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|∼
from
P
(
F
),
the power set of formulae, to
F
, the set of formulae, satisfying the following axioms.
These axioms are generalizations of that of the classical notion of consequence [21, 17].
(GC1) If
A graded consequence relation [7] is characterized as a fuzzy relation
ʱ
∈
|∼
ʱ
X
then
gr
(
X
) =1.
ↆ
|∼
ʱ
≤
|∼
ʱ
(GC2) If
X
Y
then
gr
(
X
)
gr
(
Y
).
(GC3)
in f
ʲ
∈
Z
gr
(
X
|∼
ʲ)
∗
gr
(
X
∪
Z
|∼
ʱ)
≤
gr
(
X
|∼
ʱ).
ʱ
is a consequence of
X
, is a member of the underlying set of the complete residuated lat-
tice. The monoidal operation
For each set of formulae
X
and formula
ʱ
,
gr
(
X
|∼
ʱ
), read as the degree to which
,oftheresiduated lattice, computes meta-level conjunc-
tion. In [7] a representation theorem is proved establishing the soundness-completeness
like connection between
∗
. There are several other papers [8, 9, 13, 14] where
GCT is developed considering other meta-logical notions, their interrelations, axiomatic
counterpart of graded consequence, its proof theory, and GCT in the context of fuzzy
sets of premises too. It may be mentioned that some other researchers have also dealt
with similar ideas and of them some contributed towards generalization of the above
mentioned notion of GCT [5, 18, 19].
Development of GCT, to date, assumes a semantic base which is an arbitrary collection
|∼
and
|≈
{
T
i
}
i
∈
I
of fuzzy sets assigning single values to the object languageformulae. Each
T
i
may
be counted as an expert whose opinion, i.e., values assigned to the object languageformu-
lae, forms the initial context or the database. Based on the collective database of
T
i
}
i
∈
I
decisions are made. The decision maker wants to decide whether a particular formula
{
ʱ
is a consequence of a set
X
of formulae, which is a matter of grade in GCT. In this paper
we shall consider interval-valued semantics for the notion of graded consequence. More
specifically, we shall consider that experts are allowed to assign an interval to the object
languageformulae, and then based on the mechanism of GCT the valuetowhich
a formula
is a consequence of a set of formulae
will be computed.
There are plenty of instances where it is impossible to claim
precisely
that
an im-
precise concept applies to an object to a specific degree
.Asaresult when an imprecise
concept is quantized by a single value, the inherent impreciseness of the concept is
somewhat lost. Assigning an interval-value, to some extent, manages to retain the un-
certainty
of understading an imprecise concept
as it only attaches a set of possible inter-
pretations to the concept. In this regard let usquote a few lines from [10]. “
IVFS theory
emerged from the observation thatinalot of cases no objective procedure is available
to select the crisp membership degrees of elements inafuzzy set. It wassuggested to
alleviate that problem byallowing to specify onlyaninterval ...to which the actual
membership degree is assumed to belong
.Thus interval mathematics and its applica-
tion in the context of imprecise reasoning is quite significant. GCT provides a general
set up for imprecise reasoning. So, developing GCT in the context of interval-valued
semantics is meaningful both from the angle of theory building and applications. In this
paper we shall present three different attitudes of decision making based on GCT. The
information coming from different sources, which may be counted as the collection
of
T
i
's, as well as the attitude (conservative, liberal, moderate) of the decision maker
play roles in the process of decision making and in the final conclusion. Keeping this
practical motivation in mind we here propose three different notions for deriving con-
clusion which satisfy the graded consequence axioms [7]. In each of these cases, the
