 Gärdenfors (2006) and more generally Dietrich and List (2008) and Dokow and Holzman (2010b) have shown that if—while relaxing completeness—we require collective judgment sets to be deductively closed (i.e., for any \(p \in X\) entailed by \(J\), it must be that \(p \in J)\), we face an impossibility result again. For the same agendas that lead to the impossibility result reviewed in  Section 5.3, there exists no judgment aggregation rule satisfying universal domain, collective consistency and deductive closure, independence, unanimity preservation, and non-oligarchy. An aggregation rule is called oligarchic if there is an antecedently fixed subset \(M \subseteq N\) (the ‘oligarchs’) such that, for any profile \(\langle J_1, J_2 , \ldots ,J_n\rangle\), \(J = \{p \in X : p \in J_i\) for all \(i \in M\}\). Unanimity rule and dictatorships are special cases with \(M = N\) and \(M = \{i\}\) for some \(i \in N\), respectively.