Skip to main content
SpringerLink
Log in

Gentzen and Jaśkowski Natural Deduction: Fundamentally Similar but Importantly Different

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

Gentzen’s and Jaśkowski’s formulations of natural deduction are logically equivalent in the normal sense of those words. However, Gentzen’s formulation more straightforwardly lends itself both to a normalization theorem and to a theory of “meaning” for connectives (which leads to a view of semantics called ‘inferentialism’). The present paper investigates cases where Jaskowski’s formulation seems better suited. These cases range from the phenomenology and epistemology of proof construction to the ways to incorporate novel logical connectives into the language. We close with a demonstration of this latter aspect by considering a Sheffer function for intuitionistic logic.

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. Belnap N. (1962) Tonk, plonk and plink. Analysis 22: 130–134

    Article  Google Scholar 

  2. Bergmann M., Moor J., Nelson J. (2008) The Logic Book, Fifth Edition, Random House, New York

    Google Scholar 

  3. Black, M. (1953) Does the logical truth that \({{\exists x (Fx \vee \neg Fx)}}\) entail that at least one individual exists?, Analysis 14:1–2.

  4. Bonevac D. (1987) Deduction. Mayfield Press, Mountain View, CA

    Google Scholar 

  5. Brandom R. (1994) Making it Explicit. Harvard University Press, Cambridge, MA

    Google Scholar 

  6. Brandom R. (2000) Articulating Reasons. Harvard UP, Cambridge, MA

    Google Scholar 

  7. Cellucci, C. (1995) On Quine’s approach to natural deduction, in P. Leonardi, and M. Santambrogio (eds.), On Quine: New Essays, Cambridge UP, Cambridge, pp. 314–335.

  8. Church, A., A formulation of the logic of sense and denotation, in P. Henle (ed.), Structure, Method and Meaning: Essays in Honor of H.M. Sheffer, LiberalArts Press, NY, 1951.

  9. Church, A., Introduction to Mathematical Logic, Princeton UP, Princeton, 1956.

  10. Cooper, N. (1953) Does the logical truth that \({{\exists x (Fx \vee \neg Fx)}}\) entail that at least one individual exists?, Analysis 14:3–5.

  11. Corcoran J., Weaver G. (1969) Logical consequence in modal logic: Natural deduction in S5. Notre Dame Journal of Formal Logic 10: 370–384

    Article  Google Scholar 

  12. Curry H. (1963) Foundations of Mathematical Logic. McGraw-Hill, New York

    Google Scholar 

  13. Došen K. (1985) An intuitionistic Sheffer function. Notre Dame Journal of Formal Logic 26: 479–482

    Article  Google Scholar 

  14. Dummett, M. (1978) The philosophical basis of intuitionistic logic, in Truth and Other Enigmas, Duckworth, London, pp. 215–247.

  15. Dummett, M. (1993) Language and truth, in The Seas of Language, Clarendon, Oxford, pp. 117–165.

  16. Michael Dunn (1993) Star and perp. Philosophical Perspectives: Language and Logic 7: 331–358

    Article  Google Scholar 

  17. Fitch F. (1952) Symbolic Logic: An Introduction. Ronald Press, NY

    Google Scholar 

  18. Fitch F. (1966) Natural deduction rules for obligation. American Philosophical Quarterly 3: 27–28

    Google Scholar 

  19. Fitting, M. (1983) Proof Methods for Modal and Intuitionistic Logics, Reidel, Dordrecht.

  20. Garson J. (2006) Modal Logic for Philosophers. Cambridge Univ. Press, Cambridge

    Book  Google Scholar 

  21. Gentzen, G. (1934) Untersuchungen über das logische Schließen, I and II, Mathematische Zeitschrift 39:176–210, 405–431. English translation “Investigations into Logical Deduction”, published in American Philosophical Quarterly 1:288–306, 1964, and 2:204–218, 1965. Reprinted in M. E. Szabo (ed.) (1969) The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam, pp. 68–131. Page references to the APQ version.

  22. Gentzen, G. (1936) Die Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen 112:493–565. English translation “The Consistency of Elementary Number Theory” published in M. E. Szabo (ed.) (1969) The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam, pp. 132–213.

  23. Hailperin T. (1953) Quantification theory and empty individual domains. Journal of Symbolic Logic 18: 197–200

    Article  Google Scholar 

  24. Hailperin, T. (1957) A theory of restricted quantification, I and II, Journal of Symbolic Logic 22:19–35 and 113–129. Correction in Journal of Symbolic Logic 25:54– 56, (1960).

