# Towards a Formal Distributional Semantics: Simulating Logical Calculi with Tensors

arXiv: https://arxiv.org/abs/1304.5823 · HTML: https://mlsanity.com/paper/1304.5823/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Apr 2013.

## Built on by (most cited first, 10 of 10)

- [Logic Embeddings for Complex Query Answering](https://mlsanity.com/paper/2103.00418/) (2021)
- [Learning Symbolic Rules for Reasoning in Quasi-Natural Language](https://mlsanity.com/paper/2111.12038/) (2021)
- [Evaluating Relaxations of Logic for Neural Networks: A Comprehensive Study](https://mlsanity.com/paper/2107.13646/) (2021)
- [Understanding the Use of Quantifiers in Mandarin](https://mlsanity.com/paper/2209.11977/) (2022)
- [Boolean matrix logic programming for active learning of gene functions in genome-scale metabolic network models](https://mlsanity.com/paper/2405.06724/) (2024)
- [Boolean Matrix Logic Programming on the GPU](https://mlsanity.com/paper/2408.10369/) (2024)
- [A vector logic for intensional formal semantics](https://mlsanity.com/paper/2602.02940/) (2026)
- [Pregroup Grammars, their Syntax and Semantics](https://mlsanity.com/paper/2109.11237/) (2021)
- [Simulating Petri nets with Boolean Matrix Logic Programming](https://mlsanity.com/paper/2405.11412/) (2024)
- [Active learning of digenic functions with boolean matrix logic programming](https://mlsanity.com/paper/2408.14487/) (2024)

## Abstract (arXiv)

The development of compositional distributional models of semantics reconciling the empirical aspects of distributional semantics with the compositional aspects of formal semantics is a popular topic in the contemporary literature. This paper seeks to bring this reconciliation one step further by showing how the mathematical constructs commonly used in compositional distributional models, such as tensors and matrices, can be used to simulate different aspects of predicate logic. This paper discusses how the canonical isomorphism between tensors and multilinear maps can be exploited to simulate a full-blown quantifier-free predicate calculus using tensors. It provides tensor interpretations of the set of logical connectives required to model propositional calculi. It suggests a variant of these tensor calculi capable of modelling quantifiers, using few non-linear operations. It finally discusses the relation between these variants, and how this relation should constitute the subject of future work.