Information Technology Reference
In-Depth Information
are crisp sets. R-implications generalize the material conditional. Of course, this will
happen with any
J
such that
J
(
0
,
0
)
=
1
,
J
(
0
,
1
)
=
1
,
J
(
1
,
0
)
=
0
,
J
(
1
,
1
)
=
1
Example 3.2.7
The immediate generalization of the Boolean conditional
a
ₒ
b
=
a
+
(μ
+
˃)(
)
=
μ
(
b
, is given by
x
,
y
x
)
+
˃(
y
)
, that is, by
J
(
a
,
b
)
=
S
(
N
(
a
),
b
)
,
for all
a
,
b
in
[
0
,
1
]
. These operators are called S-implications (S shortens 'strong').
With,
•
S
=
max,
N
=
N
0
,is
J
(
a
,
b
)
=
max
(
1
−
a
,
b
)
, called the Kleene-Diennes
conditional.
pr od
∗
,
N
•
S
=
=
N
0
,is
J
(
a
,
b
)
=
1
−
a
+
ab
, called the Reichenbach conditional.
W
∗
,
N
•
S
=
=
N
0
,is
J
(
a
,
b
)
=
min
(
1
,
1
−
a
+
b
)
, called the Łukasiewicz
conditional.
Notice that the MP inequality
T
0
(
S
(
N
(
a
),
b
))
a
, is verified on the last three cases
with
T
0
=
W
:
•
W
(
a
,
max
(
1
−
a
,
b
))
=
max
(
0
,
a
+
b
−
1
)
=
W
(
a
,
b
)
b
•
W
(
a
,
1
−
a
+
a
·
b
)
=
max
(
0
,
a
·
b
)
=
a
·
b
b
•
W
(
a
,
min
(
1
,
1
−
a
+
b
))
=
max
(
0
,
min
(
a
,
b
))
=
min
(
a
,
b
)
b
,
hence, the three cases are W-conditionals.
Example 3.2.8
Let us see how is
J
T
, when
T
is, respectively, the continuous t-norm
min,
pr od
˕
,
W
˕
.
1
,
if
a
b
•
T
=
min,
J
min
(
a
,
b
)
=
sup
{
z
∈[
0
,
1
];
min
(
z
,
a
)
b
}=
(
G
odel
¨
b
,
if
a
>
b
implication).
•
T
=
pr od
˕
,
J
T
(
a
,
b
)
=
sup
{
z
∈[
0
,
1
];
˕(
a
).˕(
z
)
˕(
b
)
}=
1
,
if
a
b
(Goguen implication).
(
˕(
b
)
˕(
˕
−
1
)
),
if
a
>
b
a
];
˕
−
1
}=
˕
−
1
•
T
=
W
,
J
T
(
a
,
b
)
=
sup
{
z
∈[
0
,
1
(
W
(˕(
a
), ˕(
z
))
b
(
min
(
1
,
1
−
˕
˕(
a
)
+
˕(
b
)))
(Łukasiewicz implication).
Since each
J
T
is a
T
-conditional,
G
odel
's is a min-conditional, Goguen's are prod
¨
-
˕
conditionals, and Łukasiewicz's are W
-conditionals. Notice that the S-implications
˕
of the form
W
˕
(
)
=
˕
−
1
W
∗
(˕(
)))
=
˕
−
1
W
∗
(
N
˕
(
a
),
b
(
N
˕
(
a
)), ˕(
b
(
1
−
˕(
a
), ˕(
b
)))
=
˕
−
1
(
min
(
1
,
1
−
˕(
a
)
+
˕(
b
))),
are exactly the Łukasiewicz's R-implications: the only R-implications that are
S-implications are the Łukasiewicz's ones. If, for instance, it were
Search WWH ::
Custom Search