Logic and Majority Voting

Journal of Philosophical Logic 51 (2):347-382 (2021)
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Abstract

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.

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Citations of this work

About Logically Probable Sentences.Adam Olszewski - forthcoming - Bulletin of the Section of Logic:33 pp..

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References found in this work

Basic proof theory.A. S. Troelstra - 1996 - New York: Cambridge University Press. Edited by Helmut Schwichtenberg.
Social Choice and Individual Values.Kenneth Joseph Arrow - 1951 - New York, NY, USA: Wiley: New York.
Arrow's theorem in judgment aggregation.Franz Dietrich & Christian List - 2007 - Social Choice and Welfare 29 (1):19-33.
Linear Logic.Jean-Yves Girard - 1987 - Theoretical Computer Science 50:1–102.

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