Notre Dame Journal of Formal Logic

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

Alexander Bochman


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.

Article information

Notre Dame J. Formal Logic Volume 39, Number 1 (1998), 47-73.

First available in Project Euclid: 7 December 2002

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Zentralblatt MATH identifier

Primary: 03B50: Many-valued logic
Secondary: 68T27: Logic in artificial intelligence


Bochman, Alexander. Biconsequence Relations: A Four-Valued Formalism of Reasoning with Inconsistency and Incompleteness. Notre Dame J. Formal Logic 39 (1998), no. 1, 47--73. doi:10.1305/ndjfl/1039293020.

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