Hostname: page-component-76c49bb84f-ckhzl Total loading time: 0 Render date: 2025-07-09T21:01:06.219Z Has data issue: false hasContentIssue false
  • English
  • Français

Canonical extensions and relational completeness of some substructural logics*

Published online by Cambridge University Press:  12 March 2014

J. Michael Dunn ,
Mai Gehrke  and
Alessandra Palmigiano
J. Michael Dunn
Affiliation:
School of Informatics, Indiana University, Bloomington, IN 47408-3912, USAE-mail:, dunn@indiana.edu
Mai Gehrke
Affiliation:
Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003., USAE-mail:, mgehrke@nmsu.edu
Alessandra Palmigiano
Affiliation:
Departament De Logica, Historia I Filosofia De La Ciencia, Universitat De Barcelona, Barcelona. E-08028, SpainE-mail:, ccl47472@cconline.es
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 3 , September 2005 , pp. 713 - 740
Copyright
Copyright © Association for Symbolic Logic 2005

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.)

Article purchase

Temporarily unavailable

Footnotes

*

The authors wish to thank an anonymous referee and M. Dunn's student, Chunlai Zhou, for their careful reading of the manuscript and for their suggestions and corrections.

References

REFERENCES

[1]Allwein, G. and Dunn, J. M.. A Kripke semantics for linear logic, this Journal, vol. 58 (1993), pp. 514–545.Google Scholar
[2]Allwein, G. and MacCaull, W., A Kripke semantics for the logic of Gelfand quantales, Studia Logica, vol. 68 (2001). no. 2. pp. 173–228.CrossRefGoogle Scholar
[3]Bucciarelli, A. and Ehrhard, T., On phase semantics and denotational semantics in multiplicativeadditive linear logic, Annals of Pure Applied Logic, vol. 102 (2000). no. 3, pp. 247–282.CrossRefGoogle Scholar
[4]Dunn, J. M., Gaggle theory, an abstraction of galois connections and residuation, with applications to negation, implication, and various logical operators, Logics in AI (European Workshop JELIA, Amsterdam) (Eijck, J. Van, editor). Lecture Notes in Artificial Intelligence. Springer Verlag, 1990, pp. 31–51.Google Scholar
[5]Dunn, J. M., Partial gaggles applied to logics with restricted structural rules, Substructural logics (Tübingen, 1990) (Schroeder-Heister, P. and Dosen, K., editors), Studies in Logic and Computation 2, Oxford University Press, New York, 1993, pp. 63–108.CrossRefGoogle Scholar
[6]Dunn, J. M. and Hardegree, G., Algebraic Methods in Philosophical Logic, Oxford University Press, 2001.CrossRefGoogle Scholar
[7]Gehrke, M. and Harding, J., Bounded lattice expansions, Journal of Algebra, vol. 238 (2001). pp. 345–371.CrossRefGoogle Scholar
[8]Gehrke, M., Harding, J., and Venema, Y, MacNeille completions and canonical extensions, Transactions of the American Mathematical Society, (2005), to appear.Google Scholar
[9]Gehrke, M. and Jónsson, B., Bounded distributive lattices with operators, Mathematica Japónica, vol. 40(1994), pp. 207–215.Google Scholar
[10]Gehrke, M., Bounded distributive lattice expansions, Mathematica Scandinavica, vol. 94 (2004), no. 2, pp. 13–45.CrossRefGoogle Scholar
[11]Gehrke, M., Nagahashi, H., and Venema, Y., A Sahlqvist theorem for distributive modal logic. Annals of Pure and Applied Logic, vol. 131 (2005), no. 1–3, pp. 65–102.CrossRefGoogle Scholar
[12]Gehrke, M. and Priestley, H., MV-algebras and beyond: Canonicity, duality, and correspondence, manuscript in preparation.Google Scholar
[13]Ghilardi, S. and Meloni, G., Constructive canonicity in non-classical logics, Annals of Pure and Applied Logic, vol. 86 (1997), pp. 1–32.CrossRefGoogle Scholar
[14]Jónsson, B. and Tarski, A., Boolean algebras with operators I, American Journal of Mathematics, vol. 73 (1951), pp. 891–939.CrossRefGoogle Scholar
[15]Jónsson, B., Boolean algebras with operators II, American Journal of Mathematics, vol. 74 (1952), pp. 127–162.CrossRefGoogle Scholar
[16]Kamide, N., Kripke semantics for modal substructural logics, Journal of Logic, Language and Information, vol. 11 (2002), no. 4, pp. 453–470.CrossRefGoogle Scholar
[17]MacCaull, W., Relational semantics and a relational proof system for full Lambek calculus, this Journal, vol. 63 (1998), no. 2, pp. 623–637.Google Scholar