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.
Similar content being viewed by others
References
Gordeev Lev: ‘On cut elimination in the presence of peirce rule’. Archive for Mathematical Logic 26(1), 147–164 (1987)
Pereira, Luiz Carlos, Edward Hermann Haeusler, and Maria da Paz Medeiros, ‘Alguns resultados sobre fragmentos da lógica proposicional clássica’, O que nos faz pensar, 23 (2008). To appear.
Prawitz, Dag, Natural Deduction, A proof-theoretical study, Almqvist & Wiksell, 1965.
Seldin Jonathan P.: ‘On the proof-theory of the intermediate logic MH’. Journal of Symbolic Logic 51(3), 626–647 (1986)
Seldin Jonathan P.: ‘Normalization and excluded middle I’. Studia Logica 48(2), 193–217 (1989)
Zimmermann Ernst: ‘Peirce’s rule in natural deduction’. Theoretical Computer Science 275(1-2), 561–574 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pereira, L.C., Haeusler, E.H., Costa, V.G. et al. A New Normalization Strategy for the Implicational Fragment of Classical Propositional Logic. Stud Logica 96, 95–108 (2010). https://doi.org/10.1007/s11225-010-9275-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-010-9275-1