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Translations Between Gentzen–Prawitz and Jaśkowski–Fitch Natural Deduction Proofs

  • Shawn Standefer
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Abstract

Two common forms of natural deduction proof systems are found in the Gentzen–Prawitz and Jaśkowski–Fitch systems. In this paper, I provide translations between proofs in these systems, pointing out the ways in which the translations highlight the structural rules implicit in the systems. These translations work for classical, intuitionistic, and minimal logic. I then provide translations for classical S4 proofs.

Keywords

Proof theory Tree systems Cascade systems 

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Notes

Acknowledgements

I would like to thank Greg Restall, Allen Hazen, Rohan French, Fabio Lampert, and the members of the Melbourne Logic Seminar for feedback on this work. I am grateful to two anonymous referees for their detailed, constructive reports, which significantly improved the presentation of this paper and helped me to correct some errors. I owe a special thanks to Valeria de Paiva, Luiz Carlos Pereira, and Edward Hermann Haeusler, whose short course on tree-style natural deduction at NASSLLI 2016 motivated this research. This research was supported by the Australian Research Council, Discovery Grant DP150103801.

References

  1. 1.
    Anderson, A. R., and N. D. Belnap, Entailment: The Logic of Relevance and Neccessity, volume 1. Princeton University Press, Princeton, 1975.Google Scholar
  2. 2.
    Barker-Plummer, D., J. Barwise, and J. Etchemendy, Language, Proof and Logic. CSLI Publications, 2nd edition, 2011.Google Scholar
  3. 3.
    Benton, N., G. Bierman, V. de Paiva, and M. Hyland, A term calculus for intuitionistic linear logic, in M. Bezem and J. F. Groote, (eds.), Typed Lambda Calculi and Applications, Springer, Berlin Heidelberg, 1993, pp. 75–90.  https://doi.org/10.1007/BFb0037099.Google Scholar
  4. 4.
    Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic. Cambridge University Press, NP, 2002.Google Scholar
  5. 5.
    Blizard, W. D., Multiset theory, Notre Dame Journal of Formal Logic 30(1): 36–66, 1988.  https://doi.org/10.1305/ndjfl/1093634995.CrossRefGoogle Scholar
  6. 6.
    Brady, R. T., Natural deduction systems for some quantified relevant logics, Logique Et Analyse 27(8): 355–377, 1984.Google Scholar
  7. 7.
    Brady, R. T., Normalized natural deduction systems for some relevant logics I: The logic DW, Journal of Symbolic Logic 71(1): 35–66, 2006.  https://doi.org/10.2178/jsl/1140641162.CrossRefGoogle Scholar
  8. 8.
    Brady, R. T., Free semantics. Journal of Philosophical Logic 39(5): 511–529, 2010.  https://doi.org/10.1007/s10992-010-9129-3.CrossRefGoogle Scholar
  9. 9.
    Fitch, F. B., Symbolic Logic. Ronald Press Co., New York, 1952.Google Scholar
  10. 10.
    Fitch, F. B., Natural deduction rules for obligation. American Philosophical Quarterly 3(1): 27–38, January 1966.Google Scholar
  11. 11.
    Garson, J. W., Modal Logic for Philosophers. Cambridge University Press, Cambridge, 2nd edition, 2013.Google Scholar
  12. 12.
    Gentzen, G., Untersuchungen über das logische Schließen, I and II, Mathematische Zeitschrift 39: 176–210, 405–431, 1934. Translated as “Investigations into Logical Deduction” parts I and II, and published in American Philosophical Quarterly 1: 288–306, 1964, and 2: 204–218, 1965, respectively. The full article was reprinted in M. E. Szabo (ed.), The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam, 1969, pp. 68–131.Google Scholar
  13. 13.
    Geuvers, H., and I. Loeb, From deduction graphs to proof nets: Boxes and sharing in the graphical presentation of deductions, in R. Královič and P. Urzyczyn (eds.), MFCS, Lecture Notes in Computer Science, vol. 4162, Springer, 2006, pp. 39–57.  https://doi.org/10.1007/11821069_4.Google Scholar
