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EARLY STRUCTURAL REASONING. GENTZEN 1932

Published online by Cambridge University Press:  18 August 2015

ENRICO MORICONI*
Affiliation:
Dipartimento di Filosofia
*
*DIPARTIMENTO DI FILOSOFIA UNIVERSITY OF PISA VIA P. PAOLI, 15 56127 PISA ITALIA E-mail: enrico.moriconi@unipi.it

Abstract

This paper is a study of the opening section of Gentzen’s first publication of 1932, Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen, a text which shows the relevance of Hertz’s work of the 1920’s for the young Gentzen. In fact, Gentzen borrowed from Hertz the analysis of the notion of consequence, which was given in terms of the rules of thinning (Verdünnung) and cut (Schnitt) on sequents (there called “sentences”(Sätze)). Moreover, following Hertz again, he also judged it necessary to justify the forms of inference of the system by providing a semantics for them, so that it became possible to make precise the informal notion of consequence, and to show that the inference rules adopted are correct and sufficient.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2015 

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References

BIBLIOGRAPHY

Arndt, M., & Tesconi, L. (2014). Principles of explicit composition. In Moriconi, E. and Tesconi, L., editors. Second Pisa Colloquium in Logic, Language and Epistemology. Pisa: Edizioni ETS, pp. 1967.Google Scholar
Bernays, P. (1965). Betrachtungen zum Sequenzen-Kalkul. In Tymieniecka, A.-T., editor. Contributions to Logic and Methodology in Honor of J.M. Bocheński. Amsterdam: North-Holland Publishing Company, pp. 144.Google Scholar
Franks, C. (2010). Cut as consequence. History and Philosophy of Logic, 31, 349379.Google Scholar
Franks, C. (2013). Logical completeness, form, and content: An archaeology. In Kennedy, J., editor. Interpreting Gödel: Critical Essays. Cambridge: Cambridge University Press, pp. 78106.Google Scholar
Gentzen, G. (1932). Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsysteme. Mathematische Annalen, 107, 329–50. English translation in Szabo (1969).CrossRefGoogle Scholar
Gentzen, G. (1934–1935). Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39, 176210, 405–431. English translation in Szabo (1969).Google Scholar
Gentzen, G. (1936). Die Widerspruchsfreiheit der reine Zahlentheorie. Mathematische Annalen, 112, 493565. English translation in Szabo (1969).CrossRefGoogle Scholar
Moriconi, E. (2014). On the source of the notion of semantic completeness. In Moriconi, E. and Tesconi, L., editors. II Pisa Colloquium on Logic, Language and Epistemology. Pisa: Edizioni ETS, pp. 213244.Google Scholar
Schroeder-Heister, P. (2002). Resolution and the origin of structural reasoning: Early proof-theoretic ideas of Hertz and Gentzen. The Bulletin of Symbolic Logic, 8, 246265.Google Scholar
Szabo, M., editor (1969). The Collected Papers of Gerhardt Gentzen. Amsterdam: North-Holland Publ. Co.Google Scholar
Tennant, N. (2015). On Gentzen’s structural completeness proof. In Wansing, H., editor. Dag Prawitz on Proofs and Meaning, Studia Logica series Outstanding Contributions to Logic, pp. 385414.Google Scholar
von Plato, J. (2009). Gentzen’s logic. In Gabbay, D. M. and Woods, J., editors. Handbook of the History of Logic, Vol. 5. Amsterdam: North-Holland, pp. 667721.Google Scholar
von Plato, J. (2012). Gentzen’s proof systems: Byproducts in a work of genius. The Bullettin of Symbolic Logic, 18(3), 313367.CrossRefGoogle Scholar