Information Technology Reference
In-Depth Information
{
}ↂ
( {
} )
C x , for all
{
}∈ A
C y and y
C z ,
Proof Since
x
C
x
,itis x
x
.If x
( {
} ),
( {
} )
{
}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 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 A
, defined by C
(
P
) ={
y
X
;∃
x
P & x
y
}
,
verifies,
1.
x
P ,itis x
x , then x
C
(
P
) :
P
C
(
P
)
2. If P
Q , and y
C
(
P
)
, there is x
P such that x
y . Since x
Q ,itis
y
C
(
Q
)
. Hence, C
(
P
)
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
))
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
 
Search WWH ::




Custom Search