A vector logic for intensional formal semantics

TL;DR

This paper demonstrates that formal and distributional semantics can be integrated by embedding Kripke-style models into vector spaces, enabling algebraic manipulation of modal operators and handling continuous parameters with measure theory.

Contribution

It introduces a framework that embeds intensional models into vector spaces, unifying formal and distributional semantics with algebraic and measure-theoretic tools.

Findings

Kripke models embed injectively into vector spaces

Modal operators are represented as linear operators

Measure-theoretic generalization for uncountable domains

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.