Abstract
Imagine a group of individuals faces a yes-no type question whose answer is logically determined by multiple premises. There are two salient types of procedures to aggregate individual judgments -- the ``premise-based way'' (PBW) and the ``conclusion-based way'' (CBW). We derive necessary and sufficient conditions under which two procedures are universally ordered. If (and only if) a decision problem takes a ``conjunctive'' form, PBW derives a positive collective judgment (i.e., ``yes'') whenever CBW does. Furthermore, if we replace ``conjunctive'' with ``disjunctive'' in the previous line, PBW derives a negative collective judgment (i.e., ``no'') whenever PBW does. These observations highlight the fact that these two procedures are a mathematical dual of each another. Asymptotic properties are also studied. Under classical Condorcetian assumptions, PBW ensures the probability that the voting outcome is correct converges to one as the size of a group tends to infinity, whereas this holds for CBW only if an additional condition is satisfied.
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