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Comparing exhaustivity operators

Paper · 2016 · Benjamin Spector

Summary

Exhaustivity operators model how a sentence is strengthened by its quantity (scalar) implicatures. They take a proposition and a set of alternatives and return a stronger proposition. The simple Krifka-style operator negates every alternative the sentence does not entail, and it breaks down on disjunctions. The literature offers two refined operators. exh mw keeps only the worlds where as few alternatives as possible are true (van Rooij & Schulz, Spector). exh ie is Fox's (2007) operator, which negates only the "innocently excludable" alternatives.

Spector compares the two purely through formal proofs. His central theorem says the operators coincide whenever the alternative set is closed under conjunction. More generally, an alternative is innocently excludable exactly when exh mw entails its negation. So exh mw always entails exh ie, and exh ie can be defined from it.

These results have two uses. First, they show which data can actually decide between the operators. Second, they let exh ie be computed through exh mw over a far smaller set of alternatives. That matters because Sauerland-style alternative sets grow exponentially with sentence size.

Key points

  • Failure of the naive operator. For p ∨ (q ∨ r), the alternative p ∨ (q ∧ r) is not entailed by the sentence. The Krifka-style operator therefore negates it, which wrongly implies ¬p.
  • When all three agree. Whenever the Krifka-style result is non-contradictory, all three operators give the same output (Fact 1).
  • Closure results:
    • Closing alternatives under ∨, ∧ or both never changes exh mw (Proposition 3).
    • Closing them under ∨ never changes exh ie (Theorem 10).
    • Closure under ∧ makes the two operators equivalent (Theorem 9).
    • If the ∨-closure is itself ∧-closed, they are also equivalent (Corollary 11).
  • Theorem 12. Take a sentence built from atomic sentences E using only unembedded ∨ and ∧. Then exh ie over its Sauerland alternatives equals exh mw over E alone.
  • Size of the alternative set. Suppose a sentence has n connectives and each atom appears once. Sauerland's procedure then yields (3^(n+1) − 1)/2 semantically distinct alternatives (proved in the appendix).

Interesting ideas

  • I1. If a small generator set has the same ∨/∧ closure as a huge alternative set, the minimal-worlds computation gives an identical result on it, so enumeration can shrink dramatically.
  • I2. Try the cheapest operator first: if negating all non-entailed alternatives stays consistent, every operator agrees, and the expensive one is only needed when that result is contradictory.
  • I3. A costly notion (innocent exclusion) can be read off a simpler one: the excludable alternatives are exactly those whose negation follows from the minimal-world result.
  • I4. Only cases where competing analyses diverge are informative; when alternative sets are ∧-closed, the data cannot decide between the operators.

For Mova

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

  • I1 (frames): the implicature component in prag could exhaustify over the atomic clause alternatives, instead of generating every Sauerland replacement of the connectives.
  • I2 (frames): a fast path for scalar implicatures ("some" → "not all") that escalates to innocent exclusion only on contradiction, with each step logged for explainability.
  • I3 (world): build the minimal states of the text's world in which the fewest alternatives hold, then list the entailed negations as explicit, traceable implicatures.
  • I4 (learn): build implicature test suites that deliberately include divergent configurations, so evaluation can tell rival hypotheses apart rather than counting cases where both trivially succeed.

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

🔬Centre National de la Recherche ScientifiqueParis · France104 works · 60,877 citations🔬École des hautes études en sciences socialesParis · France8 works · 2,159 citations🔬Institut Jean NicodParis · France6 works · 1,359 citations

📚 Shelves 5 Computer Science · 9,086 works Artificial Intelligence · 8,889 works Arts and Humanities · 9,296 works Language and Linguistics · 9,206 works Computational Theory and Mathematics · 116 works

Cite

Cite as Benjamin Spector. 2016. Comparing exhaustivity operators. Semantics and Pragmatics.

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.3765/sp.9.11

@article{spector2016comparing,
  title = {{Comparing exhaustivity operators}},
  author = {Benjamin Spector},
  journal = {Semantics and Pragmatics},
  year = {2016},
  doi = {10.3765/sp.9.11},
  url = {https://doi.org/10.3765/sp.9.11}
}

References

16 references, 1 of them with a card of ours. ↗ leads to the work.

