\(\hbox {NL}_\lambda \) as the Logic of Scope and Movement

  • Chris BarkerEmail author


Lambek elegantly characterized part of natural language. As is well-known, his substructural logic L, and its non-associative version NL, handle basic function/argument composition well, but not scope taking and syntactic displacement—at least, not in their full generality. In previous work, I propose \(\text {NL}_\lambda \), which is NL supplemented with a single structural inference rule (“abstraction”). Abstraction closely resembles the traditional linguistic rule of quantifier raising, and characterizes both semantic scope taking and syntactic displacement. Due to the unconventional form of the abstraction inference, there has been some doubt that \(\text {NL}_\lambda \) should count at a legitimate substructural logic. This paper argues that \(\text {NL}_\lambda \) is perfectly well-behaved. In particular, it enjoys cut elimination and an interpolation result. In addition, perhaps surprisingly, it is decidable. Finally, I prove that it is sound and complete with respect to the usual class of relational frames.


Lambek Substructural logic Scope Quantifier raising Syntactic movement Continuations Decidability 



  1. Barker, C. (2007). Parasitic scope. Linguistics and Philosophy, 30(4), 407–444.CrossRefGoogle Scholar
  2. Barker, C. (2013). Scopability and sluicing. Linguistics and Philosophy, 36(3), 187–223.CrossRefGoogle Scholar
  3. Barker, C. (2015a). Scope. In S. Lappin & C. Fox (Eds.), The handbook of contemporary semantics (2d ed., pp. 47–87). London: Wiley-Blackwell.Google Scholar
  4. Barker, C. (2015b). Scope as syntactic abstraction. In T. Murata, K. Mineshima, & D. Bekki (Eds.), New frontiers in artificial intelligence (pp. 184–199). Berlin: Springer.CrossRefGoogle Scholar
  5. Barker, C. (In prep.). The logic of scope. [working title] NYU manuscript.Google Scholar
  6. Barker, C., & Shan, C. (2014). Continuations and natural language. Oxford: Oxford University Press.CrossRefGoogle Scholar
  7. Buszkowski, W. (2005). Lambek calculus with nonlogical axioms. In Casadio, C., Scott, P. J., & Seely, R. A. G. (Eds.), Language and Grammar. Studies in Mathematical Linguistics and Natural Language. CSLI Lecture Notes (Vol. 168, pp. 77–93). Stanford: CSLI Press.Google Scholar
  8. Buszkowski, W. (2010). Lambek calculus and substructural logics. Linguistic Analysis, 36, 15–48.Google Scholar
  9. Craig, W. (1957). Three uses of the Herbrand–Gentzen theorem in relating model theory and proof theory. The Journal of Symbolic Logic, 22(3), 269–285.CrossRefGoogle Scholar
  10. de Groote, P. (2002). Towards abstract categorial grammars. In Proceedings of the 40th annual meeting of the Association for Computational Linguistics (pp. 148–155). San Francisco, CA: Morgan Kaufmann.Google Scholar
  11. Heim, I., & Kratzer, A. (1998). Semantics in generative grammar. Oxford: Blackwell.Google Scholar
  12. Jäger, G. (2005). Residuation, structural rules, and context freeness. Journal of Logic, Language, and Information, 13(1), 49–57.Google Scholar
  13. Kubota, Y. (2015). Nonconstituent coordination in Japanese as constituent coordination: An analysis in hybrid type-logical categorial grammar. Linguistic Inquiry, 46(1), 1–42.CrossRefGoogle Scholar
  14. Lambek, J. (1958). The mathematics of sentence structure. The American Mathematical Monthly, 65, 154–170.CrossRefGoogle Scholar
  15. Lambek, J. (1961). On the calculus of syntactic types. In R. Jakobson (Ed.), Structure of language and its mathematical aspects (pp. 166–178). Providence: American Mathematical Society.CrossRefGoogle Scholar
  16. Lambek, J. (1988). Categorial and categorical grammars. In R. T. Oehrle, et al. (Eds.), Categorial grammars and natural language structures (pp. 297–317). Dordrecht: Reidel.CrossRefGoogle Scholar
  17. May, R. (1977). The Grammar of Quantification. Ph.D. thesis, MIT. Reprinted by New York: Garland, 1991.Google Scholar
  18. May, R. (1985). Logical Form: Its Structure and Derivation. Cambridge: MIT Press.Google Scholar
  19. Moortgat, M. (1996). Generalized quantifiers and discontinuous type constructors. In H. Bunt & A. van Horck (Eds.), Discontinuous constituency (pp. 181–208). Berlin: Mouton de Gruyter.Google Scholar
  20. Moortgat, M. (1997). Categorial type logics. In J. F. A. K. van Benthem & B. ter Meulen (Eds.), Handbook of logic and language (pp. 93–178). Amsterdam: Elsevier.CrossRefGoogle Scholar
  21. Morrill, G., Valentín, O., & Fadda, M. (2011). The displacement calculus. Journal of Logic, Language and Information, 20(1), 1–48.CrossRefGoogle Scholar
  22. Muskens, R. (2001). \(\lambda \)-grammars and the syntax–semantics interface. In R. van Rooy, M. Stokhof, et al. (Eds.), Proceedings of the 13th Amsterdam colloquium. Amsterdam: Institute for Logic, Language and Computation, Universiteit van Amsterdam.Google Scholar
  23. Oehrle, R. T. (1994). Term-labeled categorial type systems. Linguistics and Philosophy, 17(6), 633–678.CrossRefGoogle Scholar
  24. Ono, H. (2003). Substructural logics and residuated lattices–an introduction. In V. F. Hendricks & J. Malinowski (Eds.), Trends in logic: 50 years of studia logica (pp. 193–228). Dordrecht: Kluwer.CrossRefGoogle Scholar
  25. Pentus, M. (1993). Lambek grammars are context free. In Proceedings of the eighth annual IEEE symposium on logic in computer science (pp. 429–433).Google Scholar
  26. Restall, G. (2000). An introduction to substructural logics. London: Routledge.CrossRefGoogle Scholar
  27. Roorda, D. (1991). Resource logics: Proof-theoretical investigations. PhD thesis, Amsterdam.Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.New York UniversityNew YorkUSA

Personalised recommendations