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
Blackburn P., De Rijke M., Venema Y.: Modal Logic. Cambridge University Press, Cambridge (2002)
Buszkowski W.: Compatibility of categorial grammar with an associated category system. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 28, 229–238 (1982)
Buszkowski W.: Some decision problems in the theory of syntactic categories. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 28, 539–548 (1982)
Buszkowski, W., Categorial grammars with negative information, in H. Wansing (ed.), Negation. A Notion in Focus, de Gruyter, Berlin, 1996, pp. 107–126.
Buszkowski W.: Interpolation and FEP for logics of residuated algebras. Logic Journal of the IGPL 19, 437–454 (2011)
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.
Došen K.: A brief survey of frames for the Lambek calculus. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 38, 179–187 (1992)
Kanazawa M.: The Lambek calculus enriched with additional connectives. Journal of Logic, Language, and Information 1, 141–171 (1992)
Kleene S.C.: Introduction to Metamathematics. North-Holland, Amsterdam (1971)
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.
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).
Mendelson E.: Introduction to Mathematical Logic. Chapman and Hall, London (2010)
Pentus M.: Models for the Lambek calculus. Annals of Pure and Applied Logic 75, 179–213 (1995)
Segerberg K., An essay in classical modal logic, Filosofiska Studier 13, 1971.
Wansing H.: A note on negation in categorial grammar. Logic Journal of the IGPL 15, 271–286 (2007)
Zimmermann E.: Full Lambek calculus in natural deduction. Mathematical Logic Quarterly 56, 85–88 (2010)
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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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DOI: https://doi.org/10.1007/s11225-013-9474-7