Paradoxes

As mentioned in the introduction, the recent interest in judgment aggregation was sparked by the observation that majority voting has some surprising properties, which have become known as the 'doctrinal paradox' and 'discursive dilemma'. These, in turn, are related to a much older paradox of majority voting: 'Condorcet's paradox'. A good part, but by no means all, of the current literature on judgment aggregation still revolves around the question of what the best response to the doctrinal paradox and discursive dilemma is. However, the theory of judgment aggregation is not confined to, and in fact goes well beyond, the study of voting paradoxes.

6The 'doctrinal paradox'

The 'doctrinal paradox' is nicely illustrated by Kornhauser and Sager's (1986) original example from the area of jurisprudence (the name 'doctrinal paradox' was introduced in Kornhauser 1992). Suppose a three-member collegial court has to reach a verdict in a breach-of-contract case. The court is required to make judgments on three propositions:

p: The defendant was contractually obliged not to do a particular action.

q: The defendant did that action.

r: The defendant is liable for breach of contract.

According to legal doctrine, propositions p and q are jointly necessary and sufficient for proposition r. Suppose now that the three judges are divided in their judgments, as shown in Box 1. The first judge holds both p and q to be true, and hence holds r to be true as well. The second holds p to be true, but q to be false, and consequently also holds r to be false. The third holds q to be true, but p to be false, and thus holds r to be false too. This seems a sufficiently realistic scenario of disagreement. But what judgments should the court as a whole make?

If the court takes a majority vote on proposition r - the 'conclusion' - the outcome is the rejection of this proposition: a 'not liable' verdict. But if majority votes are taken on each of p and q instead - the 'premises' - then both of these propositions are accepted and hence the relevant legal doctrine requires that r should be accepted as well: a 'liable' verdict. The court's decision thus appears to depend on which aggregation rule it uses. If it uses the first of the two approaches, the so-called 'conclusion-based procedure', it will reach a 'not liable' verdict. If it uses the second, the 'premise-based procedure', it will reach a 'liable' verdict. The 'doctrinal paradox' consists in the fact that the conclusion-based and premise-based procedures may yield opposite outcomes for the same combination of individual judgments. (Kornhauser and Sager referred to 'case-by-case' and 'issue-by-issue' voting, respectively.)

But the example allows us also to make a more general observation, not made explicit in the original jurisprudence literature on this subject. Relative to the given legal doctrine - which states that r is true if and only if both p and q are true - the majority judgments across the three propositions are inconsistent. In precise terms, the set of propositions accepted by a majority, namely {p, q, not q}, is logically inconsistent relative to the constraint that r if and only if p and q. This problem - namely that majority voting may yield an inconsistent set of accepted propositions - generalizes well beyond this example and does not depend on the presence of any legal doctrine or other exogenous constraint; nor does it depend on the partition of the relevant propositions into premises and conclusions. We can describe this as a more general problem of 'majority inconsistency' (as discussed in Pettit 2001, Brennan 2001, List and Pettit 2002).

4The 'discursive dilemma' or the problem of majority inconsistency

4How general is this problem?

4Condorcet's paradox