Studia Logica

, Volume 102, Issue 1, pp 1–27 | Cite as

Cut-elimination for Weak Grzegorczyk Logic Go

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Abstract

We present a syntactic proof of cut-elimination for weak Grzegorczyk logic Go. The logic has a syntactically similar axiomatisation to Gödel–Löb logic GL (provability logic) and Grzegorczyk’s logic Grz. Semantically, GL can be viewed as the irreflexive counterpart of Go, and Grz can be viewed as the reflexive counterpart of Go. Although proofs of syntactic cut-elimination for GL and Grz have appeared in the literature, this is the first proof of syntactic cut-elimination for Go. The proof is technically interesting, requiring a deeper analysis of the derivation structures than the proofs for GL and Grz. New transformations generalising the transformations for GL and Grz are developed here.

Keywords

Proof theory Syntactic cut-elimination Go Weak Grzegorczyk logic 

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References

  1. 1.
    Amerbauer M.: Cut-free tableau calculi for some propositional normal modal logics. Studia Logica 57(2–3), 359–372 (1996)CrossRefGoogle Scholar
  2. 2.
    Belnap N.D. Jr.: Display logic. Journal of Philosophical Logic 11(4), 375–417 (1982)CrossRefGoogle Scholar
  3. 3.
    Borga M., Gentilini P.: On the proof theory of the modal logic Grz. Z. Math. Logik Grundlag. Math. 32(2), 145–148 (1986)CrossRefGoogle Scholar
  4. 4.
    Borga M.: On some proof theoretical properties of the modal logic GL. Studia Logica 42(4), 453–459 (1983)CrossRefGoogle Scholar
  5. 5.
    Esakia L.: The modalized Heyting calculus: a conservative modal extension of the intuitionistic logic. Journal of Applied Non-Classical Logics 16(3–4), 349–366 (2006)CrossRefGoogle Scholar
  6. 6.
    Gabelaia D. Topological, Algebraic and Spatio-Temporal Semantics for Multi-Dimensional Modal Logics, Ph.D. thesis, Department of Computer Science, King College London,2005Google Scholar
  7. 7.
    Gentzen G. The collected papers of Gerhard Gentzen, in M. E. Szabo (ed.), Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1969.Google Scholar
  8. 8.
    Goré, R., Tableau methods for modal and temporal logics, in Handbook of Tableau Methods, Kluwer, Dordrecht, 1999, pp. 297–396.Google Scholar
  9. 9.
    Goré, R., and R. Ramanayake, Valentini’s cut-elimination for provability logic resolved, in C. Areces, and R. Goldblatt (eds.), Advances in Modal Logic, Vol. 7 (Nancy, 2008), College Publications, 2008, pp. 91–111.Google Scholar
  10. 10.
    Litak T.: The non-reflexive counterpart of Grz, Bull. Sect. Logic Univ. ódź 36(3–4), 195–208 (2007)Google Scholar
  11. 11.
    Mints, G., Cut elimination for provability logic, in Collegium Logicum 2005: Cut-Elimination, 2006Google Scholar
  12. 12.
    Sambin , G. , Valentini S.: The modal logic of provability. The sequential approach, Journal of Philosophical Logic 11(3), 311–342 (1982)CrossRefGoogle Scholar
  13. 13.
    Sasaki K.: Löb’s axiom and cut-elimination theorem. Journal of Nanzan Academic Society Mathematical Sciences and Information Engineering 1, 91–98 (2001)Google Scholar
  14. 14.
    Troelstra, A. S., Schwichtenberg H. Basic proof theory, vol. 43 of Cambridge Tracts in Theoretical Computer Science, 2nd edn., Cambridge University Press, Cambridge, 2000Google Scholar
  15. 15.
    Valentini S.: The modal logic of provability: cut-elimination. Journal of Philosophical Logic 12(4), 471–476 (1983)CrossRefGoogle Scholar
  16. 16.
    von Plato J.: A proof of Gentzen’s Hauptsatz without multicut. Archive for Mathematical Logic 40(1), 9–18 (2001)CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Logic and Computation Group, Research School of Computer ScienceThe Australian National UniversityCanberraAustralia
  2. 2.CNRS LIX, École PolytechniquePalaiseauFrance

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