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1.2.3 Antonyms and Negations
With pairs of antonyms
, it should always be taken into account that con-
ditional statements like “If the bottle is empty, then it is not full”, conduct to the
inequality
(
P
,
aP
)
full
, showing that
not P could be taken as the
biggest antonym of P
. It is not often the case in which
aP
μ
aP
μ
not P
,
with
P
=
not P
, practically it
only happens when a
P
is such that there is not any linguistic term
aP
in the language.
When
aP
and
not P
are not coincidental, it is said that
aP
is a
strict antonym
of
P
.
When modeling
=
μ
aP
=
μ
P
ⓦ
A
, and
μ
not P
=
N
ⓦ
μ
P
, with a symmetry
A
of
X
and a strong negation
N
in
[
0
,
1
]
, it results the
condition of coherence
:
μ
P
ⓦ
A
N
ⓦ
μ
P
,
between
A
and
N
, that should be always verified. If
A
is known,
N
should be chosen
to satisfy this coherence's condition, and if what is known is
N
, then
A
should be
chosen to verify such condition.
Example 1.2.4
With
N
0
(
a
)
=
1
−
a
, and
A
(
x
)
=
10
−
x
in
X
=[
0
,
10
]
,if
⊧
⊨
0
,
if
x
∈[
0
,
4
]
x
−
4
μ
big
(
x
)
=
,
if
∈[
4
,
8
]
4
⊩
1
,
if
x
∈[
8
,
10
]
,
results
⊧
⊨
0
,
if
x
∈[
6
,
10
]
6
−
x
μ
small
(
x
)
=
μ
big
(
10
−
x
)
=
,
∈[
,
]
if
2
6
4
⊩
1
,
if
x
∈[
0
,
2
]
,
⊧
⊨
1
,
if
x
∈[
0
,
4
]
8
−
x
μ
not big
(
x
)
=
1
−
μ
big
(
x
)
=
,
if
∈[
4
,
8
]
⊩
4
0
,
if
x
∈[
8
,
10
]
,
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