We attempt to elucidate the conditions required on Girard\u27s candidates of reducibility (in French, candidats de reductibilité ) in order to establish certain properties of various typed lambda calculi, such as strong normalization and Church-Rosser property. We present two generalizations of the candidates of reducibility, an untyped version in the line of Tait and Mitchell, and a typed version which is an adaptation of Girard\u27s original method. As an application of this general result, we give two proofs of strong normalization for the second-order polymorphic lambda calculus under βη-reduction (and thus under β-reduction). We present two sets of conditions for the typed version of the candidates. The first set consists of conditions similar to those used by Stenlund (basically the typed version of Tait\u27s conditions), and the second set consists of Girard\u27s original conditions. We also compare these conditions, and prove that Girard\u27s conditions are stronger than Tait\u27s conditions. We give a new proof of the Church-Rosser theorem for both β-reduction and βη-reduction, using the modified version of Girard\u27s method. We also compare various proofs that have appeared in the literature (see section 11). We conclude by sketching the extension of the above results to Girard\u27s higher-order polymorphic calculus Fω, and in appendix 1, to Fω with product types
