将语义的可能世界模型嵌入向量空间，实现形式语义与分布语义的统一。
A vector logic for intensional formal semantics
- 用向量空间重构克里普克语义，使意义由线性映射表示。
- 模态算子转化为阈值判断，支持连续参数下的非经典逻辑。
- 适用于多维度索引（如时空）的语义建模，适合自然语言处理研究者。
形式语义学与分布语义学是两种不同的语言意义建模方式：前者通过模型论结构定义意义的指称；后者将意义表示为由使用经验塑造的高维向量空间中的向量。本文证明这两种框架在内涵语义层面具有结构性相容性。我们建立克里普克风格的内涵模型可单射嵌入向量空间，语义函数可升格为保持组合性的（多）线性映射。通过复合索引空间容纳多个索引类型（如世界、时间、位置），将内涵表示为线性算子。模态算子由代数推导得出：可达关系化为线性算子，模态条件退化为累积值的阈值检查。针对不可数索引域，我们发展测度论推广，其中必然性对应几乎处处为真，可能性对应正测度集上为真，该非经典逻辑天然适用于连续参数。
原文摘要 · Abstract (English)
Formal semantics and distributional semantics are distinct approaches to linguistic meaning: the former models meaning as reference via model-theoretic structures; the latter represents meaning as vectors in high-dimensional spaces shaped by usage. This paper proves that these frameworks are structurally compatible for intensional semantics. We establish that Kripke-style intensional models embed injectively into vector spaces, with semantic functions lifting to (multi)linear maps that preserve composition. The construction accommodates multiple index sorts (worlds, times, locations) via a compound index space, representing intensions as linear operators. Modal operators are derived algebraically: accessibility relations become linear operators, and modal conditions reduce to threshold checks on accumulated values. For uncountable index domains, we develop a measure-theoretic generalization in which necessity becomes truth almost everywhere and possibility becomes truth on a set of positive measure, a non-classical logic natural for continuous parameters.
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