Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-02-05T17:40:18.843Z Has data issue: false hasContentIssue false
  • English
  • Français

A SAHLQVIST THEOREM FOR SUBSTRUCTURAL LOGIC

Published online by Cambridge University Press:  18 March 2013

TOMOYUKI SUZUKI
TOMOYUKI SUZUKI*
Affiliation:
Department of Computer Science, University of Leicester
*
*DEPARTMENT OF INFORMATICS, UNIVERSITY OF PISA, LARGO PONTECORVO 3C, 56127 PISA, ITALY, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we establish the first-order definability of sequents with consistent variable occurrence on bi-approximation semantics by means of the Sahlqvist–van Benthem algorithm. Then together with the canonicity results in Suzuki (2011), this allows us to establish a Sahlqvist theorem for substructural logic. Our result is not limited to substructural logic but is also easily applicable to other lattice-based logics.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 2 , June 2013 , pp. 229 - 253
Copyright
Copyright © Association for Symbolic Logic 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Allwein, G., & Dunn, M. (1993). Kripke models for linear logic. The Journal of Symbolic Logic, 58, 514–545.CrossRefGoogle Scholar
Birkhoff, G. (1973). Lattice Theory, Vol. XXV (third edition). American Mathematical Society Colloquium Publications. Providence, RI: American Mathematical Society.Google Scholar
Blackburn, P., de Rijke, M., & Venema, Y. (2002). Modal Logic, Vol. 53 Cambridge Tracts in Theoretical Computer Science. Cambridge, UK: Cambridge University Press.Google Scholar
Conradie, W., & Palmigiano, A. (2012). Algorithmic correspondence and canonicity for distributive modal logic. Annals of Pure and Applied Logic, 163(3), 338–376.CrossRefGoogle Scholar
de Rijke, M., & Venema, Y. (1995). Sahlqvist’s theorem for Boolean algebras with operators with an application to cylindric algebras. Studia Logica, 54, 61–78.Google Scholar
Fine, K. (1975). Some connections between elementary and modal logic. In Kanger, S., editor. Proceedings of the Third Scandinavian Logic Symposium. Amsterdam, the Netherland: North-Holland, pp. 15–31.CrossRefGoogle Scholar
Galatos, N., & Jipsen, P. (2013). Residuated frames with applications to decidability. Transactions of the AMS, 365, 1219–1249.CrossRefGoogle Scholar
Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Vol. 151 Studies in Logics and the Foundation of Mathematics. Amsterdam, the Netherland: Elsevier.Google Scholar
Gehrke, M. (2006). Generalized Kripke frames. Studia Logica, 84, 241–275.Google Scholar
Gehrke, M., Nagahashi, H., & Venema, Y. (2005). A Sahlqvist theorem for distributive modal logic. Annals of Pure and Applied Logic, 131, 65–102.Google Scholar
Ghilardi, S., & Meloni, G. (1997). Constructive canonicity in non-classical logics. Annals of Pure and Applied Logic, 86, 1–32.Google Scholar
Goldblatt, R. (1974). Semantic analysis of orthologic. Journal of Philosophical Logic, 3, 19–35.Google Scholar
Goldblatt, R. (2006). Mathematical modal logic: a view of its evolution. In Gabbay, D. M., and Woods, J., editors. Handbook of the History of Logic, Vol. 7. Amsterdam, the Netherland: Elsevier.Google Scholar
Goldblatt, R., Hodkinson, I., & Venema, Y. (2003). On canonical modal logics that are not elementarily determined. Logique et Analyse, 46(181), 77–101.Google Scholar
Goranko, V., & Vakarelov, D. (2006). Elementary canonical formulae: Extending Sahlqvist’s theorem. Annals of Pure and Applied Logic, 141, 180–217.Google Scholar
Hartonas, C. (1997). Duality for lattice-ordered algebras and normal algebraizable logics. Studia Logica, 58(3), 403–450.Google Scholar
Hartonas, C., & Dunn, J. M. (1997). Stone duality for lattices. Algebra Universalis, 37, 391–401.Google Scholar
Kripke, S. A. (1959). A completeness theorem in modal logic. The Journal of Symbolic Logic, 24, 1–14.CrossRefGoogle Scholar
Ono, H. (2003). Substructural logics and residuated lattices—an introduction. In Hendricks, V. F., and Malinowski, J., editors. 50 Years of Studia Logica: Trends in Logic. Dordrecht, the Netherland: Kluwer Academic Publishers, pp.193–228.CrossRefGoogle Scholar
Sahlqvist, H. (1975). Completeness and correspondence in the first and second order semantics for modal logic. In Kanger, S., editor. Proceedings of the Third Scandinavian Logic Symposium. Amsterdam, the Netherland: North-Holland, pp. 110–143.Google Scholar
Sambin, G., & Vaccaro, V. (1989). A new proof of Sahlqvist’s theorem on modal definability and completeness. The Journal of Symbolic Logic, 54, 992–999.Google Scholar
Seki, T. (2003). A Sahlqvist theorem for relevant modal logics. Studia Logica, 73, 383–411.Google Scholar
Suzuki, T. (2010a). Bi-approximation semantics for substructural logic. In Advances in Modal Logic, Vol. 8. College Publications, pp. 411–433.Google Scholar
Suzuki, T. (2010b). Canonicity and Bi-Approximation in Non-Classical Logics. PhD Thesis, University of Leicester, Leicester.Google Scholar
Suzuki, T. (2011). Canonicity results of substructural and lattice-based logics. The Review of Symbolic Logic, 4(01), 1–42.CrossRefGoogle Scholar