this paper but which have some intriguing connections to some of our results and techniques, are [32] and [20]. We believe that the concept of prelogical relation would have a beneficial impact on the presentation and understanding of their results

. The idea of parametric polymorphism is that of a single operator that can be used for different data types and whose behaviour is somehow uniform for each type. Reynolds [Reynolds, 1983] uses binary relations to define 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 quantification over type variables appears. A simple categorical calculation indicates what is the appropriate way to obtain the right adjoint to weakening that models universal quantification. The resul...

We develop a notion of equivalence between interpretations of the simply typed -calculus together with an equationally dened abstract data-type, and we show that two interpretations are equivalent if and only if they are linked by a logical relation. We show that our construction generalises from the simply typed -calculus to include the linear -calculus and calculi with additional type and term constructors, such as those given by sum types or by a strong monad for modelling phenomena such as partiality or nondeterminism. This is all done in terms of category theoretic structure, using - brations to model logical relations following Hermida, and adapting Jung and Tiuryn's logical relations of varying arity to provide the completeness results, which form the heart of the work.