We investigate the problem of reasoning with
imprecise quantitative information.  We give formal semantics to a
notion of approximate observations, and define two types of entailment
for a knowledge base with imprecise information: a cautious notion,
which allows only completely justified conclusions, and a bold one,
which allows jumping to conclusions.  Both versions of the entailment
relation are shown to be decidable. We investigate the behavior of the
two alternatives on various examples, and show that the answers
obtained are intuitively desirable. The behavior of these two
entailment relations is completely characterized for a certain
sublanguage, in terms of the logic of true equality.  We demonstrate
various properties of the full logic, and show how it applies to many
situations of interest.<p>