The Naturality of Natural Deduction

Abstract

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.