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Therefore, it is natural to think that it could have many implications for artificial intelligence and computer science as well. Specifically, argumentation may be considered a most primitive capability for interaction among computational agents.\u003cBR\u003e In this paper we present an argumentation framework based on the four-valued paraconsistent logic. Tolerance and acceptance of inconsistency that this logic has as its logical feature allow for arguments on inconsistent knowledge bases with which we are often confronted. We introduce various concepts for argumentation, such as arguments, attack relations, argument justification, preferential criteria of arguments based on social norms, and so on, in a way proper to the four-valued paraconsistent logic. Then, we provide the fixpoint semantics and dialectical proof theory for our argumentation framework. 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