Any logical system that models human reasoning needs to address the possibility of error and the subsequent belief change once this error is recognized.  The need to deal with error-prone reasoning has only been widely acknowledged in the last thirty years or so; witnesses the popularity of the AGM postulates for Belief Revision.  Despite the variety of syntactical and semantical offerings, all seem to agree that we need to model a concept of minimal change by choosing the most similar epistemic state to the present one.  The favorite choice mechanisms are preferential orderings and, their generalization, distance maps.  Preferential orderings provide satisfactory representation results but fail to model iteration. Distance maps model iteration but fail to provide satisfactory completeness results. In this talk, I will introduce a third semantical approach using geodesic distance (length of shortest path on a graph) that lies between the two and combines their best features: geodesic semantics provide satisfactory completeness results like preferential orderings do and deal with iteration, as distance maps do. Further, and perhaps more important, geodesic semantics offer a novel, more natural representation of similarity using distinguishability.