Fuzzy Modelling - GeoComputation - page 285

(2007) book on Geographic Uncertainty in Environmental Security , 11 deal with fuzzy set con-
cepts. In addition to topics, there have been a number of articles on fuzzy modelling in journals. A
special issue of Fuzzy Sets and Systems (Cobb et al., 2000) served as a precursor to papers appear-
ing in that journal illustrated by GISci problems (e.g. Brandtberg, 2002; Anile et al., 2003; Bjorke,
2004; Bordogna et al., 2012). A special issue of Transactions in GIS was devoted to fuzzy sets in
geographic information systems (GIS) (Robinson et al., 2003) after which papers with fuzzy set
content appeared regularly. In fact, since 2000 rarely does a year go by without at least one paper
exhibiting fuzzy set content appearing in major journals such as the International Journal of
Geographic Information Science or Transactions in GIS .
In this chapter, we will briefly examine the nature of vagueness. Discussion will then move on to
examine classical set theory to show what can be judged to be problematic with it in terms of vague-
ness, and then we will examine fuzzy sets as an alternative. The most important topic in fuzzy set
theory will follow, namely, the definition of fuzzy memberships. Next, fuzzy logic is examined, and
discussion of extensions of fuzzy set theory and of dissenting opinions follows. In conclusion, some
other types of soft set theories are indicated, which may prove useful to GC in the future.
12.2 SORITES PARADOX
The process of argument used in the preceding section is well known in philosophy and is called
the sorites paradox (Williamson, 1994; Sainsbury, 1995). In its original formulation, the argument
went like this:
If one grain of sand is placed on a surface, is there a heap?
If a second grain is placed with the first, is there a heap?
If a third grain is placed with the first two, is there a heap?
…
If a ten millionth grain is placed with the 9,999,999 grains, is there a heap?
In the first three instances, where n = 1-3, the answer is clearly 'No'. If the last question, when
n = 10,000,000, were asked on its own, the answer would almost certainly be 'Yes', but if the process
of questioning progresses from the first to the last question, with an increment of one grain each
time, the logical answer is still 'No'; it is never the case that a non-heap is turned into a heap by the
addition of a single grain. That answer, however, is clearly wrong, and that is why it is a paradox.
The person answering will undoubtedly feel increasingly uncomfortable as the process advances,
but at no point will they, logically, be able to answer 'Yes, there is now a heap when there was not
one before'. The heap to non-heap argument by the removal of grains works the same way; there
is never not a heap, until the very last grain is removed. The core of the problem is that heap is a
vague concept. The same process of argument characterises the definition of very many geographi-
cal phenomena. Concepts such as the heap, which are sorites susceptible, are vague (Russell, 1923).
The exact causes of vagueness need not be discussed here but are comprehensively reviewed by
Williamson (1994).
Aspects of the problem of vagueness in the philosophical literature, and of the possibility of
vague objects, have for long been illustrated with geographical examples, such as where is Snowdon
(or Everest) and how long is the Thames (Sainsbury, 1989; Williamson, 1994; Varzi, 2001a,b). Fisher
and Wood (1998) present a discussion of the vagueness (or fuzziness) of mountains with attention to
the story and film of The Englishman Who Went Up a Hill but Came Down a Mountain (Monger,
1995) where the story is based on a threshold height value supposedly used by the Ordnance Survey
of Great Britain in separating a hill from a mountain. Such threshold heights, as well as measures
of separation and prominence, are indeed used by mountaineers and hikers to define peaks to be
climbed; a comprehensive list of such peaks covering much of the globe can be found at peakbagger.
com (2012). These are, however, lists of summits and as such are limited in their ability to be used