Skip to main content
SpringerLink
Log in

A logic for reasoning with inconsistency

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

Most known computational approaches to reasoning have problems when facing inconsistency, so they assume that a given logical system is consistent. Unfortunately, the latter is difficult to verify and very often is not true. It may happen that addition of data to a large system makes it inconsistent, and hence destroys the vast amount of meaningful information. We present a logic, called APC (annotated predicate calculus; cf. annotated logic programs of [4, 5]), that treats any set of clauses, either consistent or not, in a uniform way. In this logic, consequences of a contradiction are not nearly as damaging as in the standard predicate calculus, and meaningful information can still be extracted from an inconsistent set of formulae. APC has a resolution-based sound and complete proof procedure. We also introduce a novel notion of ‘epistemic entailment’ and show its importance for investigating inconsistency in predicate calculus as well as its application to nonmonotonic reasoning. Most importantly, our claim that a logical theory is an adequate model of human perception of inconsistency, is actually backed by rigorous arguments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Apt, K. R., Blair, H., and Walker, A., ‘Towards a theory of declarative knowledge’, in Foundations of Deductive Databases and Logic Programming, J. Minker (ed.), Morgan-Kaufmann, pp. 89–148 (1988).

  2. Belnap, N. D., ‘A useful four-valued logic’, in Modern Uses of Multiple-Valued Logic, J. M. Dunn and G. Epstein (eds.), D. Reidel Publishing Co. (1975).

  3. Belnap, N. D., ‘How a computer should think’, in Contemporary Aspects of Philosophy, G. Ryle (ed.), Oriel Press, pp. 30–56 (1976).

  4. Blair, H. A. and Subrahmanian, V. S., ‘Paraconsistent logic programming’, Conf. on Foundations of Software Technology and Theoretical Computer Science (LNCS 287), pp. 340–360 (1987).

  5. Blair, H. A. and Subrahmanian, V. S., ‘Paraconsistent foundations for logic programming’, Journal of Non Classical Logic, to appear (1989).

  6. Chang, C. L. and Lee, R. C. T., Symbolic Logic and Mechanical Theorem Proving, Academic Press (1973).

  7. Chen, W., Kifer., M., and Warren, D. S., ‘Hilog: A first-order semantics for higher-order logic programming constructs’, Proceedings of North American Conference on Logic Programming, Cleveland, Ohio (1989).

  8. da Costa, N. C. A., ‘On the theory of inconsistent formal systems’, Notre Dame J. of Formal Logic 15, pp. 497–510 (1974).

    Google Scholar 

  9. Enderton, H. B., A Mathematical Introduction to Logic, Academic Press (1972).

  10. Epstein, G., ‘An equational axiomatization for the disjoint system of post algebras’, IEEE Trans. Comp. (1973).

  11. Fagin, R. and Halpern, J. Y., ‘Belief, awareness, and limited reasoning’, Artificial Intelligence 34, pp. 39–76 (1988).

    Google Scholar 

  12. Fitting, M., ‘First-order modal tableaux’, Journal of Automated Reasoning (1988).

  13. Fitting, M., ‘Destructive modal resolution’, manuscript (1988).

  14. Fitting, M., ‘Negation as refutation’, Fourth Intl. Symposium on Logic in Computer Science, pp. 63–70 (1989).

  15. Fitting, M., ‘Bilattices and the semantics of logic programming’, Journal of Logic Programming to appear (1991).

  16. Geissler, C. and Konolige, K., ‘A resolution method for quantified modal logics of knowledge and belief, in Proceedings of Theoretical Aspects of Reasoning about Knowledge, J. Y. Halpern (ed.), pp. 309–324 (1986).

  17. Ginsberg, M. L., ‘Multivalued logics’, in Readings in Non-Montonic Reasoning, M. L. Ginsberg (ed.), pp. 251–255 (1987).

  18. Ginsberg, M. L., ‘Multivalued logics: A uniform approach to reasoning in artificial intelligence’, Computational Intelligence 4, pp. 265–316 (1988).

    Google Scholar 

  19. Haack, S., in Philosophy of Logic, Cambridge University Press, Cambridge, U.K. (1978).

    Google Scholar 

  20. Kifer, M. and Li, A., ‘On the semantics of rule-based expert systems with uncertainty’, in 2nd Int. Conf. on Database Theory (LNCS 326), M. Gyssens, J. Paradaens and D. V. Gucht (ed.), Springer Verlag, Bruges, Belgium, pp. 102–117 (1988).

