构建可比较理解程度的逻辑框架，用于分析人与人工智能的理解差异。
Better Understanding, Understanding Better
- 引入分级解释结构与比较连接词，量化个体对命题的理解深度。
- 证明有限层级片段具有可判定性，整体系统满足强完备性。
- 适用于认知科学、哲学与人工智能的可信推理研究。
「凡人皆知，关键在于理解。」这句常被归于爱因斯坦的名言揭示了一个普遍直觉：理解远超单纯知晓。然而，尽管理解在当代认识论、科学哲学及近期关于人工智能的争论中占据核心地位，其在认识逻辑领域却长期未受重视。哲学文献中反复出现的主题是：理解具有程度之分——某人可能更深刻地理解某一命题，且一个人的理解可能优于另一个人。本文提出一种比较性认识逻辑框架，包含层级索引的理解模态和用于表达某人在某命题上的理解优于另一人的比较连接词。语义上，我们通过引入代理索引的分级解释结构和类似证明项代数的构造，扩展了多智能体认识模型。该框架统一刻画了最小理解、普通理解、更高要求的理解乃至理想理解，并支持同一命题下不同代理间的理解比较。区分了有限有界层级演算与无穷全语言伴生系统。证明了系统的正确性与强完备性，并表明每个固定有限层级片段均具可判定性。
原文摘要 · Abstract (English)
"Any fool can know; the point is to understand." A well-known remark often attributed to Einstein captures a widely shared intuition: understanding is more than merely knowing. Yet epistemic logic has paid relatively little attention to understanding, despite its central role in contemporary epistemology, philosophy of science, and recent debates about AI. A recurring theme in the philosophical literature is that, unlike knowledge, understanding comes in degrees: one may understand something more or less well, and one's understanding may be better than another's. We introduce a comparative epistemic logic of understanding with level-indexed understanding modalities and a comparative connective for saying that one agent understands why a proposition better than another agent does. Semantically, we enrich multi-agent epistemic models with agent-indexed graded explanation structures and a justification-style term algebra. This yields a unified framework for representing minimal, ordinary, more demanding, and ideal understanding, together with comparisons between agents with respect to the same formula at issue. We distinguish a finitary bounded-level calculus from an infinitary full-language companion system. We establish soundness and strong completeness, and show that each fixed finite-level fragment is decidable.
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