An Analysis of Gödel'sdialecticaInterpretation via Linear Logic

Abstract

This article presents an analysis of Godel's dialectica interpretation via a refinement of intuitionistic logic known as linear logic. Linear logic comes naturally into the picture once one observes that the structural rule of contraction is the main cause of the lack of symmetry in Godel's interpretation. We use the fact that the dialectica interpretation of intuitionistic logic can be viewed as a composition of Girard's embedding of intuitionistic logic into linear logic followed by de Paiva's dialectica interpretation of linear logic. We then investigate the various properties of the dialectica interpretation, such as the characterisation theorem, and variants of Godel's interpretation within the linear logic context. The role of contraction in extensions to classical logic, arithmetic and analysis is also discussed.