Cut-Admissibility as a Corollary of the Subformula Property

  • Ori Lahav
  • Yoni ZoharEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10501)


We identify two wide families of propositional sequent calculi for which cut-admissibility is a corollary of the subformula property. While the subformula property is often a simple consequence of cut-admissibility, our results shed light on the converse direction, and may be used to simplify cut-admissibility proofs in various propositional sequent calculi. In particular, the results of this paper may be used in conjunction with existing methods that establish the subformula property, to obtain that cut-admissibility holds as well.



This research was supported by The Israel Science Foundation (grant no. 817-15). We thank Arnon Avron, João Marcos and the TABLEAUX’17 reviewers for their helpful feedback.


  1. 1.
    Anderson, A.R., Belnap, N.D.: Entailment: The Logic of Relevance and Necessity, vol. I. Princeton University Press, Princeton (1975)zbMATHGoogle Scholar
  2. 2.
    Arieli, O., Avron, A.: The value of the four values. Artif. Intell. 102(1), 97–141 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Arieli, O., Avron, A.: Three-valued paraconsistent propositional logics. In: Beziau, J.-Y., Chakraborty, M., Dutta, S. (eds.) New Directions in Paraconsistent Logic: 5th WCP. Kolkata, India, pp. 91–129. Springer, New Delhi (2015). doi: 10.1007/978-81-322-2719-9_4 CrossRefGoogle Scholar
  4. 4.
    Avron, A.: Simple consequence relations. Inf. Comput. 92(1), 105–139 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Avron, A.: Gentzen-type systems, resolution and tableaux. J. Autom. Reason. 10(2), 265–281 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Avron, A.: Classical Gentzen-type methods in propositional many-valued logics. In: Fitting, M., Orłowska, E. (eds.) Beyond Two: Theory and Applications of Multiple-Valued Logic. STUDFUZZ, vol. 114, pp. 117–155. Physica, Heidelberg (2003). doi: 10.1007/978-3-7908-1769-0_5 CrossRefGoogle Scholar
  7. 7.
    Avron, A.: A non-deterministic view on non-classical negations. Stud. Log.: Int. J. Symb. Log. 80(2/3), 159–194 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Avron, A.: Non-deterministic semantics for families of paraconsistent logics. Handb. Paraconsist. 9, 285–320 (2007)zbMATHGoogle Scholar
  9. 9.
    Avron, A., Konikowska, B., Zamansky, A.: Modular construction of cut-free sequent calculi for paraconsistent logics. In: Proceedings of the 27th Annual IEEE/ACM Symposium on Logic in Computer Science, LICS 2012, pp. 85–94. IEEE Computer Society (2012)Google Scholar
  10. 10.
    Avron, A., Lev, I.: Non-deterministic multi-valued structures. J. Log. Comput. 15, 241–261 (2005). Conference version: Avron, A., Lev, I.: Canonical propositional Gentzen-type systems. In: Proceedings of the International Joint Conference on Automated Reasoning, IJCAR 2001. LNAI, vol. 2083, pp. 529–544. Springer, Heidelberg (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Avron, A., Zamansky, A.: Non-deterministic semantics for logical systems. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic. HALO, vol. 16, pp. 227–304. Springer, Dordrecht (2011). doi: 10.1007/978-94-007-0479-4_4 CrossRefGoogle Scholar
  12. 12.
    Beklemishev, L., Gurevich, Y.: Propositional primal logic with disjunction. J. Log. Comput. 24(1), 257–282 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Béziau, J.-Y.: Sequents and bivaluations. Logique Anal. 44(176), 373–394 (2001)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Comon-Lundh, H., Shmatikov, V.: Intruder deductions, constraint solving and insecurity decision in presence of exclusive or. In: 2003 Proceedings of 18th Annual IEEE Symposium on Logic in Computer Science, pp. 271–280, June 2003Google Scholar
  15. 15.
    Cotrini, C., Gurevich, Y.: Basic primal infon logic. J. Log. Comput. 26(1), 117–141 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Gentzen, G.: Investigations Into Logical Deduction (1934). (in German). An English translation appears in ‘The Collected Works of Gerhard Gentzen’, edited by Szabo, M.E., North-Holland (1969)Google Scholar
  17. 17.
    Kamide, N.: A hierarchy of weak double negations. Stud. Log. 101(6), 1277–1297 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Lahav, O.: Studying sequent systems via non-deterministic multiple-valued matrices. Mult.-Valued Log. Soft Comput. 21(5–6), 575–595 (2013)MathSciNetGoogle Scholar
  19. 19.
    Lahav, O., Avron, A.: A unified semantic framework for fully structural propositional sequent systems. ACM Trans. Comput. Log. 14(4), 271–273 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Lahav, O., Zohar, Y.: On the construction of analytic sequent calculi for sub-classical logics. In: Kohlenbach, U., Barceló, P., Queiroz, R. (eds.) WoLLIC 2014. LNCS, vol. 8652, pp. 206–220. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-44145-9_15 Google Scholar
  21. 21.
    Lahav, O., Zohar, Y.: SAT-based decision procedure for analytic pure sequent calculi. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 76–90. Springer, Cham (2014). doi: 10.1007/978-3-319-08587-6_6 Google Scholar
  22. 22.
    Metcalfe, G., Olivetti, N., Gabbay, D.: Proof Theory for Fuzzy Logics. Applied Logic Series, vol. 36. Springer, Netherlands (2009). doi: 10.1007/978-1-4020-9409-5 zbMATHGoogle Scholar
  23. 23.
    Nelson, D.: Constructible falsity. J. Symb. Log. 14(1), 16–26 (1949)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Pinto, L., Uustalu, T.: Proof search and counter-model construction for bi-intuitionistic propositional logic with labelled sequents. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS (LNAI), vol. 5607, pp. 295–309. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02716-1_22 CrossRefGoogle Scholar
  25. 25.
    Poggiolesi, F.: Gentzen Calculi for Modal Propositional Logic. Trends in Logic, vol. 32. Springer, Netherlands (2011). doi: 10.1007/978-90-481-9670-8 CrossRefzbMATHGoogle Scholar
  26. 26.
    Schütte, K.: Beweistheorie. Springer, Berlin (1960)zbMATHGoogle Scholar
  27. 27.
    Suszko, R.: Remarks on Łukasiewicz’s three-valued logic. Bull. Sect. Log. 4(3), 87–90 (1975)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Takeuti, G.: Proof Theory. Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam (1975)zbMATHGoogle Scholar
  29. 29.
    Wansing, H.: The Logic of Information Structures. LNCS, vol. 681. Springer, Heidelberg (1993). doi: 10.1007/3-540-56734-8 zbMATHGoogle Scholar
  30. 30.
    Wansing, H.: Sequent systems for modal logics. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic. HALO, vol. 8, 2nd edn, pp. 61–145. Springer, Dordrecht (2002). doi: 10.1007/978-94-010-0387-2_2 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Max Planck Institute for Software Systems (MPI-SWS)KaiserslauternGermany
  2. 2.School of Computer ScienceTel Aviv UniversityTel AvivIsrael

Personalised recommendations