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1
−
b
,
if
a
b
2. With
J
pr od
, since
W
(
1
−
b
,
J
pr od
(
a
,
b
))
=
1
−
a
.
b
1
−
b
a
,
if
a
>
b
3. With
J
W
, since
W
(
1
−
b
,
min
(
1
,
1
−
a
+
b
))
=
min
(
1
−
b
,
1
−
a
)
1
−
a
.
4. With
J
(
a
,
b
)
=
1
−
a
+
ab
, since
W
(
1
−
b
,
min
(
1
,
1
−
a
+
ab
))
=
(
1
−
b
)(
1
−
a
)
1
−
a
.
5. With
J
(
a
,
b
)
=
max
(
1
−
a
,
min
(
a
,
b
))
, since
W
(
1
−
b
,
max
(
1
−
a
,
min
(
a
,
b
)))
W
(
1
−
b
,
max
(
1
−
a
,
b
))
1
−
a
.
pr od
∗
(
a
2
b
, since
W
pr od
∗
(
6. With
J
(
a
,
b
)
=
1
−
a
,
a
·
b
)
=
1
−
a
+
(
1
−
b
,
1
−
a
,
pr od
∗
(
a
·
b
))
W
(
1
−
b
,
1
−
a
,
b
))
=
W
(
1
−
b
,
1
−
a
+
a
·
b
)
1
−
a
.
Nevertheless, the case
J
(
a
,
b
)
=
T
(
a
,
b
)
is, actually, negative. To have
T
1
(
1
−
b
,
T
(
a
,
b
))
1
−
a
, it is necessary (take
a
=
1) that
T
1
(
1
−
b
,
b
)
=
0, that is,
T
1
=
W
,
butitis
W
(
1
−
b
,
T
(
a
,
b
))
=
max
(
0
,
T
(
a
,
b
)
−
b
)
=
0, since
T
(
a
,
b
)
b
implies
T
(
a
,
b
)
−
b
0. Hence, by one side from
W
(
1
−
b
,
T
(
a
,
b
))
=
0
1
−
a
, it seems
˃
,μ
ₒ
˃
:
μ
”, what
that backwards inference is possible. But, given the scheme: “
˃
·
(μ
ₒ
˃)
=
μ
0
, that forces
Conj
(
{
˃
,μ
ₒ
˃
}
)
=∅
results is
. In conclusion,
Mamdani-Larsen conditionals don't allow backwards inference.
Remark 3.2.16
It should be pointed out that except
J
min
,
J
pr od
˕
,
and
J
(
a
,
b
)
=
T
(
a
,
b
)
, most of the functions
J
are
T
0
-conditionals for
T
0
=
W
, and almost all
do also verify backwards inference also with
T
0
W
. And the t-norms in the
Łukasiewicz's family show the disturbing problem of having zero-divisors!
=
Remark 3.2.17
The name R-implication, or residuated implication, comes from the
idea of 'residuum' that clearly appear in the case of
J
pr od
when
b
a
.
If
a
>
b
,
then
J
pr od
(
a
,
b
)
=
Remark 3.2.18
In the same vein under which it was proven that R-implications
J
with
T
W
˕
are not S-implications, it is easy to show that they are not expressible
in material protoform, that is, by an expression with logical connectives. Take the
perhaps more general material protoform
=
μ
·
(˃
+
˃
)
+
μ
·
˃
. Is it possible that
J
T
0
(
,
)
=
S
1
(
T
1
(
N
1
(
),
S
2
(
,
N
2
(
)),
T
2
(
,
))),
a
b
a
b
b
a
b
for
T
0
=
W
˕
,
S
1
and
S
2
continuous t-conorms,
T
1
,
T
2
continuous t-norms, and
N
1
,
N
2
strong negations? With
b
=
0, it follows
J
T
0
(
a
,
0
)
=
sup
{
z
∈[
0
,
1
];
T
(
z
,
a
)
0
}=
0
S
1
(
T
1
(
N
1
(
a
),
S
2
(
0
,
1
)),
0
)
=
T
1
(
N
1
(
a
),
1
)
=
N
1
(
a
),
or,
S
1
(
0. This means that
S
1
is not a t-conorm. Hence, the decision of
representing an R-implication can't be taken from a material protoform interpretation
of it.
N
1
(
a
),
0
)
=
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