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μ
·
μ
=
μ
0
holds if and only
Then, the (restricted) non-contradiction principle
=
˕
if
T
W
˕
and
N
N
˕
, for any order-automorphism
and any t-conorm
S
.For
example, it holds (with
˕
=
id) if
T
=
W
,
S
=
max,
N
=
N
0
, and it does not hold
provided
T
=
min, or
T
=
pr od
.
˕
µ
+
µ
=
µ
1
2.2.8.4 Restricted Excluded-Middle Principle
A
c
μ
+
μ
=
μ
1
for fuzzy
With classical sets it always holds
A
∪
=
X
. When is it
sets? When it does hold the equation
S
(
a
,
N
(
a
))
=
1 for all
a
∈[
0
,
1
]
?
Theorem 2.2.42
If S is a continuous t-conorm, and N is a strong negation, it is
W
ˈ
S
(
a
,
N
(
a
))
=
1
for all a
∈[
0
,
1
]
, if and only if S
=
and N
ˈ
N.
W
ˈ
Proof
With
N
(
n
)
=
n
∈
(
0
,
1
)
, it follows
S
(
n
,
n
)
=
1. That is
S
=
, and 1
=
W
ˈ
(
))
=
ˈ
−
1
a
,
n
(
a
(
min
(
1
, ˈ(
a
)
+
ˈ(
N
(
a
))))
,or1
=
min
(
1
, ˈ(
a
)
+
ˈ(
N
(
a
)))
.
)
=
ˈ
−
1
Hence, 1
ˈ(
a
)
+
ˈ(
N
(
a
))
,or
N
ˈ
(
a
(
1
−
ˈ(
a
)
N
(
a
))
. That is,
N
ˈ
N
.
The reciprocal is a simple calculation.
μ
+
μ
Then, the (restricted) excluded-middle principle
=
μ
1
holds if and only
W
ˈ
if
S
=
and
N
ˈ
N
, for any order automorphism
ˈ
and any t-norm
T
.For
W
∗
,
example, it holds (with
ˈ
=
id) if
S
=
T
=
min
,
N
=
N
0
, but it does not hold
pr od
∗
.
provided
S
=
max or
S
=
2.2.8.5 Both Restricted Principles of Non-contradiction
and Excluded-Middle
From last theorems it immediately follows that,
Theorem 2.2.43
In a standard algebra of fuzzy sets with a triplet
(
T
,
S
,
N
)
, it holds
μ
·
μ
=
μ
0
and
μ
+
μ
=
μ
1
if and only if T
W
ˈ
=
W
˕
,S
=
, and N
ˈ
N
N
˕
.
W
∗
, and
N
x
2
In particular, they hold if
T
=
W
,
S
=
=
N
0
, or with
˕(
x
)
=
and
ˈ(
x
)
=
x
, they hold with the triplet:
max
T
(
x
,
y
)
=
(
0
,
x
2
+
y
2
−
1
),
S
(
x
,
y
)
=
min
(
1
,
x
+
y
),
1
1
−
x
N
(
x
)
−
x
2
,
for all
x
,
y
in
[
0
,
1
]
.
W
˕
Of course, with
˕
=
ˈ
, the principles hold with
W
˕
,
, and
N
=
N
.
˕
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