# A vector logic for intensional formal semantics

arXiv: https://arxiv.org/abs/2602.02940 · HTML: https://mlsanity.com/paper/2602.02940/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 83 references that are in the map.
- First posted on arXiv 3 Feb 2026.

## Builds on (most cited first, 8 of 8)

- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Distributional Semantics and Linguistic Theory](https://mlsanity.com/paper/1905.01896/) (2019)
- [A comparative evaluation and analysis of three generations of Distributional Semantic Models](https://mlsanity.com/paper/2105.09825/) (2021)
- [Lambek vs. Lambek: Functorial Vector Space Semantics and String Diagrams for Lambek Calculus](https://mlsanity.com/paper/1302.0393/) (2013)
- [Towards a Formal Distributional Semantics: Simulating Logical Calculi with Tensors](https://mlsanity.com/paper/1304.5823/) (2013)
- [Syntax-semantics interface: an algebraic model](https://mlsanity.com/paper/2311.06189/) (2023)
- [A Generalised Quantifier Theory of Natural Language in Categorical Compositional Distributional Semantics with Bialgebras](https://mlsanity.com/paper/1602.01635/) (2016)
- [Vector spaces as Kripke frames](https://mlsanity.com/paper/1908.05528/) (2019)

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

- [Exact semantic readout from compressed vector representations](https://mlsanity.com/paper/2609.18047/) (2026)

## Abstract (arXiv)

Formal semantics and distributional semantics are distinct approaches to linguistic meaning: the former models meaning as reference via model-theoretic structures; the latter as vectors in high-dimensional spaces shaped by usage. This paper establishes which part of intensional formal semantics admits a linear vector-space encoding. Kripke-style intensional models, with any finite collection of index sorts collected in a compound index space, embed injectively into vector spaces: primitive domains go to free carriers; intensions and other functions go to linear operators. Semantic functions lift to unique multilinear maps on the free carriers, and composition is preserved. The operator encoding of a function domain compresses its free carrier, and we characterize the functionals: a Boolean-valued functional of a power set acts linearly on operator encodings exactly when it is constant, an ultrafilter indicator, or the complement of one, so that on finite domains the nonconstant ones are Montague's individuals and their negations. Determiners over a restrictor of two or more elements, modal operators over two or more accessible indices, and attitude operators over two or more alternatives are outside the linear regime, and take, instead, the form of a linear accumulation followed by a decision. Modality is defined uniformly over measure frames, in which counting measure recovers Kripke semantics at every cardinality, and continuous measures make necessity truth almost everywhere; we give the correspondence conditions for the axioms D, T, B, 4, and 5 under measures, the Kronecker factorization of accessibility over compound indices, and the reading of measure-based modality as graded modality.