Gentzen's “Untersuchungen” [1] gave a translation from natural deduction to sequent calculus with the property that normal derivations may translate into derivations with cuts. Prawitz in [8] gave a translation that instead produced cut-free derivations. It is shown that by writing all elimination rules in the manner of disjunction elimination, with an arbitrary consequence, an isomorphic translation between normal derivations and cut-free derivations is achieved. The standard elimination rules do not permit a full normal form, which explains the cuts in (...) Gentzen's translation. Likewise, it is shown that Prawitz' translation contains an implicit process of cut elimination. (shrink)

The structure of derivations in natural deduction is analyzed through isomorphism with a suitable sequent calculus, with twelve hidden convertibilities revealed in usual natural deduction. A general formulation of conjunction and implication elimination rules is given, analogous to disjunction elimination. Normalization through permutative conversions now applies in all cases. Derivations in normal form have all major premisses of elimination rules as assumptions. Conversion in any order terminates.Through the condition that in a cut-free derivation of the sequent Γ⇒C, no inactive weakening (...) or contraction formulas remain in Γ, a correspondence with the formal derivability relation of natural deduction is obtained: All formulas of Γ become open assumptions in natural deduction, through an inductively defined translation. Weakenings are interpreted as vacuous discharges, and contractions as multiple discharges. In the other direction, non-normal derivations translate into derivations with cuts having the cut formula principal either in both premisses or in the right premiss only. (shrink)

Gentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents have (...) at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933. (shrink)

Gentzens natural deduction. Thereby the appearance of the cuts in translation is explained.

The framework of natural deduction is extended by permitting rules as assumptions which may be discharged in the course of a derivation. this leads to the concept of rules of higher levels and to a general schema for introduction and elimination rules for arbitrary n-ary sentential operators. with respect to this schema, (functional) completeness "or", "if..then" and absurdity is proved.

Gentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents have (...) at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933. (shrink)

A new set of conversions for derivations in the system of sequents for intuitionistic predicate logic will be defined. These conversions will be some modifications of Zucker's conversions from the system of sequents from [11], which will have the following characteristics: (1) these conversions will be sufficient for transforming a derivation into a cut-free one, and (2) in the natural deduction the image of each of these conversions will be either in the set of conversions for normalization procedure, or an (...) identity of derivations. This will be used to give a new proof of the normalization theorem for natural deduction, as a consequence of the cut-elimination theorem for the system of sequents. (shrink)