Abstract
A protogroup is an ordered monoid in which each element a has both a left proto-inverse a ℓ such that a ℓa ≤ 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 ℓ and 1 ≤ a ra, rendering a ℓ 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 ℓℓ ≠ a in general. Free pregroups may be used to recognize the same sentences as free protogroups.
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References
V.M. Abrusci, Sequent calculus and phase semantics for pure noncommutative classical propositional linear logic, Journal of Symbolic Logic 56 (1991), 1403–1451.
....., 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.
K. Ajdukiewicz, Die syntaktische Konnexität, Studia Philosophica 1 (1937), 1–27. Translated in: S. McCall, Polish Logic 1920-1939, Claredon Press, Oxford 1967.
Y. Bar-Hillel, A quasiarithmetical notation for syntactic description, Language 29 (1953), 47–58.
M. Barr, *-Autonomous categories, Springer LNM 752 (1979).
M. Brame and Youn Gon Kim, Directed types: an algebraic theory of production and recognition, Preprint 1997.
W. Buszkowski, W. Marciszewski and J. van Benthem, Categorial grammar, John Benjamins Publ. Co., Amsterdam 1988.
C. Casadio, A categorical approach to cliticization and agreement in Italian, Editrice Bologna 1993.
....., Unbounded dependencies in noncommutative linear logic, Proceedings of the conference: Formal grammar, ESSLLI 1997, Aix en Provence.
....., Noncommutative linear logic in linguistics, Preprint 1997.
N. Chomsky, Syntactic structures, Mouton, The Hague 1957.
....., Lectures on government and binding, Foris Publications, Dordrecht 1982.
J.R.B. Cockett and R.A.G. Seely, Weakly distributive categories, J. Pure and Applied Algebra 114 (1997), 133–173.
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.
C. Dexter, The second Inspector Morse omnibus, Pan Books, London 1994.
K. Došen and P. Schroeder-Heister (eds), Substructural logics, Oxford University Press, Oxford 1993.
G. Gazdar, E. Klein, G. Pullum and I. Sag, Generalized phrase structure grammar, Harvard University Press, Cambridge Mass. 1985.
J.-Y. Girard, Linear Logic, J. Theoretical Computer Science 50 (1987), 1–102.
V.N. Grishin, On a generalization of the Ajdukiewicz-Lambek system, in: Studies in nonclassical logics and formal systems, Nauka, Moscow 1983, 315–343.
R. Jackendo., x Syntax, a study of phrase structure, The MIT Press, Cambridge Mass. 1977.
M. Kanazawa, The Lambek calculus enriched with additional connectives, J. Logic, Language & Information 1 (1992), 141–171.
J. Lambek, The mathematics of sentence structure, Amer. Math. Monthly 65 (1958), 154–169.
....., Contribution to a mathematical analysis of the English verb-phrase, J. Canadian Linguistic Assoc. 5 (1959), 83–89.
....., On the calculus of syntactic types, AMS Proc. Symposia Applied Mathematics 12 (1961), 166–178.
....., On a connection between algebra, logic and linguistics, Diagrammes 22 (1989), 59–75.
....., Grammar as mathematics, Can. Math. Bull. 32 (1989), 257–273.
....., Production grammars revisited, Linguistic Analysis 23 (1993), 205–225.
....., From categorial grammar to bilinear logic, in: K. Došen et al. (eds) (1993), 207–237.
....., Some lattice models of bilinear logic, Algebra Universalis 34 (1995), 541–550.
....., 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.
....., Bilinear logic and Grishin algebras in: E. Orlowska (editor), Logic at work, Essays dedicated to the memory of Helena Rasiowa, Kluwer, Dordrecht, to appear.
S. Mikulás, The completeness of the Lambek calculus with respect to relational semantics, ITLT Prepublications for Logic, Semantics and Philosophy of Language, Amsterdam 1992.
M. Moortgat, Categorial investigations, Foris Publications, Dordrecht 1988.
....., Categorial type logic, in: J. van Benthem et al. 1997, 93–177.
G. Morrill, Type logical grammar, Kluwer, Dordrecht 1994.
R.T. Oehrle, Substructural logic and linguistic inference, Preprint 1997.
R.T. Oehrle, E. Bach and D. Wheeler (eds), Categorial grammar and natural language structures, D. Reidel Publishing Co., Dordrecht 1988.
M. Pentus, Lambek grammars are context free, Proc. 8th LICS conference (1993), 429–433.
M. Pentus, Models for the Lambek calculus, Annals Pure & Applied Logic 75 (1995), 179–213.
J. van Benthem, Language in action, Elsevier, Amsterdam 1991.
J. van Benthem and A. ter Meulen (eds), Handbook of Logic and Language, Elsevier, Amsterdam 1997.
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Lambek, J. (1999). Type Grammar Revisited. In: Lecomte, A., Lamarche, F., Perrier, G. (eds) Logical Aspects of Computational Linguistics. LACL 1997. Lecture Notes in Computer Science(), vol 1582. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48975-4_1
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