A vector logic for extensional formal semantics

TL;DR

This paper establishes a mathematical link between extensional formal semantics and distributional vector space semantics, enabling hybrid models that combine symbolic and sub-symbolic language understanding.

Contribution

It constructs an injective homomorphism demonstrating structural compatibility between formal and distributional semantics, providing a foundation for hybrid cognitive language models.

Findings

Semantic functions correspond to vector space operations

Preserves compositionality in hybrid models

Supports multimodal language processing

Abstract

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).