Linguistics and Philosophy

, Volume 36, Issue 1, pp 1–29 | Cite as

Constraints on the lexicalization of logical operators

  • Roni Katzir
  • Raj Singh
Research Article


We revisit a typological puzzle due to Horn (Doctoral Dissertation, UCLA, 1972) regarding the lexicalization of logical operators: in instantiations of the traditional square of opposition across categories and languages, the O corner, corresponding to ‘nand’ (= not and), ‘nevery’ (= not every), etc., is never lexicalized. We discuss Horn’s proposal, which involves the interaction of two economy conditions, one that relies on scalar implicatures and one that relies on markedness. We observe that in order to express markedness and to account for a bigger typological puzzle, namely the absence of lexicalizations of ‘XOR’ (= exclusive or), ‘all-or-none’, and many other imaginable logical operators, one must restrict the basic lexicalizable elements to a small set of primitives. We suggest that an ordering based perspective, following Keenan and Faltz (Boolean semantics for natural language, 1985), makes the stipulated primitives that we arrive at more natural. We also propose a modification to Horn’s proposal, based on recent work on implicatures, in which only the implicature condition is operative and in which markedness is part of the definition of the alternatives for scalar implicatures rather than an independent condition.


Logical operators Negation Lexicalization Ordering Scalar implicature Contradiction Markedness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abels K., Martí L. (2010) A unified approach to split scope. Natural Language Semantics 18: 435–470CrossRefGoogle Scholar
  2. Abrusán M. (2011) Presuppositional and negative islands: A semantic account. Natural Language Semantics 19: 257–321CrossRefGoogle Scholar
  3. Barwise J., Cooper R. (1981) Generalized quantifiers and natural language. Linguistics and Philosophy 4: 159–219CrossRefGoogle Scholar
  4. Béziau J.-Y. (2003) New light on the square of oppositions and its nameless corner. Logical Investigations 10: 218–233Google Scholar
  5. Blanché R. (1953) Sur l’opposition des concepts. Theoria 19: 89–130CrossRefGoogle Scholar
  6. Blanché, R. (1969). Structures intellectuelles. Paris: J. Vrin.Google Scholar
  7. Bresnan J. (1973) Syntax of the comparative clause construction in English. Linguistic Inquiry 4: 275–343Google Scholar
  8. Chemla, E. (2009). Similarity: Towards a unified account of scalar implicatures, free choice permission and presupposition projection. Under revision for Semantics and Pragmatics.Google Scholar
  9. Davey, B. A., & Priestley, H. A. (2002). Introduction to lattices and order (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
  10. de Swart, H. (2000). Scope ambiguities with negative quantifiers. In K. von Heusinger & U. Egli (Eds.), Reference and anaphoric relations (pp. 109–132). Dordrecht: Kluwer.Google Scholar
  11. Farkas D. F., Kiss K. E. (2000) On the comparative and absolute readings of superlatives. Natural Language and Linguistic Theory 18: 417–455CrossRefGoogle Scholar
  12. Fox, D. (2007). Free choice disjunction and the theory of scalar implicatures. In U. Sauerland & P. Stateva (Eds.), Presupposition and implicature in compositional semantics (pp. 71–120). New York: Palgrave-Macmillan.Google Scholar
  13. Fox D., Hackl M. (2006) The universal density of measurement. Linguistics and Philosophy 29: 537–586CrossRefGoogle Scholar
  14. Fox D., Katzir R. (2011) On the characterization of alternatives. Natural Language Semantics 19: 87–107CrossRefGoogle Scholar
  15. Gajewski J. (2010) Superlatives, NPIs and most. Journal of Semantics 27: 125–137CrossRefGoogle Scholar
