Software and Data Table 1: A nonspatial example comparing Boolean logic with fuzzy logic. In Boolean logic, truth is “crisp,” zero or one. In fuzzy logic, truth has degrees between zero and one. Fuzzy tallness and fuzzy oldness are the membership in the concepts tallness and oldness. Boolean tallness and Boolean oldness are binary memberships in these concepts. Thus in Boolean logic, a person is either tall or not; whereas in fuzzy logic, a person can be somewhat tall. The operators AND and OR are used for combining evidence in both methods. Evidence Person Fred Mike Sally Marge John Sue Height 3’ 2’’ 5’ 5’’ 5’ 9” 5’ 10” 6’ 1’’ 7’ 2’’ Fuzzy Tallness 0.00 0.21 0.28 0.42 0.54 1.00 Boolean Tallness 0 0 0 0 1 1 Age 27 30 32 41 45 65 Fuzzy Oldness 0.21 0.29 0.33 0.54 0.64 1.00 Boolean Oldness 0 0 0 1 1 1 Nonspatial Models Boolean Logic Truth (Marge is tall) = 0 Truth (Fred is old) = 1 Truth (Sally is tall and old) = 0 Truth (John is tall or old) = 1 Fuzzy Logic Truth (Marge is tall) = 0.42 Truth (Fred is old) = 1 Truth (Sally is tall and old) = 0.21 Truth (John is tall or old) = 0.54 situations that can be true or false. Fuzzy logic allows degrees of truth (expressed as a membership function) in the range of zero to one. In this example, an expert uses fuzzy membership values to define the importance of two characteristics of people (tallness and oldness) to be used as predictive evidence (values between 0 and 1). The expert also defines how the evidence is combined, in this example using fuzzy AND and OR operators. Probability is a special case of fuzzy membership. If the probability of truth is 0.8, then the probability of false is 0.2 (i.e., if the probability of an event occurring is x, then the probability of the event not occurring is always 1-x). This additive-inverse property of probability statements is not required in fuzzy logic. Fuzzy membership can be thought of as the “possibility” that the statement is true. In a Boolean model, the height of Marge (listed in Table 1) is absolutely not tall or tallness is zero, whereas in a fuzzy logic model, Marge’s height is somewhat short with a tallness of 0.42. Generally, Fuzzy logic provides an approach that allows expert semantic descriptions to be converted into a numerical spatial model to predict the location of something of interest. a membership of 0.5 indicates an ambiguous situation that is neither true nor false. An example of a membership function with semantic descriptors (e.g., possibly short, possibly tall) is shown graphically in Figure 1. Fuzzy membership thus provides sensitivity to the subtle aspects of the process being modeled. In addition, a variety of fuzzy combination operators are available that greatly extend the simple AND and OR operators used in Boolean logic and allow the flexibility and complexity incorporated in making many real-world decisions to be modeled. Continued on page 10 ArcUser Spring 2010 9 www.esri.com