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 'discursive dilemma' or the problem of majority inconsistency
Consider any set of propositions with some non-trivial logical connections; in a separate subsection below, I say more about the precise notion of 'non-triviality' required. Take, for instance, the following three propositions on which an expert panel may seek to make collective judgments:
p: Annual global carbon dioxide emissions are above threshold X.
if p then q: If annual global carbon dioxide emissions are above threshold X, then there will be a global temperature increase of at least 2 degrees Celsius over the next two decades.
q: There will be a global temperature increase of at least 2 degrees Celsius over the next two decades.
Suppose now that one third of the experts accept all three propositions, a second third accept p but reject each of if p then q and q, and the last third accept if p then q but reject p as well as q, as shown in Box 2.
Then each expert holds individually consistent judgments on the three propositions, and yet there are majorities for p, for if p then q and for not q, a logically inconsistent set of propositions. The fact that majority voting may generate inconsistent collective judgments is sometimes called the 'discursive dilemma', but it is perhaps best described as the problem of 'majority inconsistency'. For early discussions of the discursive dilemma, see Pettit (2001), List and Pettit (2002) and Brennan (2001).