Information Technology Reference
In-Depth Information
If
x
is
P
, then
y
is
Q
,
where
x
Y
,
P
is a predicate (precise or imprecise) in
X
, and
Q
is a predicate
(precise or imprecise) in
Y
. For example, with
X
∈
X
,
y
∈
=[
0
,
1
]
,
Y
=[
0
,
10
]
,
If
x
is small, then
y
is big,
or,
If
x
is small and
y
is big, then
z
is not small,
with
x
.
What it means to represent a conditional statement like “If
x
is
P
, then
y
is
Q
”?
It means to translate it in fuzzy terms. For example, '
x
is
P
' and '
y
is
Q
' will be
translated by
,
y
∈[
0
,
1
]
and
z
∈[
0
,
10
]
X
, adequate
μ
P
(
x
)
and
μ
Q
(
y
)
with adequate fuzzy sets
μ
P
,μ
Q
∈[
0
,
1
]
in the sense that they capture the use of
P
on
X
and
Q
on
Y
.
But how to represent the full statement “If
x
is
P
, then
y
is
Q
”
:=
μ
P
(
x
)
ₒ
μ
Q
(
y
)
? It is always done, in fuzzy logic, by means of a function
J
:[
0
,
1
]×[
0
,
1
]ₒ
[
0
,
1
]
, such that
μ
P
(
x
)
ₒ
μ
Q
(
y
)
=
J
(μ
P
(
x
), μ
Q
(
y
))
∈[
0
,
1
]
for all
x
Y
. But, which function
J
can be taken? It depends on the
'meaning' of the conditional statement, and this requires to look at what happens in
general.
∈
X
, and
y
∈
Remark 3.2.1
Imprecise conditionals are very useful in ordinary life, for instance,
•
If the turn is not so far, and the car's speed is not high, then press softly the break,
•
If the the food was well-cooked and of quality, and the service was good, the the
tip should be higher than 15%.
Both in common life and in technology, a lot of imprecise conditionals (rules) are
considered. The need for its representation will be obvious in the next section.
3.2.2 The Case of Boolean Algebras
Let us consider, in the first place, the case in which the statements are crisp and
belong to a Boolean algebra
·
,
+
,
stand, respectively, for
the intersection (and), the union (or), and the complement (negation, not), 0 is the
minimum element, and 1 is the maximum. As it is well known, this Boolean algebra
is naturally ordered by means of the partial order
,
·
,
+
,
;
(
B
0
,
1
)
, where
a
·
a
b
⃔
a
·
b
=
a
⃔
a
+
b
=
b
⃔
b
=
a
+
b
.
Representing 'If
a
, then
b
'by
a
ₒ
b
, what is important is that from the set of
={
,
ₒ
}
premises
P
a
a
b
should follow the statement
b as a logical consequence
.
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