  16. Gazdar G. (1979) Pragmatics: Implicature, presupposition and logical form. Academic Press, New YorkGoogle Scholar
  17. Gazdar G. (1980) A cross-categorial semantics for coordination. Linguistics and Philosophy 3: 407–409CrossRefGoogle Scholar
  18. Gazdar, G., & Pullum G. K. (1976). Truth functional connectives in natural language. In Papers from the regional meeting of the Chicago Linguistic Society (Vol. 12, pp. 220–234), Chicago, IL.Google Scholar
  19. Geurts B. (1996) On no. Journal of Semantics 13: 67–86CrossRefGoogle Scholar
  20. Grice P. (1989) Studies in the way of words. Harvard University Press, CambridgeGoogle Scholar
  21. Groenendijk, J., & Stokhof, M. (1984). Studies in the semantics of questions and the pragmatics of answers. Doctoral Dissertation, Universiteit van Amsterdam, Amsterdam.Google Scholar
  22. Hackl M. (2009) On the grammar and processing of proportional quantifiers: Most versus more than half. Natural Language Semantics 17: 63–98CrossRefGoogle Scholar
  23. Heim, I. (1982). The semantics of definite and indefinite noun phrases. Doctoral Dissertation, University of Massachusetts, Amherst.Google Scholar
  24. Heim, I. (1985). Notes on comparatives and related matters. Ms., University of Texas at Austin.Google Scholar
  25. Heim I. (1990) E-type pronouns and donkey anaphora. Linguistics and Philosophy 13: 137–178CrossRefGoogle Scholar
  26. Heim, I. (1999). Notes on superlatives. MIT lecture notes. Ms., MIT.Google Scholar
  27. Higginbotham J., May R. (1981) Questions, quantifiers and crossing. The Linguistic Review 1: 1–41CrossRefGoogle Scholar
  28. Hirschberg, J. (1985/1991). A theory of scalar implicature. New York: Garland.Google Scholar
  29. Hoeksema, J. (1999). Blocking effects and polarity sensitivity. In J. Gerbrandy, M. Marx, M. de Rijke, & Y. Venema (Eds.), Jfak: Essays dedicated to Johan van Benthem on the occasion of his 50th birthday. Amsterdam: Amsterdam University Press.Google Scholar
  30. Horn, L. (1972). On the semantic properties of the logical operators in English. Doctoral Dissertation, UCLA.Google Scholar
  31. Horn, L. (1984). Toward a new taxonomy for pragmatic inference: Q-based and R-based implicatures. In D. Schiffrin (Ed.), Meaning, form, and use in context (pp. 11–42). Washington: Georgetown University Press.Google Scholar
  32. Horn L. (1989) A natural history of negation. University of Chicago Press, ChicagoGoogle Scholar
  33. Horn, L. (1990). Hamburgers and truth: Why Gricean inference is Gricean. In BLS (Vol. 16, pp. 454–471).Google Scholar
  34. Horn L. (2000) From IF to IFF: Conditional perfection as pragmatic strengthening. Journal of Pragmatics 32: 289–326CrossRefGoogle Scholar
  35. Horn, L. (2011). Histoire d’*O: Lexical pragmatics and the geometry of opposition (pp. 383–416). Bern: Peter Lang.Google Scholar
  36. Hunter, T., & Lidz, J. (2012). Conservativity and learnability of determiners. Journal of Semantics, 29(3).Google Scholar
  37. Hunter, T., Lidz, J., Wellwood, A., & Conroy, A. (2009). Restrictions on the meaning of determiners: Typological generalisations and learnability. In E. Cormany & S. Ito (Eds.), Proceedings of SALT XIX. Ithaca, NY: CLC Publications.Google Scholar
  38. Ionin T., Matushansky O. (2006) The composition of complex cardinals. Journal of Semantics 23: 315–360CrossRefGoogle Scholar
  39. Jacobs J. (1980) Lexical decomposition in Montague grammar. Theoretical Linguistics 7: 121–136CrossRefGoogle Scholar
  40. Jaspers, D. (2005). Operators in the lexicon: On the negative logic of natural language. Doctoral Dissertation, University of Leiden, Leiden.Google Scholar
