Formulas-as-types for a Hierarchy of Sublogics of Intuitionistic Propositional Logic

This paper contains a detailed presentation of Howard's [1969] ‘forrrmlas-as-types notion of construction’ for fragments of various subsystems of intuitionistic propositional logic. Buszkowski's [1987, 1988] distinction between two kinds of lambda-abstractors is taken up and suitable fragments of typed terms are singled out (cf. also [van Benthem 1986], [Buszkowski 1987]). The relationship between cut-elimination and normalization of terms is dealt with. Amongst other things, it is shown that in certain cases in which applications of the (cut)-rule can be eliminated from an implicational fragment of the logics considered, cut-elimination and normalization of terms wrt β-reduction are homomorphic images of each other.