A protogroup is an ordered monoid in which each element a has both a left proto-inverse a such that a a ≤ 1 and a right proto-inverse a r such that aa r ≤ 1. We explore the assignment of elements of a free protogroup to English words as an aid for checking which strings of words are well-formed sentences, though ultimately we may have to relax the requirement of freeness. By a pregroup we mean a protogroup which also satisfies 1 ≤ aa and 1 ≤ a r a, rendering a a left adjoint and a r a right adjoint of a. A pregroup is precisely a poset model of classical non-commutative linear logic in which the tensor product coincides with it dual. This last condition is crucial to our treatment of passives and Wh-questions, which exploits the fact that a ℓℓa in general. Free pregroups may be used to recognize the same sentences as free protogroups.


Noun Phrase Relative Clause Linear Logic Type Assignment Mass Noun 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. V.M. Abrusci, Sequent calculus and phase semantics for pure noncommutative classical propositional linear logic, Journal of Symbolic Logic 56 (1991), 1403–1451.zbMATHCrossRefMathSciNetGoogle Scholar
  2. ....., Lambek calculus, cyclic multiplicative — additive linear logic, non-commutative multiplicative — additive linear logic: language and sequent calculus; in: V.M. Abrusci and C. Casadio (eds), Proofs and linguistic categories, Proceedings 1996 Roma Workshop, Cooperativa Libraria Universitaria Editrice Bolognia (1996), 21–48.Google Scholar
  3. K. Ajdukiewicz, Die syntaktische Konnexität, Studia Philosophica 1 (1937), 1–27. Translated in: S. McCall, Polish Logic 1920-1939, Claredon Press, Oxford 1967.Google Scholar
  4. Y. Bar-Hillel, A quasiarithmetical notation for syntactic description, Language 29 (1953), 47–58.CrossRefGoogle Scholar
  5. M. Barr, *-Autonomous categories, Springer LNM 752 (1979).Google Scholar
  6. M. Brame and Youn Gon Kim, Directed types: an algebraic theory of production and recognition, Preprint 1997.Google Scholar
  7. W. Buszkowski, W. Marciszewski and J. van Benthem, Categorial grammar, John Benjamins Publ. Co., Amsterdam 1988.zbMATHGoogle Scholar
  8. C. Casadio, A categorical approach to cliticization and agreement in Italian, Editrice Bologna 1993.Google Scholar
  9. ....., Unbounded dependencies in noncommutative linear logic, Proceedings of the conference: Formal grammar, ESSLLI 1997, Aix en Provence.Google Scholar
  10. ....., Noncommutative linear logic in linguistics, Preprint 1997.Google Scholar
  11. N. Chomsky, Syntactic structures, Mouton, The Hague 1957.Google Scholar
  12. ....., Lectures on government and binding, Foris Publications, Dordrecht 1982.Google Scholar
  13. J.R.B. Cockett and R.A.G. Seely, Weakly distributive categories, J. Pure and Applied Algebra 114 (1997), 133–173.zbMATHCrossRefMathSciNetGoogle Scholar
  14. H.B. Curry, Some logical aspects of grammatical structure, in: R. Jacobson (editor), Structure of language and its mathematical aspects, AMS Proc. Symposia Applied Mathematics 12 (1961), 56–67.Google Scholar
  15. C. Dexter, The second Inspector Morse omnibus, Pan Books, London 1994.Google Scholar
  16. K. Došen and P. Schroeder-Heister (eds), Substructural logics, Oxford University Press, Oxford 1993.Google Scholar
  17. G. Gazdar, E. Klein, G. Pullum and I. Sag, Generalized phrase structure grammar, Harvard University Press, Cambridge Mass. 1985.Google Scholar
  18. J.-Y. Girard, Linear Logic, J. Theoretical Computer Science 50 (1987), 1–102.zbMATHCrossRefMathSciNetGoogle Scholar
  19. V.N. Grishin, On a generalization of the Ajdukiewicz-Lambek system, in: Studies in nonclassical logics and formal systems, Nauka, Moscow 1983, 315–343.Google Scholar
  20. R. Jackendo., x Syntax, a study of phrase structure, The MIT Press, Cambridge Mass. 1977.Google Scholar
  21. M. Kanazawa, The Lambek calculus enriched with additional connectives, J. Logic, Language & Information 1 (1992), 141–171.zbMATHMathSciNetGoogle Scholar
  22. J. Lambek, The mathematics of sentence structure, Amer. Math. Monthly 65 (1958), 154–169.zbMATHCrossRefMathSciNetGoogle Scholar
  23. ....., Contribution to a mathematical analysis of the English verb-phrase, J. Canadian Linguistic Assoc. 5 (1959), 83–89.Google Scholar
  24. ....., On the calculus of syntactic types, AMS Proc. Symposia Applied Mathematics 12 (1961), 166–178.Google Scholar
  25. ....., On a connection between algebra, logic and linguistics, Diagrammes 22 (1989), 59–75.zbMATHMathSciNetGoogle Scholar
  26. ....., Grammar as mathematics, Can. Math. Bull. 32 (1989), 257–273.zbMATHMathSciNetGoogle Scholar
  27. ....., Production grammars revisited, Linguistic Analysis 23 (1993), 205–225.Google Scholar
  28. ....., From categorial grammar to bilinear logic, in: K. Došen et al. (eds) (1993), 207–237.Google Scholar
  29. ....., Some lattice models of bilinear logic, Algebra Universalis 34 (1995), 541–550.zbMATHCrossRefMathSciNetGoogle Scholar
  30. ....., On the nominalistic interpretation of natural languages, in: M. Marion et al. (eds), Québec Studies in Philosophy of Science I, 69–74, Kluwer, Dordrecht 1995.Google Scholar
  31. ....., Bilinear logic and Grishin algebras in: E. Orlowska (editor), Logic at work, Essays dedicated to the memory of Helena Rasiowa, Kluwer, Dordrecht, to appear.Google Scholar
  32. S. Mikulás, The completeness of the Lambek calculus with respect to relational semantics, ITLT Prepublications for Logic, Semantics and Philosophy of Language, Amsterdam 1992.Google Scholar
  33. M. Moortgat, Categorial investigations, Foris Publications, Dordrecht 1988.Google Scholar
  34. ....., Categorial type logic, in: J. van Benthem et al. 1997, 93–177.Google Scholar
  35. G. Morrill, Type logical grammar, Kluwer, Dordrecht 1994.zbMATHGoogle Scholar
  36. R.T. Oehrle, Substructural logic and linguistic inference, Preprint 1997.Google Scholar
  37. R.T. Oehrle, E. Bach and D. Wheeler (eds), Categorial grammar and natural language structures, D. Reidel Publishing Co., Dordrecht 1988.Google Scholar
  38. M. Pentus, Lambek grammars are context free, Proc. 8th LICS conference (1993), 429–433.Google Scholar
  39. M. Pentus, Models for the Lambek calculus, Annals Pure & Applied Logic 75 (1995), 179–213.zbMATHCrossRefMathSciNetGoogle Scholar
  40. J. van Benthem, Language in action, Elsevier, Amsterdam 1991.zbMATHGoogle Scholar
  41. J. van Benthem and A. ter Meulen (eds), Handbook of Logic and Language, Elsevier, Amsterdam 1997.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • J. Lambek
    • 1
  1. 1.McGill UniversityUSA

Personalised recommendations