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˄
=
[
,
]
˄
=
μ
˄
Take
true
on
0
1
, with
, and the degree
as a non-decreasing
[
,
]ₒ[
,
]
μ
˄
(
)
=
μ
˄
(
)
=
function
0
1
0
1
, such that
0
0,
1
1, once accepting that '0 is
˄
' is false, and '1 is
˄
' is true. Taking
μ
˄
(
x
)
=
x
, as it is usual in fuzzy logic, it is
Degree up to which
x
is
P
is true
=
μ
˄
(μ
P
(
x
))
=
μ
P
(
x
),
for all
x
in
X
,
that allows to accept, as it is usual,
Degree of true of
x
is
P
=
μ
P
(
x
).
1.4.2 Linguistic Modifiers
Linguistic modifiers or linguistic hedges,
m
, are adverbs acting on
P
just in the con-
catenated form
mP
. For example, with
m
very tall
.
A characteristic that linguistically distinguishes imprecise predicates from pre-
cise ones, is that in the first case and once
P
and
m
are given,
mP
is immediately
understood. If
P
is precise (for example,
P
=
very
and
P
=
tall
,itis
mP
=
even
in the set of natural numbers),
mP
needs of a new definition to be understood (what it means
very even
?). Modifiers
only modify, but do not change abruptly imprecise predicates.
If
P
in
X
is with the use
=
(
P
, μ
P
)
, and
m
in
μ
P
(
x
)
ↂ
L
is with
ↂ
m
and
μ
m
, provided
mP
ↂ
P
, it can be taken the degree
μ
mP
=
μ
m
ⓦ
μ
P
,
since,
x
mP
y
⃒
x
P
y
⃒
μ
P
(
x
)
μ
P
(
y
)
⃒
μ
P
(
x
)
m
μ
P
(
y
)
⃒
μ
m
(μ
P
(
).
Among linguistic modifiers there are two specially interesting types:
x
))
μ
m
(μ
P
(
y
)),
or
(μ
m
ⓦ
μ
P
)(
x
)
(μ
m
ⓦ
μ
P
)(
y
•
Expansive modifiers
, verifying id
)
μ
m
,
μ
P
(
x
•
Contractive modifiers
, verifying
μ
m
id
)
.
μ
P
(
x
With the expansive, it results
id
)
(μ
P
(
x
))
=
μ
P
(
x
)
μ
m
(μ
P
(
x
))
=
μ
mP
(
x
)
:
μ
P
(
x
)
μ
mP
(
x
),
μ
(
x
P
for all
x
in
X
.
With the contractive, it results
μ
mP
(
x
)
=
μ
m
(μ
P
(
x
))
id
μ
P
(
x
)
(μ
P
(
x
))
=
μ
P
(
x
)
:
μ
mP
(
x
)
μ
P
(
x
),
for all
x
in
X
.
This is what happens in
L
=[
0
,
1
]
with the Zadeh's old definitions,
)
=
√
a
a
2
μ
more or less
(
, μ
v
er y
(
)
=
.
a
a
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