Abstract
We motivate and formalize the idea of sameness by default: two objects are considered the same if they cannot be proved to be different. This idea turns out to be useful for a number of widely different applications, including natural language processing, reasoning with incomplete information, and even philosophical paradoxes. We consider two formalizations of this notion, both of which are based on Reiter’s Default Logic. The first formalization is a new relation of indistinguishability that is introduced by default. We prove that the corresponding default theory has a unique extension, in which every two objects are indistinguishable if and only if their non-equality cannot be proved from the known facts. We show that the indistinguishability relation has some desirable properties: it is reflexive, symmetric, and, while not transitive, it has a transitive “flavor.” The second formalization is an extension (modification) of the ordinary language equality by a similar default: two objects are equal if and only if their non-equality cannot be proved from the known facts. It appears to be less elegant from a formal point of view. In particular, it gives rise to multiple extensions. However, this extended equality is better suited for most of the applications discussed in this paper.
Similar content being viewed by others
References
Bry F., Yahya A. (2000). Positive unit hyperresolution tableaux and their application to minimal model generation. Journal of Automated Reasoning 25: 35–82
Charniak E. (1988). Motivation analysis, abductive unification and nonmonotonic equality. Artificial Intelligence 34: 275–295
Cohen, A. (2005). Vagueness and indiscriminability by failure. Presented at the 33rd Annual Meeting of the Society for Exact Philosophy.
Cohen, A. (2006). Anaphora resolution and minimal models. In J. Bos, & A. Koller (Eds.), Proceedings of the 5th International Conference on Inference in Computational Semantics–ICOS-5, pp. 7–16.
Cohen, A., Kaminski, M., & Makowsky, J. A. (2005). Indistinguishability by default. In S. Artemov, H. Barringer, A. S. d’Avila Garcez, L. C. Lamb, & J. Woods (Eds.), We will show them: Essays in honour of Dov Gabbay (Vol. 1, pp. 415–428). London: College Publications.
Cohen, A., & Makowsky, J. A. (1993). Two approaches to nonmonotonic equality. Technical Report 9317, Technion—Israel Institute of Technology.
Davis M. (1980). The mathematics of non-monotonic reasoning. Artificial Intelligence 13: 73–80
Gabbay, D., & Moravcsik, J. M. E. (1973). Sameness and individuation. Journal of Philosophy, 70, 16. Reprinted in Pelletier, F. J. (Ed.), (1979). Mass terms: Some philosophical problems, Dordrecht: Reidel.
Guerreiro, R., & Casanova, M. (1990). An alternative semantics for default logic. Presented at the 3rd International Workshop on Nonmonotonic Reasoning.
Kaminski M. (1995). A comparative study of open default theories. Artificial Intelligence 77: 285–319
Kaminski M. (1997). A note on the stable model semantics for logic programs. Artificial Intelligence 96: 467–479
Kaminski M. (1999). Open default theories over closed domains. Logic Journal of the IGPL 7: 577–589
Kaminski M., Makowsky J., Tiomkin M. (1998). Extensions for open default theories via the domain closure assumption. Journal of Logic and Computation 8: 169–187
Kaminski M.,Rey G. (2000). First-order non-monotonic modal logics. Fundamenta Informaticae 42: 303–333
Kryszkiewicz M. (1999). Rules in incomplete information systems. Information Sciences 113: 271–292
Lifschitz V. (1990). On open defaults. In: Lloyd J. (eds) Computational Logic: Symposium Proceedings. Springer-Verlag, Berlin, pp 80–95
Lloyd J. (1993). Foundation of logic programming, second extended edition. Springer-Verlag, Berlin
Marek W., Truszczyński M. (1993). Nonmonotonic logic. Springer-Verlag, Berlin
McCarthy, J. (1977). Epistemological problems of artificial intelligence. In Proceedings of the 5th International Joint Conference on Artificial Intelligence (pp. 1038–1044). Los Angeles, CA: Kaufmann.
Mendelson E. (1997). Introduction to mathematical logic. Chapman and Hall, London
Moore R. (1985). Semantical considerations on non–monotonic logics. Artificial Intelligence 25: 75–94
Pawlak Z. (1982). Rough sets. International Journal of Computer and Information Sciences 11: 341–356
Poole D. (1988). A logical framework for default reasoning. Artificial Intelligence 36: 27–47
Reiter R. (1980a). Equality and domain closure in first order databases. Journal of the ACM 27: 235–249
Reiter R. (1980b). A logic for default reasoning. Artificial Intelligence 13: 81–132
Williams E. (1997). Blocking and anaphora. Linguistic Inquiry 28: 577–628
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cohen, A., Kaminski, M. & Makowsky, J.A. Notions of Sameness by Default and their Application to Anaphora, Vagueness, and Uncertain Reasoning. J of Log Lang and Inf 17, 285–306 (2008). https://doi.org/10.1007/s10849-008-9057-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10849-008-9057-6