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How fuzzy is a fuzzy set? - It could be worthwhile to attempt Italian physicists
Settimo Termini and Aldo de Luca thought in the early 1970s it could have been
worthwhile to attempt to “measure fuzziness”. Termini recalled in an interview in
2010 “The very important thing was the truth functionality of fuzzy sets. It was very
appealing that there was a way of approaching the representation of uncertainty in
a truth functional way. In a single word, this appeared to me, simply: Wonder-
ful! However, it was important having ways of controlling this uncertainty, and this
should appear clearly by just looking to some properties of fuzzy sets. In a sense
it should be something that each fuzzy set carries with it, independently from any
other things.” [87]
(a)
(b)
(c)
Fig. 3.11 (a): Aldo de Luca; (b): Settimo Termini; (c): their article [21]
In 1972 the two proposed “the introduction of a 'measure of the degree of fuzzi-
ness' or 'entropy' of a generalized set” [21], “starting from the information provided
by a fuzzy set, although all the tools of probability had been defined. We asked our-
selves also what kind of properties (hoping to find strange things) this new concept
should satisfy [...] at the same moment it was absolutely clear to us that these 'en-
tropies' were 'measures of fuzziness', or, ways of measuring how much fuzzy a
fuzzy set was. The central point of thus firm conviction was the introduction of the
'sharpened order'. It is this the central idea in which the theory of 'measures of
fuzziness' is based, and in my view it corroborated the naive conviction that 'fuzzi-
ness' was not only an interesting concept but that it was really a true new scientific
concept .” [87]
When de Luca and Termini considered the entropy e of a fuzzy set of 2 X
they
beared in mind a measure that gives a value in the interval
[
0
, ]
and that satisfies
the following conditions:
(
)=
e
A
0if A is a crisp set.
(
)
(
)=
/
e
A
is maximal if A is the constant fuzzy set A
x
1
2forall x
X .
e
(
A
)
e
(
B
)
if A is 'more fuzzy' than B by the 'sharpen' order
S .
A c
e
(
A
)=
e
(
)
.
where the 'sharpen' order
S is defined as follows:
 
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