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Some Instances of Graded Consequence
in the Context of Interval-Valued Semantics
Soma Dutta
1
,Benjamın R.C. Bedregal
2
, and Mihir Kr. Chakraborty
3
1
MIMUW, University of Warsaw, Poland
somadutta9@gmail.com
2
UFRN, Natal, Brazil
bedregal@dimap.ufrn.br
3
Jadavpur University, Kolkata, India
mihirc4@gmail.com
Abstract.
This paper proposes some instances of graded consequence relation
where the object languageformulae are interpreted by sub-intervals of [0, 1].
These instances represent different attitudes of decision making that may be
called conservative, liberal, and moderate.
Keywords:
Graded consequence, Interval semantics, Imprecise reasoning.
1
Introduction
The theory of graded consequence (GCT) [6, 7] was introduced as a general meta-theory
where for any set of formulae
X
and formula
follows from
X
is a
matter of grade. Let us explain two main features of the theory of graded consequence.
(i) Classically,
X
ʱ
, that a formula
ʱ
|
=
ʱ
iff for all states of affair
T
i
if
X
ↆ
T
i
then
ʱ
∈
T
i
. Formally
this meta-linguistic sentence turns outtobe
∀
T
i
{
(
X
ↆ
T
i
)
→
ʱ
∈
T
i
}
,where
X
ↆ
T
i
is again a meta-level sentence representing
T
i
).Ingraded context,
T
i
sarefuzzy sets assigning values to the object level formulae; and the meta-linguistic
connective
∀
x
∈
F
(
x
∈
X
→
x
∈
,arecomputed by a fuzzy implication and the lattice
'infimum' operator respectively. Thus the sentence, '
→
and quantifier
∀
ʱ
(semantically) follows from
X
'
becomes graded; the grade is denoted by
gr
(
X
|≈
ʱ
). It is to be noted that following
and extending [20],
T
i
}
i
∈
I
is taken to be any arbitrary collection of fuzzy sets over
formulae; that is considering the whole collection of
T
i
's is not a necessity here. In [7],
a complete residuated lattice is considered for interpreting the meta-linguistic entities
of the notion of graded consequence. So, given any collection of fuzzy sets
{
{
}
T
i
I
,the
i
∈
|≈
ʱ
meta-linguistic sentence viz.,
X
gets the value,
|≈
ʱ
)=in
i
{
→
}
,
gr
(
X
in f
x
∈
X
T
i
(
x
)
f
T
i
(
ʱ
)
where
→
f
is the residuum operator of the complete residuated lattice.
(ii) GCT, thus, proposes a meta-theoretic set up where derivation is a graded notion.
As a part of the programme of building the meta-theory, some of its areas of concern are
(a) axiomatizing the notion of consequence (
|∼
) (b) defining its semantic counterpart
(
), (c) studying their interrelations, and also (d) studying other meta-logical notions
and their interrelations.
|≈
