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μ
·
μ
(μ, μ
)
μ
μ
·
μ
(μ, μ
)
μ
; from the
Proof
It is
min
, and
min
μ
(μ
·
μ
)
, and then with the second follows
μ
·
μ
first inequality it follows
(μ
·
μ
)
.
and the complement
,
are needed for the proof of this theorem. In the algebras of fuzzy sets the non-
contradiction principle is a theorem: the algebra's axioms imply the principle. It
is not true, as it is sometimes stated, that fuzzy sets do not verify the principle of
non-contradiction in which science is based.
The classical principle of Excluded-Middle, “It is always P or not P”, can be
interpreted as “Not (P or Not P) is a self-contradiction”and it is verified by all algebra
of fuzzy sets.
Notice that no additional hypotheses on the connective
·
,
,
·
,
+
,
)
X
Theorem 2.2.14
If
(
[
0
,
1
]
is an algebra of fuzzy sets, it holds the principle
(μ
+
μ
)
((μ
+
μ
)
)
for all
X
. That is,
of Excluded-Middle stated by
:
μ
∈[
0
,
1
]
X
,
(μ
+
μ
)
is self-contradictory.
for all
μ
∈[
0
,
1
]
Proof
It is,
(μ, μ
)
μ
+
μ
⃒
(μ
+
μ
)
μ
⃒
(μ
)
((μ
+
μ
)
)
•
μ
max
•
μ
(μ, μ
)
μ
+
μ
⃒
(μ
+
μ
)
(μ
)
max
(μ
+
μ
)
((μ
+
μ
)
)
.
then
, and the complement
,
are needed for the proof of this theorem: In the algebra of fuzzy sets the excluded-
middle principle is a theorem. In conclusion,
+
Notice that no additional hypotheses on the connective
X
,
·
,
+
,
)
In all algebra of fuzzy sets
(
[
0
,
1
]
,
•
The logic principles of non-contradiction and excluded-middle are theorems, once
stated through the concept of self-contradiction.
A very different situation appears if these two principles are stated as it is currently
done within logic and classical set theory, that is, by stating
•
“P and not P” is false
•
“P or not P” is true,
or,
•
There is no
x
in
X
such that “
x
is
P
and
x
is not
P
”
•
For all
x
in
X
it is “
x
is
P
or
x
is not
P
”
translated into
•
(μ
P
·
μ
not P
)(
x
)
=
0
,
for all
x
in
X
•
(μ
P
+
μ
not P
)(
x
)
=
1
,
for all
x
in
X
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