Logic and majority voting
Logic and majority voting
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逻辑和多数表决
DOI：
10.1007/s10992-021-09631-7
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发表时间：
2022
影响因子：
1.5
通讯作者：
Ryo Takemura
中科院分区：
文献类型：
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作者：
Jeremiah Alberg;Eve Grace;Christopher Kelly;Ryo Takemura
为了研究逻辑推理与多数投票之间的关系，我们以根岑的序列演算的方式引入了Lg组的逻辑，其中每个序列都由一组个体索引。我们还引入了Lg的集合理论语义，其中每个公式都被解释为某个闭群的集合，其成员接受该公式。给出了Lg的切消定理、完备性定理和语义完备性定理。然后，我们引入了一个表示多数人投票给Lg的推理规则，引入了多数人投票给Lv的逻辑。将判断聚合理论中的话语悖论形式化，证明了吕氏是不一致的。基于基于前提和基于结论的方法来避免悖论，我们引入了公理Lva的多数投票逻辑，其中多数投票仅应用于非逻辑公理作为构造Lg中的证明的前提，以及结论Lvc的多数投票逻辑，其中多数投票仅应用于Lg中的证明的结论。我们证明了lva和Lvc在句法上是完备和一致的，并分别基于lva和Lvc的可证明性构造了集体判断。然后，我们讨论了这些系统如何避免话语悖论。
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 Lvaand Lvcare syntactically complete and consistent, and we construct collective judgments based on the provability in Lvaand Lvc, respectively. Then, we discuss how these systems avoid the discursive paradox.