  We propose an inequality paradigm for probabilistic reasoning based on a
logic of upper and lower bounds on conditional probabilities. We investigate a
family of probabilistic logics, generalizing the work of Nilsson [14]. We
develop a variety of logical notions for probabilistic reasoning, including
soundness, completeness justification; and convergence: reduction of a theory
to a simpler logical class. We argue that a bound view is especially useful for
describing the semantics of probabilistic knowledge representation and for
describing intermediate states of probabilistic inference and updating. We show
that the Dempster-Shafer theory of evidence is formally identical to a special
case of our generalized probabilistic logic. Our paradigm thus incorporates
both Bayesian "rule-based" approaches and avowedly non-Bayesian "evidential"
approaches such as MYCIN and DempsterShafer. We suggest how to integrate the
two "schools", and explore some possibilities for novel synthesis of a variety
of ideas in probabilistic reasoning.
