Appendix: On the Reduction of Syllogisms

book_5 Before you read

Reduction is the restating of a syllogism of the Second, Third, or Fourth Figure in the form of the First Figure, which is the standard and most transparent form of deduction. The mnemonic names encode reduction instructions in their consonants: 's' (simple conversion of the preceding proposition), 'p' (conversion per accidens), 'm' (transposition of the premises), and 'c' (indirect reduction by reductio ad impossibile — applicable only to Baroco and Bocardo). The initial consonant (B, C, D, or F) identifies the target First Figure mood. The procedure is: (1) identify the Figure by the position of the middle term; (2) identify the Mood by the proposition-type vowels; (3) find the mnemonic name for that Figure and Mood; (4) apply the reduction instructions given by its consonants. Worked examples for each Figure illustrate the procedure.

To reduce a syllogism means to restate it in the shape of the First Figure. Only syllogisms of the Second, Third, and Fourth Figures are reducible. The First Figure has four valid moods:

Name	Mood	Example
Barbara	AAA	All men are mortal / John is a man / Therefore, John is mortal.
Celarent	EAE	No man is a spirit / John is a man / Therefore, John is not a spirit.
Darii	AII	All men are mortal / Some rational beings are men / Therefore, some rational beings are mortal.
Ferio	EIO	No man is a spirit / Some rational beings are men / Therefore, some rational beings are not spirits.

(Note: the singular proposition “John is a man” is equivalent to a universal A-proposition, since the singular term is taken in full Extension.)

The Mnemonic Names of All Valid Moods

All valid moods in all four Figures are given Latin proper names whose vowels indicate the proposition types (A, E, I, O) in order:

Barbara, Celarent, Darii, Ferio (First Figure) Cesare, Camestres, Festino, Baroco (Second Figure) Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison (Third Figure) Bramantip, Camenes, Dimaris, Fesapo, Fresison (Fourth Figure)

The Reduction Consonants

Certain consonants in the mnemonic names carry instructions for reduction:

Consonant	Instruction
s	Convert simply the premiss indicated by the preceding vowel
p	Convert per accidens (accidentally) the premiss indicated
m	Transpose (exchange) the major and minor premisses
c	Reduction is indirect (applies to Baroco and Bocardo)

(Note: the initial consonant — B, C, D, or F — indicates which mood of the First Figure is the target: B → Barbara, C → Celarent, D → Darii, F → Ferio.)

Procedure for Reduction

Step 1. Identify the Figure by the position of the middle term:

Position of Middle Term	Figure
Subject of major, predicate of minor	First
Predicate of both	Second
Subject of both	Third
Predicate of major, subject of minor	Fourth

Step 2. Identify the Mood by the vowels in the propositions (major premiss vowel, minor premiss vowel, conclusion vowel). Find the mnemonic name in the line for that Figure which contains those vowels in order.

Step 3. Apply the reduction instructions given by the consonants in the name.

Worked Examples

Example 1: Third Figure, Datisi → Darii

Syllogism to reduce:

(A) All men are mortal beings. (I) Some men are wise beings. (I) Therefore, some wise beings are mortal beings.

Step 1. Middle term men is the subject of both premisses → Third Figure.

Step 2. Propositions are A, I, I. In the Third Figure line, Datisi contains the vowels A-I-I in order. Datisi begins with D → target is Darii.

Step 3. Consonants in Datisi after the vowels: the s follows the second vowel (the I of the minor premiss) → convert the minor premiss simply.

Simply convert “Some men are wise beings” → “Some wise beings are men.”

Reduced syllogism (Darii, First Figure):

(A) All men are mortal beings. (I) Some wise beings are men. (I) Therefore, some wise beings are mortal beings.

Example 2: Fourth Figure, Bramantip → Barbara

Syllogism to reduce:

(A) All men are mortal beings. (A) All mortal beings are bodily beings. (I) Therefore, some bodily beings are men.

Step 1. Middle term mortal beings is the predicate of the major premiss and the subject of the minor → Fourth Figure.

Step 2. Propositions are A, A, I. In the Fourth Figure line, Bramantip contains the vowels A-A-I in order. Bramantip begins with B → target is Barbara.

Step 3. Consonants in Bramantip: m → transpose the premisses; p (following the conclusion vowel I) → may be ignored as it refers to the conclusion, not a premiss.

Transpose the premisses:

(A) All mortal beings are bodily beings. (A) All men are mortal beings. (A) Therefore, all men are bodily beings.

Reduced syllogism (Barbara, First Figure). Note: the conclusion changes from I to A in the reduction — the I-conclusion of Bramantip was a weakened form of the A-conclusion that Barbara justifies.

A Note on Baroco and Bocardo

Syllogisms of the Second Figure in Baroco and the Third Figure in Bocardo contain O-propositions, which, as we have learned, cannot be directly converted. These cannot be reduced by the direct method described above. Their reduction, if attempted, requires an indirect method — as indicated by the consonant c in their names. This indirect reduction is an advanced topic; the student who has followed the direct method closely will have little difficulty understanding it on further study.

Practical Note

As Reinstadler observes, the practical value of reduction is inconsiderable. Lepidi has gone further: “If this matter is found difficult for beginners, let it be omitted; their loss will not be great.” The value of the exercise is primarily as mental training — rigorous practice in the analysis of argument. For the student who can master its complexities, it is splendid gymnastics for the mind. For the student whose time is limited, the Eight General Laws of the Syllogism are the indispensable equipment; reduction is secondary.