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.

想象一下，一群人面对一个是不是类型的问题，其答案在逻辑上是由多个前提决定的。汇总个人判断的程序有两种主要类型：“基于前提的方式”（PBW）和“基于结论的方式”（CBW）。我们推导出了两个程序被普遍订购的必要和充分条件。如果（且仅当）决策问题采取“合取”形式，则每当CBW采取行动时，PBW都会得出肯定的集体判断（即“是”）。此外，如果我们在上一行中用“析取性”替换“合取性”，则每当PBW做出时，PBW都会得出否定的集体判断（即“否”）。这些观察结果突出了以下事实：这两个过程是数学上的对偶。还研究了渐近性质。根据经典的Condorcetian假设，