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Expanding the Realm of Systematic Proof Theory

Abstract : This paper is part of a general project of developing a sys- tematic and algebraic proof theory for nonclassical logics. Generaliz- ing our previous work on intuitionistic-substructural axioms and single- conclusion (hyper)sequent calculi, we define a hierarchy on Hilbert ax- ioms in the language of classical linear logic without exponentials. We then give a systematic procedure to transform axioms up to the level P3 of the hierarchy into inference rules in multiple-conclusion (hy- per)sequent calculi, which enjoy cut-elimination under a certain con- dition. This allows a systematic treatment of logics which could not be dealt with in the previous approach. Our method also works as a heuristic principle for finding appropriate rules for axioms located at levels higher than P3 . The case study of Abelian and Lukasiewicz logic is outlined.
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Agata Ciabattoni, Lutz Straßburger, Kazushige Terui. Expanding the Realm of Systematic Proof Theory. Computer Science Logic, CSL'09, 2009, Coimbra, Portugal. ⟨10.1007/978-3-642-04027-6_14⟩. ⟨inria-00436420⟩

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