---
title: "At least and ignorance: A reply to Coppock & Brochhagen 2013"
description: "This commentary examines how \"at least\" with a numeral (e.g. 29 references. Among them: 5."
authors: ["Bernhard Schwarz"]
year: 2016
venue: "Semantics and Pragmatics"
source: "https://semprag.org/index.php/sp/article/view/sp.9.10"
doi: "10.3765/sp.9.10"
licence: "CC BY (Semantics and Pragmatics, journal policy)"
licence_url: "https://creativecommons.org/licenses/by/3.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/schwarz-2016-least-ignorance-reply-coppock-brochhagen/"
---
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# At least and ignorance: A reply to Coppock & Brochhagen 2013

**Authors:** [Bernhard Schwarz](https://md.research.elamur.ai/free/en/authors/bernhard-schwarz/) · **Year:** 2016 · **Venue:** [Semantics and Pragmatics](https://md.research.elamur.ai/free/en/venues/semantics-and-pragmatics/) ([Semantics and Pragmatics 2016](https://md.research.elamur.ai/free/en/venues/semantics-and-pragmatics/2016/)) · **Publisher:** [Linguistic Society of America](https://md.research.elamur.ai/free/en/publishers/linguistic-society-of-america/) · **Licence:** [CC BY (Semantics and Pragmatics, journal policy)](https://creativecommons.org/licenses/by/3.0/)

**Links:** [Publisher's page](https://doi.org/10.3765/sp.9.10) · [PDF](https://semprag.org/index.php/sp/article/download/sp.9.10/pdf) · [DOI 10.3765/sp.9.10](https://doi.org/10.3765/sp.9.10) · [OpenAlex](https://openalex.org/W2344985504)

**Citations:** 5 citations (OpenAlex, 2026-10-14)

## Summary
This commentary examines how "at least" with a numeral (e.g. "At least two apples fell") signals that the speaker does not know the exact number. Coppock & Brochhagen (2013) explained this in inquisitive semantics. They treat the sentence as denoting a set of propositions and add a new principle, the Maxim of Interactive Sincerity (MIS): once a plural denotation is intersected with the speaker's belief state, more than one consistent option must remain.

Schwarz compares the full pragmatic meanings predicted by MIS with those of a neo-Gricean quantity-implicature account (the "Q analysis", building on Büring 2008). He shows that:
- the two accounts make different predictions;
- MIS fails to yield all the desired inferences for "at least";
- MIS also fails for disjunctions.

A hybrid that adds quantity implicature (CB-IQ) does match the Q analysis. But then MIS becomes redundant: it can be dropped entirely if "at least two" is given the two-member denotation {exactly 2, 3 or more}. The conclusion is that quantity implicature is the more economical explanation.

## Key points
- Both accounts give "At least two apples fell" the truth-conditional content "two or more apples fell" and assume the speaker believes it (a quality inference).
- The Q analysis treats [2] ("exactly two") and [3,...) ("three or more") as stronger alternatives. Quantity reasoning concludes the speaker believes neither. Combined with quality, this means the speaker considers both possible, which is the ignorance inference.
- C&B's denotation for "at least two" is an infinite set: {[2,...), [3,...), ...} or the two-sided {[2], [3], ...}. MIS applied to these does not reproduce all of the Q analysis's inferences, and it does not deliver ignorance for disjunctions.
- The proposed IQ analysis keeps inquisitive semantics but drops MIS. With the denotation {[2], [3,...)} it derives exactly the same inferences as the Q analysis.
- The related Maxim of Attentive Sincerity (Pruitt & Roelofsen 2011) gives inferences that are too weak in some cases and too strong in others. For instance, it predicts total ignorance, which wrongly rules out a speaker who knows the count isn't four.

## Interesting ideas
- Compare competing theories by spelling out each one's complete predicted meaning and checking how they relate logically, rather than accepting that both "derive ignorance."
- Ignorance follows from two parts: "speaker doesn't believe the stronger alternative" plus "speaker believes what was said," which together yield "speaker thinks it is possible."
- Prefer one general mechanism over a special-purpose principle that only does work for the case it was built for.
- An inference can be too strong, not just missing: total ignorance over every number contradicts acceptable uses, such as a speaker who knows the count isn't four.
- How the alternative set is chosen ({exactly 2, 3+} versus an infinite list) determines the predictions.

## For Mova

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

- **I1** (frames): When two implicature rules seem to agree, compute their full outputs on test sentences and diff them before choosing one.
- **I2** (frames): The implicature module can model "at least N" as two alternatives, exactly N and more than N. It then marks both as speaker-possible, giving an explainable epistemic state.
- **I3** (learn): When self-induced repair rules multiply, test whether a rule only fires on its original motivating case and prune it if a general rule already covers it.
- **I4** (learn): Evaluation should include negative-control sentences that check the reader does NOT over-infer ignorance or exclusions.
- **I5** (world): Store "at least N" quantities in the world model as a lower bound with an open upper bound, without enumerating or excluding specific values the text never ruled out.

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

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## Cite

Cite as: Bernhard Schwarz. 2016. At least and ignorance: A reply to Coppock & Brochhagen 2013. Semantics and Pragmatics.

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

```bibtex
@article{schwarz2016at,
  title = {{At least and ignorance: A reply to Coppock \& Brochhagen 2013}},
  author = {Bernhard Schwarz},
  journal = {Semantics and Pragmatics},
  year = {2016},
  doi = {10.3765/sp.9.10},
  url = {https://doi.org/10.3765/sp.9.10}
}
```

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## References

29 references, 4 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*