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(
,
(
,
(
,
))) =
(
,
(
+
,
(
,
))) =
W
a
max
b
1
max
a
b
max
0
max
a
b
1
a
max
a
b
W
1
2
(
a
,
b
),
if b
a or b
>
a and b
>
b . It is a W-conditional.
1
2
0
,
if b
>
a and b
Example 3.2.11 The protoform
μ ˃ = μ · ˃
(coming from the classical con-
junctive conditional a
b
=
a
·
b
)
,gives
J
(
a
,
b
) =
T
(
a
,
b
),
functions with the inconvenience of the property J
(
a
,
b
) =
J
(
b
,
a
)
, but verifying,
T 0 (
a
,
J
(
a
,
b
)) =
T 0 (
a
,
T
(
a
,
b
))
T
(
a
,
b
)
min
(
a
,
b
)
b
that is, all of them are conditionals for any t-norm T 0 and in particular, for the greatest
of them. They are always taken as min-conditionals. For example,
If T
=
min, J
(
a
,
b
) =
min
(
a
,
b
)
, is called the Mamdani conditional
) = ˕ 1
If T
=
prod
, J
(
a
,
b
(˕(
a
) · ˕(
b
))
, are called Larsen conditionals
˕
T
=
W ˕
is never used, since it can be J
(
a
,
b
) =
0 with a
>
0 and b
>
0.
x
(
1
+
x
)
˕ 1
For example, with
˕(
x
) =
(an order automorphism), it is
(
x
) =
2
8 x
+
1
1
, and
2
a
(
1
+
a
)
b
(
1
+
b
)
ab
(
1
+
a
)(
1
+
b
)
) = ˕ 1
)) = ˕ 1
) = ˕ 1
J
(
a
,
b
(˕(
a
) · ˕(
b
(
.
(
)
2
2
4
ab ( 1 + a )( 1 + b ) + 1 1
2
=
that, of course, is a min-conditional.
Remark 3.2.12 A way of avoiding the undesirable symmetry J
(
a
,
b
) =
J
(
b
,
a
)
in
the case of Mamdani-Larsen min-conditionals, is taking
a r
b s
J
(
a
,
b
) =
T
(
,
),
a r
b s
a r
b s
b s
with real numbers r
,
s with 1
>
s . Then T 0 (
a
,
T
(
,
))
min
(
min
(
,
),
)
b s
b , and J is a min-conditional.
Example 3.2.13 Once given a conditional statement 'If x is P, then y is Q', and
represented 'x is P' by
μ P (
x
)
, and 'y is Q' by
μ Q (
y
)
, it remains to be understood
what it is meant by the 'statement'
μ P (
x
) μ Q (
y
)
. It could be, or not to be,
μ P (
x
) μ Q (
y
) = P
μ Q )(
x
,
y
)
, with
μ P
μ Q a fuzzy set in X
×
Y ,
( )
identified with some expression involving the connectives and
( · )
, or
( + )
, not
.
μ P
μ Q is expressible in material form, for
In the affirmative case, it is said that
μ P μ Q = μ P + μ Q ,or
μ P μ Q = μ P + μ P · μ Q , etc. These material
example,
 
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