Measurable Majorities Are Not Finitely Axiomatizable

2026-06-24

AI Summary

This study addresses the question of whether strict majority reasoning in the finite social decision framework introduced by Moss and Pedersen admits a finite axiomatization, with particular focus on whether its coherence criterion can be replaced by a bounded finite fragment. By constructing a class of maximal canonical frames and employing techniques involving orthogonality and dimension analysis in rational vector spaces together with an extension method based on symmetric semi-scale voting coalitions, the authors demonstrate that for every positive integer \(k\), there exists an instance whose shortest coherence violation has length \(2k+2\). This result rules out the possibility of finite axiomatizability, thereby resolving Conjecture 5.7 and confirming the Intermediate Layer Family Conjecture B.25. It also establishes that no uniform finite upper bound exists for incoherence indices within the minimal Moss–Pedersen logical language.

Abstract

This theoretical note studies the finite axiomatizability of strict majority reasoning in finite social decision frames. Moss and Pedersen (2026) <doi: 10.48550/arXiv.2606.23853> introduce a coherence criterion that characterizes exactly when qualitative majority judgments are representable by a finitely additive measure. The question addressed here is whether that coherence criterion can be replaced, in the finite setting, by any bounded finite fragment. We prove that it cannot. For every $k\ge 1$, we construct a maximal standard frame whose shortest coherence violation has length exactly $2k+2$. Hence there is no uniform finite bound on the incoherence index of social decision frames, resolving Conjecture 5.7 stated by Moss and Pedersen (2026). The construction is geometric, in the sense that it proceeds via orthogonality and dimension in rational vector spaces, and self-contained: it isolates a symmetric family of half-sized voting blocs and extends it to a maximal frame in which every shorter balanced obstruction is excluded. Along the explicit infinite sequence of universe sizes obtained in the construction, this also establishes the middle-layer family predicted by Conjecture B.25 by Moss and Pedersen (2026). Together with the soundness and completeness theorem for the Moss-Pedersen minimal logic for strict majorities, this establishes that measurable social decision frames are not finitely axiomatizable in that language.

Problem

• finite axiomatizability
• strict majority
• social decision frames
• coherence criterion
• measurable majorities

Innovation

• finite axiomatizability
• strict majority
• coherence criterion
• geometric construction
• social decision frames