Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-q7jt5 Total loading time: 0.228 Render date: 2021-02-25T11:34:58.783Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

A SAHLQVIST THEOREM FOR SUBSTRUCTURAL LOGIC

Published online by Cambridge University Press:  18 March 2013

TOMOYUKI SUZUKI
Affiliation:
Department of Computer Science, University of Leicester
Corresponding

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
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.

References

Allwein, G., & Dunn, M. (1993). Kripke models for linear logic. The Journal of Symbolic Logic, 58, 514545.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), 338376.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, 6178.CrossRefGoogle 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. 1531.CrossRefGoogle Scholar
Galatos, N., & Jipsen, P. (2013). Residuated frames with applications to decidability. Transactions of the AMS, 365, 12191249.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, 241275.CrossRefGoogle Scholar
Gehrke, M., Nagahashi, H., & Venema, Y. (2005). A Sahlqvist theorem for distributive modal logic. Annals of Pure and Applied Logic, 131, 65102.CrossRefGoogle Scholar
Ghilardi, S., & Meloni, G. (1997). Constructive canonicity in non-classical logics. Annals of Pure and Applied Logic, 86, 132.CrossRefGoogle Scholar
Goldblatt, R. (1974). Semantic analysis of orthologic. Journal of Philosophical Logic, 3, 1935.CrossRefGoogle 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), 77101.Google Scholar
Goranko, V., & Vakarelov, D. (2006). Elementary canonical formulae: Extending Sahlqvist’s theorem. Annals of Pure and Applied Logic, 141, 180217.CrossRefGoogle Scholar
Hartonas, C. (1997). Duality for lattice-ordered algebras and normal algebraizable logics. Studia Logica, 58(3), 403450.CrossRefGoogle Scholar
Hartonas, C., & Dunn, J. M. (1997). Stone duality for lattices. Algebra Universalis, 37, 391401.CrossRefGoogle Scholar
Kripke, S. A. (1959). A completeness theorem in modal logic. The Journal of Symbolic Logic, 24, 114.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.193228.Google 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. 110143.CrossRefGoogle Scholar
Sambin, G., & Vaccaro, V. (1989). A new proof of Sahlqvist’s theorem on modal definability and completeness. The Journal of Symbolic Logic, 54, 992999.CrossRefGoogle Scholar
Seki, T. (2003). A Sahlqvist theorem for relevant modal logics. Studia Logica, 73, 383411.CrossRefGoogle Scholar
Suzuki, T. (2010a). Bi-approximation semantics for substructural logic. In Advances in Modal Logic, Vol. 8. College Publications, pp. 411433.Google Scholar
Suzuki, T. (2010b). Canonicity and Bi-Approximation in Non-Classical Logics. PhD Thesis, University of Leicester, Leicester.
Suzuki, T. (2011). Canonicity results of substructural and lattice-based logics. The Review of Symbolic Logic, 4(01), 142.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 33 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 25th February 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A SAHLQVIST THEOREM FOR SUBSTRUCTURAL LOGIC
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

A SAHLQVIST THEOREM FOR SUBSTRUCTURAL LOGIC
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

A SAHLQVIST THEOREM FOR SUBSTRUCTURAL LOGIC
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *