David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Journal of Symbolic Logic 72 (3):738 - 754 (2007)
Consider a general class of structural inference rules such as exchange, weakening, contraction and their generalizations. Among them, some are harmless but others do harm to cut elimination. Hence it is natural to ask under which condition cut elimination is preserved when a set of structural rules is added to a structure-free logic. The aim of this work is to give such a condition by using algebraic semantics. We consider full Lambek calculus (FL), i.e., intuitionistic logic without any structural rules, as our basic framework. Residuated lattices are the algebraic structures corresponding to FL. In this setting, we introduce a criterion, called the propagation property, that can be stated both in syntactic and algebraic terminologies. We then show that, for any set R of structural rules, the cut elimination theorem holds for FL enriched with R if and only if R satisfies the propagation property. As an application, we show that any set R of structural rules can be "completed" into another set R*, so that the cut elimination theorem holds for FL enriched with R*, while the provability remains the same
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Agata Ciabattoni, Nikolaos Galatos & Kazushige Terui (2012). Algebraic Proof Theory for Substructural Logics: Cut-Elimination and Completions. Annals of Pure and Applied Logic 163 (3):266-290.
Nikolaos Galatos & Hiroakira Ono (2010). Cut Elimination and Strong Separation for Substructural Logics: An Algebraic Approach. Annals of Pure and Applied Logic 161 (9):1097-1133.
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