Logic Journal of the IGPL 10 (6):653-678 (2002)

Abstract
We introduce an extended intuitionistic linear logic with strong negation and modality. The logic presented is a modal extension of Wansing's extended linear logic with strong negation. First, we propose three types of cut-free sequent calculi for this new logic. The first one is named a subformula calculus, which yields the subformula property. The second one is termed a dual calculus, which has positive and negative sequents. The third one is called a triple-context calculus, which is regarded as a natural extension or generalization of Hodas and Miller's dual-context calculus appearing in a linear logic programming language. Second, we present a concurrent process calculus based on the logic. This calculus is an extension of Okada's process calculus. Third, we introduce a Kripke type semantics for a fragment of the logic, and show the completeness theorems with respect to the semantics. Finally, we mention a logic programming language based on the triple-context calculus
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DOI 10.1093/jigpal/10.6.653
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Sequent Calculi for Some Trilattice Logics.Norihiro Kamide & Heinrich Wansing - 2009 - Review of Symbolic Logic 2 (2):374-395.
Gentzen-Type Methods for Bilattice Negation.Norihiro Kamide - 2005 - Studia Logica 80 (2-3):265-289.
A Relationship Between Rauszer's HB Logic and Nelson's Logic'.Norihiro Kamide - 2004 - Bulletin of the Section of Logic 33 (4):237-249.
Phase Semantics and Petri Net Interpretation for Resource-Sensitive Strong Negation.Norihiro Kamide - 2006 - Journal of Logic, Language and Information 15 (4):371-401.

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