Published online by Cambridge University Press: 12 March 2014
The notion of strong reduction is introduced in Curry and Feys' book Combinatory logic  as an analogue, in the theory of combinatore, to reduction (more exactly, βη-reduction) in the theory of λ-conversion. The existence of an analogue and its possible importance are suggested by an equivalence between the theory of combinatore and λ-conversion, and the Church-Rosser theorem in λ-conversion. This theorem implies that if a formula X is convertible to a formula X* which cannot be further reduced—is irreducible, or in normal form—then X is convertible to X* by a reduction alone. Moreover, the reduction may be performed in a certain prescribed order.
Presented, in a somewhat different version, to the Association for Symbolic Logic on December 27, 1963. This work was supported in part by The Research Foundation of State University of New York on fellowship FRF 64–40–30.