Abstract
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.
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19 February 2022
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Acknowledgements
Many thanks to the referees for the many suggestions that substantially contributed to improve the paper. Many thanks also to Gilda Ferreira, Fernando Ferreira and José Espírito Santo for several discussions on atomic polymorphism.
Funding
Open Access funding enabled and organized by Projekt DEAL. Mattia Petrolo gratefully acknowledges the support of the São Paulo Research Foundation (FAPESP) through the Project Auxlio Pesquisa Jovem Pesquisador no.2016/25891-3 and the support of the French National Research Agency (ANR) through the Project ANR-20-CE27-0004. Luca Tranchini gratefully acknowledges the support of the Deutsche Forschungsgemeinschaft (DFG) through the project “Falsity and Refutation. On the negative side of logic” (DFG TR1112/4-1).
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Pistone, P., Tranchini, L. & Petrolo, M. The Naturality of Natural Deduction (II): On Atomic Polymorphism and Generalized Propositional Connectives. Stud Logica 110, 545–592 (2022). https://doi.org/10.1007/s11225-021-09964-z
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DOI: https://doi.org/10.1007/s11225-021-09964-z