To Resolve the Problems of Judgment Aggregation from the Perspective of Logic

Abstract

It is a research hotspot to study the social choice theory from the perspective of logic in recent years. The judgment aggregation model based on logic supports a new way for the research of social choice theory. We attempt to analyze judgment aggregation dilemmas based on the judgment aggregation model, explore different aggregation problem solutions. KEYWORD: Logic; Judgment aggregation; Escape routs International Conference on Social Science, Education Management and Sports Education (SSEMSE 2015) © 2015. The authors Published by Atlantis Press 917 ΦA. We sometimes call a member of A an agenda item. (2) consistent: A set A⊆L (L) is consistent unless A⊦LΦΦ for some ΦL(L). (3) complete: A set A⊆L (L) is complete if either ΦA or ΦA for every ΦA which does not start with negation. (4) judgment set: A judgment set is a complete and consistent subset A⊆A. (5) judgment profile: An judgment profile is an ntuple (A1,...,An) , where Ai(1≤i≤n) is the individual judgment set of agent i. J(A,L) denotes the set of all individual (complete and L-consistent) judgment sets over A. When J(A,L) n , we use i to denote the ith element of , i.e., agent i's individual judgment set in judgment profile . (6) judgment aggregation rule: A judgment aggregation rule is a function f: J(A,L) n J(A,L). The judgment aggregation rule should usually meet the following conditions: (1) Independence: The group’s judgment on any agenda item in an agenda only depends on individuals’ judgment on this agenda item and has nothing to do with other factors. (2) Unanimity: If all individuals agree with certain agenda item in the agenda, then the group holds the same view. (3) Non-dictatorship: We should not ignore others’ different judgment and make a fixed individual’s judgment always be the group’s. Discursive Dilemma: If there are at least three subjects need to make a judgment on at least two different formulas p, q, and pq, then we may find out an inconsistent group judgment if we adopt majority vote aggregation rule. Similar to Arrow’s Theorem and many other impossibility theorems, there are also several impossibility theorems in judgment aggregation. Because of the advantages of judgment aggregation which preference aggregation does not have, the social choice theory researchers gradually focus on judgment aggregation. Judgment aggregation model based on logic explores a new road for researchers on social choice theory and also provides new ideas to solve problems in social choice theory. 3 ESCAPE ROUTES FROM THE PROBLEMS OF JUDGMENT AGGREGATION The impossibility theorems in judgment aggregation not only prove that there is no perfect rule for aggregation, but also make us realize that we should relax these requirements to find an aggregation rule which can be accepted. 3.1 Restricting the domain The impossibility theorems in judgment aggregation take the aggregation rule to be defined on the domain of all judgment profiles which consist of any consistent and complete individual judgment sets and place no restriction. Dietrich and List propose that whether there is admissible restriction of domain to make sure that majority judgment of agenda items is consistent.[10] Some restrictions, which are discussed, are similar to some classic domain restrictions in preference aggregation(such as single-peakedness). The method of unidimensionally aligned is based on the idea to restrict the domain. A judgment profile is unidimensionally aligned if the individuals in N can be ordered from left to right such that the individuals who agree p for every proposition p in the agenda. Consider the following example, the agenda contains the following propositions and their negations: p: Carbon emissions are above a given threshold; pq: If Carbon emissions are above a given threshold, there will be a critical temperature increase; q: There will be a critical temperature increase. As is shown in Table 2: the result is unidimensionally aligned. Any proposition in the agenda is agreed by majority if and only if it is agreed by the median individual on the given leftright order. Here the majority judgments(group judgments) are those of individual 3. For any given judgment set of all individuals consistent, thus the majority judgment set is consistent. The method of unidimensionally aligned is strong domain-restriction condition in judgment aggregation. Only if the number of individuals is odd number, this method may be effective and must be invalid when the number of individuals is even number. Table 2 A unidimensionally aligned judgment profile Ind.1 Ind.2 Ind.3 Ind.4 Ind.5