## Notre Dame Journal of Formal Logic

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

Alexander Bochman

#### 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.

#### Article information

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

Dates
First available: 7 December 2002

http://projecteuclid.org/euclid.ndjfl/1039293020

Mathematical Reviews number (MathSciNet)
MR1671730

Digital Object Identifier
doi:10.1305/ndjfl/1039293020

Zentralblatt MATH identifier
0967.03019

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

#### Citation

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

#### References

• [1] Arieli, O., and A. Avron, Reasoning with logical bilattices,'' Journal of Logic, Language, and Information, vol. 5 (1996), pp. 25--63.
• [2] Avron, A., Natural 3-valued logics: Characterization and proof theory,'' The Journal of Symbolic Logic, vol. 56, (1991), pp. 276--94.
• [3] Belnap, N. D., A useful four-valued logic,'' pp. 8--41 in Modern Uses of Multiple-Valued Logic, edited by M. Dunn and G. Epstein, D. Reidel, Dordrecht, 1977.
• [4] Blamey, S., Partial logic,'' pp. 1--70 in Handbook of Philosophical Logic, vol. 3, edited by D. M. Gabbay and F. Guenthner, D. Reidel, Dordrecht, 1986.
• [5] Bochman, A., Mereological semantics, Ph.D. thesis, Tel-Aviv University, Tel-Aviv, 1992.
• [6] Bochman, A., Biconsequence relations for nonmonotonic reasoning,'' pp. 482--92 in Proceedings of the Fifth International Conference on Principles of Knowledge Representation and Reasoning, edited by L. C. Aiello, J. Doyle, and S. C. Shapiro, M. Kaufmann, San Mateo, 1996.
• [7] Bochman, A., On a logical basis of general logic programs,'' pp 37--56 in Proceedings of the Sixth International Workshop on Nonmonotonic Extensions of Logic Programming, Lecture Notes in Artificial Intelligence,'' edited by J. G. Carbonell, J. Siekmann, Springer-Verlag, Berlin, 1996.
• [8] Bochman, A., A logical foundation for logic programming, I and II,'' The Journal of Logic Programming, vol. 35, (1998), pp. 151--94. % I % I % II % II
• [9] Carnielli, W. A., On sequents and tableaux for many-valued logics,'' Journal of Applied Non-Classical Logics, vol. 8 (1991), pp. 59--76.
• [10] Dunn, J. M., Intuitive semantics for first-degree entailment and coupled trees,'' Philosophical Studies, vol. 29, (1976), pp. 149--68.
• [11] Fitting, M. C., Bilattices and the theory of truth,'' The Journal of Philosophical Logic, vol. 18, (1989), pp. 225--56.
• [12] Fitting, M. C., Bilattices and the semantics of logic programming,'' The Journal of Logic Programming, vol. 11, (1991), pp. 91--116.
• [13] Fitting, M. C., Kleene's three-valued logics and their children,'' Fundamenta Informaticae, vol. 20, (1994), pp. 113--31.
• [14] Gabbay, D. M., Semantical Investigations in Heyting's Intuitionistic Logic, D. Reidel, Dordrecht, 1981.
• [15] Ginsberg, M. L., Multivalued logics: A uniform approach to reasoning in AI,'' Computational Intelligence, vol. 4, (1988), pp. 256--316.
• [16] Jaspars, J. O. M., Partial up and down logic,'' Notre Dame Journal of Formal Logic, vol. 36, (1995), pp. 134--57.
• [17] Rasiowa, H., An Algebraic Approach to Non-Classical Logics, North-Holland, Amsterdam, 1974.
• [18] Rousseau, G., Sequents in many-valued logics I,'' Fundamenta Mathematicae, vol. 60, (1967), pp. 23--33.
• [19] Schröter, K., Methoden zur axiomatisierung beliebiger aussagen- und pradikaten- kalkule,'' Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 1, (1955), pp. 241--51.
• [20] Scott, D., Advice on modal logic,'' pp. 152--88 in Philosophical Problems in Logic. Some Recent Developments, edited by K. Lambert, D. Reidel, Dordrecht, 1970.
• [21] Segerberg, K., Classical Propositional Operators, Clarendon Press, Oxford, 1982.
• [22] Takahashi, M., Many-valued logics of extended Gentzen style II,'' The Journal of Symbolic Logic, vol. 35, (1970), pp. 493--528.
• [23] van Benthem, J., A Manual of Intensional Logic, 2d ed., CSLI, Stanford, 1988.
• [24] Zach, R., Proof theory of finite-valued logics., Ph.D. thesis, Technishe Universität Wien, Vienna, 1993.