Areal Differentiation Using Fuzzy Set Membership and Support Logic

Qualitative thematic maps are graphic representations of the process of areal differentiation, a process recognized as one of the hallmarks of the geographer’s craft and one of geography’s grand traditions. Traditionally the process can best be described as classification based on Boolean Set Theory wherein the area bounded by the cartographic line is considered to be in that class for the target polygon, or stated differently it is considered to be contained within the boundary of the polygon. Conversely then, any content not contained in the polygon is not of that class of polygon. There are two fundamental confounding issues with the use of Boolean logic for such classification of geographic areas however. First is that the determination of class membership itself is often less than 100% obvious, either because of the lack of visual cues in interpretation from a source document, or by the subjective nature of the linguistic variables and classes selected. This suggests that the process more closely resembles fuzzy set membership than it does Boolean set membership. Second, the categories, whether discretely defined or more subjective will, when cartographically depicted result in polygonal adjacency, where the line represents two or more categories simultaneously, and where any attempt at adding something like an epsilon band of uncertainty for such lines results in conflicts because of assumed complementarity of the two set values. This paper hypothesizes that one alternate method of dealing with the uncertainty of such lines be based first on Fuzzy Set membership values to account for the classification uncertainty itself. Because Fuzzy Set Memberships do not assume complementarity, by extension support logic – an interval-based probabilistic reasoning is applied to the fuzzy set memberships to create a modified version of the epsilon band normally applied to the uncertainty of the graphic line itself. The support logic extends the fuzzy set membership of polygonal containment to include the degree of logical support there is for the category. In this support logic epsilon band, which I call an SLE-Band, you are able to include both graphic line uncertainty (the Epsilon part) with the maximum support of membership and the minimum necessary support of membership. While still at a conceptual stage of development, the paper proposes the conceptual mathematical framework supporting the development of this Fuzzy Set based support logic strategy that will enable the determination of fuzzy set membership (polygonal inclusion) as well as providing the support logic for the membership value of any polygon. Further, while not the major thrust of this paper I will propose some possible methods of graphic display and briefly describe how such cartographic layers may be compared through alternative overlay operations not currently implemented in contemporary commercial GIS software.