---
title: "On the Optimality of Vagueness — paper by Egre et al. (2020)"
description: "Égré, Spector, Mortier and Verheyen argue that vagueness helps a cooperative speaker who knows only part of the facts. 43 references."
authors: ["Paul Egre", "Benjamin Spector", "Adele Mortier", "Steven Verheyen"]
year: 2020
venue: "arXiv:2008.11841"
source: "https://arxiv.org/abs/2008.11841"
doi: "10.48550/arxiv.2008.11841"
arxiv: "2008.11841"
licence: "arXiv non-exclusive distribution licence (read only, no redistribution)"
licence_url: "http://arxiv.org/licenses/nonexclusive-distrib/1.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/egre-2020-optimality-vagueness-around-between-gricean/"
---
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# On the Optimality of Vagueness: "Around", "Between", and the Gricean Maxims

**Authors:** Paul Egre, [Benjamin Spector](https://md.research.elamur.ai/free/en/authors/benjamin-spector/), Adele Mortier and Steven Verheyen · **Year:** 2020 · **Venue:** arXiv:2008.11841 · **Licence:** [arXiv non-exclusive distribution licence (read only, no redistribution)](http://arxiv.org/licenses/nonexclusive-distrib/1.0/)

**Links:** [Publisher's page](https://arxiv.org/abs/2008.11841) · [PDF](https://arxiv.org/pdf/2008.11841) · [DOI 10.48550/arxiv.2008.11841](https://doi.org/10.48550/arxiv.2008.11841) · [arXiv 2008.11841](https://arxiv.org/abs/2008.11841) · [OpenAlex](https://openalex.org/W4287684800)

**Citations:** 1 citations (OpenAlex, 2026-10-14)

## Summary
The paper asks why ordinary language keeps vague expressions rather than replacing them with precise ones. Égré, Spector, Mortier and Verheyen argue that vagueness helps a cooperative speaker who knows only part of the facts. Such a speaker must balance Grice's Quality (assert only what you have grounds for) against Quantity (be informative enough). For this speaker, an approximator like "around n" can strike that balance better than any precise interval "between i and j".

They model "around n" probabilistically. The hearer is unsure of the tolerance margin, so the literal reading becomes a weighted blend of intervals centred on n, which concentrates belief near n. They place this inside the Rational Speech Act (RSA) framework. There, a message is scored by how closely the listener's updated distribution matches the speaker's private one, measured with KL-divergence.

The central result is that a vague message can be strictly more informative than every precise competitor. What drives the choice is the shape of the speaker's belief, not just its support. The standard Lexical Uncertainty model cannot capture this, so the paper treats vagueness as a rational tool rather than a defect.

## Key points
- The paper is by Paul Égré, Benjamin Spector, Adèle Mortier and Steven Verheyen (arXiv:2008.11841, v4 2022), forthcoming in *Linguistics and Philosophy*.
- In their semantics, "around n" carries a tolerance variable y that is independent of the quantity x. The hearer's posterior averages the prior conditioned on each window [n−y, n+y], a derivation they call BIR′, and this yields a "Ratio Inequality" favouring values near n.
- In the speaker model, utility is the negative KL-divergence between the speaker's distribution and the listener's posterior, minus a message cost c(m). Message choice then follows a softmax with temperature λ.
- In a toy domain of values 0–8, a speaker can rule out 0 and 8, so "between 1 and 7" is available. If her beliefs cluster around 4, she nevertheless prefers "around 4" to any "between" message.
- Appendix A proves a limitation of the standard Lexical Uncertainty model (Bergen et al. 2012, 2016): the speaker's choice depends only on the support of her beliefs. This limitation disappears in the authors' model and in the variants in Appendix C.

## Interesting ideas
- An imprecise expression need not be weaker: a loose phrase can carry a whole belief profile, while a wide exact interval says less even though it is precise.
- Modelling a vague word as a hidden, uncertain parameter (the tolerance) and averaging over its possible values turns fuzzy meaning into an explicit distribution.
- Message quality can be judged by how closely the hearer's resulting belief matches the speaker's own (KL-divergence), not by whether the message is true or false.
- When comparing two models, test whether one ignores some property of its input entirely. Here the Lexical Uncertainty model reacts to the support of the speaker's beliefs but never to their shape.
- Rational vagueness shows up as a choice among competing alternatives, so a vague term should be understood against the precise options the speaker skipped.

## For Mova

How these ideas could serve Mova, our small open language model — its files, stages and tests:

- **I1** (frames): implicature handling could treat "around 45" as signalling a peaked estimate, whereas "between 2 and 97" signals near-ignorance, so the two are not merely loose versus strict ranges.
- **I2** (world): quantities stated with "about", "around" or "roughly" could be stored as distributions peaked at the round number instead of hard intervals, keeping every inference step inspectable.
- **I3** (back): during generation from the meaning graph, candidate phrasings of a number (exact, "around", "between") could be ranked by this divergence, so output reflects the uncertainty recorded in the world model.
- **I4** (learn): evaluations could add negative controls in which only the shape of an underlying estimate changes, checking that answers or paraphrases actually respond to it.
- **I5** (answer): on word problems with approximate figures, a reader could reason about why the text said "around" rather than an exact range, and avoid over-committing to a crisp interval.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

## Institutions

- [Massachusetts Institute of Technology](https://md.research.elamur.ai/free/en/universities/massachusetts-institute-of-technology/) — University, Cambridge, United States · 159 works · 65,157 citations
- [Institut Jean Nicod](https://md.research.elamur.ai/free/en/labs/institut-jean-nicod/) — Research institute, Paris, France · 6 works · 1,359 citations
- [Erasmus University Rotterdam](https://md.research.elamur.ai/free/en/universities/erasmus-university-rotterdam/) — University, Rotterdam, Netherlands · 2 works · 288 citations

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🧭 Topics (1): [😉 Implicature](https://md.research.elamur.ai/free/en/topics/implicature/) · 72 works

## Cite

Cite as: Paul Egre, Benjamin Spector, Adele Mortier and Steven Verheyen. 2020. On the Optimality of Vagueness: "Around", "Between", and the Gricean Maxims. arXiv:2008.11841.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@misc{egre2020optimality,
  title = {{On the Optimality of Vagueness: "Around", "Between", and the Gricean Maxims}},
  author = {Paul Egre, Benjamin Spector, Adele Mortier and Steven Verheyen},
  year = {2020},
  doi = {10.48550/arxiv.2008.11841},
  eprint = {2008.11841},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.2008.11841}
}
```

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## References

43 references, 1 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

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*Updated 24 October 2026*