Judgment Aggregation. Gabriella Pigozzi

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 assume that Prem = {p, p → q} and Conc = {q}, the premise-based rule (fpb) generates a collective judgment accepting all items, and the conclusion-based rule (fcb) rejects the conclusion q, and does not accept any other item of the agenda.

      We give two examples of quota rules. The first is a quota rule that requires majority over the premises and their negations, but requires a unanimous vote to collectively accept the positive conclusion and one individual to reject it. That is:

.⁹ This quota rule accepts both premises but rejects the conclusion. The second one requires majority on all atomic issues and unanimity on the implicative issues.¹⁰ That is:

and

.¹¹ This rule then accepts one premise but rejects the implicative premise and the conclusion. Figure 2.1 recapitulates the outputs just discussed.

      It is no accident that all aggregation rules in the above example either fail to yield a judgment set (all except fpb and ft″, whose output is consistent and complete) or output sets that are inconsistent with one another (fpb accepts q while fcb and ft″ reject it). The reasons for such failure are deep and we will probe them in Chapter 3. The remainder of the present chapter sets the stage for those investigations.

2.2 AGENDA CONDITIONS

      We introduce here three conditions on agendas, which capture the sort of logical interdependence possibly arising between their elements.

2.2.1 HOW INTERCONNECTED IS AN AGENDA?

      We define and illustrate the agenda conditions known as non-simplicity, even-negatability and path-connectedness. We also introduce the auxiliary notion of conditional entailment.

       Non-simplicity

      The first agenda condition is almost self-explanatory, and is usually referred to as non-simplicity [NP07].

Definition 2.7 Non-simple agendas. An agenda A is non-simple (NS) iff it contains at least one set X s.t.:

      • 3 ≤ |X|;

      • X is minimally inconsistent, i.e.:

      – X is inconsistent;

      – ∀Y S.T. Y ⊂ X : Y is consistent.

      An agenda is called simple if it is not non-simple.

      It is easy to see that agenda ±{p, q, p ∧ q} is non-simple as the set {p, q, ¬(p ∧ q)} is clearly minimally inconsistent. Notice that if X is minimally inconsistent then, for some φ ∈ X, it is not only the case that X − {φ} is consistent, but also that X − {φ} ⊨ ¬φ. Non-simplicity is the minimal level of complexity for an agenda to run into problems when attempting aggregation.

      On the other hand, we will see that if an agenda is simple then aggregations of a non-degenerate kind are possible. In fact, the propositionwise majority rule can be proven to be the unique aggregation function that satisfies some highly desirable properties.¹² By Definition 2.7, simple agendas are agendas where minimally inconsistent sets have cardinality of at most two.¹³ Examples are agendas whose issues consist of logically unrelated formulae (e.g., {p, q, r}),¹⁴ or agendas whose issues can be ordered by logical strength like ±{p, p ∧ q, p ∧ q ∧ r}.

       Conditional entailment

      From non-simplicity we move now to the related notion of conditional entailment [DL13a]. This will be needed later to define the condition of path-connectedness.

Definition 2.8 Conditional entailment. Let φ, ψ ∈ A. We say that φ conditionally entails ψ (notation: φ ⊨_c ψ) if for some (possibly empty) X ⊆ A, which is consistent with φ and with ¬ψ, {φ} ∪ X ⊨ ψ.

      Conditional entailment expresses that the acceptance of ψ follows from the acceptance of φ either directly—by logical consequence—or indirectly once a set of formulae X is also accepted, which is compatible with both the acceptance of φ and the rejection of ψ. Intuitively, the fact that φ conditionally implies ψ captures a specific dependency within the structure of the agenda whereby if, on the one hand, it is possible to accept both φ and the formulae in X or both X and ¬ψ, on the other hand, accepting φ and X would compel one to also accept ψ.

      A few observations are in order. If φ ⊨ ψ (i.e., ψ is a logical consequence of φ) then φ ⊨_c ψ since ψ follows from {φ} ∪ Ø. If an agenda contains φ ≠ ψ such that φ ⊨_c ψ, then the agenda must have been generated by a set of issues containing at least two formulae. We conclude with the following observation relating conditional entailment to the property of non-simplicity:

      Fact 2.9 Conditional entailment and NS Let A be an agenda and φ, ψ ∈ A. If (i) φ ⊨_c ψ and (ii) φ ⊭ ψ, then A satisfies NS.

      Proof. By (i), (ii) and Definition 2.8 it follows that there exists X ≠ Ø such that {φ} ∪ X ⊨ ψ and hence such that X ∪ {φ, ¬ψ} is inconsistent. By the compactness¹⁵ of propositional logic there exists a smallest non-empty X′ such that X′ ∪ {φ, ¬ψ} is inconsistent. Set X′ ∪ {φ, ¬ψ} is therefore minimally inconsistent and has cardinality bigger or equal to 3. Hence A satisfies NS.

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       Even negations

      The second agenda condition is known as even negatability or even number negations property, and is slightly more involved:

      Definition 2.10 Evenly negatable agendas. An agenda A satisfies the even negations condition (EN) iff:

      • A contains a