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A "de-Fregean" semantics (and neo-Gricean pragmatics) for modified and unmodified numerals

Paper · 2015 · Christopher Kennedy

Summary

Kennedy asks why numerals with superlative modifiers ("at least six", "at most six") suggest that the speaker does not know the exact number, while comparative ones ("more than six", "fewer than six") do not, even though both only give a bound. This is a theoretical paper with no corpus or experiment. It works through constructed examples and formal semantics. Kennedy first examines Nouwen's 2010 account, which uses blocking plus a silent epistemic modal, and argues it fails for several modal interactions. He then treats bare and modified numerals alike as quantifiers over degrees. Bare numerals mean "exactly n" as part of their literal content. Modifiers differ only in whether their ordering is strict or non-strict. Under this view, a non-strict form is weaker than both its strict counterpart and the bare numeral. Ordinary Quantity reasoning then yields an ignorance implicature, just as it does for disjunction. How these numerals interact with modals, including cases where ignorance disappears, follows from scope. The result is a single pragmatic account of a widely studied puzzle.

Key points

  • Following Nouwen 2010, modifiers fall into two classes:
    • Class A uses a strict ordering: more/fewer than, over, under, between.
    • Class B uses a non-strict ordering: at least/most, minimally, maximally, from, up to, "or more/fewer".
  • Bare numerals are generalized quantifiers over degrees with two-sided (exact) truth conditions, following Kennedy 2013.
  • The alternatives for a Class B numeral are its Class A counterpart and the bare numeral. Both asymmetrically entail it, which produces the ignorance inference.
  • Büring's 2008 "authoritative readings", where ignorance vanishes under modals, are explained by the relative scope of the numeral and the modal.
  • "No more/fewer than" patterns with Class B. Differential "no" excludes a positive difference, so it allows equality. This is why it sounds strange (#) to describe a hexagon's side count with "no more than ten".

Interesting ideas

  • I1. Whether a speaker sounds uncertain is predicted by one feature of the modifier: does its bound include the named number or exclude it?
  • I2. Bare numerals are treated as exact in their literal meaning, not as lower bounds that are later strengthened by implicature.
  • I3. Ignorance from "at least n" is computed by the same alternative-and-entailment machinery as ignorance from "A or B".
  • I4. Whether ignorance appears under modals ("you may invite at most three") is a matter of scope, not lexical ambiguity.
  • I5. Sentences that state a known fact with a Class B form, like the hexagon examples, are diagnostic oddities.

For Mova

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

  • I1 (world): Store quantities from text as intervals marked open or closed, and attach a speaker-uncertainty flag whenever the bound is closed.
  • I2 (world): Entity counts from "three apples" can be recorded as exact values, which keeps arithmetic in maths word problems simple and auditable.
  • I3 (frames): The prag implicature module can reuse one alternative generator for disjunctions and numeral modifiers instead of hand-coding special cases.
  • I4 (frames): When a numeral sits under a deontic modal, resolve the scope and read the bound as a permission limit rather than as speaker ignorance.
  • I5 (learn): Use such minimal pairs as negative controls, checking that the system flags the odd variant and accepts the plain negated comparative.

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

🎓University of ChicagoChicago · United States28 works · 49,773 citations

📚 Shelves 5The 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 worksPhilosophy · 211 works

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

Cite

Cite as Christopher Kennedy. 2015. A "de-Fregean" semantics (and neo-Gricean pragmatics) for modified and unmodified numerals. Semantics and Pragmatics.

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.3765/sp.8.10

@article{kennedy2015de,
  title = {{A "de-Fregean" semantics (and neo-Gricean pragmatics) for modified and unmodified numerals}},
  author = {Christopher Kennedy},
  journal = {Semantics and Pragmatics},
  year = {2015},
  doi = {10.3765/sp.8.10},
  url = {https://doi.org/10.3765/sp.8.10}
}

