Journal of Symbolic Logic

Glivenko theorems for substructural logics over FL

Nikolaos Galatos and Hiroakira Ono

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Abstract

It is well known that classical propositional logic can be interpreted in intuitionistic propositional logic. In particular Glivenko’s theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko’s theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via a double negation interpretation. Our presentation is algebraic and is formulated in the context of residuated lattices. In the last part of the paper, we also discuss some extended forms of the Kolmogorov translation and we compare it to the Glivenko translation.

Article information

Source
J. Symbolic Logic, Volume 71, Issue 4 (2006), 1353-1384.

Dates
First available in Project Euclid: 20 November 2006

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1164060460

Digital Object Identifier
doi:10.2178/jsl/1164060460

Mathematical Reviews number (MathSciNet)
MR2275864

Zentralblatt MATH identifier
1109.03016

Subjects
Primary: Primary: 06F05, Secondary: 08B15, 03B47, 03G10, 03B05, 03B20

Keywords
Glivenko’s theorem, substructural logic, involutive, pointed residuated lattice

Citation

Galatos, Nikolaos; Ono, Hiroakira. Glivenko theorems for substructural logics over FL. J. Symbolic Logic 71 (2006), no. 4, 1353--1384. doi:10.2178/jsl/1164060460. https://projecteuclid.org/euclid.jsl/1164060460


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References

  • P. Bahls, J. Cole, N. Galatos, P. Jipsen, and C. Tsinakis, Cancellative residuated lattices, Algebra Universalis, vol. 50 (2003), no. 1, pp. 83--106.
  • W. J. Blok and D. Pigozzi, Algebraizable logics, Memoirs of the American Mathematical Society, vol. 77 (1989), no. 396.
  • K. Blount and C. Tsinakis, The structure of residuated lattices, International Journal of Algebra and Computation, vol. 13 (2003), no. 4, pp. 437--461.
  • R. Cignoli, I. D'Otaviano, and D. Mundici, Algebraic Foundations of Many-valued Reasoning, Trends in Logic -- Studia Logica Library, vol. 7, Kluwer Academic Publishers, Dordrecht, 2000.
  • R. Cignoli and A. Torrens, Hájek basic fuzzy logic and Łukasievicz infinite-valued logic, Archive for Mathematical Logic, vol. 42 (2003), pp. 361--370.
  • --------, Glivenko like theorems in natural expansions of BCK-logic, Mathematical Logic Quarterly, vol. 50 (2004), no. 2, pp. 111--125.
  • J. Czelakowski and W. Dziobiak, The parameterized local deduction theorem for quasivarieties of algebras and its application, Algebra Universalis, vol. 35 (1996), no. 3, pp. 373--419.
  • J. M. Font, R. Jansana, and D. Pigozzi, A survey of abstract algebraic logic, Studia Logica, vol. 74 (2003), no. 1--2, pp. 13--97, Special issue on \em Abstract Algebraic Logic, Part II (Barcelona, 1997).
  • N. Galatos, Varieties of residuated lattices, Ph.D. thesis, Vanderbilt University, 2003.
  • --------, Minimal varieties of residuated lattices, Algebra Universalis, vol. 52 (2005), no. 2, pp. 215--239.
  • N. Galatos and H. Ono, Algebraization, parametrized local deduction theorem and interpolation for substructural logics over $\mathbfFL$, Studia Logica, vol. 83 (2006), pp. 279--308.
  • --------, Cut elimination and strong separation for substructural logics: An algebraic approach, manuscript.
  • N. Galatos and C. Tsinakis, Generalized MV-algebras, Journal of Algebra, vol. 283 (2005), no. 1, pp. 254--291.
  • V. Glivenko, Sur quelques points de la logique de M. Brouwer, Bulletins de la classe des sciences, Academie Royale de Belgique, vol. 15 (1929), pp. 183--188.
  • P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Dordrecht-Boston-London, 1998.
  • S. Odintsov, Negative equivalence of extensions of minimal logic, Studia Logica, vol. 78 (2004), pp. 417--442.
  • H. Ono, Semantics for substructural logics, Substructural Logics (K. Došen and P. Schroeder-Heister, editors), Oxford University Press, New York, 1993, pp. 259--291.