arXiv:2603.24527cs.AI2026-03

用结构化形式揭示自指语句的语义矛盾本质,为理解知识表达提供新框架。

From Liar Paradox to Incongruent Sets: A Normal Form for Self-Reference

  • 将自指语句转化为一组局部可满足但整体不可满足的非自指句子
  • 证明不完备性在结构上表现为局部成立但全局冲突的语义承诺
  • 提出语义能量模型,定量揭示信息量与确定性的权衡关系

我们引入了不一致标准型(INF),一种自指语义句子的结构化表示。INF 将一个自指句子替换为一组有限的非自指句子,这些句子各自可满足但整体不可满足。该变换隔离了自指带来的语义障碍,同时保持经典语义的局部一致性,并附有正确性定理,刻画了全局不一致如何源于局部相容承诺。随后我们研究不一致作为语义信息性的结构性来源的作用。通过一个最小模型论的信息性概念——即句子区分可接受模型的能力——我们证明语义完备性会排除信息性,而不一致则能保留它。此外,不一致不仅限于悖论构造:任何一致的不完备一阶理论都存在由不相容完全扩展产生的有限不一致族。从这个角度看,不完备性在结构上表现为局部可实现但全局不兼容的语义承诺,为语义知识提供了最小形式基础。最后,我们引入一个量化语义框架。在一个典型的有限语义状态设置中,我们将语义承诺建模为布尔函数,并基于总影响定义了一个傅里叶分析式的语义能量。我们推导出类似不确定性的界限,关联语义确定性、信息性和频谱简单性,并建立了一个矩阵不等式,将总体语义方差限制在总语义能量内。这些结果定量表明,语义信息性无法坍缩为单一确定状态而不付出无界能量代价,从而将不一致识别为语义表征的基本结构性与量化特征。

原文摘要 · Abstract (English)

We introduce incongruent normal form (INF), a structural representation for self-referential semantic sentences. An INF replaces a self-referential sentence with a finite family of non-self-referential sentences that are individually satisfiable but not jointly satisfiable. This transformation isolates the semantic obstruction created by self-reference while preserving classical semantics locally and is accompanied by correctness theorems characterizing when global inconsistency arises from locally compatible commitments. We then study the role of incongruence as a structural source of semantic informativeness. Using a minimal model-theoretic notion of informativeness-understood as the ability of sentences to distinguish among admissible models-we show that semantic completeness precludes informativeness, while incongruence preserves it. Moreover, incongruence is not confined to paradoxical constructions: any consistent incomplete first-order theory admits finite incongruent families arising from incompatible complete extensions. In this sense, incompleteness manifests structurally as locally realizable but globally incompatible semantic commitments, providing a minimal formal basis for semantic knowledge. Finally, we introduce a quantitative semantic framework. In a canonical finite semantic-state setting, we model semantic commitments as Boolean functions and define a Fourier-analytic notion of semantic energy based on total influence. We derive uncertainty-style bounds relating semantic determinacy, informativeness, and spectral simplicity, and establish a matrix inequality bounding aggregate semantic variance by total semantic energy. These results show quantitatively that semantic informativeness cannot collapse into a single determinate state without unbounded energy cost, identifying incongruence as a fundamental structural and quantitative feature of semantic representation.

自指语义学逻辑信息性

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