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μ
Q
|
P
,
μ
not P
,
Remark 1.4.1
The decomposability or functional expressibility of
μ
aP
,
μ
PandQ
, and
μ
Por Q
, is not a general property of the L-degrees of the predi-
cates
Q
P
,
not P
,
aP
,
PandQ
, and
Por Q
. What has been shown at such respect
with functions
J
|
, respectively, is just to be taken as examples of
the existence of L-degrees. Although in the applications of fuzzy logic is currently
accepted that all these predicates are functionally expressible, that is, expressed
through numerical functions
,
N
,
A
,
∗
,
and
↕
J
,
∗
,
↕:[
0
,
1
]×[
0
,
1
]ₒ[
0
,
1
]
,
N
:[
0
,
1
]ₒ[
0
,
1
]
,
A
:
X
ₒ
X
,
it should not be considered that this is always the case.
Remark 1.4.2
Q
|
P
is an example of a
relational predicate
, that is, a predicate
R
on
larger,
implies, around, etc. Of course, once either
x
or
y
are fixed, what results is a predicate
(unary) in
Y
or
X
, respectively, as it is with '
x
is around
y
', if
X
X
×
Y
such that
(
x
,
y
)
∈
R
, with
x
∈
X
, and
y
∈
Y
. For example,
R
=
=
Y
=[
0
,
10
]
,
where with
y
.
Relational, or binary, predicates can be either precise or imprecise. In the first
case, they originate a crisp subset of
X
=
5 it results the unary predicate
around five
in
[
0
,
10
]
×
Y
defined by
ˉ,
(
,
)
∈
if
x
y
R
μ
R
(
x
,
y
)
=
ʱ,
otherwise
.
In the second, they originate an L-set in
X
×
Y
defined by
μ
R
(
x
,
y
)
=
Degree in
L
up to which it is
(
x
,
y
)
∈
R
,
once an L-degree for
R
is known.
Remark 1.4.3
In the case
L
=[
0
,
1
]
, functions
J
:[
0
,
1
]×[
0
,
1
]ₒ[
0
,
1
]
allowing
to represent
μ
Q
|
P
by
J
ⓦ
(μ
P
×
μ
Q
),
are called
fuzzy relations
, and if the predicate
Q
|
P
interprets a rule, these relations are called
fuzzy conditionals
.
1.4.4 Group Meaning
The meaning of words is not fixed for all people and all context. For example,
in a dinner with three commensals the deliciousness of the dessert plates could
easily result in three different orderings of such plates. Since language is a social
phenomenon, also meaning is such, and it is possible to talk on the meaning of
predicates for a group of people in, of course, a given context.
For a group of people
G
={
p
1
,...,
p
m
}
, a predicate
P
on
X
can show
m
primary
meanings
P
,
i
,1
i
m
. Since
m
(
P
,
i
)
=
P
,
G
i
=
1
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