# Vector spaces as Kripke frames

arXiv: https://arxiv.org/abs/1908.05528 · HTML: https://mlsanity.com/paper/1908.05528/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Aug 2019.

## Built on by (most cited first, 1 of 1)

- [A vector logic for intensional formal semantics](https://mlsanity.com/paper/2602.02940/) (2026)

## Abstract (arXiv)

In recent years, the compositional distributional approach in computational linguistics has opened the way for an integration of the lexical aspects of meaning into Lambek's type-logical grammar program. This approach is based on the observation that a sound semantics for the associative, commutative and unital Lambek calculus can be based on vector spaces by interpreting fusion as the tensor product of vector spaces. In this paper, we build on this observation and extend it to a `vector space semantics' for the general Lambek calculus, based on algebras over a field mathbbK (or mathbbK-algebras), i.e. vector spaces endowed with a bilinear binary product. Such structures are well known in algebraic geometry and algebraic topology, since they are important instances of Lie algebras and Hopf algebras. Applying results and insights from duality and representation theory for the algebraic semantics of nonclassical logics, we regard mathbbK-algebras as `Kripke frames' the complex algebras of which are complete residuated lattices. This perspective makes it possible to establish a systematic connection between vector space semantics and the standard Routley-Meyer semantics of (modal) substructural logics.