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object languageformulae are interpreted by closed sub-intervals of [0, 1], but the no-
tion of consequence is made single-valued. This value assignment is done taking either
the left-hand end point or the right-hand end point or some value in between from the
final interval that is computed as an outcome. It is not completely unrealistic to think
that experts i.e.,
T
i
's are entitled to assignarangeofvalues, but the decision maker is
constrained to conclude a single value, and such a practice of precisification in final re-
sult prevails in the literature of fuzzy set theory, especially in the area of application of
the theory. The meta-linguistic notions, e.g. consequence, consistency, inconsistency,
could also get interval-values, and this direction of research will be taken upinour
future work.
2
Interval Mathematics: Some Basic Notions
Assigning a specific grade to an imprecise sentence often pushes usintoasituation
where from a range of possible values we are to choose a single one for the sake of the
mathematical ease of computation. Lifting the whole mathematics of fuzzy set theory
in the context of interval-valued fuzzy set theory, researchers [1-4, 10, 12, 11, 15, 16]
to a great extent could manage to resolve this problem. In this section we present some
part of the development [2-4, 10, 12, 11, 16] according to the purpose of this paper.
Let us consider
U
=
{
[
a
,
b
]
/
0
≤
a
≤
b
≤
1
}
along with two order relations
≤
I
and
ↆ
, defined by: [
x
1
,
x
2
]
≤
I
[
y
1
,
y
2
] iff
x
1
≤
y
1
and
x
2
≤
y
2
and
y
2
.
(
U
,≤
I
) forms a complete lattice, and (
U
,ↆ
) forms a poset. Let
be the lattice meet
corresponding to the order relation
[
x
1
,
x
2
]
ↆ
[
y
1
,
y
2
] iff
y
1
≤
x
1
≤
x
2
≤
≤
I
.
Definition 2.1
[2]
.
An interval t-norm is a mapping
T
:
U
×
U
→
U
such that
T
is
commutative, associative, monotonic with respect to
≤
I
and
ↆ
,and[1
,
1] is the identity
element with respect to
T
.
Definition 2.2.
[16] Let
T
be an interval t-norm.
T
is called t-representable if there ex-
ists t-norms
t
1
,
t
2
on [0
,
1] such that
T
([
x
1
,
x
2
]
,
[
y
1
,
y
2
]) = [
t
1
(
x
1
,
y
1
)
,
t
2
(
x
2
,
y
2
)].
Definition 2.3.
[16] For any t-norm
∗
on [0
,
1] and
a
∈
[0
,
1],
T
∗,
a
is defined below.
T
∗,
a
([
x
1
,
x
2
]
,
[
y
1
,
y
2
]) = [
x
1
∗
y
1
,
max
((
x
2
∗
y
2
)
∗
a
,
x
1
∗
y
2
,
x
2
∗
y
1
)].
[0
,
1],
T
∗,
a
is
an interval t-norm. Moreover, for
a
=1,
T
∗,
a
becomes a t-representable t-norm [16]; i.e.
T
∗,
1
([
x
1
,
x
2
]
,
[
y
1
,
y
2
]) = [
x
1
∗
In [11] it has been shown that for any t-norm
∗
on [0
,
1] and any
a
∈
y
2
].Forthepurpose of this paper we shall consider
such an
T
∗,
1
, and denote this interval t-norm based on
y
1
,
x
2
∗
∗
as
∗
I
.
→
I
:
U
×
U
→
U
is
Definition 2.4.
Given
→
,theresiduumof
∗
on [0
,
1],and1
∈
[0
,
1],
defined by: [
x
1
,
x
2
]
→
I
[
y
1
,
y
2
] = [
min
{
x
1
→
y
1
,
x
2
→
y
2
}
,
min
{
(
x
2
∗
1)
→
y
2
,
x
1
→
y
2
}
].
= [
min
{
x
1
→
y
1
,
x
2
→
y
2
}
,
min
{
x
2
→
y
2
,
x
1
→
y
2
}
].
In [16] it is shown that
→
I
is an interval fuzzy implication with the adjoint pair
→
I
)on
U
.For
I
1
,
I
2
,
I
∈
U
, the following properties of (
∗
I
,→
I
) are of particular
interest here.
(
∗
I
,