  14. 14.
    Geuvers, H., and I. Loeb, Natural deduction via graphs: Formal definition and computation rules, Mathematical Structures in Computer Science 17(03): 485–526, 2007.  https://doi.org/10.1017/S0960129507006123.CrossRefGoogle Scholar
  15. 15.
    Hawthorn, J., Natural deduction in normal modal logic. Notre Dame Journal of Formal Logic 31(2): 263–273, 1990.  https://doi.org/10.1305/ndjfl/1093635420.CrossRefGoogle Scholar
  16. 16.
    Hazen, A., Natural deduction and Hilbert’s \(\varepsilon \)-operator, Journal of Philosophical Logic. 16(4): 411–421, 1987.  https://doi.org/10.1007/BF00431186.CrossRefGoogle Scholar
  17. 17.
    Hazen, A. P., and F. J. Pelletier, Gentzen and Jaśkowski natural deduction: Fundamentally similar but importantly different. Studia Logica 102(6): 1103–1142, 2014.  https://doi.org/10.1007/s11225-014-9564-1.CrossRefGoogle Scholar
  18. 18.
    Hughes, G. E., and M. J. Cresswell, A New Introduction to Modal Logic. Routledge, London, 2012.Google Scholar
  19. 19.
    Humberstone, L., Philosophical Applications of Modal Logic. College Publications, London, 2016.Google Scholar
  20. 20.
    Jacinto, B., and S. Read, General-elimination stability, Studia Logica 105(2): 361–405, 2016.  https://doi.org/10.1007/s11225-016-9692-x.CrossRefGoogle Scholar
  21. 21.
    Jaśkowski, S., On the rules of suppositions in formal logic. Studia Logica 1: 5–32, 1934. Reprinted in S. McCall, (ed.), Polish Logic 1920–1939, Oxford University Press, 1967, pp. 232–258.Google Scholar
  22. 22.
    Kozhemiachenko, D., A simulation of natural deduction and Gentzen sequent calculus, Logic and Logical Philosophy 27(1): 67–84, 2017.  https://doi.org/10.12775/LLP.2017.009.
  23. 23.
    Lampert, F., Natural deduction for diagonal operators, in M. Zack and D. Schlimm, (eds.), Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta, Cham: Birkhäuser, 2017, pp. 39–51.CrossRefGoogle Scholar
  24. 24.
    Leivant, D., Assumption classes in natural deduction. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25(1–2): 1–4, 1979.  https://doi.org/10.1002/malq.19790250102.CrossRefGoogle Scholar
  25. 25.
    Medeiros, M. P. N., A new S4 classical modal logic in natural deduction, Journal of Symbolic Logic 71(3): 799–809, 2006.  https://doi.org/10.2178/jsl/1154698578.CrossRefGoogle Scholar
  26. 26.
    Mints, G., A Short Introduction to Modal Logic. Center for the Study of Language and Information, 1992.Google Scholar
  27. 27.
    Negri, S., Proof theory for modal logic, Philosophy Compass 6(8): 523–538, 2011.  https://doi.org/10.1111/j.1747-9991.2011.00418.x.CrossRefGoogle Scholar
  28. 28.
    Negri, S., and J. von Plato, Sequent calculus in natural deduction style, Journal of Symbolic Logic 66(4): 1803–1816, 2001.  https://doi.org/10.2307/2694976.CrossRefGoogle Scholar
  29. 29.
    Negri, S., and J. von Plato, Structural Proof Theory. Cambridge University Press, Cambridge, NP 2001.CrossRefGoogle Scholar
  30. 30.
    Pelletier, F. J., A brief history of natural deduction. History and Philosophy of Logic 20(1): 1–31, 1999.  https://doi.org/10.1080/014453499298165.CrossRefGoogle Scholar
  31. 31.
    Pelletier, F. J., A history of natural deduction and elementary logic textbooks, in J. Woods and B. Brown, (eds.), Logical Consequence: Rival Approaches, vol. 1, Oxford University Press, 2000, pp. 105–138.Google Scholar
  32. 32.
    Pelletier, F. J., and A. Hazen, Natural deduction, in D. Gabbay and J. Woods, (eds.), Handbook of the History of Logic, vol. 11, Dordrecht, 2012, pp. 341–414.Google Scholar
  33. 33.
    Prawitz, D., Natural Deduction: A Proof-Theoretical Study. Almqvist and Wicksell, Stockholm, 1965.Google Scholar
  34. 34.
