Spatial Vagueness and Second-Order Vagueness

This paper presents a geometric characterization of geographic objects with vague boundaries. Since the framework is based on ordering geometry and does not rely on numerical concepts, it can be applied to qualitative spatial reasoning. The characterization describes vague regions and gradual transitions between two vague regions. This approach takes into account that the region of all points that definitely belong to an object is also vague, called higher-order vagueness. Higher-order vagueness is usually neglected by theories modeling spatial vagueness. The approach distinguishes firstand second-order vague regions and characterizes the gradual transitions of second-order vague regions. These gradual transitions allow transitions within transitions. In general, it is not possible to reduce higher-order vagueness to first-order vagueness. Using a spatial ordering structure the article shows when second-order vagueness is useful and examines the relationship between firstand second-order vagueness. An example shows how second-order vagueness can be used to represent geographic objects.