  25. Harman G. (1973) Thought. Princeton UP, Princeton

    Google Scholar 

  26. Hazen A. P. (1990) Actuality and quantification. Notre Dame Journal of Formal Logic 41: 498–508

    Article  Google Scholar 

  27. Hazen, A. P. (1992) Subminimal negation, Tech. rep., University of Melbourne. University of Melbourne Philosophy Department Preprint 1/92.

  28. Hazen A. P. (1995) Is even minimal negation constructive?. Analysis 55: 105–107

    Article  Google Scholar 

  29. Hazen A.P. (1999) Logic and analyticity. European Review of Philosophy 4:79–110 Special issue on “The Nature of Logic”, A. Varzi (ed.). This special issue is sometimes characterized as a separate book under that title and editor.

  30. Hendry, H., and G. Massey (1969) On the concepts of Sheffer functions, in K. Lambert (ed.), The Logical Way of Doing Things, Yale UP, New Haven, CT, pp. 279–293.

  31. Herbrand, J. (1928) Sur la théorie de la démonstration, Comptes rendus hebdomadaires des séances de l’Académie des Sciences (Paris) 186:1274–1276.

  32. Herbrand, J. (1930) Recherches sur la théorie de la démonstration, Ph.D. thesis, University of Paris. Reprinted in W. Goldfarb (ed. & trans.) (1971) Logical Writings, D. Reidel, Dordrecht.

  33. Hintikka J. (1959) Existential presuppositions and existential commitments. Journal of Philosophy 56: 125–137

    Article  Google Scholar 

  34. Hughes G., Cresswell M. (1968) An Introduction to Modal Logic. Methuen, London

    Google Scholar 

  35. Humberstone L. (2011) The Connectives. MIT Press, Cambridge, MA

    Google Scholar 

  36. Indrzejczak A. (2010) Natural Deduction, Hybrid Systems and Modal Logics. Springer, Berlin

    Book  Google Scholar 

  37. Jaśkowski, S. (1929) Teoria dedukcji oparta na regułach założeniowych (Theory of deduction based on suppositional rules), in Ksiȩga pamia̧tkowa pierwszego polskiego zjazdu matematycznego (Proceedings of the First Polish Mathematical Congress), 1927, Polish Mathematical Society, Kraków, p. 36.

  38. Jaśkowski, S. (1934) On the rules of suppositions in formal logic, Studia Logica1:5– 32. Reprinted in S. McCall (1967) Polish Logic 1920–1939 Oxford UP, pp. 232–258.

  39. Johansson, I. (1936) The minimal calculus, a reduced intuitionistic formalism, Compositio Mathematica 4:119–136. Original title Der Minimalkalkul, ein reduzierter intuitionistischer Formalismus.

  40. Kapp A. (1953) Does the logical truth that \({{\exists x (Fx \vee \neg Fx)}}\) entail that at least one individual exists?. Analysis 14: 2–3

    Article  Google Scholar 

  41. Kleene S. (1967) Elementary Logic. Wiley, NY

    Google Scholar 

  42. Kolmogorov, A. (1925) On the principle of the excluded middle, in J. van Heijenoort (ed.), From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879–1931, Harvard UP, Cambridge, MA, pp. 414–437. Originally published as “O principe tertium non datur”, Matematiceškij Sbornik 32:646–667.

  43. Kuznetsov A. (1965) Analogi ‘shtrikha sheffera’ v konstruktivnoĭ logike. Doklady Akademii Nauk SSSR 160: 274–277

    Google Scholar 

  44. Lambert K. (1963) Existential import revisited. Notre Dame Journal of Formal Logic 4: 288–292

    Article  Google Scholar 

  45. Leblanc H., Hailperin T. (1959) Nondesignating singular terms. Philosophical Review 68: 129–136

    Article  Google Scholar 

  46. Lemmon E. J. (1965) Beginning Logic. Nelson, London

    Google Scholar 

  47. Leonard H. S. (1957) The logic of existence. Philosophical Studies 7: 49–64

    Article  Google Scholar 

  48. Lewis D. (1973) Counterfactuals. Blackwell, Oxford

    Google Scholar 

  49. Mates B. (1965) Elementary Logic. Oxford UP, NY

    Google Scholar 

  50. Mostowski A. (1951) On the rules of proof in the pure functional calculus of the first order. Journal of Symbolic Logic 16: 107–111

    Article  Google Scholar 

  51. Pelletier F. J. (1999) A brief history of natural deduction. History and Philosophy of Logic 20: 1–31

    Article  Google Scholar 

  52. Pelletier, F. J. (2000) A history of natural deduction and elementary logic textbooks, in J. Woods, and B. Brown (eds.), Logical Consequence: Rival Approaches, Vol. 1, Hermes Science Pubs., Oxford, pp. 105–138.

