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A SEQUENT CALCULUS ISOMORPHIC TO GENTZEN’S NATURAL DEDUCTION

  • JAN VON PLATO (a1)

Abstract

Gentzen’s thesis proved the equivalence of natural deduction, sequent calculus, and axiomatic logic through a cycle of translations. Mysteriously, even normal derivations in natural deduction got translated into sequent derivations with cuts. It is shown that the insertion of special cuts, whenever left conjunction, left implication, or left universal quantification in sequent calculus is used, results in sequent calculus derivations isomorphic to those in Gentzen’s natural deduction. Thereby the appearance of the cuts in translation is explained.

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*DEPARTMENT OF PHILOSOPHY, UNIVERSITY OF HELSINKI, 00014 U. OF HELSINKI, FINLAND. E-mail:jan.vonplato@helsinki.fi

References

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Gentzen, G. (1934–1935). Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39, 176210 and 405–431.
Gentzen, G. (1938). Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie. Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, 4, 1944.
Gentzen, G. (1969). In Szabo, M., editor. The Collected Papers of Gerhard Gentzen. North-Holland.
Gentzen, G. (2008). The normalization of derivations. The Bulletin of Symbolic Logic, 14, 245257.
Menzler-Trott, E. (2007). Logic’s Lost Genius: The Life of Gerhard Gentzen. Providence, RI: AMS.
Negri, S., & von Plato, J. (2001). Structural Proof Theory. Cambridge, UK.
von Plato, J. (2001a). Natural deduction with general elimination rules. Archive for Mathematical Logic, 40, 541567.
von Plato, J. (2001b). A proof of Gentzen’s Hauptsatz without multicut. Archive for Mathematical Logic, 40, 918.
von Plato, J. (2003). Translations from natural deduction to sequent calculus. Mathematical Logic Quarterly, 49, 435443.
von Plato, J. (2008). Gentzen’s proof of normalization for intuitionistic natural deduction. The Bulletin of Symbolic Logic, 14, 240244.
von Plato, J. (2009). Gentzen’s logic. In Gabbay, D., and Woods, J., editors. Handbook of the History of Logic, Vol. 5. pp. 667721.
Pottinger, G. (1977). Normalization as a homomorphic image of cut-elimination. Annals of Mathematical Logic, 12, 323357.
Prawitz, D. (1965). Natural Deduction: A Proof-Theoretical Study. Stockholm, Sweden: Almqvist & Wicksell.
Troelstra, A., & Schwichtenberg, H. (2000). Basic Proof Theory (second edition). Cambridge, UK.
Zucker, J. (1974). Cut-elimination and normalization. Annals of Mathematical Logic, 7, 1112.
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A SEQUENT CALCULUS ISOMORPHIC TO GENTZEN’S NATURAL DEDUCTION

  • JAN VON PLATO (a1)

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