## Notre Dame Journal of Formal Logic

### On Partial and Paraconsistent Logics

Reinhard Muskens

#### Abstract

In this paper we consider the theory of predicate logics in which the principle of bivalence or the principle of noncontradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove model existence. For , the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalize to partial and paraconsistent logics once the right setup is chosen. Our logic has a semantics that also underlies Belnap's logic and is related to the logic of bilattices. is in focus most of the time, but it is also shown how results obtained for can be transferred to several variants.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 40, Number 3 (1999), 352-374.

Dates
First available in Project Euclid: 28 May 2002

https://projecteuclid.org/euclid.ndjfl/1022615616

Digital Object Identifier
doi:10.1305/ndjfl/1022615616

Mathematical Reviews number (MathSciNet)
MR1845625

Zentralblatt MATH identifier
1007.03029

Subjects
Primary: 03B53: Paraconsistent logics
Secondary: 03B60: Other nonclassical logic

#### Citation

Muskens, Reinhard. On Partial and Paraconsistent Logics. Notre Dame J. Formal Logic 40 (1999), no. 3, 352--374. doi:10.1305/ndjfl/1022615616. https://projecteuclid.org/euclid.ndjfl/1022615616

#### References

• [1] Arieli, O., and A. Avron, Reasoning with logical bilattices,'' The Journal of Logic, Language, and Information, vol. 5 (1996), pp. 25--63.
• [2] Baaz, M., C. G. Fermüller, and R. Zach, Elimination of cuts in first-order finite-valued logics,'' The Journal of Information Processing and Cybernetics, vol. 29 (1994), pp. 333--55.
• [3] Barwise, J., and J. Perry, Situations and Attitudes, The MIT Press, Cambridge, 1983.
• [4] Belnap, N. D., Jr., A useful four-valued logic,'' pp. 8--37 in Modern Uses of Multiple-Valued Logic, edited by J. M. Dunn and G. Epstein, Reidel, Dordrecht, 1977.
• [5] Blamey, S., Partial logic,'' pp. 1--70 in Handbook of Philosophical Logic, vol. 3, edited by D. M. Gabbay and F. Guenthner, Reidel, Dordrecht, 1986.
• [6] Bochman, A., A logical foundation for logic programming I: biconsequence relations and nonmonotonic completion,'' The Journal of Logic Programming, vol. 35 (1998), pp. 151--70.
• [7] Carnielli, W. A., Systematization of finite many-valued logics through the method of tableaux,'' The Journal of Symbolic Logic, vol. 52 (1987), pp. 473--93.
• [8] Carnielli, W. A., On sequents and tableaux for many-valued logics,'' The Journal of Non-Classical Logic, vol. 8 (1991), pp. 59--78.
• [9] Cleave, J. P., The notion of logical consequence in the logic of inexact predicates,'' Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 20 (1974), pp. 307--24.
• [10] D'Agostino, M., Investigations into the complexity of some propositional calculi,'' Technical Report, Oxford University Computing Laboratory, 1990.
• [11] Dunn, J. M., Intuitive semantics for first-degree entailments and coupled trees','' Philosophical Studies, vol. 29 (1976), pp. 149--68.
• [12] Feferman, S., Toward useful type free theories I,'' The Journal of Symbolic Logic, vol. 49 (1984), pp. 75--111.
• [13] Fitting, M, Bilattices and the semantics of logic programming,'' The Journal of Logic Programming, vol. 11 (1991), pp. 91--116.
• [14] Fitting, M., First-Order Logic and Automated Theorem Proving, Springer, New York, 1996.
• [15] Gilmore, P. C., The consistency of partial set theory without extensionality,'' pp. 147--53 in Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics 13, Part II, American Mathematical Society, Providence, 1974.
• [16] Ginzburg, M. L., Multivalued logics: a uniform approach to reasoning in A,'' Computer Intelligence, vol. 4 (1988), pp. 256--316.
• [17] Hähnle, R., Automated Deduction in Multiple-valued Logics, Oxford University Press, Oxford, 1993.
• [18] Holden, M., Weak logic theory,'' Theoretical Computer Science, vol. 79 (1991), pp. 295--321.
• [19] Jaspars, J., Calculi for Constructive Communication, Ph.D thesis, Tilburg University, Tilburg, 1994.
• [20] Kleene, S. C., Introduction to Metamathematics, Van Nostrand, Amsterdam, 1952.
• [21] Langholm, T., Partiality, Truth and Persistence, CSLI Lecture Notes, Stanford, 1988.
• [22] Langholm, T., How different is partial logic?,'' pp. 3--43 in Partiality, Modality, and Nonmonotonicity, edited by P. Doherty, CSLI, Stanford, 1996.
• [23] Muskens, R. A., Going partial in Montague grammar,'' pp. 175--220 in Semantics and Contextual Expression, Proceedings of the Sixth Amsterdam Colloquium, edited by R. Bartsch, J. van Benthem, and P. van Emde Boas, Foris, Dordrecht, 1989.
• [24] Muskens, R. A., Meaning and Partiality, CSLI, Stanford, 1995.
• [25] Rousseau, G., Sequents in many-valued logic I,'' Fundamenta Mathematicæ, vol. 60 (1967), pp. 23--33.
• [26] Schröter, K., Methoden zur Axiomatisierung beliebiger Aussagen- und Prädikatenkalküle,'' Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 241--51.
• [27] Takeuti, G., Proof Theory, North-Holland, Amsterdam, 1975.
• [28] Thijsse, E., Partial Logic and Knowledge Representation, Ph.D thesis, Tilburg University, Tilburg, 1992.
• [29] Visser, A., Four-valued semantics and the Liar,'' The Journal of Philosophical Logic, vol. 13 (1984), pp. 181--212.
• [30] Woodruff, P. W., Truth and logic, part I: paradox and truth,'' The Journal of Philosophical Logic, vol. 13 (1984), pp. 213--32.
• [31] Zach, R., `Proof theory of finite-valued logics.'' Technical Report TUW-E185.2-Z.1-93, Institut für Computersprachen, Technische Universität Wien, Vienna, 1993.