## Notre Dame Journal of Formal Logic

### Semantics-based Nonmonotonic Inference

Heinrich Wansing

#### Abstract

In this paper we discuss Gabbay's idea of basing nonmonotonic deduction on semantic consequence in intuitionistic logic extended by a consistency operator and Turner's suggestion of replacing the intuitionistic base system by Kleene's three-valued logic. It is shown that a certain counterintuitive feature of these approaches can be avoided by using Nelson's constructive logic N instead of intuitionistic logic or Kleene's system. Moreover, in N a more general notion of consistency can be defined and nonmonotonic deduction can thus be based on a logical system satisfying the Deduction Theorem.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 36, Number 1 (1995), 44-54.

Dates
First available in Project Euclid: 19 December 2002

https://projecteuclid.org/euclid.ndjfl/1040308828

Digital Object Identifier
doi:10.1305/ndjfl/1040308828

Mathematical Reviews number (MathSciNet)
MR1359107

Zentralblatt MATH identifier
0839.03012

#### Citation

Wansing, Heinrich. Semantics-based Nonmonotonic Inference. Notre Dame J. Formal Logic 36 (1995), no. 1, 44--54. doi:10.1305/ndjfl/1040308828. https://projecteuclid.org/euclid.ndjfl/1040308828

#### References

• [1] Almukdad, A., and D. Nelson, Constructible falsity and inexact predicates," Journal of Symbolic Logic, vol. 49 (1984), pp. 231--233.
• [2] Clarke, M., Intuitionistic Non-Monotonic Reasoning---further results," pp. 525--527 in ECAI 88. Proceedings of the 8th European Conference on Artificial Intelligence, edited by Y. Kodratoff, Pitman, London, 1988.
• [3] Clarke, M., and D. Gabbay, An Intuitionistic Basis for Non-Monotonic Reasoning," pp. 163--178 in Non-Standard Logics for Automated Reasoning, edited by P. Smets et al., Academic Press, London, 1988.
• [4] Došen, K., Intuitionistic Double Negation as a Necessity Operator," Publications de L'Institute Mathematique, vol. 49 (1984), pp. 15--20.
• [5] Gabbay, D., Intuitionistic Basis for Non-Monotonic Logic," pp. 260--273 in Proceedings of the 6th Conference on Automated Deduction. Lecture Notes in CS 138, Springer-Verlag, Berlin, 1982.
• [6] Gurevich, Y., Intuitionistic Logic with Strong Negation," Studia Logica, vol. 36 (1977), pp. 49--59.
• [7] Jaspars, J., Calculi for Constructive Communication, PhD Thesis, University of Tilburg, 1994.
• [8] Kripke, S., Semantical analysis of intuitionistic logic I," pp. 92--129 in Formal Systems and Recursive Functions, edited by J. Crossley and M. Dummett, North-Holland, Amsterdam, 1965.
• [9] \mboxŁukaszewicz, W., Non-Monotonic Reasoning. Formalization of Commonsense Reasoning, Ellis Horwood, Chichester, 1990.
• [10] McDermott D., and J. Doyle, Non-Monotonic Logic I," Journal of Artificial Intelligence, vol. 13 (1980), pp. 41--72.
• [11] Pearce, D., Answer Sets and Constructive Logic, II: Extended Logic Programs and Related Non-monotonic Formalisms," pp. 457--475 in Logic Programming and Non-Monotonic Reasoning, edited by L. Pereira and A. Nerode, MIT Press, Cambridge (Massachusetts), 1993.
• [12] Pearce D., and G. Wagner, Logic Programming with Strong Negation," pp. 311--326 in Proceedings of the Workshop on Extensions of Logic Programming. Lecture Notes in AI 475, edited by P. Schroeder-Heister, Springer-Verlag, Berlin, 1990.
• [13] Turner, R., Logics for Artificial Intelligence, Ellis Horwood, Chichester, 1984.
• [14] Wagner G., Logic Programming with Strong Negation and Inexact Predicates," Journal of Logic and Computation, vol. 1 (1991), pp. 835--859.
• [15] Wagner G., Vivid Logic. Knowledge-Based Reasoning with Two Kinds of Negation. Lecture Notes in AI 764, Springer-Verlag, Berlin, 1994.
• [16] Wansing, H., The Logic of Information Structures. Lecture Notes in AI 681, Springer-Verlag, Berlin, 1993.