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In-Depth Information
5 Logic for Other Types of Relations
Although the logic LNIS is confined to capture the approximations of concepts
relative to indiscernibility, similarity and inclusion relations only, but it can
easily be extended to capture other types of relations as well. In this section, we
briefly sketch the extensions to capture the indistinguishability relations defined
in Section 1. The syntax and semantics can be modified in a natural way to
accommodate these relations. As far as axiomatization is concerned, main task
is to come up with the axioms relating the relations with the attributes and
attribute-values of the objects. We list below the axioms for the relations defined
in Section 1. These axioms along with the axioms 1-6, 14-18 will give us the
desired sound and complete deductive system.
Negative Similarity Relation
-
C
ʱ
→
B
ʱ
for
C
ↆ
B
ↆA
.
→
a
(
v∈V
a
(
-
i
¬
(
a, v
)
∧
∅
(
i
→¬
(
a, v
)))).
-
¬
(
b,v
)
∧
B∪{b}
ʱ
→
B
(
¬
(
b,v
)
→
ʱ
).
Complementarity Relation
-
C
ʱ
→
B
ʱ
for
C
ↆ
B
ↆA
.
-
(
a, v
)
→
a
(
¬
(
a, v
)).
-
¬
(
a, v
)
→
a
((
a, v
)).
→
B
v∈V
b
(
b,v
))
ʱ
.
-
i
∧
B∪{b}
ʱ
¬
(
b,v
)
ₔ
∅
(
i
→
→
Weak Indiscernibility Relation
-
B
ʱ
→
C
ʱ
for
∅
=
C
ↆ
B
ↆA
.
→
B
b∈B
v∈V
b
(
b,v
)
(
b,v
))
,
B
-
i
ₔ
1
∅
(
i
→
∅
=
.
-
∅
↥
.
Weak Similarity Relation
-
B
ʱ
→
C
ʱ
for
∅
=
C
ↆ
B
ↆA
.
→
B
b∈B
v∈V
b
(
b,v
)
(
b,v
))
,
B
1
∅
-
i
∧
(
i
→
=
∅
.
-
∅
↥
.
Weak Inclusion Relation
-
B
ʱ
→
C
ʱ
for
∅
=
C
ↆ
B
ↆA
.
→
B
b∈B
v∈V
b
d
(
b,v
)
,
B
1
∅
-
i
(
i
→
(
b,v
))
→
=
∅
.
-
∅
↥
.
