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that the painfulness or severity of symptoms such as headaches or cyanosis could be
represented by membership functions or by the degree of abnormality of a clinical or
diagnostic test result [24, 51], or they proposed a nearly linear membership function
for a fuzzy set abnormal cholesterol C, expressed in mg/100 ml of serum [24, 25],
they used the modifiers Zadeh had introduced to calculate the degree of membership
of a test result in a fuzzy set
S
2
of the degree of membership of the test result in fuzzy
set
S
1
[51].
Also Philip Smets and his co-authors [80] emphasized the fuzziness of diagnostic
terms such as arteriosclerosis or angina pectoris. Such diseases are not clearly or
sharply defined, which is why it was often not possible to determine precisely the
symptoms that clearly stand for a disease. However, diagnoses could also be defined
as fuzzy sets whose elements are symptoms. These are assigned a membership
value that indicates the intensity with which the symptom belongs to the fuzzy set
representing the disease in question.
Moon, Jordanov, Perez-Ojeda and Türksen [51] had attempted to represent symp-
tom combinations by means of logical conjunctions (
)offuzzysets.
A short time later Elie Sanchez chose for this purpose the concept of the fuzzy
relation
R
∧
and
∨
D
between the symptom set
S
and the diagnosis set
D
. In doing so,
he assumed that a doctor translates his knowledge and his experience into degrees of
association between symptoms and diagnoses. This suggestion by Sanchez resulted
in the successful application of Fuzzy Set Theory in the field of medical diagnosis.
⊆
S
×
3.4.2
Medical Decision-Making
Elie Sanchez considered relationships between symptoms and diseases mathemat-
ically and he intended to expand Zadeh's theory of fuzzy relations towards medi-
cal aspects. The shape this plan would take can be gleaned from his contributions
Compositions of Fuzzy Relations
[73] and
Medical Diagnosis and Composite Fuzzy
Relations
[74] in a volume about advances in Fuzzy Set Theory and its applications,
which was published in 1979. In these two papers, he demonstrated how the max-
min composition rules Zadeh had introduced in his seminal article [103] could be
used as a rule of inference, in particular in medical diagnostics.
He referred to the fact that medical diagnoses often had to be made without any
precise analysis being possible. One or more illnesses then had to be inferred from a
patient's symptoms, which most often be cannot be described in any exact way. In so
doing, neither the set of diseases taken into consideration nor the conclusion about
the disease(s) drawn from the symptoms can be precise. Sanchez introduced the
relationships between the set of symptoms and the set of diseases as fuzzy relations.
“In a given pathology, we denote by
S
a set of symptoms,
D
a set of diagnosis
and
P
a set of patients. What we call “medical knowledge” is a fuzzy relation,
generally denoted by
R
, from
S
to
D
expressing associations between symptoms, or
syndromes, and diagnosis, or groups of diagnosis. [74, p. 438]
