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International Workshop on Logic, Language, Information, and Computation

WoLLIC 2012: Logic, Language, Information and Computation pp 56–69Cite as

Admissible Rules: From Characterizations to Applications

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7456))

Abstract

The admissible rules of a logic (understood as a structural consequence relation) may be described as rules that can be added to the logic without producing any new theorems, or, equivalently, as rules such that any substitution making the premises into theorems, also makes the conclusion into a theorem. However, this equivalence collapses once multiple-conclusion or other, more exotic, admissible rules are considered. The first aim of this paper is to explain how such distinctions can be explained and characterized. The second aim is to explore how these rules can be useful in determining properties of classes of algebras.

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References

  1. Avron, A.: A constructive analysis of RM. Journal of Symbolic Logic 52(4), 939–951 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baaz, M., Zach, R.: Hypersequents and the Proof Theory of Intuitionistic Fuzzy Logic. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 187–201. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  3. Babenyshev, S., Rybakov, V., Schmidt, R.A., Tishkovsky, D.: A tableau method for checking rule admissibility in S4. In: Proceedings of UNIF 2009. ENTCS, vol. 262, pp. 17–32 (2010)

    Google Scholar 

  4. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Graduate Texts in Mathematics, vol. 78. Springer, New York (1981)

    Book  MATH  Google Scholar 

  5. Cabrer, L.M., Metcalfe, G.: Admissibility via unifiability (preprint)

    Google Scholar 

  6. Chang, C., Keisler, H.: Model Theory. Studies in Logic and the Foundations of Mathematics, vol. 73. Elsevier (1977)

    Google Scholar 

  7. Ciabattoni, A., Metcalfe, G.: Density elimination. Theoretical Computer Science 403, 328–346 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cintula, P., Metcalfe, G.: Admissible rules in the implication-negation fragment of intuitionistic logic. Annals of Pure and Applied Logic 162(10), 162–171 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier (2007)

    Google Scholar 

  10. Gentzen, G.: Untersuchungen über das Logische Schliessen. Math. Zeitschrift 210, 176–210, 405–431 (1935)

    Article  MathSciNet  Google Scholar 

  11. Ghilardi, S.: Unification in intuitionistic logic. Journal of Symbolic Logic 64(2), 859–880 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ghilardi, S.: Best solving modal equations. Annals of Pure and Applied Logic 102(3), 184–198 (2000)

    Article  MathSciNet  Google Scholar 

  13. Ghilardi, S.: A resolution/tableaux algorithm for projective approximations in IPC. Logic Journal of the IGPL 10(3), 227–241 (2002)

    Article  MathSciNet  Google Scholar 

  14. Iemhoff, R.: On the admissible rules of intuitionistic propositional logic. Journal of Symbolic Logic 66(1), 281–294 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Iemhoff, R.: Intermediate logics and Visser’s rules. Notre Dame Journal of Formal Logic 46(1), 65–81 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Iemhoff, R., Metcalfe, G.: Proof theory for admissible rules. Annals of Pure and Applied Logic 159(1-2), 171–186 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jeřábek, E.: Admissible rules of modal logics. Journal of Logic and Computation 15, 411–431 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jeřábek, E.: Admissible rules of Łukasiewicz logic. Journal of Logic and Computation 20(2), 425–447 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jeřábek, E.: Bases of admissible rules of Łukasiewicz logic. Journal of Logic and Computation 20(6), 1149–1163 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lorenzen, P.: Einführung in die operative Logik und Mathematik. Grundlehren der mathematischen Wissenschaften, vol. 78. Springer (1955)

    Google Scholar 

  21. Metcalfe, G.: Proof theory of mathematical fuzzy logic. In: Handbook of Mathematical Fuzzy Logic, ch. 3, vol. I, pp. 209–282. King’s College Publications (2011)

    Google Scholar 

  22. Metcalfe, G., Montagna, F.: Substructural fuzzy logics. Journal of Symbolic Logic 72(3), 834–864 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Metcalfe, G., Olivetti, N., Gabbay, D.: Proof Theory for Fuzzy Logics. Springer (2008)

    Google Scholar 

  24. Metcalfe, G., Röthlisberger, C.: Unifiability and Admissibility in Finite Algebras. In: Cooper, S.B., Dawar, A., Löwe, B. (eds.) CiE 2012. LNCS, vol. 7318, pp. 485–495. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  25. Rozière, P.: Admissible and derivable rules in intuitionistic logic. Mathematical Structures in Computer Science 2(3), 129–136 (1993)

    Article  Google Scholar 

  26. Rybakov, V.: Admissibility of Logical Inference Rules. Studies in Logic and the Foundations of Mathematics, vol. 136. Elsevier, Amsterdam (1997)

    Book  MATH  Google Scholar 

  27. Takeuti, G., Titani, T.: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. Journal of Symbolic Logic 49(3), 851–866 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  28. Whitman, P.: Free lattices. Annals of Mathematics 42, 325–329 (1941)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics Institute, University of Bern, Sidlerstrasse 5, Bern, 3012, Switzerland

    George Metcalfe

Authors
  1. George Metcalfe
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Luke Ong

  2. Centro de Informática, Universidade Federal de Pernambuco, Av. jornalista Anibal Fernandes, s/n, 50740-560, Recife, PE, Brazil

    Ruy de Queiroz

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Metcalfe, G. (2012). Admissible Rules: From Characterizations to Applications. In: Ong, L., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2012. Lecture Notes in Computer Science, vol 7456. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32621-9_4

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  • DOI: https://doi.org/10.1007/978-3-642-32621-9_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32620-2

  • Online ISBN: 978-3-642-32621-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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