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|≈
ʣ
2
|≈
ʣ
2
∗
∪
(i) becomes, inf
ʲ
∈
Z
gr
(
X
ʲ
)
gr
(
X
Z
ʱ
)
≤
∩
∧
ʲ
∈
Z
{∧
→
}
∗
∩
{∧
→
}
r
(
I
[
X
T
i
(
x
)
I
T
i
(
ʲ
)
])
r
(
I
[
Z
T
i
(
x
)
I
T
i
(
ʱ
)
])
i
∈
x
∈
i
∈
x
∈
X
∪
∩
{∧
→
∧
ʲ
∈
Z
T
i
(
}
∗
∩
{∧
→
}
])
Now following the steps below the inequality (1) of the proof of Theorem 3.3 we get:
inf
ʲ
∈
Z
gr
(
X
=
r
(
I
[
X
T
i
(
x
)
ʲ
)
])
r
(
I
[
Z
T
i
(
x
)
I
T
i
(
ʱ
)
i
∈
x
∈
I
i
∈
x
∈
X
∪
|≈
ʣ
2
|≈
ʣ
2
|≈
ʣ
ʲ
)
∗
gr
(
X
∪
Z
ʱ
)
≤
r
[
∩
{∧
X
T
i
(
x
)
→
I
T
i
(
ʱ
)
}
] =
gr
(
X
ʱ
).
i
∈
I
x
∈
|≈
ʣ
1
|≈
ʣ
1
|≈
ʣ
Similar is the argument for inf
ʲ
∈
Z
gr
(
X
ʲ
)
∗
gr
(
X
∪
Z
ʱ
)
≤
gr
(
X
ʱ
).
4Con lu ion
In this paper we have studied some possible ways of obtaining the notion of graded
consequence incorporating interval semantics for the object language only. One natural
direction is to extend the idea when both object languageformulae and the notion of
consequence assume intervals. This step is yet to be developed in ourfurther work, and
the attempt made in this paper would work as a basis for this future plan.
From the development made in this paper we observe that, in the context of interval
semantics, GCT is simultaneously exploiting two different lattice structures (
ↆ
e
)
over the set of sub-intervals of [0
,
1]. This adds an important dimension. Having en-
dowed with both the notions of
intervallying belowaset of intervals
(
≤
I
,
≤
I
)and
interval
lying in theintersection of a set of intervals
(
ↆ
e
)wemanage to address different atti-
tudes of decision making. Given a databse based on a set of experts opinion, different
notions of
|≈
are introduced to address the following aspects of decision making.(i)
(
) proposes a set up where the interval lying below all the experts' opinion would be
counted. (ii) (
ʣ
ʣ
) proposes a set up where the interval lying in the common consensus
zone would be counted. (iii) (
ʣ
1
) proposes a set up where the interval, considering ex-
pert's opinion, can be revised equally (finitely) many times, and then the interval lying
in the common consensus zone would be counted. (iv) (
ʣ
2
) proposes a set upwherethe
interval, taking care of expert's opinion, are revised (following a specific method viz.,
I
ʵ
) till they reach a concrete value, and then the common consensus zone is consid-
ered. We call the approach (i) as
conservative
, (ii) as liberal, and (iii), (iv) as moderate.
Thusthisstudy provides a theoretical framework where a decision maker having some
of the above attitudes derives, with certain degree, a decision from a set of imprecise
information.
Acknowledgements.
The authors acknowledgethevaluable comments of the anony-
mousreviewers.Thefirstauthor of this paper also acknowledges the support obtained
from The Institute of Mathematical Sciences, Chennai, India, during the initial phase of
preparation of this paper; the final preparation of this paper has been carried outduring
the tenure of an ERCIM 'Alain Bensoussan' fellowship.
