On Plotkin-Abadi Logic for Parametric Polymorphism Towards a Categorical Understanding

The idea of parametric polymorphism is that of a single operator that can be used for di erent data types and whose behaviour is somehow uniform for each type. Reynolds [Reynolds, 1983] uses binary relations to de ne a uniformity condition for parametric polymorphism. In [Plotkin & Abadi, 1993] the authors proposed a second order logic for second order lambda-calculus; this logic is able to handle parametric polymorphism in the binary relational sense of Reynolds. In this paper we examine a categorical framework for this logic. This framework is based on the notion of categorical model of second order lambda-calculus as given, for example, in [Pitts, 1987, Seely, 1987, Robinson, 1992, Jacobs, 1991]. Going through the categorical constructions of the model, an unexpected property of quanti cation over type variables appears. A simple categorical calculation indicates what is the appropriate way to obtain the right adjoint to weakening that models universal quanti cation. The result coincides with a construction given by Plotkin and Abadi to de ne the parametricity axiom schema.