Arrow-Type Impossibility for Genuinely Modal Judgments

2026-05-22

AI Summary

This study investigates whether Arrow-type dictatorship results persist when aggregating only modal judgments rather than factual propositions. By constructing agendas generated from a single propositional variable and modal operators, and analyzing them within a sparse modal framework, the paper establishes—for the first time without relying on factual grounding—that modal semantic structure alone can induce strong logical dependencies sufficient to yield dictatorship. The core contributions include a semantic reduction theorem and a local-to-global aggregation mechanism. Integrating modal logic, frame semantics, and combinatorial covering techniques, the authors transform consistency checking into an efficiently computable problem. Impossibility theorems are shown to hold in both simple cyclic and symmetric frame families, while an efficient non-dictatorial aggregation algorithm is also devised.

Abstract

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.

Problem

• judgment aggregation
• modal logic
• Arrow-type impossibility
• dictatorship
• logical interconnections

Innovation

• modal judgment aggregation
• Arrow-type impossibility
• semantic reduction
• frame geometry
• combinatorial covering