  41. Kamp, H. (1981). A theory of truth and semantic representation. In J. Groenendijk (Ed.), Formal methods in the study of language. Amsterdam: Mathematical Center.Google Scholar
  42. Katzir R. (2007) Structurally-defined alternatives. Linguistics and Philosophy 30: 669–690CrossRefGoogle Scholar
  43. Katzir, R., & Singh R. (2009). On the absence of XOR in natural language. In P. ÉgrÉ & G. Magri (Ed.), Presuppositions and implicatures: Proceedings of the MIT-Paris workshop (pp. 118–129). Cambridge, MA: MITWPL.Google Scholar
  44. Keenan, E. L., & Faltz, L. (1978). Logical types for natural language. UCLA Occasional Papers in Linguistics. Los Angeles, CA: UCLA.Google Scholar
  45. Keenan E. L., Faltz L. (1985) Boolean semantics for natural language. Reidel, DordrechtGoogle Scholar
  46. Keenan E. L., Stavi J. (1986) A semantic characterization of natural language determiners. Linguistics and Philosophy 9: 253–326CrossRefGoogle Scholar
  47. Kneale, W., & Kneale, M. (1962). The development of logic. Oxford: Oxford University Press.Google Scholar
  48. Kotek, H., Sudo, Y., Howard, E., & Hackl, M. (2011). Three readings of most. In Proceedings of SALT (Vol. 21, pp. 353–372).Google Scholar
  49. Krasikova, S. (2012). Definiteness in superlatives. In Logic, language and meaning (pp. 411–420). Berlin: Springer.Google Scholar
  50. Kratzer, A. (1981). The notional category of modality. In H. Eickmeyer & H. Rieser (Eds.), Words, worlds, and contexts (pp. 38–74). Berlin: Walter de Gruyter.Google Scholar
  51. Kratzer, A. (1986). Conditionals. (Reprinted from Semantics: An international handbook of contemporary research, by A. von Stechow & D. Wunderlich, Eds., 1991, Berlin: Walter de Gruyter.)Google Scholar
  52. Kratzer, A. (1998). Scope or pseudoscope? Are there wide-scope indefinites? In S. Rothstein (Ed.), Events and grammar (pp. 163–196). Dordrecht: Kluwer.Google Scholar
  53. Landman F. (1989a) Groups, I. Linguistics and Philosophy 12: 559–605CrossRefGoogle Scholar
  54. Landman F. (1989b) Groups, II. Linguistics and Philosophy 12: 723–744CrossRefGoogle Scholar
  55. Landman, F. (2004). Indefinites and the type of sets. Oxford: Blackwell.Google Scholar
  56. Lewis, D. (1975). Adverbs of quantification. In E. Keenan (Ed.), Formal semantics of natural language. Cambridge: Cambridge University Press.Google Scholar
  57. Link, G. (1983). The logical analysis of plurals and mass terms: A lattice-theoretic approach. In R. Bäuerle, C. Schwarze, & A. von Stechow (Eds.), Meaning, use, and interpretation of language (pp. 303–323). Berlin: De Gruyter.Google Scholar
  58. Löbner, S. (1983). Phase quantification: A uniform treatment of some quantifiers from different categories. In Proceedings of the second Japanese-Korean Joint workshop on formal grammar at Kyoto, pp. 127–140, The Logico-Linguistic Society of Japan, Japan.Google Scholar
  59. Ludlow P., Neale S. (1991) Indefinite descriptions: In defense of Russell. Linguistics and Philosophy 14: 171–202CrossRefGoogle Scholar
  60. Matthewson L. (2001) Quantification and the nature of crosslinguistic variation. Natural Language Semantics 9: 145–189CrossRefGoogle Scholar
  61. Matthewson, L. (2011). Strategies of quantification in st’át’imcets and the rest of the world. Ms., University of British Columbia.Google Scholar
  62. McCawley, J. D. (1972). A program for logic. In Semantics of natural language (pp. 157–212). Dordrecht: Reidel.Google Scholar
  63. Montague, R. (1974). The proper treatment of quantification in English. In R. H. Thomason (Ed.), Formal philosophy: Selected papers of Richard Montague. New Haven, CT: Yale University Press.Google Scholar
