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Valentini (1983) has presented a proof of cut-elimination for provability logic GL for a sequent calculus using sequents built from sets as opposed to multisets, thus avoiding an explicit contraction rule. From a formal point of view, it is more syntactic and satisfying to explicitly identify the applications of the contraction rule that are ‘hidden’ in proofs of cut-elimination for such sequent calculi. There is often an underlying assumption that the move to a proof of cut-elimination for sequents built from multisets is straightforward. Recently, however, it has been claimed that Valentini’s arguments to eliminate cut do not terminate when applied to a multiset formulation of the calculus with an explicit rule of contraction. The claim has led to much confusion and various authors have sought new proofs of cut-elimination for GL in a multiset setting.

Here we refute this claim by placing Valentini’s arguments in a formal setting and proving cut-elimination for sequents built from multisets. The use of sequents built from multisets enables us to accurately account for the interplay between the weakening and contraction rules. Furthermore, Valentini’s original proof relies on a novel induction parameter called “width” which is computed ‘globally’. It is difficult to verify the correctness of his induction argument based on “width.” In our formulation however, verification of the induction argument is straightforward. Finally, the multiset setting also introduces a new complication in the case of contractions above cut when the cut-formula is boxed. We deal with this using a new transformation based on Valentini’s original arguments.

Finally, we discuss the possibility of adapting this cut-elimination procedure to other logics axiomatizable by formulae of a syntactically similar form to the GL axiom.

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Avron, A. (1984). On modal systems having arithmetical interpretations. Journal of Symbolic Logic, 49(3), 935942.
Avron, A. (1996). The method of hypersequents in the proof theory of propositional non-classical logics. In Logic: From Foundations to Applications (Staffordshire, 1993). Oxford Sci. Publ. New York, NY: Oxford University Press, pp. 132.
Belnap, N. D Jr. (1982). Display logic. Journal of Philosophical Logic, 11(4), 375417.
Bimbó, K. (2007). , , LK, and cutfree proofs. Journal of Philosophical Logic, 36(5), 557570.
Borga, M. (1983). On some proof theoretical properties of the modal logic GL. Studia Logica: An International Journal for Symbolic Logic, 42(4), 453459.
Borga, M., & Gentilini, P. (1986). On the proof theory of the modal logic Grz. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 32(2), 145148.
Borisavljević, M., Došen, K., & Petrić, Z. (2000). On permuting cut with contraction. Mathematical Structures in Computer Science, 10(2), 99136. The Lambek Festschrift: mathematical structures in computer science (Montreal, QC, 1997).
Dawson, J. E., & Goré, R. (2010). Generic methods for formalising sequent calculi applied to provability logic. In Fermüller, C. G., and Voronkov, A., editors. LPAR (Yogyakarta), Vol. 6397 of Lecture Notes in Computer Science. Berlin Heidelberg, Germany: Springer, pp. 263277.
Demri, S., & Goré, R. (2002). Theoremhood-preserving maps characterizing cut elimination for modal provability logics. Journal of Logic and Computation, 12(5), 861884.
Došen, K. (1985). Sequent-systems for modal logic. Journal of Symbolic Logic, 50(1), 149168.
Gentzen, G. (1969). The collected papers of Gerhard Gentzen. In Szabo, M. E., editor. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland Publishing Co.
Goré, R. (1999). Tableau methods for modal and temporal logics. In Handbook of Tableau Methods. Dordrecht, The Netherlands: Kluwer Academic Publishers, pp. 297396.
Goré, R., Heinle, W., & Heuerding, A. (1997). Relations between propositional normal modal logics: An overview. Journal of Logic and Computation, 7(5), 649658.
Goré, R., & Ramanayake, R. (2008). Valentini’s cut-elimination for provability logic resolved. In Areces, C., and Goldblatt, R., editors. Advances in Modal Logic, Vol. 7. London: College Publications, pp. 91111.
Goré, R., & Ramanayake, R. (2012). Cut-elimination for weak Grzegorczyk logic Go. Accepted subject to minor revisions by Studia Logica.
Guglielmi, A. (2007). A system of interaction and structure. ACM Transactions on Computational Logic, 8(1), Art. 1, 64.
Indrzejczak, A. (1996). Cut-free sequent calculus for S5. University of Łódź, Department of Logic: Bulletin of the Section of Logic, 25(2), 95102.
Leivant, D. (1981). On the proof theory of the modal logic for arithmetic provability. Journal of Symbolic Logic, 46(3), 531538.
Mints, G. (2005). Cut elimination for provability logic. In Collegium Logicum 2005: Cut-Elimination.
Moen, A. (2001). The Proposed Algorithms for Eliminating Cuts in the Provability Calculus GLS do not Terminate. NWPT 2001, Norwegian Computing Center, 2001-12-10.
Negri, S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic, 34(5–6), 507544.
Negri, S., & von Plato, J. (2001). Structural Proof Theory. Cambridge, UK: Cambridge University Press. Appendix C by Aarne Ranta.
Poggiolesi, F. (2008). A cut-free simple sequent calculus for modal logic S5. The Review of Symbolic Logic, 1(1), 315.
Poggiolesi, F. (2009). A purely syntactic and cut-free sequent calculus for the modal logic of provability. Review of Symbolic Logic, 2(4), 593611.
Pottinger, G. (1983). Uniform, cut-free formulations of T, S4 and S5 (abstract). Journal of Symbolic Logic, 48, 900901.
Sambin, G., & Valentini, S. (1982). The modal logic of provability. The sequential approach. Journal of Philosophical Logic, 11(3), 311342.
Sasaki, K. (2001). Löb’s axiom and cut-elimination theorem. Journal of the Nanzan Academic Society of Mathematical Sciences and Information Engineering, 1, 9198.
Sato, M. (1980). A cut-free Gentzen-type system for the modal logic S5. Journal of Symbolic Logic, 45(1), 6784.
Shimura, T. (1991). Cut-free systems for the modal logic S4.3 and S4.3Grz. Reports on Mathematical Logic, 25, 5773.
Shimura, T. (1992). Cut-free systems for some modal logics containing S4. Reports on Mathematical Logic, 26, 3965.
Solovay, R. M. (1976). Provability interpretations of modal logic. Israel Journal of Mathematics, 25(3–4), 287304.
Švejdar, V. (1999). On provability logic. Nordic Journal of Philosophical Logic, 4(2), 95116.
Takeuti, G. (1987). Proof Theory (second edition), Vol. 81 of Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland Publishing Co.
Troelstra, A. S., & Schwichtenberg, H. (2000). Basic Proof Theory (second edition), Vol. 43 of Cambridge Tracts in Theoretical Computer Science. Cambridge, UK: Cambridge University Press.
Valentini, S. (1983). The modal logic of provability: Cut-elimination. Journal of Philosophical Logic, 12(4), 471476.
Valentini, S. (1986). A syntactic proof of cut-elimination for GLlin. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 32(2), 137144.
von Plato, J. (2001). A proof of Gentzen’s Hauptsatz without multicut. Archive for Mathematical Logic, 40(1), 918.
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The Review of Symbolic Logic
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