Notre Dame Journal of Formal Logic

On Partial and Paraconsistent Logics

Reinhard Muskens


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 $ \bf L_4$, 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 $ \bf L_4$ has a semantics that also underlies Belnap's logic and is related to the logic of bilattices. $ \bf L_4$ is in focus most of the time, but it is also shown how results obtained for $ \bf L_4$ can be transferred to several variants.

Article information

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

First available in Project Euclid: 28 May 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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  • Arieli, O., and A. Avron, “Reasoning with logical bilattices,” The Journal of Logic, Language, and Information, vol. 5 (1996), pp. 25–63. Zbl 0851.03017 MR 97c:03094
  • 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. Zbl 0821.03013
  • Barwise, J., and J. Perry, Situations and Attitudes, The MIT Press, Cambridge, 1983. Zbl 0946.03007 MR 2001h:03051
  • 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. Zbl 0424.03012 MR 58:5021
  • 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. Zbl 0875.03023
  • 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. Zbl 0905.68032 MR 2000e:68028
  • 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. Zbl 0633.03008 MR 88g:03034
  • Carnielli, W. A., “On sequents and tableaux for many-valued logics,” The Journal of Non-Classical Logic, vol. 8 (1991), pp. 59–78. Zbl 0774.03006 MR 94b:03046
  • 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. Zbl 0299.02015 MR 51:10028
  • D'Agostino, M., “Investigations into the complexity of some propositional calculi,” Technical Report, Oxford University Computing Laboratory, 1990.
  • Dunn, J. M., “Intuitive semantics for first-degree entailments and `coupled trees',” Philosophical Studies, vol. 29 (1976), pp. 149–68. MR 58:10311
  • Feferman, S., “Toward useful type free theories I,” The Journal of Symbolic Logic, vol. 49 (1984), pp. 75–111. Zbl 0574.03043 MR 85i:03068
  • Fitting, M, “Bilattices and the semantics of logic programming,” The Journal of Logic Programming, vol. 11 (1991), pp. 91–116. Zbl 0757.68028 MR 92c:68119
  • Fitting, M., First-Order Logic and Automated Theorem Proving, Springer, New York, 1996. Zbl 0848.68101 MR 97b:68197
  • 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. Zbl 0309.02065 MR 50:12721
  • Ginzburg, M. L., “Multivalued logics: a uniform approach to reasoning in A,” Computer Intelligence, vol. 4 (1988), pp. 256–316.
  • Hähnle, R., Automated Deduction in Multiple-valued Logics, Oxford University Press, Oxford, 1993. Zbl 0798.03010 MR 96a:68107
  • Holden, M., “Weak logic theory,” Theoretical Computer Science, vol. 79 (1991), pp. 295–321. Zbl 0724.03020 MR 92f:68111
  • Jaspars, J., Calculi for Constructive Communication, Ph.D thesis, Tilburg University, Tilburg, 1994.
  • Kleene, S. C., Introduction to Metamathematics, Van Nostrand, Amsterdam, 1952. Zbl 0047.00703 MR 14:525m
  • Langholm, T., Partiality, Truth and Persistence, CSLI Lecture Notes, Stanford, 1988. Zbl 0665.03024
  • Langholm, T., “How different is partial logic?,” pp. 3–43 in Partiality, Modality, and Nonmonotonicity, edited by P. Doherty, CSLI, Stanford, 1996. Zbl 0904.03012 MR 1427283
  • 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. Zbl 0773.03020
  • Muskens, R. A., Meaning and Partiality, CSLI, Stanford, 1995.
  • Rousseau, G., “Sequents in many-valued logic I,” Fundamenta Mathematicæ, vol. 60 (1967), pp. 23–33. Zbl 0154.25504 MR 35:1451
  • Schr öter, K., “Methoden zur Axiomatisierung beliebiger Aussagen- und Prädikatenkalküle,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 241–51. Zbl 0066.25604 MR 17:1038g
  • Takeuti, G., Proof Theory, North-Holland, Amsterdam, 1975. Zbl 0681.03039 MR 58:27366b
  • Thijsse, E., Partial Logic and Knowledge Representation, Ph.D thesis, Tilburg University, Tilburg, 1992.
  • Visser, A., “Four-valued semantics and the Liar,” The Journal of Philosophical Logic, vol. 13 (1984), pp. 181–212. Zbl 0546.03007 MR 86f:03042
  • Woodruff, P. W., “Truth and logic, part I: paradox and truth,” The Journal of Philosophical Logic, vol. 13 (1984), pp. 213–32.
  • Zach, R., “Proof theory of finite-valued logics.” Technical Report TUW-E185.2-Z.1-93, Institut für Computersprachen, Technische Universität Wien, Vienna, 1993.