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Robust reasoning with rules that have exceptions: From second-order probability to argumentation via upper envelopes of probability and possibility plus directed graphs

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Rules having rare exceptions may be interpreted as assertions of high conditional probability. In other words, a rule If X then Y may be interpreted as meaning that Pr(Y∣X) 1. A general approach to reasoning with such rules, based on second-order probability, is advocated. Within this general approach, different reasoning methods are needed, with the selection of a specific method being dependent upon what knowledge is available about the relative sizes, across rules, of upper bounds on each rule's exception probabilities Pr(¬Y∣X). A method of reasoning, entailment with universal near surety, is formulated for the case when no information is available concerning the relative sizes of upper bounds on exception probabilities. Any conclusion attained under these conditions is robust in the sense that it will still be attained if information about the relative sizes of exception probability bounds becomes available. It is shown that reasoning via entailment with universal near surety is equivalent to reasoning in a particular type of argumentation system having the property that when two subsets of the rule base conflict with each other, the effectively more specific subset overrides the other. As stepping stones toward attaining this argumentation result, theorems are proved characterizing entailment with universal near surety in terms of upper envelopes of probability measures, upper envelopes of possibility measures, and directed graphs. In addition, various attributes of entailment with universal near surety, including property inheritance, are examined.

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Authors and Affiliations

  1. SPAWARSYSCEN 246205 (PL-BK), 53345 Engineer Street, San Diego, CA, 92152-7260, USA

    Donald Bamber & I. R. Goodman

  2. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003, USA

    Hung T. Nguyen

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  1. Donald Bamber
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  2. I. R. Goodman
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  3. Hung T. Nguyen
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Correspondence to Donald Bamber.

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Bamber, D., Goodman, I.R. & Nguyen, H.T. Robust reasoning with rules that have exceptions: From second-order probability to argumentation via upper envelopes of probability and possibility plus directed graphs. Ann Math Artif Intell 45, 83–171 (2005). https://doi.org/10.1007/s10472-005-9008-8

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