A vector logic for intensional formal semantics
Daniel Quigley
TL;DR
The paper addresses unifying truth-conditional formal semantics with distributional, vector-based representations for intensional phenomena. It proves an intensional homomorphism: Kripke-style frames with multiple index sorts embed injectively into vector spaces, with intensions represented as linear operators and semantic composition preserved by lifted multilinear maps. Modal operators are derived algebraically, recasting accessibility as linear operators and quantification over worlds, times, and locations as accumulation with thresholds or measures; for uncountable domains, necessity becomes truth almost everywhere and possibility truth on sets of positive measure. The work provides a principled interface between symbolic and sub-symbolic semantics, enabling potential neural-symbolic implementations and paving the way for richer cognitively plausible models that handle modality, tense, and propositional attitudes within a unified vector-logic framework.
Abstract
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.