There is a fundamental tension between natural variation in the real world and the dominant data models of GIS, which focus on mutually exclusive, homogeneous classes of objects with abrupt spatial boundaries. We have also seen how some researchers such as Bouille (1982) and Burrough (1986a) were from an early stage calling for essentially fuzzy phenomena to be modeled as such. Initial moves in this direction were to use probabilities as we have seen in the overlay of categorical data. But, as pointed out by Ehrliholzer (1995), the more interpretive and complex the data in a coverage, the more suitable are qualitative methods of assessment likely to be.

Starting in the early 1990s, there was a move toward researching the treatment of probabilities as fuzzy measures (Heuvelink and Burrough, 1993; van Gaans and Burrough, 1993) where the Boolean selection is replaced by fuzzy logic in which intersection (AND) and union (OR) are instead based on MIN and MAX functions, respectively. The resulting probabilities thus reflect the degree of class membership in the final product. By the late 1990s, a considerable body of literature had developed in which fuzzy concepts (fuzzy sets, fuzzy logic, and fuzzy numbers) had been used as a means of accounting for variability and as a means of propagating that variability and any uncertainty to the analytical products.

Zadeh (1965) first introduced, as a concept, fuzzy sets and their associated logic. Whereas traditional mathematics and logic have assumed precise symbols with equally precise meanings, fuzzy sets are used to describe classes of inexact objects. Thus, though Boolean logic relies on a binary (0, 1) (termed crisp), fuzzy sets have a continuum of membership. Because imprecisely defined classes are an important element in human thinking, fuzzy sets have found early application in knowledge engineering.

In other words, for an intersection AND the minimum membership of all elements x are taken and in the union OR the maximum membership of all elements x are taken (examples are given below). This has important parallels with Formula (8.3) and Formula (8.4). Thus, fuzzy sets can be propagated through analyses typical of those carried out in GIS where overlay is combined with Boolean selection. Fuzzy set operations and the use of fuzzy sets in a geographical context have been reviewed by Macmillan (1995). The term fuzzy has been introduced into GIS for handling uncertainty, though for the most part it has been loosely applied to any nonbinary treatment of data, such as probabilities. However, there are important differences between probabilities and fuzzy sets.

First, probabilities are still crisp numbers in the way they are formally defined. Second, probability of A and ¬A must sum to unity, but not necessarily so for fuzzy sets where there can be some unknown or unquantified residual. Thus, fuzzy sets, from this perspective, are easier to use than probabilities. Nevertheless, the use of fuzzy set theory proper has thus far been quite restricted (Unwin, 1995) and is reviewed in a spatial analysis context by Altman (1994). One area of application has been the “fuzzification” of data, database queries, and classification schemes through the use of fuzzy membership functions, as a means of overcoming the uncertainty implicit in the binary handling of data (Kollias and Voliotis, 1991; Burrough et al., 1992; Guesgen and Albrecht, 2000). Another area of application has been to quantify verbal assessments of data quality from image interpreters and as a consequence of expert evaluations (Hadipriono et al., 1991; Gopal and Woodcock, 1994; Brimicombe, 1997; 2000a). Use of fuzzy numbers for recording and propagating geometric uncertainty is given in Brimicombe (1993; 1998). However, we are still some way off from seeing fuzzy concepts as part of mainstream GIS software. The theoretical dryness of fuzzy sets (above) can be brought to life and illustrated through a GIS example.

**Example of Fuzzy Sets in GIS**

In Figure 8.4, I illustrated the problems of interpreting natural variation into discrete classes. Suppose we could express our certainty of class membership linguistically, somehow store that against the appropriate polygon in GIS, and propagate that linguistically expressed uncertainty to any analytical products.

Nice? Well, entirely feasible. Before we look at a GIS example, first we need to consider how fuzzy sets are used to describe linguistic terms. Verbal assessments or linguistic hedges are a common qualitative indicator of data accuracy and reliability. For example, a set of linguistic hedges (certain, reliable, well-defined, poorly defined) for features and boundaries have been defined and encouraged for use in API for terrain evaluation by the Geological Society Working Party on Land Surface Evaluation for Engineering Practice (Edwards et al., 1982). The problem that arises, however, is that these types of “standard” linguistic hedges (Table 8.3) are defined in terms of yet other hedges that, to each individual, may have different nuances and interpretations. When more than one language is considered, the problem of “meaning” of linguistic hedges is compounded apparently to the point of impossibility. One of the earliest and main applications of fuzzy sets has been to represent qualifying adjectives such as “tall” or “short.” Empirical studies of fuzzy set equivalents of linguistic hedges (Zadeh, 1972; Lakoff, 1973; Kaufmann, 1975a) have shown a general pattern of reduced spread in the fuzzy sets as they tend toward the more definite boundaries of 0 and 1.