Ein Syntaktischer Beweis für die Zulässigkeit der Schnittregel im Kalkül von Schütte für die Intuitionistische Typenlogik

Abstract

In [1] Girard gives a procedure, by which all derivations in the “calculus of natural deductions” of Prawitz [4] are transformed into normal derivations. Exploiting his idea we give a syntactical proof of the admissibility of the cut rule in Schütte's formal system of intuitionistic type theory. We obtain a normal form theorem but not a normalization theorem. Our “Berechnungsprädikate” are different from the “candidats de reductibilite” of Girard. In the case of second order logic “Berechnungsprädikate für Terme t^(O)” are not defined for derivations ending with r^O e t^(O) which are normalizable, but for finite formula sets Γ with the property, that Γ→r^O e t^(O) is derivable without cut.