Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-11T18:55:03.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivalence of consequence relations: an order-theoretic and categorical perspective

Published online by Cambridge University Press:  12 March 2014

Nikolaos Galatos  and
Constantine Tsinakis
Nikolaos Galatos
Affiliation:
Department of Mathematics, University of Denver, 2360 S. Gaylord St. Denver, Co 80208, USA, E-mail: ngalatos@du.edu, URL: http://www.math.du.edu/~ngalatos
Constantine Tsinakis
Affiliation:
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, Tn 37240, USA, E-mail: constantine.tsinakis@vanderbilt.edu, URL: http://www.math.vanderbilt.edu/people/tsinakis
Get access Rights & Permissions [Opens in a new window]

Abstract

Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be denned syntactically, in terms of translations of formulas, and order-theoretically, in terms of the associated lattices of theories. W. Blok and D. Pigozzi proved in [4] that the two definitions coincide in the case of an algebraizable sentential deductive system. A refined treatment of this equivalence was provided by W. Blok and B. Jónsson in [3]. Other authors have extended this result to the cases of κ-deductive systems and of consequence relations on associative, commutative, multiple conclusion sequents. Our main result subsumes all existing results in the literature and reveals their common character. The proofs are of order-theoretic and categorical nature.

Keywords

Consequence relationclosure operatorequivalencealgebraizableresiduated latticemoduleprojective
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 780 - 810
Copyright
Copyright © Association for Symbolic Logic 2009

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

REFERENCES

[1]Avron, A., The method of hypersequents in proof theory of propositional nonclassical logics, Logic: From foundations to applications (Hodges, W.et al., editor), Oxford University Press, 1996, pp. 1–32.Google Scholar
[2]Birkhoff, G., Lattice theory, third ed., American Mathematical Society Colloquium Publications, vol. XXV, American Mathematical Society, 1967.Google Scholar
[3]Blok, W.J. and Jónsson, B., Equivalence of consequence operations, Stadia Logica, vol. 83 (2006), no. 1–3, pp. 91–110.CrossRefGoogle Scholar
[4]Blok, W.J. and Pigozzi, D., Algebraizable logics, Memoirs of the AMS, vol. 77 (1989), no. 396.CrossRefGoogle Scholar
[5]Blount, K. and Tsinakis, C., The structure of residuated lattices, International Journal of Algebra and Computation, vol. 13 (2003), no. 4, pp. 437–461.CrossRefGoogle Scholar
[6]Blyth, T.S., Lattices and ordered algebraic structures, Universitext, Springer-Verlag, London, 2005.Google Scholar
[7]Bolc, L. and Borowik, P., Many-valued logics 2: Automated reasoning and practical applications, Springer-Verlag, 2004.Google Scholar
[8]Galatos, N., Minimal varieties of residuated lattices, Algebra Universalis, vol. 52 (2005), no. 2, pp. 215–239.CrossRefGoogle Scholar
[9]Galatos, N., Jipsen, P., Kowalski, T., and Ono, H., Residuated lattices: an algebraic glimpse at substructural logics, Studies in Logic and the Foundations of Mathematics, vol. 151, Elsevier, 2007.Google Scholar
[10]Galatos, N. and Ono, H., Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL, Studia Logica, vol. 83 (2006), pp. 279–308.CrossRefGoogle Scholar
[11]Galatos, N., Cut elimination and strong separation for substructural logics: an algebraic approach, Annals of Pure and Applied Logic, accepted for publication.Google Scholar
[12]Jipsen, P. and Tsinakis, C., A survey of residuated lattices, Ordered algebraic structures (Martinez, J., editor), Kluwer, Dordrecht, 2002, pp. 19–56.CrossRefGoogle Scholar
[13]Pynko, A., Definitional equivalence and algebraizability of generalized logical systems, Annals of Pure and Applied Logic, vol. 98 (1999), pp. 1–68.CrossRefGoogle Scholar
[14]Raftery, J., Correspondences between Gentzen and Hilbert systems, this Journal, vol. 71 (2006), no. 3, pp. 903–957.Google Scholar
[15]Rebagliato, J. and Verdú, V., On the algebraization of some Gentzen systems, Fundamenta Informaticae, vol. 18 (1993), no. 2-4, pp. 319–338.CrossRefGoogle Scholar