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should always be true according to classical logic. Thus, it seems that vagueness is
in conflict with classical logic and vice versa. By the way, this fact has already been
discovered in 1923 by one of the founders of classical logic himself [7].
The indeterminacy of being or not being a member of a vague class brings with
it that objects can be members of a vague class to different extents. So, in charac-
terizing their membership, linguistic hedges such as
more or less, very, very very,
and others come into play such that an object
x
may be stronger than another ob-
ject
y
a member of a vague class
C
. While, for example, nobody has a brother of
whom we could say
“he is a very brother”,
many of us have read some topics of
which one can say “this is a very interesting book”. Thus, the class of
brothers
is
not vague, whereas the class of
interesting topics
is vague. The susceptibility to
linguistic hedges is a typical characteristic of vague classes. It renders a vague class
granular. We can thus say that a vague class is susceptible to linguistic hedges
and
granular.
As mentioned previously, fuzzy logic is a conceptualization and precise theory
of vagueness, specifically, of ontically vague classes, including relations. The key
notion of the entire system of fuzzy logic has been and still remains the concept
of
fuzzy set
. Fuzziness is the property of a fuzzy set to be fuzzy. Therefore, it
is appropriate to differentiate between
vagueness,
as sketched thus far, on the one
hand; and
fuzziness
, on the other. Vagueness is not fuzziness, and fuzziness is not
vagueness. But what is fuzziness exactly?
2.4
The Nature of Fuzziness
After we have approved the postulate that
vagueness
exists in the world out there,
we can now ask the corresponding question whether
fuzziness
tooexistsintheworld
out there. To answer this question, I shall first make the notion of
fuzziness
a little
bit more precise by constructing what I shall call a
fuzzy structure
. To this end we
recall the concept of
fuzzy set
.
To obtain a fuzzy set, we need three things:
Ω
= a collection of objects
μ
= a function that maps
Ω
to [0, 1]
=
x
)
|
∈
Ω
.
A
,
μ
(
x
x
We need, first, a finite or infinite collection of objects, here symbolized by
and
also called a base set or universe of discourse; second, a function, here symbolized
by
Ω
μ
, which maps
Ω
to the unit interval; third, an emerging subset
A
in, or over,
Ω
which consists of pairs of objects such that the first object,
x,
is an element of
Ω
und
the second object is the function value
μ
(
x
)
, referred to as the degree of membership
of
x
in fuzzy set
A
.
We can use this basic terminology to introduce a new concept which will help us
decide whether there exists fuzziness in the world:
Definition 2.
An object
ξ
is a fuzzy structure
if and only if there are
Ω
, A, and
μ
such that:
