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Gentzen's Proof Systems: Byproducts in a Work of Genius

Published online by Cambridge University Press:  15 January 2014

Jan von Plato
Jan von Plato*
Affiliation:
University of Helsinki, Finland, 00014 Helsinki, FinlandE-mail: [email protected]
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Abstract

Gentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of these calculi. In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934–35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG. After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 18 , Issue 3 , September 2012 , pp. 313 - 367
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

Ackermann, W. [1924], Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit, Mathematische Annalen, vol. 93, pp. 1–36.CrossRefGoogle Scholar
Andou, J. [1995], A normalization-procedure for the first order classical natural deduction with full logical symbols, Tsukuba Journal of Mathematics, vol. 19, pp. 153–162.CrossRefGoogle Scholar
Bernays, P. [1918], Beiträge zur axiomatischen Behandlung des Logik-Kalküls, Manuscript Hs. 973:193, Bernays collection, ETH-Zurich.Google Scholar
Bernays, P. [1926], Axiomatische Untersuchung des Aussagen-Kalkuls der “Principia mathematica, Mathematische Zeitschrift, vol. 25, pp. 305–320.CrossRefGoogle Scholar
Bernays, P. [1936], Logical calculus, Lectures at the Institute for Advanced Study, Princeton 1935–36.Google Scholar
Bernays, P. [1945], Review of Ketonen (1944), The Journal of Symbolic Logic, vol. 10, pp. 127–130.Google Scholar
Bernays, P. [1965], Betrachtungen zum Sequenzen-kalkul, Contributions to logic and methodology in honor of J. M. Bochenski, North-Holland, pp. 1–44.Google Scholar
Beth, E. [1955], Semantic entailment and formal derivability, Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Afd. Letterkunde, vol. 18, pp. 309–342.Google Scholar
Brouwer, L. [1927], Über Definitionsbereiche von Funktionen, Mathematische Annalen, vol. 97, pp. 60–75.Google Scholar
Brouwer, L. [1928], Intuitionistische Betrachtungen über den Formalismus, Sitzungsberichte der Preussischen Akademie der Wissenschaften, pp. 48–52.Google Scholar
van Dalen, D. [2005], Mystic, geometer, and intuitionist: The life of L. E. J. Brouwer, vol. II, Oxford University Press.Google Scholar
van Dalen, D. [2011], The selected correspondence of L. E. J. Brouwer, Springer, Berlin.CrossRefGoogle Scholar
Dyckhoff, R. [1988], Implementing asimpleproof assistant, Workshop on programming for logic teaching, Proceedings 23/1988, University of Leeds: Centre for Theoretical Computer Science, pp. 49–59.Google Scholar
Ernstein, A. [1905], Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in Flüssigkeiten suspendierte Teilchen, Annalen der Physik, vol. 17, pp. 549–560.Google Scholar
Ernstein, A. [1911], Bemerkungen zu den P. Hertzschen Arbeiten: mechanische Grundlagen der Thermodynamik, Annalen der Physik, vol. 34, pp. 175–176.Google Scholar
Gentzen, G. [1932], Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsysteme, Mathematische Annalen, vol. 107, pp. 329–350.CrossRefGoogle Scholar
Gentzen, G. [1933], Über das Verhältnis zwischen intuitionistischer und klassischer Arithmetik, submitted for publication March 15, 1933 but withdrawn, published in Archiv für mathematische Logik, vol. 16 (1974), pp. 119–132.CrossRefGoogle Scholar
Gentzen, G. [1934–1935], Untersuchungen Über das logische Schliessen, Mathematische Zeitschrift, vol. 39, pp. 176–210 and 405–431.Google Scholar
Gentzen, G. [1936], Die Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen, vol. 112, pp. 493–565.CrossRefGoogle Scholar
Gentzen, G. [1938], Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie, Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, vol. 4, pp. 19–44.Google Scholar
Gentzen, G. [1943], Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie, Mathematische Annalen, vol. 120, pp. 140–161.Google Scholar
Gentzen, G. [1964–1965], Investigations into logical deduction, American Philosophical Quarterly, vol. 1, pp. 288–306 and vol. 2, pp. 204–218.Google Scholar
Gentzen, G. [1969], The collected papers of Gerhard Gentzen, Szabo, M. (editor), North-Holland.Google Scholar
Gentzen, G. [2008], The normalization of derivations, this Bulletin, vol. 14, pp. 245–257.Google Scholar
Gödel, K. [1931], Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38, pp. 173–198, also in Gödel 1986.CrossRefGoogle Scholar
Gödel, K. [1933], Zur intuitionistischen Arithmetik und Zahlentheorie, as reprinted in Godel (1986), pp. 286–295.Google Scholar
Gödel, K. [1958], Über eine noch nicht benutzte Erweiterung des finiten Standpunktes, Dialectica, vol. 12, pp. 280–287, also in Gödel 1986.Google Scholar
Gödel, K. [1986], Collected works, vol. 1, Oxford University Press.Google Scholar
Hertz, P. [1923], Über Axiomensysteme für beliebige Satzsysteme. Teil II, Mathematische Annalen, vol. 89, pp. 76–102.Google Scholar
Hertz, P. [1929], Über Axiomensysteme für beliebige Satzsysteme, Mathematische Annalen, vol. 101, pp. 457–514.Google Scholar
Heyting, A. [1930], Die formalen Regeln der intuitionistischen Logik, Sitzungsberichte der Preussischen Akademie der Wissenschaften, pp. 42–56.Google Scholar