References

59 references, 2 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

  1. Alrenga, Peter & Christopher Kennedy. 2014. No more shall we part: Quantifiers in English comparatives. Natural Language Semantics 22. 1–53. http://dx.doi.org/10.1007/s11050-013-9099-4. ↗
  2. Brasoveanu, Adrian. 2013. Modified numerals as post-suppositions. Journal of Semantics 30(2). 155–209. http://dx.doi.org/10.1093/jos/ffs003. ↗
  3. Breheny, Richard. 2008. A new look at the semantics and pragmatics of numerically quantified noun phrases. Journal of Semantics 25. 93–139. http://dx.doi.org/10.1093/jos/ffm016. ↗
  4. Bultinck, Bert. 2005. Numerous meanings: The meaning of English cardinals and the legacy of Paul Grice. Amsterdam: Elsevier. find
  5. Büring, Daniel. 2007. More or less. Chicago Linguistics Society (CLS) 43. 3–17. find
  6. Büring, Daniel. 2008. The least at least can do. West Coast Conference on Formal Linguistics (WCCFL) 26. 114–120. find
  7. Carston, Robyn. 1998. Informativeness, relevance and scalar implicature. In Robyn Carston & S. Uchida (eds.), Relevance theory: Applications and implications, 179–236. Amsterdam: John Benjamins. ↗
  8. Chierchia, Gennaro. 2004. Scalar implicatures, polarity phenomena and the syntax/pragmatics interface. In Adriana Belletti (ed.), Structures and beyond. Oxford, UK: Oxford University Press. ↗
  9. Chierchia, Gennaro. 2006. Broaden your views: Implicatures of domain widening and the “logicality” of language. Linguistic Inquiry 37(4). 535–590. http://dx.doi.org/10.1162/ling.2006.37.4.535. ↗
  10. Chierchia, Gennaro, Danny Fox & Benjamin Spector. 2012. The grammatical view of scalar implicature and the relationship between semantics and pragmatics. In Claudia Maienborn, Klaus von Heusinger & Paul Portner (eds.), Semantics: An international handbook of natural language meaning, vol. 3. Boston, MA: de Gruyter. find
  11. Chung, Sandra & William A. Ladusaw. 2004. Restriction and saturation. Cambridge, MA: MIT Press. ↗
  12. Cohen, Ariel & Manfred Krifka. 2010. Superlative quantifiers as modifiers of meta-speech acts. The Baltic International Yearbook of Cognition, Logic and Communication 6. 1–56. http://dx.doi.org/10.4148/biyclc.v6i0.1578. ↗
  13. Coppock, Elizabeth & Thomas Brochhagen. 2013. Raising and resolving issues with scalar modifiers. Semantics and Pragmatics 6(3). 1–57. http://dx.doi.org/10.3765/sp.6.3. A “de-Fregean” semantics for modified and unmodified numerals Cresswell, M. J. 1976. The semantics of degree. In Barbara Partee (ed.), Montague grammar, 261–292. New York, NY: Academic Press. ↗
  14. Cummins, Chris & Napoleon Katsos. 2010. Comparative and superlative quantifiers: Pragmatic effects of comparison type. Journal of Semantics 27. 271–305. http://dx.doi.org/10.1093/jos/ffq006. von Fintel, Kai & Sabine Iatridou. 2007. Anatomy of a modal construction. Linguistic Inquiry 38. 445–483. http://dx.doi.org/10.1162/ling.2007.38.3. 445. ↗
  15. Fox, Danny. 2007. Free choice disjunction and the theory of scalar implicatures. In Uli Sauerland & Penka Stateva (eds.), Presupposition and implicature in compositional semantics, 71–120. London: Palgrave MacMillan. find
  16. Fox, Danny & Martin Hackl. 2007. The universal density of measurement. Linguistics and Philosophy 29. 157–186. http://dx.doi.org/10.1007/s10988-006-9004-4. ↗
  17. Franke, Michael. 2011. Quantity implicatures, exhaustive interpretation, and rational conversation. Semantics and Pragmatics 4(1). 1–82. http://dx.doi.org/10.3765/sp.4.1. ↗
  18. Frege, Gottlob. 1980 [1884]. The foundations of arithmetic (Grundlagen der Arithmetik). Evanston, IL: Northwestern University Press. find
  19. Gazdar, Gerald. 1977. Implicature, presupposition and logical form. Sussex, UK: University of Sussex PhD thesis. find
