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3.3 Short Note on Other Modes of Reasoning
The mode of reasoning given by the scheme
{
μ, μ
ₒ
˃
}
˃
is classically called
Modus Ponendo Ponens
(from the Latin, mode of starting the
truth (of
˃
) by placing the truth (of
μ
)), or, for short
Modus Ponens
. That given by
{
˃
,μ
ₒ
˃
}
μ
is classically called
Modus Tollendo Tollens
(from
the Latin, mode of stating the falsity (of
the scheme
)), or, for
short
Modus Tollens
. They correspond to what we called forwards and backwards
reasoning. But there are again other
modes
of reasoning that can be considered, for
example,
μ
) by placing the falsity (of
˃
{
μ
,μ
+
˃
}
˃
•
Modus Tollendo Ponens
, given by the scheme
and also calledMode
μ
·
(μ
+
˃)
=
μ
·
μ
+
μ
·
˃
=
of Disjunctive Reasoning, and classically proven by
μ
·
˃
μ
(in a Boolean algebra).
{
μ, (μ
·
˃)
}
˃
, classically proven
•
Modus Ponendo Tollens
, given by the scheme
μ
·
(μ
·
˃)
=
μ
·
(μ
+
˃
)
=
μ
·
˃
˃
(in a Boolean algebra).
by
•
Constructive Dilemma
, given by the scheme
{
μ
+
ʻ, μ
ₒ
˃, ʻ
ₒ
ʷ
}
˃
+
ʷ
,
(μ
+
ʻ)
·
(μ
ₒ
˃)
·
(ʻ
ₒ
ʷ)
=
(μ
+
ʻ)
·
(μ
+
˃)
·
(ʻ
+
ʷ)
=
μ˃ ʻ
+
ʷʻμ
+
ʷμ˃
+
ʷʻ˃
˃
+
ʷ
classically proven by
a
+
(in a Boolean algebra where
a
ₒ
b
=
b
).
{
μ
+
˃
,ʻ
ₒ
μ, ʷ
ₒ
˃,
}
ʻ
+
˃
,
•
Destructive Dilemma
, given by the scheme
(μ
+
˃
)(ʻ
+
μ)(ʷ
+
˃)
=
ʻ
(μ
·
ʷ
+
μ
˃)
+
ʷ
(˃
·
ʻ
+
μ
·
˃
)
ʻ
+
ʷ
(in a Boolean algebra where
a
classically proven by
a
+
ₒ
b
=
b
).
What in the fuzzy case? For example, in the case of the Disjunctive Mode we
need to find all the possibilities for
μ
·
(μ
+
˃)
˃
, that is, to solve the functional
equation
T
(
N
(
a
),
S
(
a
,
b
))
b
for all
a
,
b
in
[
0
,
1
]
, in the three variables
T
,
S
,
N
. With
b
=
0, it follows
T
(
N
(
a
),
a
)
=
0, or
T
=
W
,
N
N
. Taking
N
=
N
it results
W
˕
(
N
˕
(
a
),
S
˕
˕
˕
))
=
˕
−
1
(
a
,
b
(
max
(
0
,
1
−
˕(
a
)
+
˕(
S
(
a
,
b
))
−
1
))
=
˕
−
1
(
max
(
0
,
+
˕(
S
(
a
,
b
))
−
˕(
a
)))
b
, implying
˕(
S
(
a
,
b
))
−
˕(
a
)
˕(
b
)
)
˕
−
1
or
˕(
S
(
a
,
b
))
˕(
a
)
+
˕(
b
)
. Hence,
S
(
a
,
b
(
min
(
1
,˕(
a
)
+
˕(
b
)))
=
W
∗
(
W
˕
a
,
b
)
. For example, it can be taken
S
=
or
S
=
max.
))
=
˕
−
1
)))
˕
−
1
•
W
˕
(
N
˕
(
a
),
max
(
a
,
b
(
max
(
0
,˕(
b
)
−
˕(
a
(˕(
b
))
=
b
W
˕
(
))
=
˕
−
1
)))
˕
−
1
•
W
˕
(
N
˕
(
a
),
a
,
b
(
min
(
1
−
˕(
a
), ˕(
b
(˕(
b
))
=
b
Hence, the disjunctive mode can be used in, for example, the cases
(
W
˕
)
W
˕
,
N
˕
,
max
)
and
(
W
˕
,
N
˕
,
.
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