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(μ
P
∗
(
),
(μ
P
(
), μ
Q
(
)))
μ(
),
∀
∈
,
∈
,
T
x
J
x
y
y
y
Y
x
X
that is,
μ
∈
Cons
(
{
μ
P
∗
,μ
P
ₒ
μ
Q
}
)
. Nevertheless, if
T
=
T
0
, the continuous
t-norm for which
J
is a
T
0
-conditional, that is, such that
T
0
(μ
P
(
x
),
J
(μ
P
(
x
), μ
Q
(
y
)))
μ(
y
)
=
,
∀
y
∈
Y
,
x
∈
X
,
it could be that when
μ
=
μ
Q
. A undesiderable situation, because
fuzzy logic must contain all classical cases.
μ
P
∗
=
μ
P
, then
For example, with the rule 'If
x
is small, then
y
is big' (
X
=
Y
=[
0
,
1
]
),
and
J
(
a
,
b
)
=
max
(
1
−
a
,
b
)
that is a
W
-conditional, taking
μ
S
(
x
)
=
1
−
x
and
μ
B
(
y
)
=
y
, follows:
•
With
T
=
W
,
μ(
y
)
=
Sup
W
((
1
−
x
),
max
(
x
,
y
))
=
Sup
(
0
,
y
−
x
)
=
y
=
x
∈[
0
,
1
]
x
∈[
0
,
1
]
μ
B
(
y
)
.
•
With
T
=
pr od
,
μ(
y
)
=
Sup
(
1
−
x
)
max
(
x
,
y
)
=
Sup
max
((
1
−
x
)
x
,(
1
−
x
∈[
0
,
1
]
x
∈[
0
,
1
]
x
)
y
)
=
y
=
max
(
1
/
4
,
y
/
2
)
, not coincidental with
μ
B
.
•
with
T
=
min,
μ(
y
)
=
Sup
min
((
1
−
x
),
max
(
x
,
y
))
=
1, or
μ
=
μ
1
, also not
x
∈[
0
,
1
]
coincidental with
μ
B
.
Hence, although with any continuous t-norm
T
, an output
μ
is obtained, if this
T
P
∗
Q
∗
.For
does not make
J
a
T
-conditional it is not sure that
P
=
implies
Q
=
this reason, it is necessary to take
T
0
with CRI!
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