Skip to main content
SpringerLink
Log in

Harmony in Multiple-Conclusion Natural-Deduction

  • Published:
Logica Universalis Aims and scope Submit manuscript

Abstract

The paper studies the extension of harmony and stability, major themes in proof-theoretic semantics, from single-conclusion natural-deduction systems to multiple-conclusions natural-deduction, independently of classical logic. An extension of the method of obtaining harmoniously-induced general elimination rules from given introduction rules is suggested, taking into account sub-structurality. Finally, the reductions and expansions of the multiple-conclusions natural-deduction representation of classical logic are formulated.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blamey S., Humberstone L.: A perspective on modal sequent logic. Publ. RIMS Kyoto Univ. 27, 763–782 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Boric̆ić B.: On sequence-conclusion natural-deduction systems. J. Philos. Log. 14, 359–377 (1985)

    Article  Google Scholar 

  3. Boric̆ić B.: On certain normalizable natural deduction formulations of some propositional intermediate logics. Notre Dame J. Form. Log. 29(4), 563–568 (1988)

    Article  Google Scholar 

  4. Boric̆ić, B., Ilic̆, M.: An alternative normalisation of the implicative fragment of classical logic. Stud. Log. (2014, to appear)

  5. Cellucci C.: Existential instantiation and normalization in sequent natural deduction. Ann. Pure Appl. Log. 58, 111–148 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Davies R., Pfenning F.: A modal analysis of staged computation. J. ACM 48(3), 555–604 (2001)

    Article  MathSciNet  Google Scholar 

  7. Dos̆en K.: Logical constants as punctuation marks. Notre Dame J. Form. Log. 30(3), 362–381 (1989)

    Article  MathSciNet  Google Scholar 

  8. Dummett, M.: The Logical Basis of Metaphysics. Harvard University Press, Cambridge (1993) (paperback). Hard copy (1991)

  9. Francez, N.: Relevant harmony. J. Log. Comput. (2013) doi:10.1093/logcom/ext026. Special issue Logic: Between Semantics and Proof Theory, in honor of Arnon Avron’s 60th birthday

  10. Francez, N.: Views of proof-theoretic semantics: Reified proof-theoretic meanings. J. Comput. Log. (2014, to appear). Special issue in honour of Roy Dyckhoff

  11. Francez N., Ben-Avi G.: Proof-theoretic semantic values for logical operators. Rev. Symb. Log. 4(3), 337–485 (2011)

    Article  MathSciNet  Google Scholar 

  12. Francez N., Dyckhoff R.: Proof-theoretic semantics for a natural language fragment. Linguist. Philos. 33(6), 447–477 (2010)

    Article  Google Scholar 

  13. Francez N., Dyckhoff R.: A note on harmony. J. Philos. Log. 41(3), 613–628 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Francez N., Dyckhoff R., Ben-Avi G.: Proof-theoretic semantics for subsentential phrases. Stud. Log. 94, 381–401 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gentzen, G.: The consistency of elementary number theory. In: Szabo, M.E. (ed.) The collected papers of Gerhard Gentzen, pp. 493–565. North-Holland, Amsterdam (1935). English translation of the 1935 paper in Mathematische Annalen (in German)

  16. Gentzen, G.: Investigations into logical deduction. In: Szabo, M.E. (ed.) The collected papers of Gerhard Gentzen, pp. 68–131. North-Holland, Amsterdam (1935). English translation of the 1935 paper in German

  17. Girard J.-Y.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hjortland, O.T.: The Structure of Logical Consequence: Proof-Theoretic Conceptions. PhD thesis, University of St Andrews, (2009)

  19. Hjortland, O.T.: Harmony and the context of deducibility. In: Novaes C.D., Hjortland, O.T. (eds.) Insolubles and Consequences: Essays in Honour of Stephen Read, pp. 135–154. College Publications, London 2013

  20. Humberstone, L.: The Connectives. MIT Press, Cambridge (2011)

  21. Milne P.: Classical harmony: rules of inference and the meaning of the logical constants. Synthese 100(1), 49–94 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Moriconi E., Tesconi L.: On inversion principles. Hist. Philos. Log. 29(2), 103–113 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Negri S.: Varieties of linear calculi. J. Philos. Log. 32(6), 569–590 (2002)

