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This work is concerned with addressing an anomoly involving probability and logic. This includes the interpretation and evaluation of implicative statements in natural language, compatible with conditional probability. One of the chief motivations for investigating this problem has been the need to formalize rigorously the appropriate connections between conditional probabilities and the underlying production rules in expert sytems. This is accomplished through the development of a comprehensive theory of conditional events and an associated logic. The results of this effort should be of prime use in the design and evaluation of inference rules in expert systems, and also, allow for a new expansion of probability to include at the syntactic level the concept of conditioning. The monograph is intended for two audiences: AI researchers who are primarily interested in the management of uncertainty in expert systems, and mathematicians in the fields of probabilistic modeling, logic, and algebra.