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Consistency preservation in Quantity implicature: The case of at least

Paper · 2016 · Bernhard Schwarz

Summary

The paper asks how a Neo-Gricean theory of Quantity implicature should handle the speaker-ignorance reading of at least. Example: "Al hired at least two cooks" suggests the speaker does not know the exact number. Building on Büring (2008), Schwarz treats this reading as a Quantity implicature, parallel to disjunction ("two or more"). He then formally tests the widely used Standard Recipe, which follows Sauerland (2004). That recipe first derives primary inferences ("speaker doesn't believe q") and then strengthens each one to a secondary inference ("speaker believes not-q"). Each strengthening is checked for consistency on its own. For unembedded at least, these individually admissible strengthenings together contradict the assertion. Two existing repairs from van Rooij & Schulz and Spector avoid the contradiction but predict epistemic inferences that are too strong. The fix that works restricts strengthening using Fox's (2007a) innocent exclusion. This matters because the argument applies to any theory that strengthens primary to secondary belief implications, not only to Neo-Gricean pragmatics.

Key points

  • The Standard Recipe has three layers. The Quality inference comes first. Primary inferences ¬◻q are added for every stronger alternative q. Secondary inferences ◻¬q are added only if each passes a separate consistency check.
  • A secondary inference amounts to the primary inference plus an assumption that the speaker is competent about q. This means the recipe treats speakers as competent by default.
  • For "at least two", the key alternatives are a symmetric pair: exactly two, [2], and more than two, [3,...). Strengthening both at once yields an inconsistent set of inferences.
  • The paper contrasts two further procedures, one grounded in exhaustivity and one grounded in innocent exclusion. They agree on bottom-line inferences, but only the innocent-exclusion version gets the epistemic inferences right.
  • Spector (2016) shows that the two Exh definitions coincide whenever the alternative set is closed under conjunction. Here Exh applied to (1) is vacuous, so the argument does not carry over to the Grammatical Theory of Implicature.

Interesting ideas

  • I1. Checking each candidate inference for consistency one at a time is not enough, because two individually safe additions can jointly contradict the assertion; consistency has to be checked over the whole set at once.
  • I2. Innocent exclusion negates only those alternatives that are excluded in every maximal consistent set of negations, so it is a principled way to choose which defaults survive.
  • I3. Keeping two layers of inference apart, speaker ignorance (¬◻q) versus speaker belief in the negation (◻¬q), makes the competence assumption explicit and cancellable.
  • I4. Two theories can agree on truth conditions yet differ in their epistemic predictions, so they must be tested on outputs about speaker belief and not only on bottom-line meaning.

For Mova

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

  • I1 (frames): When prag strengthens "not known" into "known false" for scalar alternatives, it should run one joint consistency check over all candidate strengthenings rather than accepting each one independently.
  • I2 (world): The same "keep only what survives in every maximal consistent choice" rule could settle conflicting default assumptions about entities and quantities in the text's world without picking one arbitrarily.
  • I3 (frames): Implicature frames could store primary and secondary inferences as separate, explainable nodes, so "at least two" yields "speaker unsure whether exactly two" instead of a false upper bound.
  • I4 (learn): Evaluation of the implicature module should include test items that target speaker-knowledge inferences, since items that check only truth conditions cannot tell competing strengthening rules apart.

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

🎓McGill UniversityMontreal · Canada51 works · 10,476 citations

📚 Shelves 4The field of science this work belongs to (OpenAlex fields).Computer Science · 9,086 worksArtificial Intelligence · 8,889 worksArts and Humanities · 9,296 worksLanguage and Linguistics · 9,206 works

🧭 Topics 1Research subjects and languages the work is about.Implicature · 72 works

Cite

Cite as Bernhard Schwarz. 2016. Consistency preservation in Quantity implicature: The case of at least. Semantics and Pragmatics.

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.3765/sp.9.1

@article{schwarz2016consistency,
  title = {{Consistency preservation in Quantity implicature: The case of at least}},
  author = {Bernhard Schwarz},
  journal = {Semantics and Pragmatics},
  year = {2016},
  doi = {10.3765/sp.9.1},
  url = {https://doi.org/10.3765/sp.9.1}
}

References

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