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≤
→
≤
→
(i) If
I
1
I
I
2
then
I
2
I
I
I
I
1
I
I
≤
→
≤
→
(ii) If
I
1
I
I
2
then
I
I
I
1
I
I
I
I
2
I
I
≥
I
I
→
(iii)
I
(iv) [1
,
1]
→
I
I
=
I
I
I
then
I
I
I
= [1
,
1]
(v) If
I
≤
→
I
((
I
1
I
2
)
(vi) (
I
1
→
I
I
2
)
∗
→
I
I
)
≤
I
(
I
1
→
I
I
).
(vii)
k
(
I
→
I
I
k
) = (
I
→
I
k
I
k
)
(viii)
I
1
∗
I
I
2
≤
I
I
3
iff
I
2
≤
I
I
1
→
I
I
3
.
3
GCT in the Context of Interval-Valued Semantics
In this section we propose a few different definitions for the semantic notion of graded
consequence. These definitions incorporate different decision making attitudes from
practical perspectives, and when the semantics for the object languageformulae is re-
stricted to the degenerate intervals, each of the notions yields the original notion of
graded consequence [7, 8].
3.1
Graded Consequence: Form (
ʣ
)
Definition 3.1.
Given a collection of interval-valued fuzzy sets, say
{
T
i
}
i
∈
I
,thegrade
) =
l
([
i
∈
I
{
x
∈
X
T
i
(
x
)
of
X
)
where
l
([
.
]) represents the left-hand end point of an interval; that is,
l
([
x
1
,
x
2
]) =
x
1
.
|≈
ʱ
,i.e.
gr
(
X
|≈
ʱ
→
I
T
i
(
ʱ
)
}
]),
...(
ʣ
The similarity and the differences between the notions of
gr
(
X
|≈
ʱ
), given in Sec-
tion 1 and form (
ʣ
), are quite visible from their respective forms. According to (
ʣ
),to
find out the degree to which
follows from
X
one has to first find outthetruth-interval
assignment to the formulae by a set of experts
ʱ
{
T
i
}
i
∈
I
. Then, the left-hand end point of
the least interval-value assigned to
x
∈
X
T
i
(
x
)
→
I
T
i
(
ʱ
) (if every member of
X
is true
then
is true) needs to be computed. In order to stick to a single value for the notion of
derivation, in this case, the left-hand end point of the resultant interval is taken.
One might think that valuefor(
ʱ
→
I
with
the component intervals, the left-hand end point of the concerned intervals are taken out,
and the corresponding implication operation for single-valued case is applied. In order
to show that (
ʣ
)would be the same if before computing
ʣ
) is not the same as computing inf
i
[
l
(
∧
x
∈
X
T
i
(
x
))
→
l
(
T
i
(
ʱ
)],letus con-
sider
l
([
.
3
,.
7]
→
I
is defined in terms of the ordinary Łukasiewicz im-
plication following the Definition 2.4. Then it can be checked that
l
([
.
3
,.
7]
→
I
[
.
2
,.
3]),where
→
I
[
.
2
,.
3])
=
.
6, and
l
([
.
3
,.
7])
→
Ł
l
([
.
2
,.
3]) =
.
9.
}
i
∈
K
) =
l
(
i
{
Lemma 3.1.
inf
i
l
(
}
i
∈
K
).
Lemma 3.2.
i
I
i
∗
I
i
I
i
≤
I
i
(
I
i
∗
I
I
i
),where
I
i
,
I
i
are intervals.
Lemma 3.3.
i
j
I
ij
=
j
i
I
ij
.
{
[
x
i
,
y
i
]
[
x
i
,
y
i
]
Lemma 3.4.
If
∗
I
is a t-representable t-norm with respect to an ordinary t-norm
∗
then
I
1
∗
I
I
2
≤
I
I
3
implies
l
(
I
1
)
∗
l
(
I
2
)
≤
l
(
I
3
).
