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DEDUCTION: Part 3 - Page a

Figures and Moods of
The Categorical Syllogism

by Jonathan Dolhenty, Ph.D.

 

The Third Figure

 
M P

M S

S P

The middle term (M) is the subject of both premises.

The rule for the Third Figure is: The minor premise must be affirmative and the conclusion must be particular.

Explanation of the Rule

First Part: The minor premise must be affirmative.

If the minor premise is negative, the major premise would have to be affirmative. This is because the premises cannot consist of two negative propositions, according to General Rule Number 6.

In that case, however, the major term (P), which is the predicate of the major premise, would be a particular term, since it is the predicate of an affirmative proposition.

According to General Rule Number 7, a negative minor premise would necessitate a negative conclusion. That would make its predicate, which is the major term (P), a universal term. This would result in an illicit major, a violation of General Rule Number 2.

This will occur every time the minor premise is negative. So the minor premise can never be negative and to avoid an illicit major, the minor premise must be affirmative.

Second Part: The conclusion must be particular.

The minor term (S) will always be a particular term in the premise since, according to the first part of this rule, the minor premise must be affirmative. In this minor premise the minor term (S) is the predicate. The predicate of an affirmative sentence is always particular.

The minor term (S) is the subject of the conclusion. It must also be a particular term in the conclusion. Otherwise General Rule Number 2 would be violated and we would have an illicit minor. Therefore, the conclusion must be particular.

The Valid Moods of the Third Figure

When we consider the moods which are valid for the syllogisms of the Third Figure, we find that six of the eight possible legitimate combinations are valid and two are invalid. Here are the diagrams:

Moods of the Third Figure
A
uM
+
pP
A
uM
+
pS

pS
+
pP

Valid
A
uM
+
pP
E
uM
-
uS

uS
-
uP

Illicit Major
A
uM
+
pP
I
pS
+
pS

pS
+
pP

Valid
A
uM
+
pP
O
pM
-
uS

uS
-
uP

Illicit Major
E
uM
-
uP
A
uM
+
pS

pS
-
uP

Valid
E
uM
-
uP
I
pM
+
pS

pS
-
uP

Valid
I
pM
+
pP
A
uM
+
pS

pS
+
pP

Valid
O
pM
-
uP
A
uM
+
pS

pS
-
uP

Valid

Clearly, there are six valid moods for the Third Figure: A A, A I, E A, E I, I A, and O A. Each one follows the rule that the minor premise must be affirmative and the conclusion must be particular.

The A E and A O moods are invalid because of an illicit major brought into the conclusion. They violate the first part of the rule: The minor premise must be affirmative.

Study the following arguments which illustrate the valid moods:

A

All birds have wings.

A

All birds are bipeds.

Therefore, Some bipeds have wings.

A

All geniuses think they are wise.

I

Some geniuses are foolish people.

Therefore, Some foolish people think they are wise.

E

No misers are generous.

A

All misers are people who abuse their riches.

Therefore, Some people who abuse their riches are not generous.

E

No animals are immortal.

I

Some animals are dogs.

Therefore, Some dogs are not immortal.

I

Some Communists are revolutionary.

A

All Communists are radicals.

Therefore, Some radicals are revolutionary.

O

Some criminals are teenagers.

A

All criminals break the law.

Therefore, Some who break the law are teenagers.

To Page 2-b of Figures and Moods of the Syllogism


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