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In-Depth Information
P
y
⃒
μ
P
(
)
μ
P
(
)
μ
P
is an
[
,
]−
verifies:
x
x
y
, and hence
0
1
degree for
P
in
[
,
]
[
,
]
that are around five.
Obviously, this degree does not perfectly reflect the primary use of
P
.
0
10
, that originates
a fuzzy set P
∼
of the numbers in
0
10
Example 1.1.8
Consider
P
=
big
in
X
=[
0
,
10
]
. Let us show several possible
degrees for it, after agreeing that '
x
P
y
if and only if
x
y
, in the linear order of
[
0
,
10
]
'.
A
[
0
,
1
]
-degree
μ
P
is any function
X
=[
0
,
10
]ₒ[
0
,
1
]
, such that
If
x
y
,
then
μ
P
(
x
)
μ
P
(
y
),
that is, any non-decreasing function (of which there are many). We can also agree
that
μ
P
(
0
)
=
0, and
μ
P
(
10
)
=
1. With this, it is clear that all degrees for
big
will
show some
family resemblance
.
Once fixed
(
L
,
)
=
(
[
0
,
1
]
,
)
, and
P
=
, the different uses of big only
depend on which function
μ
P
is chosen to reflect the meaning of the predicate in
[
0
,
10
]
. Of course, it is
P
=
μ
P
if and only if function
μ
P
is strictly non-decreasing,
x
2
as it is the case either with
μ
P
(
x
)
=
x
/
10, or with
μ
P
(
x
)
=
/
100. Provided
μ
P
is not strictly non-decreasing as, for example, with
P
4 (strictly), but
μ
P
(
)
=
μ
P
(
)
=
.
μ
P
does not perfectly reflect the
since 6
6
4
0
5,
primary use of
big
in
[
0
,
10
]
. In the same way, the crisp degree
1
,
if
x
>
8
μ
P
(
x
)
=
0
,
otherwise
,
does not perfectly reflect the primary use of
big
when translated into 'after eight'.
Another possible model for
μ
P
is
⊧
⊨
⊩
0
,
if
x
∈[
0
,
2
]
x
−
2
μ
P
(
x
)
=
,
if
x
∈[
2
,
8
]
6
1
if
x
∈[
8
,
10
]
,
with graphic.
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