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•
ↂ
(
)
ↂ
(
)
If
P
Q
, then
C
Q
C
P
,
C
is anti-monotonic.
•
ↂ
(
)
(
)
(
)
ↂ
If
P
Q
, then
C
P
and
C
Q
are not comparable, that is, it is neither
C
P
C
(
Q
)
, nor
C
(
Q
)
ↂ
C
(
P
)
,
C
is non-monotonic.
Most common conclusive reasonings are not deductive, but of a conjectural type, that
is, the conclusion
is provisionally accepted because, simply, it is non contradictory
with all or with part of the information. It should be noticed that the lost of the
monotonicity of
C
motivates the lost of the transitive property of the preorder
˃
C
.
is a conjectural kind of conclusive reasoning if there exists
an operator of consequences
C
, such that
We will say that
P
˃
˃
⃔
˃
∈
P
C
(
P
)
⃔
N
ⓦ
˃
∈
C
(
P
),
for some strong negation
N
. Analogously, if what we have is a preorder
(instead
of
C
),
˃
is a conjecture of the information contained in
P
, when
:
μ
˃
.
P
˃
⃔∀
μ
∈
P
In what follows, we will only take into account the operator
Cons
(
P
)
={
˃
;
μ
∧
˃
}
μ
∧
=
μ
0
. Consequently, the set of the conjectures that, through
Cons
, is associated to any finite set
P
, provided
={
μ
1
,...,μ
n
}
of premises such that
μ
∧
=
μ
1
···
μ
n
=
T
ⓦ
(μ
1
×
...
×
μ
n
)
=
μ
0
, for some continuous t-norm
T
,is
X
;
μ
∧
˃
}
c
(
)
={
˃
∈[
,
]
,
Conj
P
0
1
˃
=
with
, for some strong negation
N
.
Notice that it is necessary to take a pair of connectives
N
ⓦ
˃
(
T
,
N
)
to define
Conj
(
P
)
.
Remark 3.1.4
Instead of
μ
∧
=
μ
0
, in what follows we will suppose that
μ
∧
is not
μ
∧
μ
∧
self-contradictory, that is
. With this hypothesis, if
˃
∈
Cons
(
P
)
⃔
μ
∧
μ
∧
˃
because of
˃
μ
∧
μ
∧
μ
∧
˃
, it can't be
implies
. Hence, it should
μ
∧
˃
or
be
˃
∈
Conj
(
P
)
. That is,
Cons
(
P
)
ↂ
Conj
(
P
)
. In addition;
Cons
is
consistent. Notice that
μ
0
∈
Conj
(
P
)
.
μ
∧
μ
∧
In fact,
is more general than
μ
∧
=
μ
0
, since
μ
∧
=
μ
0
implies
μ
∧
=
μ
0
μ
1
=
μ
0
=
μ
∧
μ
∧
μ
∧
. Hence, if it is
,itisalso
μ
∧
=
μ
0
.
With this change,
Conj
(
P
)
=
Cons
(
P
)
∪
Hyp
(
P
)
∪
Sp
(
P
),
where
•
Hyp
(
P
)
={
˃
∈
Conj
(
P
)
;
μ
0
<˃ <μ
∧
}
, is the set of hypotheses of
P
.
•
Sp
(
P
)
={
˃
∈
Conj
(
P
)
;
μ
∧
NC
˃
}
, is the set of speculations of
P
, where NC
means non-comparable under
,
and verifying
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