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 β and η 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 βη-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.

An anti-realist theory of meaning suitable for both logical and proper axioms is investigated. As opposed to other anti-realist accounts, like DummettPrawitz verificationism, the standard framework of classical logic is not called into question. In particular, semantical features are not limited solely to inferential ones, but also computational aspects play an essential role in the process of determination of meaning. In order to deal with such computational aspects, a relaxation of syntax is shown to be necessary. This leads to a general kind of proof theory, where the objects of study are not typed objects like deductions, but rather untyped ones, in which formulas have been replaced by geometrical configurations.

A comparison is given between two conditions used to define logical constants: Belnap's uniqueness and Hacking's deducibility of identicals. It is shown that, in spite of some surface similarities, there is a deep difference between them. On the one hand, deducibility of identicals turns out to be a weaker and less demanding condition than uniqueness. On the other hand, deducibility of identicals is shown to be more faithful to the inferentialist perspective, permitting definition of genuinely proof-theoretical concepts. This kind of analysis is driven by exploiting the Curry-Howard correspondence. In particular, deducibility of identicals is shown to correspond to the computational property of eta expansion, which is essential in the characterization of propositional identity.1 Note also that Belnap requires the deducibility relation to satisfy a set of structural properties that contains the cut rule and the weakening rule(s); however, unlike Hacking, he does not explicitly ask for the admissibility of these rules (cf. Belnap, 1962, p. 132).3 The difference between uniqueness and strict uniqueness (Došen and Schroeder-Heister, 1985, p. 166) is not taken into account here: in a sequent calculus setting such a difference is difficult to identify in a very neat way. 4 Belnap's original definition of UNI is slightly different from the one we presented here. However, in presence of transitivity of the deducibility relation -i.e., in presence of the cut rule -our own definition entails Belnap's one. The latter will be discussed in more details in section 5.2.

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