    Google Scholar 

  21. Kifer, M. and Subrahmanian, V. S., ‘On the expressive power of annotated logic programs’, North American Conference of Logic Programming, Cleveland, Ohio (1989).

  22. Kifer, M. and Li, A., ‘A theory of rule-based expert systems’, in preparation, 1990.

  23. Levesque, H. J., ‘A logic of implicit and explicit belief’, AAAI-84, pp. 198–202 (1984).

  24. Lifschitz, V., ‘On the declarative semantics of logic programs with negation’, in Foundations of Deductive Databases and Logic Programming, J. Minker (ed.), Morgan-Kaufmann, Los Altos, CA, pp. 177–192 (1988).

    Google Scholar 

  25. Maier, D. and Warren, D. S., Computing with Logic, Benjamin Cummings, Menlo Park, CA (1988).

    Google Scholar 

  26. McCarthy, J., ‘Applications of circumscription to formalizing common-sense knowledge’, Artificial Intelligence 28, pp. 89–116 (1986).

    Google Scholar 

  27. Minker, J. (ed.), Foundations of Deductive Databases and Logic Programming, Morgan-Kaufmann, Los Altos, CA (1988).

    Google Scholar 

  28. Moore, R. C., ‘A formal theory of knowledge and action’, in Formal Theories of the Common-Sence World, J. R. Hobbs and R. C. Moore (ed.), pp. 319–358 (1985).

  29. Perlis, D., ‘Circumscripton as introspection’, Univ. of Maryland Tech. Report (1987).

  30. Priest, G., ‘Reasoning about truth’, Artificial Intelligence 39 pp. 231–244 (1989).

    Google Scholar 

  31. Przymusinski, T. C., ‘On the declarative semantics of deductive databases and logic programs’, in Foundations of Deductive Databases and Logic Programming, J. Minker (ed.), Morgan-Kaufmann, Los Altos, CA, pp. 193–216 (1988).

    Google Scholar 

  32. Rasiowa, H. and Epstein, G., ‘Approximate reasoning and Scott's information systems’, in Proc. of the 2nd Int. Symposium on Methodologies for Intelligence Systems, Z. W. Ras and M. Zemankova (ed.), North-Holland, pp. 33–42 (1987).

  33. Reiter, R., ‘A logic for default reasoning’, Artificial Intelligence 13, pp. 81–132. (1980).

    Google Scholar 

  34. Rescher, N. and Brandom, R., The Logic of Inconsistency, Billing and Sons, Ltd., Oxford, U.K. (1980).

    Google Scholar 

  35. Rine, D. C., ‘Some relationships between logic programming and multiple-valued logic’, Symp. on Multiple-Valued Logic, pp. 160–163 (1986).

  36. Sandewall, E., ‘A functional approach to non-monotonic logic’, IJCAI-85, pp. 100–106 (1985).

  37. Subrahmanian, V. S., ‘On the semantics of quantitative logic programs’, IEEE Symposium on Logic Programming, pp. 173–182 (1987).

  38. Subrahmanian, V. S., ‘Mechanical proof procedures for many-valued lattice-based logic programming’, Contemporary Mathematics, to appear (1989).

  39. Touretzky, D. S., Horty, J. F., Thomason, R. H., ‘A clash of intuitions: the current state of nonmonotonic multiple inheritance systems’, IJCAI-87, pp. 476–482 (1987).

  40. van Emden, M. H. ’Quantitative deduction and its fixpoint theory’, The Journal of Logic Programming, pp. 37–53 (1986).

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, SUNY at Stony Brook, 11794, Stony Brook, NY, U.S.A.

    Michael Kifer

  2. Department of Computer Science, The Hebrew University, 91904, Jerusalem, Israel

    Eliezer L. Lozinskii

Authors
  1. Michael Kifer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eliezer L. Lozinskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

A preliminary report on this research appeared in LICS'89.

Work of M. Kifer was supported in part by the NSF grants DCR-8603676, IRI-8903507.

Work of E. L. Lozinskii was supported in part by Israel National Council for Research and Development under the grants 2454-3-87, 2545-2-87, 2545-3-89 and by Israel Academy of Science, grant 224-88.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kifer, M., Lozinskii, E.L. A logic for reasoning with inconsistency. J Autom Reasoning 9, 179–215 (1992). https://doi.org/10.1007/BF00245460

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00245460

Key words

Search

Navigation