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Remark 3.2.2
If the Boolean algebra
B
is complete, that is, for any
A
B
,
A
,
∈
it exists
Sup A
B
, then
a
+
a
+
Sup
{
z
∈
B
;
a
·
z
b
}=
Sup
{
z
∈
B
;
z
b
}=
b
.
Remark 3.2.3
The character of conditional of
a
+
b
is exclusive of Boolean algebras.
a
+
That is, in any ortholattice, the validity of
a
·
(
b
)
b
,
for all
a
,
b
, forces the
ortholattice to be a Boolean algebra.
a
+
Remark 3.2.4 a
b
, is a property that only holds in Boolean algebras, that
is, in ortholattices the equivalence
a
ₒ
b
a
+
b
, is not valid. It only holds
in Boolean algebras. For example, in orthomodular lattices, both
a
·
z
b
⃔
z
a
+
ₒ
1
b
=
a
·
b
,
b
+
a
·
b
(that verify
a
b
ₒ
1
a
), are conditionals, but is
and
a
ₒ
2
b
=
ₒ
2
b
=
neither
a
ₒ
1
b
a
ₒ
2
b
nor
a
ₒ
2
b
a
ₒ
1
b
.
a
+
The conditional
a
ₒ
1
b
=
a
·
b
is called the
Sasaki hook
, and
a
ₒ
2
b
=
b
+
b
is the
Dishkant hook
, and, of course, only in Boolean algebras are both
coincidental with
a
+
a
·
b
. The Sasaki and the Dishkant hooks are used as models for
the conditional statements in the reasoning in Quantum Logic.
Remark 3.2.5
The scheme of
Modus Ponens
If
a
,
then
b
a
b
,
corresponds to
forwards reasoning
, that is, goes from the antecedent
a
to the con-
sequent
b
thanks to the conditional
a
b
.
Backwards
reasoning
goes from the consequent to the antecedent (also thanks to
a
ₒ
b
, through
a
·
(
a
ₒ
b
)
ₒ
b
), ad it
is modeled by the
Modus Tollens
scheme.
If
a
,
then
b
not
b
not
a
,
that is translated by
b
·
(
a
b
)
+
a
a
+
a
ₒ
b
)
⃔
a
ₒ
b
(
=
b
. Thus,
a
+
in Boolean algebras,
a
ₒ
b
=
b
, also allows backwards reasoning, provided
b
·
(
b
·
(
a
+
a
·
b
a
ₒ
b
)
=
b
)
=
=
0, or
a
+
b
=
1. Nevertheless, although
b
·
(
a
, it is clear that the conjunctive conditional
a
a
·
b
)
=
0
ₒ
b
=
a
·
b
does
not allow backwards reasoning since
b
·
(
a
ₒ
b
)
=
0.
3.2.3 Fuzzy Conditionals
Let us return to the case of fuzzy logic, that is, to a conditional linguistic expression,
or rule, like 'If
x
is
P
, then
y
is
Q
', represented in fuzzy terms by
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