Biconsequence Relations: A Four-Valued Formalism of Reasoning with Inconsistency and Incompleteness

Abstract
We suggest a general formalism of four-valued reasoning, called biconsequence relations, intended to serve as a logical framework for reasoning with incomplete and inconsistent data. The formalism is based on a four-valued semantics suggested by Belnap. As for the classical sequent calculus, any four-valued connective can be defined in biconsequence relations using suitable introduction and elimination rules. In addition, various three-valued and partial logics are shown to be special cases of this formalism obtained by imposing appropriate additional logical rules. We show also that such rules are instances of a single logical principle called coherence. The latter can be considered a general requirement securing that the information we can infer in this framework will be classically coherent
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