Information Technology Reference
In-Depth Information
Representation Theorems with Respect to
(
ʣ
)
Theorem 3.1.
For any
{
T
i
}
i
∈
I
,
|≈
in the sense of (
ʣ
)isagraded consequence relation.
X
,
x
∈
X
T
i
(
x
)
Proof.
(GC1) For
ʱ
∈
≤
I
T
i
(
ʱ
).So,using (v) we have
gr
(
X
|≈
ʱ
) =1.
Y
,
x
∈
Y
T
i
(
x
)
I
x
∈
X
T
i
(
x
). Hence by (i) GC2 is immediate.
(GC2) For
X
ↆ
≤
|≈
ʲ
∗
∪
|≈
ʱ
(GC3) inf
Z
gr
(
X
)
gr
(
X
Z
)
ʲ
∈
Z
l
[
i
∈
I
{
x
∈
X
T
i
(
x
)
l
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
→
}
∗
→
}
=inf
I
T
i
(
ʲ
)
]
I
T
i
(
ʱ
)
]
ʲ
∈
=
l
[
ʲ
∈
Z
{
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
T
i
(ʲ))
}
]
∗
l
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
→
I
T
i
(ʱ)
}
] (Lemma 3.1)
=
l
[
i
∈
I
{
ʲ
∈
Z
(
x
∈
X
T
i
(
x
)
→
I
T
i
(ʲ))
}
]
∗
l
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
→
I
T
i
(ʱ)
}
] (Lemma 3.3)
=
l
[
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
l
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
))
}
]
∗
→
I
T
i
(
ʱ
)
}
] (by(vii))...(1)
{
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
}∗
I
{
x
∈
X
∪
Z
T
i
(
x
)
Now for each
T
i
,
ʲ
)
→
I
T
i
(
ʱ
)
}
{
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
[(
x
∈
X
T
i
(
x
))
(
x
∈
Z
T
i
(
x
))]
=
ʲ
)
}∗
I
{
→
I
T
i
(
ʱ
)
}
≤
I
{
x
∈
X
T
i
(
x
)
→
I
T
i
(
ʱ
)
}
. (by (vi)).
Therefore,
i
∈
I
[
{
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
}∗
I
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
)
→
I
T
i
(
ʱ
)
}
]
≤
I
i
∈
I
[
x
∈
X
T
i
(
x
)
→
I
T
i
(
ʱ
)].
...(2)
Also,
i
∈
I
[
{
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
∗
I
i
∈
I
[
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
)
}
]
→
I
T
i
(
ʱ
)
}
]
≤
I
i
∈
I
[
{
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
}∗
I
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
)
→
i
T
i
(
ʱ
)
}
] (Lemma 3.2)
≤
I
i
∈
I
[
x
∈
X
T
i
(
x
)
→
I
T
i
(
ʱ
)]
(by (2))
l
[
i
∈
I
(
x
∈
X
T
i
(
x
)
→
I
ʲ
∈
Z
T
i
(
l
[
i
∈
I
{
x
∈
X
∪
Z
T
i
(
x
)
ʲ
))
}
]
∗
→
I
T
i
(
ʱ
)
}
]
l
[
i
∈
I
{
x
∈
X
T
i
(
x
)
≤
→
I
T
i
(
ʱ
)
}
] (Lemma 3.4 as
∗
I
is t-representable)
...(3)
Hence from (1) and (3) we have, inf
ʲ
∈
Z
gr
(
X
|≈
ʲ
)
∗
gr
(
X
∪
Z
|≈
ʱ
)
≤
gr
(
X
|≈
ʱ
).
Theorem 3.2.
For any graded consequence relation
|∼
there is a collection of interval-
valued fuzzy sets such that
|≈
generated in the sense of (
ʣ
) coincides with
|∼
.
|∼
{
}
X
∈
P
(
F
)
Proof.
Given
,agraded consequence relation, let us consider the collection
T
X
of interval-valued fuzzy sets over formulae such that
T
X
(
)]
BIR
,where[
x
]
BIR
represents the best interval representation [3] of
x
,i.e.theinterval[
x
,
x
].
We want to prove
gr
(
X
ʱ
)=[
gr
(
X
|∼
ʱ
|∼
ʱ) =
l
[
Y
∈
P
(
F
)
{
x
∈
X
T
Y
(
x
)
→
I
T
Y
(ʱ)
}
].
By (GC3) and Lemma 3.1, we have
l
(
T
Y
(ʲ))
≥
l
(
∧
ʱ
∈
X
T
Y
(ʱ)).
As
T
Y
'saredegenerate intervals by proposition 4.1 of [16] every true identity express-
ible in [0, 1] is expressible in
U
.So,
T
X
∪
Y
(ʲ)
∗
I
∧
ʱ
∈
X
T
Y
(ʱ)
≤
I
T
Y
(ʲ). Then following
the proof in [7] we can show,
T
X
(
l
(
T
X
∪
Y
(ʲ))
∗
ʲ
)
≤
I
∧
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)] ...(i)
∧
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)]
≤
I
∧
X
ↆ
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)] ...(ii)
Now for
X
ↆ
Y
,by(GC2),
T
X
(
ʱ
)
≤
I
T
Y
(
ʱ
). Then by (GC1) we have
[1
,
1] =
∧
ʱ
∈
X
T
X
(
ʱ
)
≤
I
∧
ʱ
∈
X
T
Y
(
ʱ
).i.e.
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
) =
T
Y
(
ʲ
).
Hence (ii) becomes
∧
Y
∈
P
(
F
)
[
∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)]
≤
I
∧
X
ↆ
Y
∈
P
(
F
)
T
Y
(
ʲ
) =
T
X
(
ʲ
) ...(iii)
Combining (i) and (iii) we have
gr
(
X
].
Remark 1.
In this new context the meta-level algebraic structuremaybeviewedas
|∼
ʲ
) =
l
[
∧
Y
∈
P
(
F
)
{∧
ʱ
∈
X
T
Y
(
ʱ
)
→
I
T
Y
(
ʲ
)
}
U
,∗
I
,→
I
,
[0
,
0]
,
[1
,
1]
,
l
, a complete residuated lattice with a function
l
:
U
→
[0
,
1].
The structure
U
,∗
I
,→
I
,
[0
,
0]
,
[1
,
1]
,
l
is formed out of a complete residuated lattice
([0
,
1]
,
∗
,
→
,
0
,
1). Specifically, the adjoint pair (
∗
I
,
→
I
) is defined in terms of the adjoint
pair (
∗
,
→
).
