Developing a suggestion by Russell, Prawitz showed how the usual natural deduction inference rules for disjunction, conjunction and absurdity can be derived using those for implication and the second order quantifier in propositional intuitionistic second order logic NI\(^2\). It is however well known that the translation does not preserve the relations of identity among derivations induced by the permutative conversions and immediate expansions for the definable connectives, at least when the equational theory of NI\(^2\) is assumed to consist only of \(\beta \)- and \(\eta \)-equations. On the basis of the categorial interpretation of NI\(^2\), we introduce a new class of equations expressing what in categorial terms is a naturality condition satisfied by the transformations interpreting NI\(^2\)-derivations. We show that the Russell–Prawitz translation does preserve identity of proof with respect to the enriched system by highlighting the fact that naturality corresponds to a generalized permutation principle. Finally we sketch how these results could be used to investigate the properties of connectives definable in the framework of higher-level rules.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Aczel, P., The Russell-Prawitz modality, Mathematical Structures in Computer Science 11: 541–554, 2001.
Altenkirch, T., P. Dybjer, M. Hoffman, and P. J. Scott, Normalization by evaluation for typed lambda calculus with coproducts, in 16th Annual IEEE Symposium on Logic in Computer Science, Boston, Massachussetts, 2001, pp. 303–310.
Bainbridge, E.S., P. J. Freyd, A. Scedrov, and P. J. Scott, Functorial polymorphism, Theoretical Computer Science 70: 35–64, 1990.
de Lataillade, J., Dinatural terms in System F, in 24th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, Los Angeles, California, USA, 2009, pp. 267–276.
Ferreira, F., and G. Ferreira, Commuting conversions vs. the standard conversions of the “good” connectives, Studia Logica 92(1): 63–84, 2009.
Ferreira, F., and G. Ferreira, Atomic polymorphism, Journal of Symbolic Logic 78(1): 260–274, 2013.
Freyd, P. J., J.-Y. Girard, A. Scedrov, and P. J. Scott, Semantic parametricity in the polymorphic lambda calculus, in LICS ’88., Proceedings of the Third Annual Symposium on Logic in Computer Science, IEEE, 1988, pp. 274–279.
Ghani, N., \(\beta \eta \)-equality for coproducts, in TLCA ’95, International Conference on Typed Lambda Calculi and Applications, vol. 902 of Lecture Notes in Computer Science, Springer-Verlag, 1995, pp. 171–185.
Girard, J.-Y., Y. Lafont, and P. Taylor, Proof and Types, Cambridge University Press, 1989.
Girard, J.-Y., A. Scedrov, and P. J. Scott, Normal forms and cut-free proofs as natural transformations, in Y. Moschovakis, (ed.), Logic from Computer Science, vol. 21 of Mathematical Sciences Research Institute Publications, Springer-Verlag, 1992, pp. 217–241.
Lindley, S., Extensional rewriting with sums, in Typed Lambda Calculi and Applications, TLCA 2007, vol. 4583 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2007, pp. 255–271.
Prawitz, D., Natural deduction, a proof-theoretical study, Almqvist & Wiskell, 1965.
Prawitz, D., Ideas and results in proof theory, in J.E. Fenstad, (ed.), Proceedings of the 2nd Scandinavian Logic Symposium (Oslo), Studies in logic and foundations of mathematics, vol. 63, North-Holland, 1971.
Prawitz, D., Proofs and the meaning and completeness of the logical constants, in J. Hintikka, I. Niiniluoto, and E. Saarinen, (eds.), Essays on Mathematical and Philosophical Logic: Proceedings of the Fourth Scandinavian Logic Symposium and the First Soviet-Finnish Logic Conference, Jyväskylä, Finland, June 29–July 6, 1976, Kluwer, Dordrecht, 1979, pp. 25–40.
Russell, B., The Principles of Mathematics, George Allen and Unwin Ltd. (2nd edition 1937), 1903.
Schroeder-Heister, P., A natural extension of natural deduction, Journal of Symbolic Logic 49(4): 1284–1300, 1984.
Schroeder-Heister, P., The calculus of higher-level rules, propositional quantification, and the foundational approach to proof-theoretic harmony, Studia Logica 102(6): 1185–1216, 2014.
Schwichtenberg, H., and A. S. Troelstra, Basic proof theory, Cambridge University Press, 2000.
Seely, R. A. G., Weak adjointness in proof theory, in Proceedings of the Durham Conference on Applications of Sheaves, vol. 753 of Springer Lecture Notes in Mathematics, Springer Berlin, 1979, pp. 697–701.
Tranchini, L., Proof-theoretic harmony: Towards an intensional account, Synthese, Online first (2016). https://doi.org/10.1007/s11229-016-1200-3.
About this article
Cite this article
Tranchini, L., Pistone, P. & Petrolo, M. The Naturality of Natural Deduction. Stud Logica 107, 195–231 (2019). https://doi.org/10.1007/s11225-017-9772-6
- Identity of proof
- Permutative conversions
- Dinaturality condition
- Functorial interpretation
- \(\eta \)-conversion
- Russell–Prawitz translation
- Second order logic