On the structure of metric spaces related to pre-rough logic

The consideration of approximate reasoning merging rough approximation and formal deduction in pre-rough logic leads to three different types of metric spaces, which we call pre-rough logic metric space, pre-rough upper logic metric space and pre-rough lower logic metric space, respectively. In this paper, we investigate the structure of these three metric spaces mainly from the perspective of topology. It is shown that the three metric spaces have no isolated points. In addition, the continuity of logic connectives w.r.t. rough upper pseudo-metric and rough lower pseudo-metric is examined and the robust analysis of rough logic is also studied. Lastly, the notion of rough divergency degree of any logic theory is proposed and its topological characterization is presented.