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.

翻译：形式语义学与分布语义学是处理语言意义的两种不同进路：前者通过模型论结构将意义建模为指称关系；后者则将意义表示为高维使用模式空间中的向量。本文证明这两种框架在内涵语义学领域具有结构兼容性。我们建立了克里普克式内涵模型到向量空间的单射嵌入，并将语义函数提升为保持组合性的（多）线性映射。该构造通过复合索引空间容纳多种索引类型（可能世界、时间、位置），将内涵表示为线性算子。模态算子通过代数方式导出：可达关系转化为线性算子，模态条件简化为累积值的阈值检测。针对不可数索引域，我们发展了测度论推广方案，其中必然性转化为几乎处处真值，可能性转化为正测度集上的真值，这种非经典逻辑天然适用于连续参数场景。