    Article  MathSciNet  Google Scholar 

  24. Negri S., von Plato J.: Sequent calculus in natural deduction style. J. Symb. Log. 66(4), 1803–1816 (2001)

    Article  MATH  Google Scholar 

  25. Olkhovikov, G.K., Schroeder-Heister, P.: On flattening general elimination rules. Rev. Symb. Log. 7(1), (2014)

  26. Pfenning F., Davies R.: A judgmental reconstruction of modal logic. Math. Struct. Comput. Sci. 11, 511–540 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Prawitz, D.: Natural Deduction: A Proof-Theoretical Study. Almqvist and Wicksell, Stockholm (1965). Soft cover edition by Dover, 2006

  28. Prawitz, D.: Ideas and results in proof theory. In: Fenstad, J. (ed.) Proceedings of the 2nd Scandinavian Symposium. North-Holland, Amsterdam (1971)

  29. Prawitz, D.: Proofs and the meaning and completeness of logical constants. In: Hintikka, J., Niiniluoto, I., Saarinen, E. (eds.) Essays in Mathematical and Philosophical Logic, pp. 25–40. Reidel, Dordrecht (1978)

  30. Prior A.N.: The runabout inference-ticket. Analysis, 21, 38–39 (1960)

    Article  Google Scholar 

  31. Read S.: Harmony and autonomy in classical logic. J. Philos. Log. 29, 123–154 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  32. Read S.: General-elimination harmony and the meaning of the logical constants. J. Philos. Log. 39, 557–576 (2010)

    Article  MATH  Google Scholar 

  33. Read, S.: General elimination harmony and higher-level rules. In: Wansing, H. (ed.) Dag Prawitz on Proofs and Meaning. Springer, Berlin (2014). Studia Logica Outstanding contributions to logic series

  34. Restall, G.: Multiple conclusions. In: In 12th International Congress on Logic, Methodology and Philosophy of Science, pp. 189–205 (2005)

  35. Rumfitt I.: ‘yes’ and ‘no’. Mind 169(436, 781–823), 169–436, 781823 (2000)

    MathSciNet  Google Scholar 

  36. Sandqvist, T.: Acceptance, inference, and the multiple-conclusion sequent. Synthese (2011). doi:10.1007/s11229-011-9909-5

  37. Schroeder-Heister, P.: The calculus of higher-level rules, propositional quantification, and the foundational approach to proof-theoretic harmony. In: Indrzejczak, A. (ed.) Gentzen’s and Jaśkowski’s Heritage: 80 Years of Natural Deduction and Sequent Calculi (2014). Special issue of Studia Logica

  38. Schroeder-Heister, P.: Harmony in proof-theoretic semantics: a reductive analysis. In: Wansing, H. (ed.) Dag Prawitz on Proofs and Meaning. Springer, Berlin (2014). Studia Logica Outstanding contributions to logic series

  39. Steinberger F.: Why conclusions should remain single. J. Philos. Log. 40, 333–355 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  40. Tennant, N.: The Taming of the True. Oxford University Press, Oxford (1997)

  41. Tesconi, L.: A Strong Normalization Theorem for natural deduction with general elimination rules. PhD thesis, University of Pisa (2004)

  42. von Plato J.: Natural deduction with general elimination rules. Arch. Math. Log. 40, 541–567 (2001)

    Article  MATH  Google Scholar 

  43. Wadler P.: There is no substitute for linear logic. In: 8th International Workshop on Mathematical Foundations of Programming Semantics. Oxford, UK (1992)

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, The Technion-IIT, Haifa, Israel

    Nissim Francez

Authors
  1. Nissim Francez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nissim Francez.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Francez, N. Harmony in Multiple-Conclusion Natural-Deduction. Log. Univers. 8, 215–259 (2014). https://doi.org/10.1007/s11787-014-0103-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11787-014-0103-7

Mathematics Subject Classification (2010)

Keywords

Search

Navigation