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.
Similar content being viewed by others
References
Amerbauer M.: Cut-free tableau calculi for some propositional normal modal logics. Studia Logica 57(2–3), 359–372 (1996)
Belnap N.D. Jr.: Display logic. Journal of Philosophical Logic 11(4), 375–417 (1982)
Borga M., Gentilini P.: On the proof theory of the modal logic Grz. Z. Math. Logik Grundlag. Math. 32(2), 145–148 (1986)
Borga M.: On some proof theoretical properties of the modal logic GL. Studia Logica 42(4), 453–459 (1983)
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)
Gabelaia D. Topological, Algebraic and Spatio-Temporal Semantics for Multi-Dimensional Modal Logics, Ph.D. thesis, Department of Computer Science, King College London,2005
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.
Goré, R., Tableau methods for modal and temporal logics, in Handbook of Tableau Methods, Kluwer, Dordrecht, 1999, pp. 297–396.
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.
Litak T.: The non-reflexive counterpart of Grz, Bull. Sect. Logic Univ. ódź 36(3–4), 195–208 (2007)
Mints, G., Cut elimination for provability logic, in Collegium Logicum 2005: Cut-Elimination, 2006
Sambin , G. , Valentini S.: The modal logic of provability. The sequential approach, Journal of Philosophical Logic 11(3), 311–342 (1982)
Sasaki K.: Löb’s axiom and cut-elimination theorem. Journal of Nanzan Academic Society Mathematical Sciences and Information Engineering 1, 91–98 (2001)
Troelstra, A. S., Schwichtenberg H. Basic proof theory, vol. 43 of Cambridge Tracts in Theoretical Computer Science, 2nd edn., Cambridge University Press, Cambridge, 2000
Valentini S.: The modal logic of provability: cut-elimination. Journal of Philosophical Logic 12(4), 471–476 (1983)
von Plato J.: A proof of Gentzen’s Hauptsatz without multicut. Archive for Mathematical Logic 40(1), 9–18 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
presented by Heinrich Wansing
Rights and permissions
About this article
Cite this article
Goré, R., Ramanayake, R. Cut-elimination for Weak Grzegorczyk Logic Go . Stud Logica 102, 1–27 (2014). https://doi.org/10.1007/s11225-012-9432-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-012-9432-9