aggregation function for J is a function

. The output set f(P), where

, is sometimes denoted J. Set J is then called a collective set. A collective set J which is a judgment set is called a collective judgment set.

      So, an aggregation function takes as input a profile of consistent and complete subsets of the agenda (i.e., judgment sets) and outputs a subset of the agenda. Such subset is neither necessarily consistent nor necessarily complete. In other words, the collective set is not necessarily a judgment set. In view of our discussion of the doctrinal paradox and the discursive dilemma this should not come as a surprise: the output of an aggregation function might not be ‘rational’ in the sense in which individual judgment sets are.

      Remark