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Some Syntactic Interpretations in Different Systems of Full Lambek Calculus

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Modality, Semantics and Interpretations

Part of the book series: Logic in Asia: Studia Logica Library ((LIAA))

Abstract

Buszkowski (Logical Aspects of Computational Linguistics, 2014 [8]) defines an interpretation of FL without 1 in its version without empty antecedents of sequents (employed in type grammars) and applies this interpretation to prove some general results on the complexity of substructural logics and the generative capacity of type grammars. Here this interpretation is extended for nonassociative logics (also with structural rules), logics with 1, logics with distributive laws for \(\wedge ,\vee \), logics with unary modalities, and multiplicative fragments.

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References

  1. Y. Bar-Hillel, Ch. Gaifman, and E. Shamir. On categorial and phrase structure grammars. Bulletin Res. Council Israel, F(9):155–166, 1960.

    Google Scholar 

  2. F. Belardinelli, P. Jipsen, and H. Ono. Algebraic aspects of cut elimination. Studia Logica, 77(2):209–240.

    Google Scholar 

  3. W. Buszkowski. Completeness results for Lambek syntactic calculus. Zeitschrift f. mathematische Logik und Grundlagen der Mathematik, 32(1):13–28, 1986.

    Google Scholar 

  4. W. Buszkowski. Mathematical linguistics and proof theory. In van Benthem and ter Meulen [33], pages 683–736.

    Google Scholar 

  5. W. Buszkowski. Finite models of some substructural logics. Mathematical Logic Quarterly, 48(1):63–72, 2002.

    Google Scholar 

  6. W. Buszkowski. Lambek calculus with nonlogical axioms. In Casadio et al. [10], pages 77–93.

    Google Scholar 

  7. W. Buszkowski. Interpolation and FEP for logics of residuated algebras. Logic Journal of the IGPL, 19(3):437–454, 2011.

    Google Scholar 

  8. W. Buszkowski. An interpretation of Full Lambek Calculus in its variant without empty antecedents of sequents. In N. Asher and S.Soloviev, editors, Logical Aspects of Computational Linguistics. LACL 2014, volume 8535 of LNCS, pages 30–43. 2014.

    Google Scholar 

  9. W. Buszkowski and M. Farulewski. Nonassociative Lambek calculus with additives and context-free languages. In Languages: From Formal to Natural, volume 5533 of LNCS, pages 45–58. 2009.

    Google Scholar 

  10. C. Casadio, P. J. Scott, and R. A. Seely, editors. Language and Grammar. Studies in Mathematical Linguistics and Natural Language. CSLI Publications, 2005.

    Google Scholar 

  11. A. Clark. A learnable representation for syntax using residuated lattices. In P. de Groote, M. Egg, and L. Kallmeyer, editors, Formal Grammar. FG 2009, volume 5591 of Lecture Notes in Artificial Intelligence, pages 183–198. Springer, 2011.

    Google Scholar 

  12. N. Galatos, P. Jipsen, T. Kowalski, and H. Ono. Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and The Foundations of Mathematics. Elsevier, 2007.

    Google Scholar 

  13. N. Galatos and H. Ono. Cut elimination and strong separation for substructural logics: an algebraic approach. Annals of Pure and Applied Logic, 161(9):1097–1133, 2010.

    Google Scholar 

  14. J.-Y. Girard. Linear logic. Theoretical Computer Science, 50:1–102, 1987.

    Google Scholar 

  15. Z. Haniková and R. Horčik. The finite embeddability property for residuated groupoids. Algebra Universalis, 72(1):1–13, 2014.

    Google Scholar 

  16. R. Horčik and K. Terui. Disjunction property and complexity of substructural logics. Theoretical Computer Science, 412:3992–4006, 2011.

    Google Scholar 

  17. M. Kanazawa. The Lambek calculus enriched with additional connectives. Journal of Logic, Language and Information, 1(2), 1992.

    Google Scholar 

  18. M. Kozak. Distributive Full Lambek Calculus has the finite model property. Studia Logica, 91(2):201–216, 2009.

    Google Scholar 

  19. S. Kuznetsov. Lambek grammars with the unit. In P. de Groote and M.-J. Nederhof, editors, Formal Grammar 2010/2011, volume 7395 of LNCS, pages 262–266. Springer, 2012.

    Google Scholar 

  20. J. Lambek. The mathematics of sentence structure. American Mathematical Monthly, 65:154–170, 1958.

    Google Scholar 

  21. J. Lambek. On the calculus of syntactic types. In R. Jakobson, editor, Structure of Language and Its Mathematical Aspects, pages 166–178. American Mathematical Society, Providence, 1961.

    Google Scholar 

  22. J. Lambek. From Word to Sentence: a computational algebraic approach to grammar. Polimetrica, 2008.

    Google Scholar 

  23. Z. Lin. Modal nonassociative Lambek calculus with assumptions: complexity and context-freeness. In Language and Automata Theory and Applications. LATA 2010, volume 6031 of LNCS, pages 414–425. Springer, 2010.

    Google Scholar 

  24. Z. Lin. Distributive Full Nonassociative Lambek Calculus with S4-modalities is context-free. In Logical Aspects of Computational Linguistics. LACL 2012, volume 7351 of LNCS, pages 161–172. Springer, 2012.

    Google Scholar 

  25. M. Ma and Z. Lin. Residuated basic logic. Paper presented at AWPL 2, Guangzhou, 2014.

    Google Scholar 

  26. M. Moortgat. Multimodal linguistic inference. Journal of Logic, Language and Information, 5(3–4):349–385, 1996.

    Google Scholar 

  27. M. Moortgat. Categorial Type Logics. In van Benthem and ter Meulen [33], pages 93–177.

    Google Scholar 

  28. M. Moortgat and R.T. Oehrle. Pregroups and type-logical grammar: Searching for convergence. In Casadio et al. [10], pages 141–160.

    Google Scholar 

  29. R. Moot and C. Retoré. The Logic of Categorial Grammars. A Deductive Account of Natural Language Syntax and Semantics, volume 6850 of LNCS. Springer, 2012.

    Google Scholar 

  30. G. Morrill. Type Logical Grammar. Kluwer, Dordrecht, 1994.

    Google Scholar 

  31. M. Okada and K. Terui. The finite model property for various fragments of intuitionistic linear logic. Journal of Symbolic Logic, 64:790–802, 1999.

    Google Scholar 

  32. H. Ono and Y. Komori. Logics without the contraction rule. Journal of Symbolic Logic, 50:169–201, 1985.

    Google Scholar 

  33. J. van Benthem and A. ter Meulen, editors. Handbook of Logic and Language. Elsevier, 1997.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań, Poznań, Poland

    Wojciech Buszkowski

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  1. Wojciech Buszkowski
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Correspondence to Wojciech Buszkowski .

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Editors and Affiliations

  1. Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China

    Shier Ju

  2. Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China

    Hu Liu

  3. Japan Advanced Institute of Science and Technology, Ishikawa, Japan

    Hiroakira Ono

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Buszkowski, W. (2015). Some Syntactic Interpretations in Different Systems of Full Lambek Calculus. In: Ju, S., Liu, H., Ono, H. (eds) Modality, Semantics and Interpretations. Logic in Asia: Studia Logica Library. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47197-5_2

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