  We set up a model for reasoning about metric spaces with belief theoretic
measures. The uncertainty in these spaces stems from both probability and
metric. To represent both aspect of uncertainty, we choose an expected distance
function as a measure of uncertainty. A formal logical system is constructed
for the reasoning about expected distance. Soundness and completeness are shown
for this logic. For reasoning on product metric space with uncertainty, a new
metric is defined and shown to have good properties.
