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It is clear that if
P
is finite,
C
is compact. This happens, for example if
X
is finite.
A consequence's operator
C
is
consistent
when '
q
q
∈
∈
(
)
⃒
(
)
C
P
C
P
'.
Example 3.1.1
In a finite lattice
(
X
,
·
,
+;
0
,
1
)
, let's consider
A
=
P
0
(
X
)
={
P
=
{
p
1
,...,
p
n
}ↂ
X
,
p
∧
=
p
1
···
p
n
=
0
}
, and the operator
Cons
:
P
0
(
X
)
ₒ
P
0
(
X
)
, defined by
Cons
(
P
)
={
q
∈
X
;
p
∧
q
}
,
with the partial order
given by
x
y
⃔
x
·
y
=
x
.
Cons
is a consequence's
operator:
1. Since
p
∧
p
i
,itis
p
i
∈
Cons
(
p
)
, for all
p
i
∈
P
. That is,
P
ↂ
Cons
(
P
)
.
2. If
P
={
p
1
,...,
p
n
}ↂ{
p
1
,...,
p
n
,
p
n
+
1
,...,
p
m
}=
Q
,
since
q
∧
=
p
1
···
p
n
·
p
n
+
1
···
p
m
p
1
···
p
n
=
p
,if
q
∈
Cons
(
P
)
,from
p
∧
q
, it follows
∧
q
∧
q
,
and
q
∈
Cons
(
Q
)
. Hence,
P
ↂ
Q
implies
Cons
(
P
)
ↂ
Cons
(
Q
)
.
3. Obviously, Min
Cons
(
P
)
=
p
∧
, hence,
Cons
(
P
)
∈
P
0
(
X
)
since Min
Cons
(
P
)
=
0. Then, if
q
∈
Cons
(
Cons
(
P
))
,orMin
Cons
(
P
)
q
,or
p
∧
q
,it
follows
q
∈
Cons
(
P
)
. Hence,
Cons
(
Cons
(
P
))
ↂ
Cons
(
P
)
.
In any finite lattice
(
X
,
·
,
+;
0
,
1
)
, it can be considered the logic
(
X
,
P
0
(
X
),
Cons
)
,
and it can be proved that if the lattice is endowed with a complement
such that
,
·
,
+
,
;
(
X
0
,
1
)
is a Boolean algebra, any operator of consequences
C
:
P
0
(
X
)
ₒ
P
0
(
X
)
verifies
C
ↂ
Cons
, that is
C
(
P
)
ↂ
Cons
(
P
)
, for all
P
∈
P
0
(
X
)
.
Cons
is
A
=
P
0
(
)
the biggest operator of consequences in a Boolean algebra with
X
.
p
∧
is not self contradictory (
p
∧
The set of
premises P
is
consistent
,if
p
∧
). If
p
∧
,itwillbealso
p
∧
∈
it were
p
∧
Cons
(
P
)
, that is absurd if
Cons
is consistent.
and
q
∈
q
(or
q
p
∧
In this cases,
q
∈
Cons
(
P
)
Cons
(
P
)
,or
p
∧
q
and
p
∧
)
p
∧
does imply
q
implies
p
∧
, that is absurd. Thus,
q
∈
Cons
(
P
)
∈
Cons
(
P
)
,
and the operator
Cons
is consistent.
X
endowed with a fuzzy
Remark 3.1.2
Instead of a lattice, let us take the set
[
0
,
1
]
intersection
μ
∧
=
μ
1
···
μ
n
=
T
ⓦ
(μ
1
×···×
μ
n
)
(
T
a continuous t-norm), the partial order
μ
˃
⃔
μ(
x
)
˃(
x
)
, for all
x
∈
X
, and
X
the empty set
μ
0
=
μ
∅
. Take the set
P
0
(
[
0
,
1
]
)
that consists of the finite subsets
X
P
={
μ
1
,...,μ
n
}ↂ[
0
,
1
]
such that
μ
∧
=
μ
0
. The definition
X
Cons
(
P
)
={
˃
∈[
0
,
1
]
;
μ
∧
˃
}
,
allows the same result as in the case before.
There is a, perhaps alternative, way of constructing a logic in a set
X
. It follows
from the following results.
•
If
(
X
, A,
C
)
is a logic in
X
, the binary relation '
x
C
y
⃔
y
∈
C
(
{
x
}
)
', defined
if
{
x
}∈
A
, is a preorder.
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