Reasoning about Uncertainty in Metric Spaces

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 structures. 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 spaces with uncertainty, a new metric is defined and shown to have good properties.