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# Raising and resolving issues with scalar modifiers

**Authors:** [Elizabeth Coppock](https://md.mova.elamur.ai/free/en/docs/papers/authors/elizabeth-coppock/) and Thomas Brochhagen · **Year:** 2013 · **Venue:** Semantics and Pragmatics 6(3) **Link:** [doi.org/10.3765/sp.6.3](https://doi.org/10.3765/sp.6.3) · **Licence:** [CC BY-NC 3.0](https://creativecommons.org/licenses/by-nc/3.0/) as printed on the article (the library card says CC BY) — summary only

[**128** citations](https://openalex.org/works?filter=cites:W1976144168) openalex · 2026-10-19[**6** citations in our library](https://md.mova.elamur.ai/free/en/docs/papers/cited-by/coppock-2013-raising-resolving-issues-scalar-modifiers/) of the 140673 papers whose bibliographies we read[Semantics and Pragmatics](https://openalex.org/S33105140)[licence](https://creativecommons.org/licenses/by/3.0/)

- [Publisher's page](https://doi.org/10.3765/sp.6.3)
- [PDF](https://semprag.org/index.php/sp/article/download/sp.6.3/pdf)
- [DOI 10.3765/sp.6.3](https://doi.org/10.3765/sp.6.3)
- [OpenAlex](https://openalex.org/W1976144168)

**Authors (2)**

1. [Elizabeth Coppock](https://openalex.org/A5074610868) [ORCID](https://orcid.org/0000-0001-9987-344X) · [Semantic Scholar](https://www.semanticscholar.org/author/3009498) · h 17 · 1130 citations · 80 works
2. [Thomas Brochhagen](https://openalex.org/A5085655949) [ORCID](https://orcid.org/0000-0002-0301-0541) · [Semantic Scholar](https://www.semanticscholar.org/author/3203093) · h 9 · 404 citations · 50 works

Citations and author metrics from openalex (2026-10-19); authors with the same name are not always told apart.

## What it is about

The paper asks what the superlative modifiers "at least" and "at most" mean, and why they differ from comparatives ("more than", "fewer than"). The authors first give a simple analysis: "at least" presupposes, "at most" asserts, and both rank answers to the current question under discussion (QUD) by pragmatic strength, as "only" does. This captures truth conditions, focus-sensitivity and distribution. It does not explain three facts: superlatives give ignorance implicatures ("#A hexagon has at least five sides" sounds odd if the speaker knows the answer), they give no scalar implicature, and they interact oddly with modals. The fix is to treat each superlative as an inquisitive proposition: a set of possibilities that draws attention to the higher-ranked answers. The method is theoretical, with worked examples and no experiment. The paper reports one data point: in Dutch judgements, about half of the speakers did not accept "four rabbits implies at least three", so the authors read the entailment as valid but pragmatically unwelcome.

## Ideas

- I1. `prag/src/infer.rs`: represent a numeral modifier as a set of alternatives ranked by a QUD scale, stored in a BTreeMap. "At least n" = the answers ranked at or above n; "at most n" = the answers ranked at or below n, excluding those above. Check against the 57-page worked examples before coding.
- I2. Ignorance rule: if the text's own facts already settle the QUD for the speaker (a lesson states the value), flag an "at least" sentence as pragmatically odd and keep the truth value. Measure on hand-made pairs from `world/src/knowledge.rs` lessons: "at least five" where the lesson says six.
- I3. Scalar-implicature gate: a bare numeral gets a "no more than" implicature, an "at least" numeral does not. Test on school multiple-choice items, with "exactly" as the negative control, which must score worse.
- I4. Missing-reading check: "at most two" under "may" grants no permission to have fewer. Encode it as a rule over the modal frame, and test it on the `world` QA items for "may" and "must".

## For Mova

The relevant chapters are [12. Pronouns, determiners and numerals](https://md.mova.elamur.ai/free/en/docs/languages/en/12-pronouns-determiners-numerals/) (numeral modifiers), [33. Implicature and presupposition](https://md.mova.elamur.ai/free/en/docs/languages/en/33-implicature-and-presupposition/) (ignorance versus scalar implicature) and [27. Negation](https://md.mova.elamur.ai/free/en/docs/languages/en/27-negation/) (superlatives resist negation). The code is `prag/src/infer.rs` and `prag/src/rsa.rs` for implicatures, and `world/src/knowledge.rs` for school items. The measure: accuracy of the SLM student on "at least" and "at most" items, run twice for determinism, plus the "exactly" negative control.

Topics: [Implicature](https://md.mova.elamur.ai/free/en/docs/papers/topics/implicature/)

## Cite

The paper itself, not this card. BibTeX from the library's record of the work. [DOI 10.3765/sp.6.3](https://doi.org/10.3765/sp.6.3)

```
@article{coppock2013raising,
  title = {{Raising and resolving issues with scalar modifiers}},
  author = {Coppock, Elizabeth and Brochhagen, Thomas},
  journal = {Semantics and Pragmatics},
  year = {2013},
  doi = {10.3765/sp.6.3},
  url = {https://doi.org/10.3765/sp.6.3}
}
```

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