  20. Geurts, Bart. 2006. Take five. In Svetlana Vogeleer & Liliane Tasmowski (eds.), Non-definiteness and plurality, 311–329. Amsterdam: Benjamins. http://dx.doi.org/10.1075/la.95.16geu. ↗
  21. Geurts, Bart & Rick Nouwen. 2007. At least et al.: The semantics of scalar modifiers. Language 83(3). 533–559. http://dx.doi.org/10.1353/lan.2007. 0115. ↗
  22. Hackl, Martin. 2000. Comparative quantifiers. Cambridge, MA: Massachusetts Institute of Technology PhD thesis. find
  23. Heim, Irene. 2000. Degree operators and scope. Semantics and Linguistic Theory (SALT) 10. 40–64. ↗
  24. Hirschberg, Julia. 1985. A theory of scalar implicature. Philadelphia, PA: University of Pennsylvania PhD thesis. find
  25. Horn, Laurence. 1972. On the semantic properties of logical operators in English. Los Angeles, CA: University of California, Los Angeles PhD thesis. find
  26. Horn, Laurence. 1992. The said and the unsaid. Semantics And Linguistic Theory (SALT) 2. 163–192. ↗
  27. Christopher Kennedy Huang, Yi Ting, Elizabeth Spelke & Jesse Snedeker. 2013. What exactly do numbers mean? Language Learning and Development 9. 105–129. http://dx.doi.org/10.1080/15475441.2012.658731. ↗
  28. Hurewitz, Felicia, Anna Papafragou, Lila Gleitman & Rochel Gelman. 2007. Asymmetries in the acquisition of numbers and quantifiers. Language Learning and Development 2(2). 77–96. http://dx.doi . org / 10 . 1207 / s15473341lld0202_1. ↗
  29. Katzir, Roni. 2007. Structurally-defined alternatives. Linguistics and Philosophy 30. 669–690. http://dx.doi.org/10.1007/s10988-008-9029-y. ↗
  30. Kennedy, Christopher. 1999. Projecting the adjective: The syntax and semantics of gradability and comparison. (1997 UCSC Ph.D Dissertation). New York, NY: Garland. find
  31. Kennedy, Christopher. 2013. A scalar semantics for scalar readings of number words. In Ivano Caponigro & Carlo Cecchetto (eds.), From grammar to meaning: The spontaneous logicality of language, 172–200. Cambridge, UK: Cambridge University Press. http://dx.doi.org/10.1017/cbo9781139519328. 010. ↗
  32. Kennedy, Christopher & Jason Stanley. 2009. On average. Mind 118(583-646). http://dx.doi.org/10.1093/mind/fzp094. ↗
  33. Koenig, Jean-Pierre. 1991. Scalar predicates and negation: Punctual semantics and interval interpretations. Chicago Linguistic Society (CLS) 27. 140–155. find
  34. Krasikova, Sveta. 2010. Comparing to what is possible and permitted. Sinn and Bedeutung 14. 256–272. find
  35. Kratzer, Angelika & Junko Shimoyama. 2002. Indeterminate pronouns: The view from Japanese. 3. find
  36. Krifka, Manfred. 1989. Nominal reference, temporal constitution and quantification in event semantics. In Renate Bartsch, Johann van Benthem & Peter van Emde Boas (eds.), Semantics and contextual expression, 75–115. Stanford, CA: CSLI Publications. ↗
  37. Krifka, Manfred. 1998. At least some determiners aren’t determiners. In Ken Turner (ed.), The semantics/pragmatics interface from different points of view, 257–291. Oxford, UK: Elsevier. find
  38. Landman, Fred. 2003. Predicate argument mismatches and the adjectival theory of indefinites. In Martine Coene & Yves d’Hulst (eds.), The syntax and semantics of noun phrases, vol. 1 (From NP to DP), 211–238. Philadelphia, PA: John Benjamins. http://dx.doi.org/10.1075/la.55.10lan. ↗
  39. Landman, Fred. 2004. Indefinites and the type of sets. Oxford, UK: Oxford University Press. http://dx.doi.org/10.1002/9780470759318. A “de-Fregean” semantics for modified and unmodified numerals Marty, Paul, Emmanuel Chemla & Benjamin Spector. 2013. Interpreting numerals and scalar items under memory load. Lingua 133. 152–162. http://dx.doi.org/10.1016/j.lingua.2013.03.006. ↗
  40. Matsumoto, Yo. 1995. The conversational condition on Horn scales. Linguistics and Philosophy 18. 21–60. http://dx.doi.org/10.1007/bf00984960. ↗
