  Belief updating schemes in artificial intelligence may be viewed as three
dimensional languages, consisting of a syntax (e.g. probabilities or certainty
factors), a calculus (e.g. Bayesian or CF combination rules), and a semantics
(i.e. cognitive interpretations of competing formalisms). This paper studies
the rational scope of those languages on the syntax and calculus grounds. In
particular, the paper presents an endomorphism theorem which highlights the
limitations imposed by the conditional independence assumptions implicit in the
CF calculus. Implications of the theorem to the relationship between the CF and
the Bayesian languages and the Dempster-Shafer theory of evidence are
presented. The paper concludes with a discussion of some implications on
rule-based knowledge engineering in uncertain domains.