  1. Büring, Daniel. 2008. The least at least can do. In Charles B. Chang & Hannah J. Haynie (eds.), West Coast Conference on Formal Linguistics (WCCFL) 26, 114–120. Somerville, MA: Cascadilla Proceedings Project. http://www.lingref.com/cpp/wccfl/26/paper1662.pdf. ↗
  2. Chierchia, Gennaro. 2004. Scalar implicatures, polarity phenomena, and the syntax/pragmatics interface. In Adriana Belletti (ed.), Structures and beyond, vol. 3, 39–103. Oxford University Press. ↗
  3. Benjamin Spector Fox, Danny. 2007. Free choice and the theory of scalar implicatures. In Uli Sauerland & Penka Stateva (eds.), Presupposition and implicature in compositional semantics, 537–586. New York: Palgrave Macmillan. http://dx.doi.org/10.1057/9780230210752_4. ↗
  4. Groenendijk, Jeroen & Martin Stokhof. 1984. Studies on the semantics of questions and the pragmatics of answers. University of Amsterdam dissertation. http://hdl.handle.net/11245/1.392528. ↗
  5. Katzir, Roni. 2007. Structurally-defined alternatives. Linguistics and Philosophy 30(6). 669–690. http://dx.doi.org/10.1007/s10988-008-9029-y. ↗
  6. Krifka, Manfred. 1993. Focus and presupposition in dynamic interpretation. Journal of Semantics 10(4). 269–300. http://dx.doi.org/10.1093/jos/10.4. 269. ↗
  7. Mascarenhas, Salvador. 2014. Formal semantics and the psychology of reasoning. NYU dissertation. http://semanticsarchive.net/Archive/TdiNTBhM/ mascarenhas-2014-dissertation.pdf. ↗
  8. Mayr, Clemens. 2011. Implicatures of modified numerals. In Ivano Caponigro & Carlo Cecchetto (eds.), From grammar to meaning: The spontaneous logicality of language, chap. 6, 139–171. Cambridge University Press. http://dx.doi.org/10.1017/CBO9781139519328.009. van Rooij, Robert & Katrin Schulz. 2004. Exhaustive interpretation of complex sentences. Journal of Logic, Language and Information 13(4). 491–519. http://dx.doi.org/10.1007/s10849-004-2118-6. ↗
  9. Sauerland, Uli. 2004. Scalar implicatures in complex sentences. Linguistics and Philosophy 27(3). 367–391. http://dx.doi.org/10.1023/b:ling.0000023378. 71748.db. ↗
  10. Schulz, Katrin & Robert van Rooij. 2006. Pragmatic meaning and non-monotonic reasoning: The case of exhaustive interpretation. Linguistics and Philosophy 29. 205–250. http://dx.doi.org/10.1007/s10988-005-3760-4. ↗
  11. Schwarz, Bernhard. 2013. At least and quantity implicature: Choices and consequences. In Amsterdam Colloquium, vol. 19, 187–194. http://maloni.humanities.uva.nl/AC2013/AC_proceedings.pdf. ↗
  12. Schwarz, Bernhard. 2016. Consistency preservation in Quantity implicature: The case of at least. Semantics and Pragmatics 9(1). 1–47. http://dx.doi.org/10.3765/sp.9.1. ↗ our card
  13. Spector, Benjamin. 2003. Scalar implicatures: Exhaustivity and Gricean reasoning. In Balder ten Cate (ed.), European Summer School in Logic, Language and Information (ESSLLI), student session, vol. 8, 277–288. Revised version in Spector 2007. Vienna, Austria. Comparing exhaustivity operators Spector, Benjamin. 2006. Aspects de la pragmatique des opérateurs logiques. Université Paris 7 dissertation. http://lumiere.ens.fr/~bspector/THESE_ SPECTOR/THESE_SPECTOR_AVEC_ANNEXE2.pdf. ↗
  14. Spector, Benjamin. 2007. Scalar implicatures: Exhaustivity and Gricean reasoning. In Maria Aloni, Paul Dekker & Alastair Butler (eds.), Questions in dynamic semantics, vol. 17 (Current Research in the Semantics/Pragmatics Interface), 225–249. Brill. http://dx.doi.org/10.1163/9780080470993_011. ↗
  15. Szabolcsi, Anna. 1983. Focussing properties, or the trap of first order. Theoretical Linguistics 10(1-3). 125–146. http://dx.doi.org/10.1515/thli.1983.10.1-3.125. ↗
  16. Wikipedia. 2016. Dedekind number - Wikipedia, The Free Encyclopedia. Online; accessed 2016-12-15. https://en.wikipedia.org/wiki/Dedekind_number. Benjamin Spector École Normale Supérieure Institut Jean Nicod 29 rue d’Ulm 75005 Paris, France Pavillon Jardin benjamin.spector@ens.fr ↗

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