  41. Meyr, Clemens. 2013. Implicatures of modified numerals. In Ivano Caponigro & Carlo Cecchetto (eds.), From grammar to meaning: The spontaneous logicality of language, 139–171. Cambridge, UK: Cambridge University Press. http://dx.doi.org/10.1017/cbo9781139519328.009. ↗
  42. Musolino, Julien. 2004. The semantics and acquisition of number words: Integrating linguistic and developmental perspectives. Cognition 93(1). 1–41. http://dx.doi.org/10.1016/j.cognition.2003.10.002. ↗
  43. Nouwen, Rick. 2008a. Directionality in numeral quantifiers: The case of up to. Semantics and Linguistic Theory (SALT) 18. 551–568. ↗
  44. Nouwen, Rick. 2008b. Upper-bounded no more: The exhaustive interpretation of non-strict comparison. Natural Language Semantics 16. 271–295. http://dx.doi.org/10.1007/s11050-008-9034-2. ↗
  45. Nouwen, Rick. 2010. Two kinds of modified numerals. Semantics and Pragmatics 3(3). 1–41. http://dx.doi.org/10.3765/sp.3.3. ↗
  46. Noveck, Ira. 2001. When children are more logical than adults: Experimental investigations of scalar implicature. Cognition 78(2). 165–188. http://dx. doi.org/10.1016/s0010-0277(00)00114-1. ↗
  47. Papafragou, Anna & Julien Musolino. 2003. Scalar implicatures: Experiments at the semantics-pragmatics interface. Cognition 86. 253–282. http://dx. doi.org/10.1016/s0010-0277(02)00179-8. ↗
  48. Partee, Barbara. 1987. Noun phrase interpretation and type shifting principles. In Geroen Groenendijk, Dik de Jongh & Martin Stokhof (eds.), Studies in discourse representation and the theory of generalized quantifiers, 115–143. Dordrecht: Foris. http://dx.doi.org/10.1002/9780470758335.ch15. ↗
  49. Rett, Jessica. 2014. Measure phrase equatives and modified numerals. Journal of Semantics. http://dx.doi.org/10.1093/jos/ffu004. ↗
  50. Sadock, Jerrold. 1984. Whither radical pragmatics? In Deborah Schiffren (ed.), Meaning, form and use in context: Linguistic applications, 139–149. Washington, DC: Georgetown University Press. find
  51. Salmon, Nathan. 1997. Wholes, parts, and numbers. Noûs Supplement: Philosophical Perspectives, 11, Mind, Causation, and World 31. 1–15. http://dx.doi.org/10.1111/0029-4624.31.s11.1. ↗
  52. Christopher Kennedy Sauerland, Uli. 2004. Scalar implicatures in complex sentences. Linguistics and Philosophy 27. 267–391. http://dx.doi.org/10.1023/b:ling.0000023378. 71748.db. ↗
  53. Scharten, Rosemarijn. 1997. Exhaustive interpretation: A discourse-semantic account. Nijmegen: Katholieke Universiteit Nijmegen PhD thesis. find
  54. Schwarz, Bernhard. 2013. At least and quantity implicature: Choices and consequences. Amsterdam Colloquium 19. 187–194. ↗
  55. Schwarz, Bernhard, Brian Buccola & Michael Hamilton. 2012. Two types of class B numeral modifiers: A reply to Nouwen 2010. Semantics and Pragmatics 5(1). 1–25. http://dx.doi.org/10.3765/sp.5.1. ↗
  56. Schwarz, Bernhard & Junko Shimoyama. 2011. Negative islands and obviation by wa in Japanese degree questions. Semantics and Linguistic Theory 20. 702–719. ↗
  57. Schwarzschild, Roger. 2005. Measure phrases as modifiers of adjectives. Recherches Linguistiques de Vincennes 35. 207–228. http://dx.doi.org/10. 4000/rlv.1401. ↗
  58. Schwarzschild, Roger & Karina Wilkinson. 2002. Quantifiers in comparatives: A semantics of degree based on intervals. Natural Language Semantics 10. 1–41. http://dx.doi.org/10.1023/a:1015545424775. von Stechow, Arnim. 1984. Comparing semantic theories of comparison. Journal of Semantics 3. 1–77. http://dx.doi.org/10.1093/jos/3.1-2.1. ↗
  59. Takahashi, Shoichi. 2006. More than two quantifiers. Natural Language Semantics 14. 57–101. http://dx.doi.org/10.1007/s11050-005-4534-9. Christopher Kennedy Department of Linguistics University of Chicago Chicago, IL 60637 USA ck@uchicago.edu ↗

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