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.

4The 'doctrinal paradox'

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

6How general is this problem?

The problem of majority inconsistency can arise as soon as the set of propositions (and their negations) on which judgments are to be made exhibits a simple combinatorial property: it has a 'minimally inconsistent' subset of three or more propositions (Dietrich and List 2007b, Nehring and Puppe 2007b). A set of propositions is called 'minimally inconsistent' if it is inconsistent and every proper subset of it is consistent. In the example, a minimally inconsistent set with these properties is {p, if p then q, not q}. As soon as there exists at least one minimally inconsistent subset of three or more propositions among the proposition-negation pairs on the agenda, combinations of judgments such as the one in Box 2 become possible, for which the majority judgments are inconsistent. Indeed, as explained separately below, Condorcet's classic paradox of cyclical majority preferences is an instance of this general phenomenon, which Guilbaud (1952) described as the 'Condorcet effect'.  

4Condorcet's paradox