Entailment with near surety of scaled assertions of high conditional probability

An assertion of high conditional probability or, more briefly, an HCP asser tion is a statement of the type: The conditional probability of B given A i s close to one. The goal of this papers is to construct logics of HCP asser tions 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 i ts intended application from conditionals to HCP assertions, and then weake ning its criterion for entailment. According to the weakened entailment cri terion, called the Criterion of Near Surety and which may be loosely interp reted as a Bayesian criterion, a conclusion is entailed it and only if near ly 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 struct than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be sclaed; this means that to each HCP a ssertion 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, if it is known t hat the conditional probabilities of C given A and of -C given B are both c lose 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 cond itional probability of C given A /\ B is close to one. The resulting logic, called NSL-S, is also nonmonotonic. Despite great difference in their defi nitions of entailment, entailment in NSl is equivalent to Lehmann and Magid or's rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSL-S is equivalent t o entailment in Goldszmidt and Pearl's System-Z+. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their random-structures method, which ass umed a price distribution similar to that of NSL, it appears possible to de fine an entailment relation equivalent to that of NSL. In their random-worl ds method, which assumed a prior distribution dramatically different from t hat of NSL, it is known that the entailment relation is different from that of NSL.