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Relational Semantics of the Lambek Calculus Extended with Classical Propositional Logic

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Abstract

We show that the relational semantics of the Lambek calculus, both nonassociative and associative, is also sound and complete for its extension with classical propositional logic. Then, using filtrations, we obtain the finite model property for the nonassociative Lambek calculus extended with classical propositional logic.

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References

  1. Blackburn P., De Rijke M., Venema Y.: Modal Logic. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  2. Buszkowski W.: Compatibility of categorial grammar with an associated category system. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 28, 229–238 (1982)

    Article  Google Scholar 

  3. Buszkowski W.: Some decision problems in the theory of syntactic categories. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 28, 539–548 (1982)

    Article  Google Scholar 

  4. Buszkowski, W., Categorial grammars with negative information, in H. Wansing (ed.), Negation. A Notion in Focus, de Gruyter, Berlin, 1996, pp. 107–126.

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

    Article  Google Scholar 

  6. Buszkowski, W., and M. Farulewski, Nonassociative Lambek calculus with additives and context-free languages, in O. Grumberg, M. Kaminski, S. Katz, and S. Wintner (eds.), Languages: From Formal to Natural, Essays Dedicated to Nissim Francez on the Occasion of His 65th Birthday, volume 5533 of Lecture Notes in Computer Science, Springer, Berlin Heidelberg, 2009, pp. 45–58.

  7. Došen K.: A brief survey of frames for the Lambek calculus. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 38, 179–187 (1992)

    Article  Google Scholar 

  8. Kanazawa M.: The Lambek calculus enriched with additional connectives. Journal of Logic, Language, and Information 1, 141–171 (1992)

    Article  Google Scholar 

  9. Kleene S.C.: Introduction to Metamathematics. North-Holland, Amsterdam (1971)

    Google Scholar 

  10. Kurtonina, N., and M. Moortgat, Relational semantics for the Lambek-Grishin calculus, in C. Ebert, G. Jäger, and J. Michaelis (eds.), The Mathematics of Language, 10th and 11th Biennial Conferences, MOL 10 and MOL 11, volume 6149 of Lecture Notes in Computer Science, Springer, Berlin Heidelberg, 2010, pp. 210–222.

  11. Lambek, J., The mathematics of sentence structure, American Mathematical Monthly 65:154–170, 1958. (Also in Categorial Grammars, W. Buszkowski, W. Marciszewski, and J. van Benthem (eds.), John Benjamins, Amsterdam, 1988).

  12. Mendelson E.: Introduction to Mathematical Logic. Chapman and Hall, London (2010)

    Google Scholar 

  13. Pentus M.: Models for the Lambek calculus. Annals of Pure and Applied Logic 75, 179–213 (1995)

    Article  Google Scholar 

  14. Segerberg K., An essay in classical modal logic, Filosofiska Studier 13, 1971.

  15. Wansing H.: A note on negation in categorial grammar. Logic Journal of the IGPL 15, 271–286 (2007)

    Article  Google Scholar 

  16. Zimmermann E.: Full Lambek calculus in natural deduction. Mathematical Logic Quarterly 56, 85–88 (2010)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Computer Science, Technion - Israel Institute of Technology, 32000, Haifa, Israel

    Michael Kaminski & Nissim Francez

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  1. Michael Kaminski
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  2. Nissim Francez
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Correspondence to Michael Kaminski.

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Kaminski, M., Francez, N. Relational Semantics of the Lambek Calculus Extended with Classical Propositional Logic. Stud Logica 102, 479–497 (2014). https://doi.org/10.1007/s11225-013-9474-7

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