Skip to main content
SpringerLink
Log in

Cut-Admissibility as a Corollary of the Subformula Property

  • Conference paper
  • First Online:
Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10501))

Abstract

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 is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Note that by defining sequents to be pairs of sets we implicitly include other standard structural rules, such as exchange and contraction.

References

  1. Anderson, A.R., Belnap, N.D.: Entailment: The Logic of Relevance and Necessity, vol. I. Princeton University Press, Princeton (1975)

    MATH  Google Scholar 

  2. Arieli, O., Avron, A.: The value of the four values. Artif. Intell. 102(1), 97–141 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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

    Chapter  Google Scholar 

  4. Avron, A.: Simple consequence relations. Inf. Comput. 92(1), 105–139 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. Avron, A.: Gentzen-type systems, resolution and tableaux. J. Autom. Reason. 10(2), 265–281 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  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

    Chapter  Google Scholar 

  7. Avron, A.: A non-deterministic view on non-classical negations. Stud. Log.: Int. J. Symb. Log. 80(2/3), 159–194 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Avron, A.: Non-deterministic semantics for families of paraconsistent logics. Handb. Paraconsist. 9, 285–320 (2007)

    MATH  Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  12. Beklemishev, L., Gurevich, Y.: Propositional primal logic with disjunction. J. Log. Comput. 24(1), 257–282 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  13. Béziau, J.-Y.: Sequents and bivaluations. Logique Anal. 44(176), 373–394 (2001)

    MATH  MathSciNet  Google Scholar 

  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 2003

    Google Scholar 

  15. Cotrini, C., Gurevich, Y.: Basic primal infon logic. J. Log. Comput. 26(1), 117–141 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  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. Kamide, N.: A hierarchy of weak double negations. Stud. Log. 101(6), 1277–1297 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lahav, O.: Studying sequent systems via non-deterministic multiple-valued matrices. Mult.-Valued Log. Soft Comput. 21(5–6), 575–595 (2013)

    MathSciNet  Google Scholar 

  19. Lahav, O., Avron, A.: A unified semantic framework for fully structural propositional sequent systems. ACM Trans. Comput. Log. 14(4), 271–273 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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

    MATH  Google Scholar 

  23. Nelson, D.: Constructible falsity. J. Symb. Log. 14(1), 16–26 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  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

    Chapter  Google Scholar 

  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

    Book  MATH  Google Scholar 

  26. Schütte, K.: Beweistheorie. Springer, Berlin (1960)

    MATH  Google Scholar 

  27. Suszko, R.: Remarks on Łukasiewicz’s three-valued logic. Bull. Sect. Log. 4(3), 87–90 (1975)

    MATH  MathSciNet  Google Scholar 

  28. Takeuti, G.: Proof Theory. Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam (1975)

    MATH  Google Scholar 

  29. Wansing, H.: The Logic of Information Structures. LNCS, vol. 681. Springer, Heidelberg (1993). doi:10.1007/3-540-56734-8

    MATH  Google Scholar 

  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

    Chapter  Google Scholar 

Download references

Acknowledgments

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.

Author information

Authors and Affiliations

  1. Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany

    Ori Lahav

  2. School of Computer Science, Tel Aviv University, Tel Aviv, Israel

    Yoni Zohar

Authors
  1. Ori Lahav
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yoni Zohar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoni Zohar .

Editor information

Editors and Affiliations

  1. University of Manchester, Manchester, United Kingdom

    Renate A. Schmidt

  2. University of Brasília, Brasília D.F., Brazil

    Cláudia Nalon

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Lahav, O., Zohar, Y. (2017). Cut-Admissibility as a Corollary of the Subformula Property. In: Schmidt, R., Nalon, C. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2017. Lecture Notes in Computer Science(), vol 10501. Springer, Cham. https://doi.org/10.1007/978-3-319-66902-1_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-66902-1_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66901-4

  • Online ISBN: 978-3-319-66902-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation