Restricted Rules of Inference and Paraconsistency
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We present here two logical systems - intuitionistic paraconsistent weak Kleene logic (IPWK) and paraconsistent pre-rough logic (PPRL) as examples of logics with some rules of inference that have variable sharing restrictions imposed on them. These systems have the same set of theorems as intuitionistic propositional logic and pre-rough logic, respectively, but are paraconsistent while the original systems are not. We discuss algebraic semantics for these logics. A contaminating element, intended to denote a state of indeterminacy, is used to extend each Heyting algebra and each pre-rough algebra. The classes of these extended Heyting algebras and the extended pre-rough algebras form models of IPWK and PPRL, respectively. We then prove the soundness and completeness results for these systems.
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