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X
,
·
,
+
,
)
. If the complement
is involutive
(μ
=
μ)
(
[
,
]
gives the new algebra
0
1
,
(μ
+
˃)
=
μ
·
˃
.
Analogously, with the operation
then
μ
·
˃
=
(μ
+
˃
)
, one has the new algebra
X
,
·
,
+
,
)
and, if
is involutive,
(μ
·
˃)
=
μ
+
˃
.
(
[
0
,
1
]
2.2.3 Non-contradiction and Excluded-Middle
An statement is self-contradictory whenever entails its negation. For example, the
only classical set that is self-contradictory is the empty one:
A
c
A
c
A
ↆ
⃒
A
∩
A
ↆ
A
∩
⃒
A
ↆ ∅ ⃒
A
= ∅
.
Perhaps, this is the reason of the difficulty children do have on accepting that
∅
is a
set!
Within an algebra of fuzzy sets there are many self-contradictory fuzzy sets. For
example, with
N
=
1
−
id it is
1
2
,
μ
μ
⃔
μ(
x
)
1
−
μ(
x
)
⃔
μ(
x
)
∀
x
∈
X
,
hence:
μ
is self-contradictory if and only if
μ
μ
. Analogously, with the strong
1
2
1
−
x
negation
N
(
x
)
=
x
,itis
1
+
1
−
μ(
x
)
μ
μ
⃔
μ(
2
x
)
)
⃔
μ(
x
)
+
2
μ(
x
)
−
1
0
1
+
μ(
x
√
2
⃔
μ(
x
)
−
1
,
∀
x
∈
X
,
μ
μ
√
2
−
1
.
Notice that 1/2 is the fixed-p
oi
nt of the strong negation
N
that is,
μ
is self-contradictory if and only id
=
1
−
id
(
1
−
n
=
, and that
√
2
n
⃔
n
=
1
/
2
)
−
1 is the fixed-point of the strong negation
√
2
1
−
id
1
−
n
N
=
id
(
=
n
⃔
n
=
−
1
)
.
1
+
1
+
n
Notice that if
μ
P
μ
aP
, since it is always supposed that
μ
aP
μ
not P
,itfollows
μ
P
μ
not P
, and
μ
P
is self-contradictory.
The classical principle of non-contradiction, “it is impossible to have both an
statement and its negation”, could be interpreted as “P and not P is impossible”, or
“P and not P is self-contradictory”. All general algebras of fuzzy sets do verify the
principle of non-contradiction once stated in this form.
X
,
·
,
+
,
)
Theorem 2.2.13
If
(
[
0
,
1
]
is an algebra of fuzzy sets, it holds the principle
μ
·
μ
(μ
·
μ
)
for all
X
. That is, for all
of non-contradiction stated by:
μ
∈[
0
,
1
]
X
,
μ
·
μ
is self-contradictory.
μ
∈[
0
,
1
]
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