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In a previous paper (of which this is a prosecution) we investigated the extraction of proof-theoretic properties of natural deduction derivations from their impredicative translation into System F. Our key idea was to introduce an extended equational theory for System F codifying at a syntactic level some properties found in parametric models of polymorphic type theory. A different approach to extract proof-theoretic properties of natural deduction derivations was proposed in a recent series of papers on the basis of an embedding of intuitionistic propositional logic into a predicative fragment of System F, called  atomic  System F. In this paper we show that this approach finds a general explanation within our equational study of second-order natural deduction, and a clear semantic justification in terms of parametricity.

The Naturality of Natural Deduction (II): On Atomic Polymorphism and Generalized Propositional Connectives