  64. Moretti, A. (2012). Why the logical hexagon? Logica Universalis (pp. 1–39). doi: 10.1007/s11787-012-0045-x.
  65. Parsons T. (1997) The traditional square of opposition: A biography. Acta Analytica 18: 23–49Google Scholar
  66. Partee, B. (1987). Noun phrase interpretation and type shifting principles. In J. Groenendijk, D. de Jongh, & M. Stokhof (Eds.), Studies in discourse representation theory and the theory of generalized quantifiers. Dordrecht: Foris.Google Scholar
  67. Partee, B. H., & Rooth, M. (1983). Generalized conjunction and type ambiguity. In R. Bäuerle, C. Schwarze, & A. von Stechow (Eds.), Meaning, use and interpretation of language (pp. 362–383). Berlin: de Gruyter.Google Scholar
  68. Reinhart T. (1997) Quantifier scope: How labor is divided between QR and choice functions. Linguistics and Philosophy 20: 335–397CrossRefGoogle Scholar
  69. Rullmann, H. (1995). Maximality in the semantics of wh-constructions. Doctoral Dissertation, University of Massachusetts at Amherst, Amherst, MA.Google Scholar
  70. Russell B. (1919) Introduction to mathematical philosophy. Allen and Unwin, LondonGoogle Scholar
  71. Sauerland, U. (1998). The meaning of chains. Doctoral Dissertation, MIT, Cambridge, MA.Google Scholar
  72. Sauerland, U. (2000). No ‘no’: On the crosslinguistic absence of a determiner ‘no’. In Proceedings of the Tsukuba workshop on determiners and quantification (pp. 415–444).Google Scholar
  73. Sauerland U. (2004) Scalar implicatures in complex sentences. Linguistics and Philosophy 27: 367–391CrossRefGoogle Scholar
  74. Seuren P. (2006) The natural logic of language and cognition. Pragmatics 16: 103–138Google Scholar
  75. Sevi, A. (2005). Exhaustivity: A semantic account of ‘quantity’ implicatures. Doctoral Dissertation, Tel-Aviv University.Google Scholar
  76. Sharvit Y., Stateva P. (2002) Superlative expressions, context, and focus. Linguistics and Philosophy 25: 453–504CrossRefGoogle Scholar
  77. Solt, S. (2011). How many Most’s? In I. Reich, E. Horch, & P. Dennis (Eds.), Proceedings of Sinn und Bedeutung 15 (pp. 565–579). Saarbrücken: Saarland Universiy Press.Google Scholar
  78. Spector, B. (2006). Aspects de la pragmatique des opérateurs logiques. Doctoral Dissertation, Université de Paris 7, Paris.Google Scholar
  79. Strawson P. F. (1952) Introduction to logical theory. Methuen, LondonGoogle Scholar
  80. Swanson E. (2010) Structurally defined alternatives and lexicalizations of XOR. Linguistics and Philosophy 33: 31–36CrossRefGoogle Scholar
  81. Szabolcsi, A. (1986). Comparative superlatives. In N. Fukui, T. Rapoport, & E. Sagey (Eds.), Papers in theoretical linguistics (Vol. 8, pp. 245–265). Cambridge, MA: MITWPL.Google Scholar
  82. Szabolcsi, A. (2010). Quantification. Cambridge: Cambridge University Press.Google Scholar
  83. Szabolcsi, A. (2012). Compositionality without word boundaries: (The) more and (the) most. In Proceedings of SALT (Vol. 22, pp. 1–25).Google Scholar
  84. Szabolcsi, A., Whang, J. D., & Zu, V. (2012). Compositionality questions: Quantifier words and their multi-functional parts. Ms., NYU, June 2012Google Scholar
  85. Szabolcsi A., Zwarts F. (1993) Weak islands and an algebraic semantics for scope taking. Natural Language Semantics 1: 235–284CrossRefGoogle Scholar
  86. Benthem J. (1984) Questions about Quantifiers. The Journal of Symbolic Logic 49: 443–466CrossRefGoogle Scholar
  87. Rooij R., Schulz K. (2004) Exhaustive interpretation of complex sentences. Journal of Logic, Language and Information 13: 491–519CrossRefGoogle Scholar
  88. Fintel K. (1999) NPI licensing, Strawson entailment, and context dependency. Journal of Semantics 16: 97–148CrossRefGoogle Scholar
  89. Fintel K., Matthewson L. (2008) Universals in semantics. The Linguistic Review 25: 139–201Google Scholar
  90. Winter Y. (1997) Choice functions and the scopal semantics of indefinites. Linguistics and Philosophy 20: 399–467CrossRefGoogle Scholar
  91. Zeijlstra H. (2011) On the syntactically complex status of negative indefinites. Journal of Comparative Germanic Linguistics 4: 111–138CrossRefGoogle Scholar
  92. Zweig, E. (2006). Nouns and adjectives in numeral NPs. In L. Bateman & C. Ussery (Eds.), Proceedings of NELS 35 (pp. 663–675). Amherst, MA: GLSA.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Linguistics and Sagol School of NeuroscienceTel Aviv UniversityTel AvivIsrael
  2. 2.Institute of Cognitive ScienceCarleton UniversityOttawaCanada

Personalised recommendations