Notions of sameness by default and their application to anaphora, vagueness, and uncertain reasoning

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.
Keywords Anaphora  Default Logic  Equality  Herbrand models  Indistinguishability  Rough Set Theory  Sorites  Vagueness
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DOI 10.1007/s10849-008-9057-6
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Semantic Considerations on Nonmonotonic Logic.Robert C. Moore - 1985 - Artificial Intelligence 25:75-94.
Sameness and Individuation.D. Gabbay & J. M. Moravcsik - 1973 - Journal of Philosophy 70 (16):513-526.
A Logic for Default Reasoning.Ray Reiter - 1980 - Artificial Intelligence 13:81-137.

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