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{
}ↂ
(
{
}
)
C
x
, for all
{
}∈
A
C
y
and
y
C
z
,
Proof
Since
x
C
x
,itis
x
x
.If
x
∈
(
{
}
),
∈
(
{
}
)
{
}ↂ
(
{
}
)
(
{
}
)
ↂ
it is
y
C
x
and
z
C
y
, hence from
y
C
x
follows
C
y
C
(
C
(
{
x
}
))
={
x
}
, and
z
∈
C
(
{
x
}
)
,or
x
C
z
.
Notice that the preorder
C
is defined only with the pairs
(
x
,
y
)
∈
X
×
X
such
that
{
x
}∈
A
and
{
y
}∈
A
.
•
Given a set
X
and
A
ↂ
P
(
X
)
,if
is a preorder in
X
such that
If
x
∈
P
, and
x
y
,itis
y
∈
Q
,forsome
Q
∈
A
,
the operator
C
:
A
ₒ
A
, defined by
C
(
P
)
={
y
∈
X
;∃
x
∈
P
&
x
y
}
,
verifies,
1.
∀
x
∈
P
,itis
x
x
, then
x
∈
C
(
P
)
:
P
ↂ
C
(
P
)
2. If
P
ↂ
Q
, and
y
∈
C
(
P
)
, there is
x
∈
P
such that
x
y
. Since
x
∈
Q
,itis
y
∈
C
(
Q
)
. Hence,
C
(
P
)
ↂ
C
(
Q
)
.
3. If
y
∈
C
(
C
(
P
)),
there is
x
∈
C
(
P
)
, such that
x
y
. But there is also
z
∈
P
such that
z
x
. Hence
z
y
, and
y
∈
C
(
P
).
That is,
C
(
C
(
P
))
ↂ
C
(
P
)
.
4. The preorder
C
coincides with the initial
, since
x
C
y
⃔
y
∈
C
(
{
x
}
)
⃔
x
y
.
Example 3.1.3
Let it be the set
X
={
1
,
2
,
3
,
4
,
5
}
endowed with the preorder
4
5
2
3
1
It is, for example
C
(
{
1
,
5
}
)
={
1
,
5
,
2
,
3
,
4
}
,
C
(
{
4
,
5
}
)
={
4
,
5
}
,
C
(
{
2
,
3
,
4
}
)
=
{
2
,
3
,
4
,
5
}
,
C
(
{
1
,
2
,
4
,
5
}
)
={
1
,
2
,
4
,
5
,
3
}=
X
,
C
(
{
5
}
)
={
5
}
,
and
C
(
{
3
}
)
=
{
3
,
.
Hence, it is possible to identify a logic in a set with a preordering of it.
4
,
5
}
3.1.2 Conjecturing
A process to pass from a set of premises
P
={
μ
1
,...,μ
n
}
to a 'conclusion'
˃
is a
conclusive reasoning
, that is sometimes symbolized by
P
˃
. A conclusive
reasoning
P
˃
is
deductive
if there exists either an operator of consequences
C
,
or a preorder
, such that
P
˃
is equivalent to
˃
∈
C
(
P
)
,orto
μ
i
˃
for all
μ
i
P
.
Anyway, not all conclusive reasonings are deductive. In common reasoning if
there is the possibility of stating
P
∈
˃
and defining
C
(
P
)
={
˃
;
P
˃
}
, the axiom
of monotonicity is not always verified by such
C
, but it could verify
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