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perception of what is a heap. Hence, in the same classical paradox it appears a
perceptive view of what is not a heap, since in the identification between heap and
set of grains, there is no any paradox as far as
h
(
)
1
is a set. What is not clear at all
(
)
is which is the first number
p
such that
h
is not a heap. In a perceptive-based
view the only that perhaps could be known is some interval
p
(
p
−
i
,
p
+
k
)
to which
p
belongs to.
From a flat three-dimensional set of grains to, for instance, a cone of them, there
are many, many possibilities for having a heap in such a way that the statements '
h
is a heap' are up to some degree. For instance, figure (a) in section 15.4.2 shows
aheapuptoadegreelike0
.
9, figure (b) up to a degree 0
.
1, and figure (c) up to
adegree0
4. Nevertheless, these degrees do be fixed in the most reasonable way
than possible, that is, it should be specified by a clearly expresable point of view
allowing such quantification through a comparative process.
.
Remark 1.
The term 'heap' appears as gradable to the layperson by, at least, their
different shapes with different basis in the floor, and the different heights they show,
but not by their number of grains that nobody is actually going to count. What is not
clear at all is if the 'degree up to which h is a heap' is, or is not always a number,
and to which universe of discourse
X
heaps do belong to have the possibility of
representing the numerical degrees (provided they exist) by means of a 'fuzzy set'
in
X
. In any case, and if it is possible, what they can be considered to be are
indices
of 'heapness'.
Remark 2.
As it will be shown in what follows, the comments made on 'heap' are
applicable to other imprecise terms like, for instance, 'small' in a numerical uni-
verse of discourse and once a membership function according with its use in the
corresponding context has been designed ([7]). In the field of medicine, that is
full of imprecise technical concepts, the Sorites' process presented in this paper,
could be useful to determine if a medical concept (C) can be specified, or not, by
a classical set. In the negative case, it is not possible to conduct reasonings involv-
ing the concept in the classical 'boolean' way. For instance, if 'diabetes'(D) and
'high blood preassure'(B) are interpreted as fuzzy sets, the statement ( (D and H) or
(D and not H)) is only equivalent to D in some algebras of fuzzy sets in which no
law of duality holds.
After a concept is designed as a fuzzy set, accordingly with its technical use, it could
be managed under the rules allowed in the corresponding algebra of fuzzy sets. In
addition, when the representation of its negation is chosen, it can be approached by
the crisp set whose elements are those in the universe that are more C than not C.
15.2
The Case of the Term 'small' in the Interval [0,10] of the
Real Line
It seems reasonable to agree that a layperson does use the term 'small' in the real
interval
[
0
,
10
]
by following the four rules,
