This paper analyses the problem of aggregating judgments when strategic voters hold private information about which propositions are true and share a common preference for true collective judgments. We go beyond previous work by introducing logical interconnections between the propositions. A voter’s private information can be inconclusive. The goal is to determine the voting rules which lead to collective judgments that efficiently incorporate all private information. We characterize the (rare) situations in which such rules exist, as well as the nature of these rules.