Measurable Majorities Are Not Finitely Axiomatizable
Authors: Lawrence S. Moss, Arthur Paul Pedersen
Organizations: (†) Dept. of Mathematics, Indiana University, Bloomington. · (‡) Dept. of Computer Science & the Intel Investigations Lab, the City College of New York; the Graduate Center & Remote Sensing Earth Systems Institute, the City University of New York.
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≥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.
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.
A common problem in social choice is to determine whether there is a social choice procedure, such as a voting method, satisfying some desired criteria. Computer-aided methods such as SAT solving can sometimes answer these questions. However, under typical encodings, a SAT solver may only synthesize a voting method on a finite domain, while we may want one on an infinite domain, such as the domain of all preference profiles for a fixed number of candidates but any finite number of voters. In this paper, we use an approach based on reasoning with constrained Horn clauses and computation with polyhedra to synthesize a voting method on an infinite domain. We then use SMT and Lean to verify its properties. Our main result is a possibility theorem about four well-known criteria from voting theory: the Condorcet winner and loser criteria, positive involvement, and resolvability. Previous work has shown that for five or more candidates, there is no voting method satisfying these axioms, and that for four candidates, there is no method satisfying these core axioms plus one more invariance axiom. Here we show that for four candidates, there does exist a method satisfying the core axioms and more.
Judgment aggregation studies how to combine individual judgments on logically related propositions into a collective judgment. Classical impossibility results show that sufficiently strong logical interconnections force dictatorship under natural aggregation axioms. In this paper, we ask whether such impossibility can still arise when the objects of aggregation are required to be genuinely modal judgments rather than plain factual propositions. Since modal logic contains propositional logic, this question is meaningful only if one excludes fact-based aggregation in disguise. We show that Arrow-type impossibility already re-emerges in a strikingly sparse modal setting. We prove an impossibility theorem on a simple cyclic frame for an agenda generated from a single propositional variable by repeated applications of a single modal operator, and we further demonstrate this phenomenon for an alternative family of frames satisfying a natural symmetry condition. Thus, even under a modal-operator requirement, semantic structure alone can generate the logical interconnections needed for dictatorship. Technically, our analysis has two layers. First, we prove a semantic reduction theorem showing that certain iterated modal patterns can be collapsed by shifting the evaluation point. Second, building on this reduction, we identify a local-to-global frame mechanism by which frame geometry yields minimally inconsistent modal judgment sets and the strong path-connectivity required for impossibility. The same reduction also turns consistency checking into a small combinatorial covering problem, which yields efficient implementations of non-dictatorial aggregation procedures.