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.

翻译：判断聚合研究如何将个体对逻辑相关命题的判断整合为集体判断。经典不可能性结果显示，在自然聚合公理下，足够强的逻辑关联必然导致独裁。本文探讨当聚合对象要求为真正模态判断而非简单事实命题时，这种不可能性是否仍然存在。由于模态逻辑包含命题逻辑，只有排除伪装成模态的事实聚合，该问题才具有意义。我们证明，在极其稀疏的模态框架中，箭型不可能性早已重现。我们针对由单个命题变量通过重复应用单一模态算子生成的议程，在简单循环框架上证明了不可能性定理，并进一步在满足自然对称条件的另一类框架中展示了该现象。因此，即使施加模态算子要求，仅凭语义结构本身就能产生导致独裁所需的逻辑关联。技术层面，我们的分析包含两个层次：首先证明语义归约定理——通过转移评价点可折叠特定迭代模态模式；其次基于此归约，识别出局部到全局的框架机制——框架几何结构既生成极小不一致模态判断集合，又产生不可能性所需的强路径连通性。同一归约还将一致性检验转化为小型组合覆盖问题，从而为非独裁聚合程序提供高效实现方案。