To investigate the relationship between logical reasoning and majority voting, we introduce logic with groups Lg in the style of Gentzen’s sequent calculus, where every sequent is indexed by a group of individuals. We also introduce the set-theoretical semantics of Lg, where every formula is interpreted as a certain closed set of groups whose members accept that formula. We present the cut-elimination theorem, and the soundness and semantic completeness theorems of Lg. Then, introducing an inference rule representing majority voting to Lg, we introduce logic with majority voting Lv. Formalizing the discursive paradox in judgment aggregation theory, we show that Lv is inconsistent. Based on the premise-based and conclusion-based approaches to avoid the paradox, we introduce logic with majority voting for axioms Lva, where majority voting is applied only to non-logical axioms as premises to construct a proof in Lg, and logic with majority voting for conclusions Lvc, where majority voting is applied only to the conclusion of a proof in Lg. We show that both Lva and Lvc are syntactically complete and consistent, and we construct collective judgments based on the provability in Lva and Lvc, respectively. Then, we discuss how these systems avoid the discursive paradox.

为了研究逻辑推理和多数投票之间的关系，我们以 Gentzen 序列演算的风格引入了带有组Lg 的逻辑，其中每个序列都由一组个体索引。我们还介绍了Lg的集合论语义，其中每个公式都被解释为一个特定的封闭组，其成员接受该公式。我们提出了切消定理，以及Lg的健全性和语义完备性定理。然后，将代表多数表决的推理规则引入Lg，我们引入具有多数表决Lv 的逻辑。将判断聚合理论中的话语悖论形式化，我们证明Lv不一致。基于基于前提和基于结论的方法来避免悖论，我们引入了对公理 Lv a进行多数投票的逻辑，其中多数投票仅适用于非逻辑公理作为前提以构建Lg 中的证明，并且逻辑与对结论 Lv c 的多数表决，其中多数表决仅适用于Lg 中证明的结论。我们证明了 Lv a和 Lv c在句法上是完整和一致的，我们分别基于 Lv a和 Lv c的可证明性构建了集体判断。然后，我们讨论这些系统如何避免话语悖论。