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In-Depth Information
Proposition 2. (V1)
W/Ind
A
ↆ{
[[
i
]]
M
:
i
∈
ʘ
}
.
W/Ind
A
∪{∅}
(V2)
{
[[
i
]]
M
:
i
∈
ʘ
}ↆ
.
(V3)
[[
i
]]
M
∩
[[
j
]]
M
=
∅
,fori,j
∈
ʘ,andi
=
j.
Ind
A
(V4)
For i,j
∈
ʘ,if
(
x, y
)
/
∈
,
[
x
]
Ind
A
=[
i
]]
M
and
[
y
]
Ind
A
=[
j
]]
M
,then
i
=
j.
From conditions (V1)-V(4), it is evident that the element of
ʘ
are used as
nominals to name the equivalence classes of
Ind
A
such that different equivalence
classes are provided with different names. We use these nominals to give the
following deductive system. Let
B
1
2
3
ↆA
,
B
∈{
B
,
B
,
B
}
,and
i
∈
ʘ
.
Axiom schema:
1. All axioms of classical propositional logic.
2.
B
(
ʱ
→
ʲ
)
→
(
B
ʱ
→
B
ʲ
).
1
∅
3.
ʱ
→
ʱ
.
1
1
∅
4.
ʱ
→
∅
♦
ʱ
.
1
1
∅
1
∅
5.
♦
∅
♦
ʱ
→
♦
ʱ
.
m
∅
ʱ
ₔ
n
∅
ʱ
,where
m, n
∈{
1
,
2
,
3
}
6.
.
7.
C
ʱ →
B
ʱ
for
C ↆ B
.
8. (
a, v
)
k
→
a
(
a, v
)for
k
∈{
1
,
3
}
.
a
9.
¬
(
a, v
)
→
¬
(
a, v
).
v∈V
a
(
a, v
)
(
a, v
))
.
a
1
∅
10.
i
→
∧
(
i
→
v∈V
b
(
b,v
)
(
b,v
))
ʱ
.
11.
i
∧
1
B∪{b}
ʱ
→
1
B
ₔ
1
∅
(
i
→
→
2
B∪{b}
2
12. (
b,v
)
∧
ʱ
→
B
((
b,v
)
→
ʱ
).
v∈V
b
(
b,v
)
ʱ
.
3
B
3
B
1
∅
13.
i
∧
ʱ
→
(
i
→
(
b,v
))
→
→
∪{
b
}
1
∅
14.
i
∧
(
a, v
)
→
(
i
→
(
a, v
)).
15.
i
∧¬
(
a, v
)
→
1
∅
(
i
→¬
(
a, v
)).
16.
i∈ʘ
i
.
17.
¬
i
∨¬
j
for distinct elements
i
and
j
of
ʘ
.
1
A
18.
i
→
i
.
Rules of inference
:
N. ʱ
MP.
ʱ
B
ʱ
ʱ
→
ʲ
ʲ
The notion of theoremhood is defined in the usual way, and we write
ʱ
to
indicate that
ʱ
is a theorem of the above deductive system.
Axioms 7-13 relate attribute, attribute-values of the objects with the relations
corresponding to the modal operators
B
,
k
. For instance, let
R
B
be
∈{
1
,
2
,
3
}
2
B
, and let us see how the
axioms 7, 10 and 12 relate attribute, attribute-values of the objects with
R
B
.
Axiom 7 for
the relation corresponding to the modal operator
2
B
corresponds to the condition
R
B
ↆ
R
C
for
C
ↆ
B
. Axiom 10