    Quispe-Cruz, M., E. H. Haeusler, and L. Gordeev, Proof graphs for minimal implicational logic, in M. Ayala-Rincón, E. Bonelli, and I. Mackie, (eds.), Proceedings 9th International Workshop on Developments in Computational Models, Buenos Aires, Argentina, 26 August 2013, vol. 144 of Electronic Proceedings in Theoretical Computer Science, Open Publishing Association, 2014, pp. 16–29.  https://doi.org/10.4204/EPTCS.144.2.CrossRefGoogle Scholar
  35. 35.
    Raggio, A., Gentzen’s Hauptsatz for the systems NI and NK, Logique Et Analyse 8: 91–100, 1965.Google Scholar
  36. 36.
    Read, S., Harmony and modality, in C. Dégremont, L. Keiff, and H. Rückert, (eds.), On Dialogues, Logics and other Strange Things, Kings College Publications, 2008, pp. 285–303.Google Scholar
  37. 37.
    Read, S., General-elimination harmony and the meaning of the logical constants. Journal of Philosophical Logic 39(5): 557–576, 2010.  https://doi.org/10.1007/s10992-010-9133-7.CrossRefGoogle Scholar
  38. 38.
    Restall, G., An Introduction to Substructural Logics. Routledge, London, 2000.CrossRefGoogle Scholar
  39. 39.
    Restall, G., Normal proofs, cut free derivations and structural rules. Studia Logica 102(6): 1143–1166, 2014.  https://doi.org/10.1007/s11225-014-9598-4.CrossRefGoogle Scholar
  40. 40.
    Rogerson, S., Natural deduction and Curry’s paradox. Journal of Philosophical Logic 36(2): 155–179, 2007.  https://doi.org/10.1007/s10992-006-9032-0.CrossRefGoogle Scholar
  41. 41.
    Schroeder-Heister, P., A natural extension of natural deduction. Journal of Symbolic Logic 49(4): 1284–1300, 1984.  https://doi.org/10.2307/2274279.CrossRefGoogle Scholar
  42. 42.
    Siemens, D. F. Jr., Fitch-style rules for many modal logics, Notre Dame Journal of Formal Logic 18(4): 631–636, October 1977.  https://doi.org/10.1305/ndjfl/1093888133.CrossRefGoogle Scholar
  43. 43.
    Slaney, J., A general logic. Australasian Journal of Philosophy 68(1): 74–88, 1990.  https://doi.org/10.1080/00048409012340183.CrossRefGoogle Scholar
  44. 44.
    Standefer, S., Trees for E. Logic Journal of the IGPL 26(3): 300–315, 2018.  https://doi.org/10.1093/jigpal/jzy003.CrossRefGoogle Scholar
  45. 45.
    Thomason, R. H., A decision procedure for Fitch’s propositional calculus. Notre Dame Journal of Formal Logic 8(1–2): 101–117, 1967.  https://doi.org/10.1305/ndjfl/1093956248.CrossRefGoogle Scholar
  46. 46.
    Thomason, R. H., A Fitch-style formulation of conditional logic. Logique Et Analyse 52: 397–412, 1970.Google Scholar
  47. 47.
    van Benthem, J., Modal Logic for Open Minds. CSLI Publications, 2010.Google Scholar
  48. 48.
    van Dalen, D., Logic and Structure. Springer, London, 5th edition, 2013.CrossRefGoogle Scholar
  49. 49.
    von Plato, J., A problem of normal form in natural deduction. Mathematical Logic Quarterly 46(1): 121–124, 2000.  https://doi.org/10.1002/(SICI)1521-3870(200001)46:1<121::AID-MALQ121>3.0.CO;2-A.
  50. 50.
    von Plato, J., Normal derivability in modal logic. Mathematical Logic Quarterly 51(6): 632–638, 2005.  https://doi.org/10.1002/malq.200410054.CrossRefGoogle Scholar
  51. 51.
    von Plato, J., From Gentzen to Jaskowski and back: Algorithmic translation of derivations between the two main systems of natural deduction. Bulletin of the Section of Logic 46(1/2): 65–73, 2017.  https://doi.org/10.18778/0138-0680.46.1.2.06.CrossRefGoogle Scholar
  52. 52.
    von Plato, J., and G. Gentzen, Gentzen’s proof of normalization for natural deduction. Bulletin of Symbolic Logic 14(2): 240–257, 2008.  https://doi.org/10.2178/bsl/1208442829.CrossRefGoogle Scholar
  53. 53.
    Wisdom, W. A., Possibility-elimination in natural deduction. Notre Dame Journal of Formal Logic 5(4): 295–298, 1964.  https://doi.org/10.1305/ndjfl/1093957978.CrossRefGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Historical and Philosophical StudiesThe University of MelbourneParkvilleAustralia

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