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 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.

The Naturality of Natural Deduction (II): On Atomic Polymorphism and Generalized Propositional Connectives

“…As suggested in the previous paper (cf. [33] Section 4.3), the results we are concerned with are not limited to the standard intuitionistic connectives, but scale smoothly to a wider class of connectives investigated in proof-theoretic semantics (see e.g. [24,26]).…”

“…In this section we introduce a formal framework for natural deduction which extends the one from [33] to a more general class of propositional connectives.…”

“…In a previous paper, of which the present one is a follow-up, we explored a solution to the equivalence-preservation problem based on the fact that the RP-translation of conjunctions and disjunctions does yield categorical products/co-products in the class of parametric models of NI 2 [1,14,25]. With the goal of making this result accessible to the proof theory community at large, in [33] we provided a purely syntactic reconstruction of it: we introduced an equational theory extending the one arising from the usual βand η-conversions for NI 2 -derivations using a new class of conversions-that we called ε-conversions-expressing a naturality condition for NI 2 -derivations which is satisfied by all parametric models of NI 2 . We then showed that the RP-translation does preserve the full equivalence of NI-derivations as soon as NI 2 -derivations are considered under this stronger equivalence.…”

Polymorphism and the Obstinate Circularity of Second Order Logic: A Victims’ Tale

“…The naturality condition appears as a very perspicuous characterization of the uniformity of polymorphic programs, as the latter should correspond to functions over types "that behave in the same way for all types" [40]. Functorial interpretations of typed -calculi and natural deduction (see [2,21]) show that terms (i.e., proofs) correspond to natural transformations, i.e., to uniform families (see [50] for a functorial view of natural deduction within natural deduction itself). §7.…”

General Proof Theory: Introduction

“…He considers this question to be the most fundamental problem of general proof theory. Tranchini, Pistone and Petrolo [23] show that the Russell-Prawitz translation of firstinto second-order logic preserves identity of proof with respect to a certain enriched system. Piecha and Schroeder-Heister [15] show that intuitionistic propositional logic is incomplete with respect to standard notions of prooftheoretic validity.…”