Proof systems combining classical and paraconsistent negations

Studia Logica 91 (2):217 - 238 (2009)
Abstract
New propositional and first-order paraconsistent logics (called L ω and FL ω , respectively) are introduced as Gentzen-type sequent calculi with classical and paraconsistent negations. The embedding theorems of L ω and FL ω into propositional (first-order, respectively) classical logic are shown, and the completeness theorems with respect to simple semantics for L ω and FL ω are proved. The cut-elimination theorems for L ω and FL ω are shown using both syntactical ways via the embedding theorems and semantical ways via the completeness theorems.
Keywords Paraconsistent negation  sequent calculus  cut-elimination  completeness
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    Ofer Arieli & Arnon Avron (1996). Reasoning with Logical Bilattices. Journal of Logic, Language and Information 5 (1):25--63.

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