Heyting, A. [1931], Die intuitionistische Grundlegung der Mathematik, Erkenntnis, vol. 2, pp. 106–115.Google Scholar
Heyting, A. [1935], Intuitionistische Wiskunde, Mathematica B, vol. 4, pp. 72–83.Google Scholar
Heyting, A. [1956], Intuitionism: An introduction, North-Holland.Google Scholar
Hilbert, D. [1923], Die logischen Grundlagen der Mathematik, Mathematische Annalen, vol. 88, pp. 151–165.Google Scholar
Hilbert, D. [1928], Die Grundagen der Mathematik, Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität, vol. 6, pp. 65–85.Google Scholar
Hilbert, D. [1931], Die Grundlegung der elementaren Zahlenlehre, Mathematische Annalen, vol. 104, pp. 485–494.CrossRefGoogle Scholar
Hilbert, D. and Ackermann, W. [1928], Grundzüge der theoretischen Logik, Springer.Google Scholar
Hilbert, D. and Bernays, P. [1934, 1939], Grundlagen der Mathematik I-II, Springer.Google Scholar
Jaskowski, S. [1934], On the rules of supposition in formal logic, as reprinted in McCall, S. (editor), Polish Logic 1920–1939, pp. 232–258, Oxford University Press, 1967.Google Scholar
Johansson, I. [1936], Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus, Compositio Mathematica, vol. 4, pp. 119–136.Google Scholar
Kanckos, A. [2010], Consistency of Heyting arithmetic in natural deduction, Mathematical Logic Quarterly, vol. 56, pp. 611–624.Google Scholar
Ketonen, O. [1944], Untersuchungen zum Prädikatenkalkül, Annales Academiae Scientiarum Fennicae.Google Scholar
Kleene, S. [1952], Introduction to metamathematics, North-Holland.Google Scholar
Kolmogorov, A. [1925], On the principle of excluded middle, translation of Russian original in van Heijenoort (1967), pp. 416–437.Google Scholar
Lopez-Escobar, E. [1999], Standardizing the N systems of Gentzen, Models, algebras, and proofs (Caicedo, X. and Montenegro, C., editors), Dekker, pp. 411–434.Google Scholar
Menzler-Trott, E. [2007], Logic's lost genius: The life of Gerhard Gentzen, American Mathematical Society.Google Scholar
Negri, S. and von Plato, J. [2001], Structural proof theory, Cambridge University Press.Google Scholar
von Plato, J. [1994], Creating modern probability, Cambridge University Press.Google Scholar
von Plato, J. [2001a], A proof of Gentzen's Hauptsatz without multicut , Archive for Mathematical Logic, vol. 40, pp. 9–18.Google Scholar
von Plato, J. [2001b], Natural deduction with general elimination rules, Archive for Mathematical Logic, vol. 40, pp. 541–567.CrossRefGoogle Scholar
von Plato, J. [2007], In the shadows of the Löwenheim–Skolem theorem: early combinatorial analyses of mathematical proofs, this Bulletin, vol. 13, pp. 189–225.Google Scholar
von Plato, J. [2008], Gentzen's proof of normalization for natural deduction, this Bulletin, vol. 14, pp. 240–244.Google Scholar
von Plato, J. [2009], Gentzen's logic, Handbook of the history of logic, vol. 5, pp. 667–721.CrossRefGoogle Scholar
von Plato, J. [2011], A sequent calculus isomorphic to Gentzen's natural deduction, The Review of Symbolic Logic, vol. 4, pp. 43–53.CrossRefGoogle Scholar
von Plato, J. and Siders, A. [2012], Normal derivability in classical natural deduction, The Review of Symbolic Logic,in press.Google Scholar
Prawitz, D. [1965], Natural deduction: A proof-theoretical study, Almqvist & Wicksell.Google Scholar
Raggio, A. [1965], Gentzen's Hauptsatz for the systems NI and NK, Logique et Analyse, vol. 8, pp. 91–100.Google Scholar
Russell, B. [1906], The theory of implication, American Journal of Mathematics, vol. 28, pp. 159–202.Google Scholar
Schroeder-Heister, P. [1984], A natural extension of natural deduction, The Journal of Symbolic Logic, vol. 49, pp. 1284–1300.Google Scholar
Schroeder-Heister, P. [2002], Resolution and the origins of structural reasoning: early proof-theoretic ideas of Hertz and Gentzen, this Bulletin, vol. 8, pp. 246–265.Google Scholar
Siders, A. [2012a], A direct Gentzen-style consistency proof for Heyting arithmetic, The quest for consistency (Baaz, M., Kahle, R., and Rathjen, M., editors), to appear.Google Scholar
Siders, A. [2012b], Gentzen s consistency proof without heightlines, submitted for publication.Google Scholar
Skolem, T. [1919], Untersuchungen über die Axiome des Klassenkalküls und über Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen, as reprinted in Skolem 1970, pp. 67–101.Google Scholar
Skolem, T. [1970], Selected works in logic, Universitetsforlaget, Oslo, Fenstad, J. E. (editor).Google Scholar
Stålmarck, G. [1991], Normalization theorems for full first order classical natural deduction, The Journal of Symbolic Logic, vol. 56, pp. 129–149.Google Scholar
Statman, R. [1974], The structural complexity of proofs, Dissertation, Stanford University.Google Scholar
Tennant, N. [1992], Autologic, Edinburgh University Press.Google Scholar
Troelstra, A. [1990], On the early history of intuitionistic logic, Mathematical logic (Petkov, P., editor), Plenum Press, pp. 3–17.Google Scholar
van Heijenoort, J. (editor) [1967], From Frege to Godel, A source book in mathematical logic, 1879–1931, Harvard University Press.Google Scholar
Zach, R. [1999], Completeness before Post: Bernays, Hilbert, and the development of propositional logic, this Bulletin, vol. 3, pp. 331–366.Google Scholar