Fuzzy Modelling - GeoComputation - page 284

summit location may itself be unknowable because a single mountain massif may have a number
of locations at the same maximum elevation. Fisher (2000) and Robinson (1988), among others,
would add that this is a fundamental and pragmatic issue for geographical information science
(GISci). The issue was referred to by Harvey (1969, p. 216) as individuation , but if an individual
is vague or poorly defined, then the identification is problematic. Such vagueness is, however,
directly addressed by modelling the phenomena as fuzzy sets; Varzi (2001b) relates this question
to philosophical vagueness.
Many geographical objects and concepts referred to in everyday statements and conversations
are vague. Yet, through necessity, people are able to accommodate such matters in their everyday
lives since having to do so forms an essential part of residing and functioning in our modern world.
Varzi (2001b) sees the vagueness as an exclusively semantic problem caused by the naming of
things. He does not believe in the existence of vague geographical objects with boundaries that
are themselves a matter of degree. In the description and analysis of geographical phenomena and
location, however, people widely use vague terms, as adjectives, nouns and verbs, which makes it
hard or impossible to clearly define the geographical individual in any meaningful way that does
not involve an arbitrary cut-off (Fisher, 2000; Varzi, 2001a,b). It is also important to understand that
different types of boundary can be recognised, including the bona fide) boundaries that we find in
the physical world and iat (or human-demarcation-induced) boundaries that much of geography has
to deal with. Mixed cases also exist: some objects can possess both bona fide) and iat boundaries,
for example, the North Sea has land-water ( bona fide) ) and water-water ( iat ) boundaries (Smith
and Varzi, 1997)!
Vagueness is many faceted. For example, if we wish to list the major cities in Europe, not an
unusual request, then we might question what is meant by the term major, what and where is
a city and where is Europe; in common usage, none is unequivocally defined, and although all
could be defined for a particular question, there would always be doubt about the cities and areas
which are both included and excluded from consideration. The extents of cities are a matter for
debate; any one city has a specific extent because it is a political entity and has iat boundaries
which are a matter of human construction and imposition (Smith, 2001; Varzi, 2001a). However,
most cities have suburbs which may or may not be included in the political definition of the
extent of the city. The question is not specific in how largeness of the cities is to be measured.
It could be the population, the economy or the extent of the metropolitan area, to name only
three possible measures, themselves not uncontroversial. Where too is Europe? Is it a set of
political entities, and if so, which are to be included in that set and which excluded? If a limiting
boundary is decided by iat , then would any city outside that geographical area to any degree be
excluded? On the other hand, the boundary could be redrawn, but then redrawing the boundar-
ies could have no end, until all land areas and all settlements are included (arguably the total
population on the Eurasian continent, if not the entire land surface of the planet, could then be
included). Europe itself is a vague noun, a geographic region without unequivocal limits. Indeed,
the recognition of any of the continents has been criticised by Lewis and Wigen (1997), since
their consideration is limited by the crisp concept of mapping, where hard boundaries between
continents are redrawn on world maps. On the other hand, a rather more creative fuzzy remap-
ping of continents has been presented by Didelon et al. (2011) using contributed opinions from
students based in a variety of countries.
Much has happened in fuzzy modelling of geographical phenomena since the first edition of
the current book was published (Fisher, 2000). Topics with an exclusive focus on fuzzy mod-
elling applied to geographic problems have appeared (Petry et al., 2005; Lodwick, 2008), and
more general topics on spatial analysis and GISci have contributions on fuzzy modelling illustrat-
ing recognition of its importance to GISci. The Handbook of Geographic Information Science
(Wilson and Fotheringham, 2008) has three chapters with significant fuzzy set content, including
that by Robinson (2008), while Robinson (2009) contributed a chapter to the Handbook of Spatial
Analysis (Fotheringham and Rogerson, 2009). Out of the 17 chapters in Morris and Kokhan's