Entailment in Probability of Thresholded Generalizations
  A nonmonotonic logic of thresholded generalizations is presented. Given
propositions A and B from a language L and a positive integer k, the
thresholded generalization A=>B{k} means that the conditional probability
P(B|A) falls short of one by no more than c*d^k. A two-level probability
structure is defined. At the lower level, a model is defined to be a
probability function on L. At the upper level, there is a probability
distribution over models. A definition is given of what it means for a
collection of thresholded generalizations to entail another thresholded
generalization. This nonmonotonic entailment relation, called "entailment in
probability", has the feature that its conclusions are "probabilistically
trustworthy" meaning that, given true premises, it is improbable that an
entailed conclusion would be false. A procedure is presented for ascertaining
whether any given collection of premises entails any given conclusion. It is
shown that entailment in probability is closely related to Goldszmidt and
Pearl's System-Z^+, thereby demonstrating that the conclusions of System-Z^+
are probabilistically trustworthy.
