Measurable Majorities Are Not Finitely Axiomatizable
For researchers in social choice theory and logic, this resolves a foundational open problem about the finite axiomatizability of majority reasoning, establishing a limitation of the Moss-Pedersen minimal logic.
The paper proves that measurable majorities in finite social decision frames are not finitely axiomatizable, resolving a conjecture by Moss and Pedersen (2026). For every k≥1, a frame with shortest coherence violation of length 2k+2 is constructed, showing no uniform finite bound exists.
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.