Bulletin of Symbolic Logic

Gentzen's proof systems: byproducts in a work of genius

Jan von Plato

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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 A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934—35.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.

Article information

Bull. Symbolic Logic, Volume 18, Issue 3 (2012), 313-367.

First available in Project Euclid: 13 August 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


von Plato, Jan. Gentzen's proof systems: byproducts in a work of genius. Bull. Symbolic Logic 18 (2012), no. 3, 313--367. doi:10.2178/bsl/1344861886. https://projecteuclid.org/euclid.bsl/1344861886

Export citation