The naturality of natural deduction

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. We show that these result generalize some facts which have gone so far unnoticed, namely that the Russell-Prawitz translation maps particular classes of instances of the equations governing disjunction (and the other definable connectives) onto equations which are already included in the $\beta\eta$ equational theory of $NI^2$. Finally, we compare our approach with the one proposed by Ferreira and Ferreira and show that the naturality condition suggests a generalization of their methods to a wider class of formulas.