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CANONICITY RESULTS OF SUBSTRUCTURAL AND LATTICE-BASED LOGICS

Published online by Cambridge University Press:  20 September 2010

TOMOYUKI SUZUKI
TOMOYUKI SUZUKI*
Affiliation:
University of Leicester
*
*DEPARTMENT OF COMPUTER SCIENCE, UNIVERSITY OF LEICESTER, LEICESTER LE1 7RH, UNITED KINGDOM. E-mail: tomoyuki.suzuki@mcs.le.ac.uk
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Abstract

In this paper, we extend the canonicity methodology in Ghilardi & Meloni (1997) to arbitrary lattice expansions, and syntactically describe canonical inequalities for lattice expansions consisting of ε-join preserving operations, ε-meet preserving operations, ε-additive operations, ε-multiplicative operations, adjoint pairs, and constants. This approach gives us a uniform account of canonicity for substructural and lattice-based logics. Our method not only covers existing results, but also systematically accounts for many canonical inequalities containing nonsmooth additive and multiplicative uniform operations. Furthermore, we compare our technique with the approach in Dunn et al. (2005) and Gehrke et al. (2005).

Type
Research Article
Information
The Review of Symbolic Logic , Volume 4 , Issue 1 , March 2011 , pp. 1 - 42
Copyright
Copyright © Association for Symbolic Logic 2010

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References

BIBLIOGRAPHY

Birkhoff, G. (1948). Lattice Theory (revised edition), Volume XXV of American Mathematical Society Colloquium Publications. Providence: American Mathematical Society.Google Scholar
Blackburn, P., de Rijke, M., & Venema, Y. (2002). Modal Logic, Volume 53 of Cambridge Tracts in Theoretical Computer Science. Cambridge: Cambridge University Press.Google Scholar
Burris, S., & Sankappanavar, H. (1981). A Course in Universal Algebra (Graduate Texts in Mathematics, editor). New York: Springer-Verlag.CrossRefGoogle Scholar
Chagrov, A., & Zakharyaschev, M. (1997). Modal Logic, Volume 35 of Oxford Logic Guides. New York: Oxford Science Publications.Google Scholar
Davey, B., & Priestley, H. (2002). Introduction to Lattices and Order (second edition). Cambridge: Cambridge University Press.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.CrossRefGoogle Scholar
Dos̆en, K. (1988). Sequent-systems and groupoid models. I. Studia Logica, 47, 353–385.CrossRefGoogle Scholar
Dos̆en, K. (1989). Sequent-systems and groupoid models. II. Studia Logica, 48, 41–65.CrossRefGoogle Scholar
Dunn, M. (1986). Relevance logic and entailment. In Gabbay, D. and Guenthner, F., editors. Handbook of Philosophical Logic, Vol. III, Chapter 3. Dordrecht: Kluwer Academic Publishers, pp. 117–224.CrossRefGoogle Scholar
Dunn, M., Gehrke, M., & Palmigiano, A. (2005). Canonical extensions and relational completeness of some substructural logics. The Journal of Symbolic Logic, 70, 713–740.CrossRefGoogle Scholar
Fine, K. (1975). Some connections between elementary and modal logic. In Kanger, S., editor. Proceedings of the Third Scandinavian Logic Symposium. Amsterdam: North-Holland, pp. 15–31.CrossRefGoogle Scholar
Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151 of Studies in Logics and the Foundation of Mathematics. Amsterdam: Elsevier.Google Scholar
Gehrke, M., & Harding, J. (2001). Bounded lattice expansions. Journal of Algebra, 239, 345–371.CrossRefGoogle Scholar
Gehrke, M., & Jónsson, B. (1994). Bounded distributive lattices with operators. Mathematica Japonica, 40, 207–215.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
Gehrke, M., & Priestley, H. (2002). Non-canonicity of MV-algebras. Houston Journal of Mathematics, 28(3), 449.Google Scholar
Gehrke, M., & Priestley, H. (2007). Canonical extensions of double quasioperator algebras: An algebraic perspective on duality for certain algebras with binary operators. Journal of Pure and Applied Algebra, 209, 269–290.Google Scholar
Ghilardi, S., & Meloni, G. (1997). Constructive canonicity in non-classical logics. Annals of Pure and Applied Logic, 86, 1–32.CrossRefGoogle Scholar
Goldblatt, R. (2005). 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: Elsevier.Google Scholar
Goranko, V., & Vakarelov, D. (2006). Elementary canonical formulae: Extending Sahlqvist’s theorem. Annals of Pure and Applied Logic, 141, 180–217.CrossRefGoogle Scholar
Harding, J. (1998). Canonical completions of lattices and ortholattices. Tatra Mountains Mathematical Publications, 15, 85–96.Google Scholar
Jónsson, B. (1994). On the canonicity of Sahlqvist identities. Studia Logica, 53, 473–491.CrossRefGoogle Scholar
Jónsson, B., & Tarski, A. (1951). Boolean algebras with operators I. American Journal of Mathematics, 73, 891–993.CrossRefGoogle Scholar
Jónsson, B., & Tarski, A. (1952). Boolean algebras with operators II. American Journal of Mathematics, 74, 127–162.CrossRefGoogle Scholar
Lambek, J., & Scott, P. (1986). Introduction to Higher Order Categorical Logic, Volume 7 of Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press.Google Scholar
Mac Lane, S. (1971). Categories for the Working Mathematician, Volume 5 of Graduate Texts in Mathematics. New York: Springer-Verlag.CrossRefGoogle Scholar
Ono, H. (2003). Substructural logics and residuated lattices—An introduction. In Hendricks, V. F. and Malinowski, J., eds. 50 Years of Studia Logica: Trends in Logic. Dordrecht: Kluwer Academic Publishers, pp. 193–228.CrossRefGoogle Scholar
Ono, H., & Komori, Y. (1985). Logics without the contraction rule. The Journal of Symbolic Logic, 50, 169–201.CrossRefGoogle Scholar
Routley, R., Plumwood, V., Meyer, R. K., & Brady, R. T. (1982). Relevant Logics and Their Rivals. Part 1. The Basic Philosophical and Semantical Theory. Atascadero: Ridgeview Publishing Company.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: North-Holland, pp. 110–143.CrossRefGoogle Scholar
Sambin, G., & Vaccaro, V. (1988). Topology and duality in modal logic. Annals of Pure and Applied Logic, 37, 249–296.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.CrossRefGoogle Scholar
Seki, T. (2003a). General frames for relevant modal logics. Notre Dame Journal of Formal Logic, 44, 93–109.CrossRefGoogle Scholar
Seki, T. (2003b). A Sahlqvist theorem for relevant modal logics. Studia Logica, 73, 383–411.CrossRefGoogle Scholar
Suzuki, T. (2007). Kripke completeness of some distributive substructural logics. Master’s Thesis, Japan Advanced Institute of Science and Technology.Google Scholar
Suzuki, T. (2008). A relational semantics for distributive substructural logics and the topological characterization of the descriptive frames. CALCO-jnr 2007 Report No.367, Department of Informatics, University of Bergen.Google Scholar
Urquhart, A. (1996). Duality for algebras of relevant logics. Studia Logica, 56, 263–276.Google Scholar