Fuzzy Modelling - GeoComputation - page 286

in defining mountains. Fisher et al. (2004, 2007a) argue that although the summit gives a focus for
defining mountains, it does not express the spatial extent of any mountain; they have attempted to
do just this using fuzzy sets.
Fuzzy set theory is an attempt to address the problem posed by phenomena that are sorites sus-
ceptible (or vague) to quantitative representation and manipulation by means of traditional sets and
logics. Fuzzy set theory as such was first suggested by Zadeh (1965). It sought to avoid the philo-
sophical problems of vagueness and the pitfalls inherent in entering that argument (Kosko, 1993).
It also furthered earlier work developed by Kaplan and Schott (1951) and Black (1937), although no
specific reference is made to such material by Zadeh.
Gale (1972), a behavioural geographer, seems to have been the first person to suggest that
fuzzy sets might be worth the attention of geographers. A subsequent paper by Pipkin (1978)
further explored the geographical application of fuzzy sets in an investigation of spatial choice,
while Ponsard (1977) examined settlement hierarchies and Leung (1979) was more interested
in land use planning and decision-making. Robinson and Strahler (1984) were among the first
to suggest that fuzzy sets may have far more to offer modern geographical methods and com-
puter models, in explaining how they present a logical basis for storing uncertainty information
about their analysis of satellite imagery and application of GIS. Following these papers, there
have been hundreds of examples of the application of fuzzy sets to many areas of geographical
endeavour.
12.3 CLASSICAL SETS AND BOOLEAN LOGIC
To understand what a fuzzy set is, it is first necessary to consider what is meant by a classical
set . A classical set is a container that wholly includes or wholly excludes any given element. For
example, the set of days of the week unquestionably includes Monday, Thursday and Saturday.
It just as unquestionably excludes butter, elephants, liberty and so on. It is called a classical set
simply because the concepts involved have been around for such a long time, originating from
the writings of Aristotle (384 BC-322 BC), who first formulated the law of excluded middle . Put
simply, X must either be in set A or in set not-A. Key characters in the development and present
understanding of set theory, as subsequently formalised by logicians and mathematicians, are
George Boole (1815-1864), John Venn (1834-1923) and Georg Cantor (1845-1918). The resultant
classical sets are hereinafter referred to as crisp sets, which are often illustrated by means of
Venn diagrams and analysed using Boolean algebra. Figure 12.1a shows a standard representation
of two crisp sets, A and B . They are shown within a rectangle, the universe of discourse (or just
universe, a term used to mean all things under consideration during a discussion, examination
or study, i.e. everything that we are talking about), and possess hard boundaries. The boundar-
ies can be in single or multiple attributes, or in space, or in time. Crisp sets can become the
basis of logical operations which involve the combination of the sets, for example, by the use of
Boolean algebra. The union (Figure 12.1b) is the region occupied by either A or B , the intersect
(Figure 12.1c) is the region occupied by both A and B , and the inverse (Figure 12.1d) is the region
occupied by neither A nor B . Whether or not an object belongs to a set can also be portrayed as
a line graph (Figure 12.2), which is effectively a cross section through the Venn diagram, where
belonging is shown by code 1 and non-belonging by code 0, a binary coding. This is indicated as
a value of either 0 or 1 or {0, 1}, using {} to identify a specific collection of elements in that set,
that is, the only permitted numbers are 0 and 1. Union and intersect can also be shown (Figure
12.2b and c, respectively).
If the belonging of an individual to a set is based on the possession of a threshold value of a par-
ticular property, and the diagnostic possession of the property is the subject of error (owing to poor
observation, faulty measurement, etc.), then there is a probability of the property being observed or
measured correctly, and so whether or not an individual is a member of a set can be assigned a prob-
ability ( p ( x ) A ), that is, the probability of x being in A , as can its probability of belonging to B ( p ( x ) B ).