Entailment with Near Surety of Scaled Assertions of High Conditional Probability
 Donald Bamber
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Abstract
An assertion of high conditional probability or, more briefly, an HCP assertion is a statement of the type: The conditional probability of B given A is close to one. The goal of this paper is to construct logics of HCP assertions whose conclusions are highly likely to be correct rather than certain to be correct. Such logics would allow useful conclusions to be drawn when the premises are not strong enough to allow conclusions to be reached with certainty. This goal is achieved by taking Adams" (1966) logic, changing its intended application from conditionals to HCP assertions, and then weakening its criterion for entailment. According to the weakened entailment criterion, called the Criterion of Near Surety and which may be loosely interpreted as a Bayesian criterion, a conclusion is entailed if and only if nearly every model of the premises is a model of the conclusion. The resulting logic, called NSL, is nonmonotonic. Entailment in this logic, although not as strict as entailment in Adams" logic, is more strict than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be scaled; this means that to each HCP assertion was associated a bound on the deviation from 1 of the conditional probability that is the subject of the assertion. Scaling of HCP assertions is useful for breaking entailment deadlocks. For example, it it is known that the conditional probabilities of C given A and of ¬ C given B are both close to one but the bound on the former"s deviation from 1 is much smaller than the latter"s, then it may be concluded that in all likelihood the conditional probability of C given A ∧ B is close to one. The resulting logic, called NSLS, is also nonmonotonic. Despite great differences in their definitions of entailment, entailment in NSL is equivalent to Lehmann and Magidor"s rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSLS is equivalent to entailment in Goldszmidt and Pearl"s SystemZ ^{+}. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their randomstructures method, which assumed a prior distribution similar to that of NSL, it appears possible to define an entailment relation equivalent to that of NSL. In their randomworlds method, which assumed a prior distribution dramatically different from that of NSL, it is known that the entailment relation is different from that of NSL.
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 Title
 Entailment with Near Surety of Scaled Assertions of High Conditional Probability
 Journal

Journal of Philosophical Logic
Volume 29, Issue 1 , pp 174
 Cover Date
 20000201
 DOI
 10.1023/A:1004650520609
 Print ISSN
 00223611
 Online ISSN
 15730433
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 nonmonotonic logic
 nonmonotonic reasoning
 entailment
 conditional probability
 secondorder probability
 Bayesian inference
 conditionals
 rulebased systems
 exceptions
 Authors

 Donald Bamber ^{(1)}
 Author Affiliations

 1. Space & Naval Warfare Systems Center, Code D44215, 53345 Ryne Road, San Diego, CA, 921527251, USA