Parametricity and Variants of Girard's J Operator

The Girard-Reynolds polymorphic -calculus is generally regarded as a calculus of parametric polymorphism in which all well-formed terms are strongly normalizing with respect to -reductions. Girard demonstrated that the additional of a simple "non-parametric" operation, J , to the calculus allows the de nition of a non-normalizing term. Since the type of J is not inhabited by any closed term, one might suspect that this may play a role in de ning a non-normalizing term using it. We demonstrate that this is not the case by giving a simple variant, J , of J whose type is otherwise inhabited and which causes normalization to fail. It appears that impredicativity is essential to the argument; predicative variants of the polymorphic -calculus admit non-parametric operations without sacri cing normalization.