This paper proves a homomorphism between extensional formal semantics and distributional vector space semantics, demonstrating structural compatibility. Formal semantics models meaning as reference, using logical structures to map linguistic expressions to truth conditions, while distributional semantics represents meaning through word vectors derived from contextual usage. By constructing injective mappings that preserve semantic relationships, we show that every semantic function in an extensional model corresponds to a compatible vector space operation. This result respects compositionality and extends to function compositions, constant interpretations, and $n$-ary relations. Rather than pursuing unification, we highlight a mathematical foundation for hybrid cognitive models that integrate symbolic and sub-symbolic reasoning and semantics. These findings support multimodal language processing, aligning `meaning as reference' (Frege, Tarski) with `meaning as use' (Wittgenstein, Firth).

翻译：本文证明了外延形式语义与分布式向量空间语义之间存在同态映射，从而揭示了两者的结构兼容性。形式语义将意义建模为指称关系，运用逻辑结构将语言表达式映射至真值条件；而分布式语义则通过基于上下文使用推导出的词向量来表征意义。通过构建保持语义关系的单射映射，我们证明了外延模型中的每个语义函数都对应着一个兼容的向量空间运算。该结果遵循组合性原则，并可推广至函数复合、常量解释及$n$元关系。相较于追求统一理论，我们着重为整合符号与亚符号推理及语义的混合认知模型提供了数学基础。这些发现支持多模态语言处理，并将"意义即指称"（弗雷格、塔斯基）与"意义即使用"（维特根斯坦、弗斯）两种范式相统一。