  53. Pelletier, F. J., and A. P. Hazen (2012) A brief history of natural deduction, in D. Gabbay, F. J. Pelletier, and J. Woods (eds.), Handbook of the History of Logic; Vol. 11: A History of Logic’s Central Concepts, Elsevier, Amsterdam, pp. 341–414.

  54. Peregrin J. (2008) What is the logic of inference?. Studia Logica 88: 263–294

    Article  Google Scholar 

  55. Prawitz, D. (1965) Natural Deduction: A Proof-theoretical Study, Almqvist & Wicksell, Stockholm.

  56. Prawitz, D. (1979) Proofs and the meaning and completeness of the logical constants, in J. Hintikka, I. Niiniluoto, and E. Saarinen (eds.), Essays on Mathematical and Philosophical Logic, Reidel, Dordrecht, pp. 25–40.

  57. Price R. (1961) The stroke function and natural deduction. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 7: 117–123

    Article  Google Scholar 

  58. Quine W. V. (1950) Methods of Logic. Henry Holt & Co., New York

    Google Scholar 

  59. Quine, W. V. (1954) Quantification and the empty domain, Journal of Symbolic Logic19:177–179. Reprinted, with correction, in Quine (1995).

  60. Quine, W. V. (1995) Selected Logic Papers: Enlarged Edition, Harvard UP, Cambridge, MA.

  61. Read S. (2010) General-elimination harmony and the meaning of the logical constants. Journal of Philosophical Logic 39: 557–576

    Article  Google Scholar 

  62. Rescher N. (1959) On the logic of existence and denotation. Philosophical Review 69: 157–180

    Article  Google Scholar 

  63. Routley R. (1975) Review of Ohnishi & Matsumoto. Journal of Symbolic Logic 40: 466–467

    Article  Google Scholar 

  64. Russell B. (1906) The theory of implication. American Journal of Mathematics 28: 159–202

    Article  Google Scholar 

  65. Russell B. (1919) Introduction to Mathematical Philosophy. Allen and Unwin, London

    Google Scholar 

  66. Schroeder-Heister, P. (1984a) Generalized rules for quantifiers and the completeness of the intuitionistic operators \({\wedge, \vee, \supset, \perp, \forall, \exists}\), in M. Richter, E. Börger, W. Oberschelp, B. Schinzel, and W. Thomas (eds.), Computation and Proof Theory: Proceedings of the Logic Colloquium Held in Aachen, July 18–23, 1983, Part II, Springer- Verlag, Berlin, pp. 399–426. Volume 1104 of Lecture Notes in Mathematics.

  67. Schroeder-Heister P. (1984) A natural extension of natural deduction. Journal of Symbolic Logic 49: 1284–1300

    Article  Google Scholar 

  68. Schroeder-Heister, P. (2014) The calculus of higher-level rules and the foundational approach to proof-theoretic harmony, Studia Logica.

  69. Sheffer H. (1913) A set of five independent postulates for Boolean algebras, with application to logical constants. Trans. of the American Mathematical Society 14: 481–488

    Article  Google Scholar 

  70. Suppes P. (1957) Introduction to Logic. Van Nostrand/Reinhold Press, Princeton

    Google Scholar 

  71. Tarski, A. (1930) Über einige fundamentalen Begriffe der Metamathematik, Comptes rendus des séances de la Société des Sciences et Lettres de Varsovie (Classe III) 23:22– 29. English translation “On Some Fundamental Concepts of Metamathematics” in Tarski, 1956, pp. 30–37.

  72. Tarski A. (1956) Logic, Semantics, Metamathematics. Clarendon, Oxford

    Google Scholar 

  73. Thomason R. (1970) A Fitch-style formulation of conditional logic. Logique et Analyse 52: 397–412

    Google Scholar 

  74. Wu, K. Johnson (1979) Natural deduction for free logic, Logique et Analyse 88:435– 445.

  75. Zucker J., Tragesser R. (1978) The adequacy problem for inferential logic. Journal of Philosophical Logic 7: 501–516

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, University of Alberta, Edmonton, Canada

    Allen P. Hazen & Francis Jeffry Pelletier

Authors
  1. Allen P. Hazen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francis Jeffry Pelletier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Allen P. Hazen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hazen, A.P., Pelletier, F.J. Gentzen and Jaśkowski Natural Deduction: Fundamentally Similar but Importantly Different. Stud Logica 102, 1103–1142 (2014). https://doi.org/10.1007/s11225-014-9564-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-014-9564-1

Keywords

Search

Navigation