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)=
r
[
Y
∈
P
(
F
)
{
ʣ
1
), and
gr
(
X
C
I
(
I
[
x
∈
X
T
Y
(
x
)
→
I
T
Y
(ʱ)]
)
}
]considering (ʣ
2
).
|∼
ʱ
considering (
) is a degenerate interval,
x
∈
X
T
Y
(
x
)
As for
ʱ
∈
F
and
Y
ↆ
F
,
T
Y
(
ʱ
→
I
T
Y
(
ʱ
) is a de-
generate interval. Hence by the construction of
I
,
I
ʵ
,
C
I
(
I
[
x
∈
X
T
Y
(
x
)
→
I
T
Y
(ʱ)]
)=
x
∈
X
T
Y
(
x
)
→
I
T
Y
(ʱ) =
C
I
(
I
[
x
∈
X
T
Y
(
x
)
→
I
T
Y
(ʱ)]
).
i.e.
r
(
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʱ)]
)
}
) =
r
(
∩
Y
∈
P
(
F
)
{
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʱ)]
)
}
)
∩
Y
∈
P
(
F
)
{
=
r
[
Y
∈
P
(
F
)
[
x
∈
X
T
Y
(
x
)
→
I
T
Y
(
ʱ
)]
Rest follows from the Theorem 3.4.
|≈
Let us distinguish the notions of graded semantic consequence by superscribing
(ʣ
)
,and
(ʣ
n
)
,
n
=1,2.
(ʣ)
,
with their respective forms. So, we have
|≈
|≈
|≈
ʣ
1
)
ʣ
)
(
ʣ
)
(
(
Theorem 3.8.
gr
(
X
|≈
ʱ
)
≤
gr
(
X
|≈
ʱ
)
≤
gr
(
X
|≈
ʱ
)
(ʣ
2
)
(ʣ
)
(ʣ)
and
gr
(
X
|≈
ʱ
)
≤
gr
(
X
|≈
ʱ
)
≤
gr
(
X
|≈
ʱ
).
Proof.
Using Lemma 3.9 we have,
l
(
∧
i
∈
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(
ʱ
)])
≤
∩
∧
→
≤
∩
∧
→
l
(
I
[
X
T
i
(
x
)
I
T
i
(
ʱ
)])
r
(
I
[
X
T
i
(
x
)
I
T
i
(
ʱ
)]) ...(i)
i
∈
x
∈
i
∈
x
∈
∩
i
∈
I
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʱ)]
),
∩
i
∈
I
C
I
(
I
j
Also
)]
)
ↆ
e
∩
i
∈
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(
ʱ
)]
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(
ʱ
∩
i
∈
I
C
I
(
I
j
and they are overlapping.So,
l
(
∩
i
∈
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(
ʱ
)])
≤
l
(
)]
))
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(
ʱ
r
(
∩
i
∈
I
C
I
(
I
j
≤
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʱ)]
))
≤
r
(
∩
i
∈
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʱ)]) ...(ii)
ʣ
1
)
ʣ
)
(
ʣ
)
(
(
|≈
≤
|≈
≤
|≈
(i) and (ii) imply,
gr
(
X
ʱ
)
gr
(
X
ʱ
)
gr
(
X
ʱ
).
I
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʱ)]
))
∩
∧
→
≤
∩
Also following Note 3.4,
l
(
I
[
X
T
i
(
x
)
I
T
i
(
ʱ
)])
r
(
i
∈
x
∈
i
∈
≤
r
(
∩
i
∈
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʱ)])
...(iii).
ʣ
2
)
ʣ
)
(
ʣ
)
(
(
|≈
≤
|≈
≤
|≈
Hence, combining (i) and (iii),
gr
(
X
ʱ
)
gr
(
X
ʱ
)
gr
(
X
ʱ
).
ʣ
1
)
I
ʵ
,
j
(of Definition 3.8) is chosen instead of
I
j
Note 3.5.
If for (
then
(ʣ
2
)
(ʣ
1
)
(ʣ
)
(ʣ)
gr
(
X
|≈
ʱ
)
≤
gr
(
X
|≈
ʱ
)
≤
gr
(
X
|≈
ʱ
)
≤
gr
(
X
|≈
ʱ
).
ʣ
n
)(
n
= 1, 2), satisfies (GC1), (GC2),
Theorem 3.9.
Given
{
T
i
}
i
∈
I
,
|≈
in the sense of (
and a variant of (GC3).
Y
,
C
I
(
I
j
≤
I
C
I
(
I
j
Proof.
(GC1) is immediate. If
X
[
∧
x
∈
Y
T
i
(
x
)
→
I
T
i
(ʱ)]
)and
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʱ)]
)
≤
I
C
I
(
I
[
∧
x
∈
Y
T
i
(
x
)
→
I
T
i
(ʱ)]
) are obtained by condition (ii) of Defini-
tion 3.5 and Theorem 3.6 respectively. Hence by Lemma 3.8, GC2 holds for (
ↆ
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʱ)]
)
ʣ
1
), (
ʣ
2
).
ʣ
n
),
n
= 1, 2, we prove that a variant form of (GC3), i.e.
Now for each of (
|≈
ʣ
n
|≈
ʣ
n
|≈
ʣ
inf
ʲ
∈
Z
gr
(
X
ʲ
)
∗
gr
(
X
∪
Z
ʱ
)
≤
gr
(
X
ʱ
) holds.
|≈
ʣ
2
|≈
ʣ
2
inf
ʲ
∈
Z
gr
(
X
ʲ
)
∗
gr
(
X
∪
Z
ʱ
)
{∩
i
∈
I
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʲ)]
)
}
]
∗
r
[
∩
i
∈
I
C
I
(
I
[
∧
x
∈
X
∪
Z
T
i
(
x
)
→
I
T
i
(ʱ)]
)
}
].
=inf
ʲ
∈
Z
[
r
∩
ʲ
∈
Z
{∩
i
∈
I
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʲ)]
)
}
]
∗
r
[
∩
i
∈
I
C
I
(
I
[
∧
x
∈
X
∪
Z
T
i
(
x
)
→
I
T
i
(ʱ)]
)
}
]. (Lemma 3.5)
=
r
[
∩
i
∈
I
{∩
ʲ
∈
Z
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʲ)]
)
}
]
∗
r
[
∩
i
∈
I
C
I
(
I
[
∧
x
∈
X
∪
Z
T
i
(
x
)
→
I
T
i
(ʱ)]
)
}
] (Lemma 3.6)
=
r
[
...(i)
∩
ʲ
∈
Z
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʲ)]
)
ↆ
e
∩
ʲ
∈
Z
{∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʲ)
}
ↆ
e
∧
ʲ
∈
Z
{∧
x
∈
X
T
i
(
x
)
Using Note 3.4,
→
I
T
i
(
ʲ
)
}
(Lemma 3.10).
∩
ʲ
∈
Z
C
I
(
I
[
∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʲ)]
)])
≤
r
(
∩
i
∈
I
[
∧
ʲ
∈
Z
{∧
x
∈
X
T
i
(
x
)
→
I
T
i
(ʲ)
}
]).
Hence,
r
(
∩
i
∈
I
[
∩
i
∈
I
[
C
I
(
I
[
∧
x
∈
X
∪
Z
T
i
(
x
)
→
I
T
i
(ʱ)]
)])
≤
r
(
∩
i
∈
I
[
{∧
x
∈
X
∪
Z
T
i
(
x
)
→
I
T
i
(ʱ)
}
]).
Similarly,
r
(
