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x
i
)
From this expression, we can derive the traditional mission of a physician: To
precisely identify symptoms and diseases and then find a suitable treatment for it.
Let's see an example expressed by a composed expert rule: “if blood glucose lev-
els are high at fasting in patient
A
, and he is usually tired, and he is usually very
thirsty (polydipsia), and he urinates frequently (polyuria), and he is usually hun-
gry (polyphagia), and he has lost weight, then patient
A
suffers from Diabetes, so
treatment is diet, exercise and insulin.”
Nevertheless, sharp classifications suffer from an important drawback: real life
is extremely complex and usually sharp partitions don't fit well with the real phys-
ical models (human beings, in this case). As an example, hypoglycemia is defined
as sugar concentrations in blood of less or equal to 70
mg
x
i
∈
S
,
y
j
∈
T
/
y
j
=
f
(
/
dl
, while severe hypo-
glycemia is given for glucose levels of less to 45
mg
/
dl
. Now, we can reflect on
the following questions: a) Is 71
mg
/
dl
a case of normal glycaemia levels? b) Is
69
mg
dl
an absolutely clear example of hypoglycemia? And even more interest-
ingly, c) can be 69
mg
/
/
dl
considered in the same category of sugar concentrations
as 46
mg
dl
?
These are questions that strongly put stress on Aristotle-type classification sys-
tems. Plato had been also interested on these ideas in his
Theory of forms
and as
we can read from his
Dialogues
, he thought that all the forms in the real world are
imperfect, being perceived as such by our senses. In his line of reasoning only ideal
forms are perfect and only through the human reason we can perceive them. While
less precise than the ideas of his disciple on this matter, from our point of view
we suspect that if only Plato had had a strong knowledge in algebra and functions,
maybe he had anticipated fuzzy sets by more than 2
/
000 years because the inherent
foundations of fuzziness were already present in his mind. Sadly it was not the case,
so classic logic appeared with Aristotle, reigning supreme until the 20th century.
The shift of paradigm about the very concept of classifications and classical sets
and then logic had a sharp launch in a seminal paper from Lofty A. Zadeh in 1965
[10], unambiguously titled “Fuzzy sets”. As it is well known at present, boundaries
of a fuzzy set are not precise, so the membership of a given element to a given fuzzy
set is not a binary, yes or no question, but it converts to a question of degree [7].
Ifwesayanelement
x
is a member of
A
, we are not speaking about something
completely true or completely false, but it may be true only to some degree, usually
expressed in the closed interval
,
, the degree to which
x
is actually a member
of
A
. For defining fuzzy-sets, a membership function maps elements from a given
universe of discourse into real numbers inside the unit interval
[
0
,
1
]
[
0
,
1
]
[6]
μ
A
:
X
−→
[
0
,
1
]
In this expression, the function
μ
A
completely defines the fuzzy set
A
.Nowletus
design some trapezoidal membership functions for the fuzzy sets “hypoglycemia”
and “normal” for sugar concentrations in blood. For hypoglycemia we shall have the
membership function
μ
h
(
x
)=
1when
x
belongs to the interval
[
0
,
60
)
and
μ
h
(
x
)=
(
60
−
x
)
/
20
+
1when
x
belongs to the interval
[
60
,
80
]
.
On the other hand, for
