Judgment Aggregation. Gabriella Pigozzi

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      I.e., φ is collectively accepted iff all individuals accept it. We will refer to this rule as the propositionwise unanimity rule or simply as the unanimity rule.

       Quota rule:

      where

      Formula 2.3 defines the class of all propositionwise threshold-based rules. Clearly, the propositionwise majority rule is a particular quota rule whose threshold has been fixed at

      It must be noted that the selection of the thresholds has an impact on the ‘rationality’ of the collective set. For instance, it is not difficult to see that the unanimity rule might return incomplete collective sets, and that a uniform quota rule imposing a common threshold lower than

      are necessary and sufficient conditions for the collective set to be complete (i.e., to contain at least one of φ or ¬φ, Formula 2.4) and, respectively, to be such that it never contains both a formula and its negation (i.e., to contain at most one of φ and ¬φ, Formula 2.5). This latter property is usually referred to as weak consistency.

      The class of all quota rules has been studied extensively in [DL07b]. We will come back to the majority rule in much more detail later in Chapter 3, and to quota rules as possible escape routes to some of the impossibility results of judgment aggregation in Chapters 4 and 5.⁸

       Premise- and conclusion-based rules

      We have already encountered the premise- and conclusion-based rules in Section 1.2. Here we give a more precise formulation of them.

       Premise-based rule:

      where: Prem ⊆ A consists of the subagenda containing the issues that are considered premises in the aggregation, and their negations; Conc ⊆ A consists of the subagenda containing the issues that are considered conclusions in the aggregation, and their negations; Prem and Conc are a partition of A; and PPrem (respectively, P^Conc) denotes the profile obtained from the restrictions of the judgment sets to the formulae in Prem (respectively, Conc). I.e., φ is collectively accepted iff it is a premise and it has been voted by the majority of the individuals or it is a conclusion entailed by the premises accepted by the majority.

       Conclusion-based rule:

      where PConc is as for the premise-based rule. I.e., φ is collectively accepted iff it is a conclusion and it has been voted by the majority of the individuals.

      Intuitively, premise- and conclusion-based rules apply propositionwise aggregation, via the majority rule, only to specific subsets of the agenda, viz., its premises or its conclusions. They have played a pivotal role in the development of the theory of judgment aggregation, and much literature has been dedicated to their analysis (see, for instance, [NP06, DM10]). We will come back to them at several places in the remaining of the book.

       An example

      It is now time to illustrate the workings of all the above rules side by side. We do that with yet another variant of the doctrinal paradox:

Example 2.6 Let A = ±{p, p → q, q}, and attach the following intuitive reading to the three issues [DL07a]:

      Figure 2.1: An illustration of several aggregation rules from Example 2.6.

      p: Current CO₂ emissions lead to global warming.

      p → q: If current CO₂ emissions lead to global warming, then we should reduce CO₂ emissions.

      q: We should reduce CO₂ emissions.

      The profile consisting of the three judgment sets

and

, once aggregated via propositionwise majority (fmaj), gives rise to an inconsistent collective judgment set

. Propositionwise unanimity (fu) does not accept any of the items of the agenda. If we СКАЧАТЬ