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(
)
∩
(
)
=
(
)
∩
(
)
=
(
)
∩
(
)
=∅
.
Cons
P
Hyp
P
Cons
P
Sp
P
Hyp
P
Sp
P
Notice that
•
Consequences follow
(in the partial order
)from
μ
∧
:
all the premises explain
the consequences
.
•
Hypotheses explain all the premises
, since
˃<μ
∧
μ
i
, all the premises follow
from each hypothesis.
•
Speculations, are the conjectures for which it is neither
μ
∧
˃
, nor
˃<μ
∧
.
With all that, processes,
•
P
˃
, with
˃
∈
Conj
(
P
)
, is a guessing, or conjectural reasoning
•
P
˃
, with
˃
∈
Cons
(
P
)
, is a deduction, or deductive reasoning
•
P
˃
, with
˃
∈
Hyp
(
P
)
, is an abduction, or abductive reasoning
•
P
˃
, with
˃
∈
Sp
(
P
)
, is an speculation, or speculative reasoning
Concerning this types of conclusive reasonings, it should be pointed out what follows.
1. The set
Conj
(
P
)
is not always in
A
=
P
0
(
X
)
. Hence, it can't be taken as a set
of premises, it has not sense to consider
Conj
(
Conj
(
P
))
,or
Cons
(
Conj
(
P
))
.
2. If
P
ↂ
Q
,itis
Conj
(
Q
)
ↂ
Conj
(
P
)
, because if
˃
∈
Conj
(
Q
)
,from
Inf Q
˃
,itis
Inf P
˃
. Hence, if
Inf P
,if
Inf Q
˃
∈
Conj
(
Q
)
it is
˃
∈
, and
Conj
is anti-monotonic.
3. The sets
Hyp
Conj
(
P
)
(
P
)
and
Sp
(
P
)
are not always in
P
0
(
X
)
. Hence, they can't be taken
as sets of premises.
4. If
P
ↂ
(
)
ↂ
(
)
˃
∈
(
)
μ
0
<˃ <
Inf Q
Q
,itis
Hyp
Q
Hyp
P
, since
Hyp
Q
,or
Inf P
. Hence, the operator
Hyp
is anti-monotonic.
5. Concerning the operator Sp, if
P
implies
μ
0
<˃ <
ↂ
Q
it can be
Sp
(
P
)
Sp
(
Q
)
, and
Sp
(
Q
)
Sp
(
P
)
. Hence, Sp is a non-monotonic operator.
3.2 Reasoning with Conditionals: Representation
3.2.1 What is a Conditional?
A conditional is an statement of the form 'If
a
, then
b
'
:=
a
ₒ
b
, with two previous
statements
a
,
b
. For example, “If it is raining, then I take an umbrella”, where
a
=
It is raining,
b
I take an umbrella, or “If the food is well cooked and well served,
and the wine is of good quality, then the tip will be higher than usual”, wit
a
=
=
The
food is well cooked and well served, and the wine is of good quality,
b
=
The tip
will be higher than usual.
Notice that the first example is a crisp conditional, but the second is an impre-
cise one. In what follows we will take into account the
representation of imprecise
conditionals
of the type
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