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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
b
a +
b , is not valid. It only holds
in Boolean algebras. For example, in orthomodular lattices, both a
·
z
b
z
a +
1 b
=
a
·
b ,
b +
a ·
b (that verify a
b 1 a ), are conditionals, but is
and a
2 b
=
2 b
=
neither a
1 b
a
2 b nor a
2 b
a
1 b .
a +
The conditional a
1 b
=
a
·
b is called the Sasaki hook , and a
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
b , through a
· (
a
b
)
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
b
)
a
b
(
=
b . Thus,
a +
in Boolean algebras, a
b
=
b , also allows backwards reasoning, provided
b · (
b · (
a +
a ·
b
a
b
) =
b
) =
=
0, or a
+
b
=
1. Nevertheless, although
b · (
a , it is clear that the conjunctive conditional a
a
·
b
) =
0
b
=
a
·
b does
not allow backwards reasoning since b · (
a
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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