A vector logic for intensional formal semantics

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.

💡 Research Summary

The paper establishes a rigorous bridge between formal intensional semantics and distributional (vector‑space) semantics. Starting from the well‑known extensional homomorphism theorem, which shows that typed domains and functions of a classical model can be injectively embedded into Hilbert spaces via linear (or multilinear) maps, the author extends this construction to Kripke‑style intensional models. An intensional model is defined by a family of index sorts (worlds, times, locations, etc.) each equipped with an accessibility relation. The Cartesian product of all index sets forms a compound index space S; an expression’s intension is a function from S to its extensional denotation.

For each semantic type τ a injective mapping h_τ sends entities, truth values, and higher‑order functions to distinct vectors or linear operators in a vector space S_{D_τ}. Truth values become orthonormal basis vectors in ℝ², and logical connectives are realized as fixed matrices (e.g., negation as