Intuitionistic phase semantics is almost classical

Authors:
Max I. Kanovich;Mitsuhiro Okada;Kazushige Terui
Affiliations:
Department of Computer Science, Queen Mary, University of London, Mile End Road, London E1 4NS and Department of Computer and Information Science, University of Pennsylvania Email: mik@dcs.qmul.ac ...;Department of Philosophy, Faculty of Letters, Keio University, 2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan Email: mitsu@abelard.flet.keio.ac.jp;National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan Email: terui@nii.ac.jp
Venue:
Mathematical Structures in Computer Science
Year:
2006

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Abstract

We study the relationship between classical phase semantics for classical linear logic (LL) and intuitionistic phase semantics for intuitionistic linear logic (ILL). We prove that (i) every intuitionistic phase space is a subspace of a classical phase space, and (ii) every intuitionistic phase space is phase isomorphic to an ‘almost classical’ phase space. Here, by an ‘almost classical’ phase space we mean an intuitionistic phase space having a double-negation-like closure operator. Based on these semantic considerations, we give a syntactic embedding of propositional ILL into LL.