A New Normalization Strategy for the Implicational Fragment of Classical Propositional Logic

Abstract

The introduction and elimination rules for material implication in natural deduction are not complete with respect to the implicational fragment of classical logic. A natural way to complete the system is through the addition of a new natural deduction rule corresponding to Peirce’s formula (((A → B) → A) → A). E. Zimmermann [6] has shown how to extend Prawitz’ normalization strategy to Peirce’s rule: applications of Peirce’s rule can be restricted to atomic conclusions. The aim of the present paper is to extend Seldin’s normalization strategy to Peirce’s rule by showing that every derivation Π in the implicational fragment can be transformed into a derivation Π′ such that no application of Peirce’s rule in Π′ occurs above applications of →-introduction and →-elimination. As a corollary of Seldin’s normalization strategy we obtain a form of Glivenko’s theorem for the classical {→}-fragment.