The naturality of natural deduction (II). Some remarks on atomic polymorphism
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In this paper (which is a prosecution of "The naturality of natural deduction", Studia Logica 2019) we investigate the exact relationship between the Russell-Prawitz translation of intuitionistic propositional logic into intuitionistc second-order propositional logic (System F), and its variant proposed by Fernando Ferreira and Gilda Ferreira into the atomic fragment of System F (System Fat). In the previous paper we investigated the Russell-Prawitz translation via an extended equational theory for System F arising from its categorical semantics. The main result of this paper is that the Russell-Prawitz translation and Ferreira and Ferreira's translation are equivalent modulo this extended equational theory. This result highlights a close connection between our previous work and that of Ferreira and Ferreira. We argue however that the approach obtained by coupling the original Russell-Prawitz translation with our extended equational theory is more satisfactory for the study of proof identity than the one based on System Fat.
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