An Inequality Paradigm for Probabilistic Knowledge: The Logic of Conditional Probability Intervals

Abstract

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.