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of continuous frames of reference, or from concepts of typicality (a situation may gen-
erally be typical of known situations and this has to be accounted for when reasoning
by analogy).
When reasoning with propositions, the uncertainty (in a wide sense) corresponds
to the inability to state whether a proposition is true or false, either because the infor-
mation is incomplete, vague, imprecise, or because the information is contradictory or
fluctuating.
In the first case, a possibilistic model makes it possible to take into account this
type of uncertainty, whereas in the second case, a probabilistic model would be well
suited.
There is another important distinction between the degree of certainty and the
degree of truth. In fuzzy logic, propositions are assigned a degree of truth, whereas
in possibilistic logic, they are usually assigned degrees of uncertainty.
8.7.1.
Fuzzy logic
In fuzzy logic [DUB 80, DUB 91], reasoning is based on elementary fuzzy propo-
sitions of the type:
X
is
P
[8.84]
where
X
is a variable with possible values in the reference space
S
and
P
is a fuzzy
subset of
S
, with the membership function
μ
P
.
The degrees of truth of such propositions are defined as values in [0
,
1] based on
μ
P
.
Logical connectives are defined in a very simple way, by using the same operators
as their set equivalents. For example, the degree of truth of a conjunction such as:
X
is
A
and
Y
is
B
is defined based on a t-norm
t
by:
μ
A
∧
B
(
x, y
)=
t
μ
A
(
x
)
,μ
B
(
y
)
.
Likewise, a disjunction such as:
X
is
A
or
Y
is
B
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