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Authors: Arthur Paul Pedersen ×
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01.
arXiv (CS.AI) 2026-06-24

The Measurable Majority

Authors:
Lawrence S. Moss↗Arthur Paul Pedersen↗

arXiv:2606.23853v1 Announce Type: cross Abstract: This paper studies strict majority reasoning in finite electorates using so-called $social decision frames$: finite sets of voters equipped with distinguished families of coalitions interpreted as those voting blocs evaluated to form a strict majority. A coherence criterion for qualitative majority judgments is identified and shown to give an exact characterization for representability of strict majorities by finitely additive measures. In addition, a minimal natural logic for reasoning about strict majorities is shown to be sound and complete. These developments motivate examination of associated combinatorial questions concerning incoherence in finite families of sets; partial results and a conjecture are given. Finally, the results of this paper are applied to correct a classical representation theorem for weak qualitative probability structures due to Patrick Suppes and to establish a May-type characterization for ordinary strict majority rule for social decision frames.

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02.
arXiv (CS.AI) 2026-06-25

Measurable Majorities Are Not Finitely Axiomatizable

Authors:
Lawrence S. Moss↗Arthur Paul Pedersen↗

arXiv:2606.25954v1 Announce Type: cross Abstract: This theoretical note studies the finite axiomatizability of strict majority reasoning in finite social decision frames. Moss and Pedersen (2026) 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.

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