These lines mean that there are four valid moods in the first figure, and four in the second, that the third figure contains six valid moods, and the fourth five. The three vowels in the name of each of the moods stand for the three propositions of the mood-the 1st for its major premiss, the 2nd for its minor premiss, and the 3rd for its conclusion. Thus the three vowels EAE in the mood Celarent signify that the major premiss is an E proposition, the minor an A proposition, and the conclusion an E proposition; and so with the rest.

There are two methods of reducing the imperfect moods, that is, the moods in the imperfect figures to the perfect; or rather of proving the truth of the conclusion of a mood in an imperfect figure by reduction to a perfect mood, that is, to a mood in the perfect figure :—(1) the one is called Direct or Ostensive Reduction, and (2) the other Indirect Reduction or Reductio per deductionem ad impossibile (i. e. Reduction by deduction to impossibility). In the first method the premisses of an imperfect mood are converted, obverted, contraposed, or transposed in order to form with them a mood in the first figure, having a conclusion which is the same as the original conclusion, or from which the original conclusion can be obtained by some process of immediate inference. In the second method, the truth of the conclusion of an imperfect mood is proved by showing, with the aid of the perfect moods and the rules of immediate inference by Opposition, that the contradictory of the conclusion is false.

§ 5. Ostensive Reduction :

The processes to be employed for reducing the imperfect moods by this method are indicated by certain letters contained in the names of the various moods. The initial letters B, C, D, F indicate that the imperfect moods are to be reduced to the perfect moods, having the same initial letters. The letter s means that the proposition signified by the vowel before it is to be converted simply. The letter p indicates that the proposition signified by the vowel before it is to be converted by limitation (per accidens). When s or p occurs after the conclusion

of an imperfect mood, i.e., after the third vowel in its name, then its signification is to be applied to the conclusion of the new syllogism, that is, this conclusion must be converted simply or by limitation in order to obtain the conclusion of the imperfect mood. The letter m means that the premisses of the imperfect syllogism are to be transposed. The letter k means that the mood containing it was reduced by the older logicians by the Indirect method. The other letters (namely l, r, n, t) are entirely meaningless, and are introduced only for phonetic purposes to make up clearly sounding words. Thus C in Camestres means that it is to be reduced to Celarent; m that the premisses are to be transposed, that is, the major premiss of this is to become the minor of the new syllogism, and the minor the major premiss; the s after the minor premiss, that that premiss is to be converted simply; and the s after the conclusion or the third vowel, that the conclusion of the new syllogism in the mood Celarent is to be converted simply in order to obtain the original conclusion; while the consonants t, r are entirely non-significant.

I. Take, for example, the mood Camestres of the 2nd figure:All metals are elements,

(A)

All A is B

(E)

No C is B

(E).. No C is A

No compounds are elements; .. No compounds are metals.

By converting simply the minor premiss, and transposing the premisses of this, we get the following new syllogism in the perfect mood Celarent:

(E)

No B is C

(A)

All A is B

(E) ... No A is C

No elements are compounds,

All metals are elements;

.. No metals are compounds.

The converse of the conclusion of the new syllogism is the same as the conclusion of the original syllogism.

II. Take the mood Festino of the 2nd figure

of the

perfect

No men are perfect,
Some beings are perfect;

.. Some beings are not men.
P

By converting simply the major premiss we get the fol

lowing:

(E) No B is A

(1)

Some C is B

(0).. Some C is not A

No perfect beings are men,

Some beings are perfect;

.. Some beings are not men.

This is in the perfect mood Ferio. Its conclusion is the

same as that of the original syllogism.

III. Take the mood Darapti of the 3rd figure

(A)

All B is A

(A)

All B is C

(I). Some C is A

All men are rational,

All men are imperfect;

.. Some imperfect beings are rational.

By converting per accidens the minor premiss we get the following:

(A) All B is A

(I)

Some C is B

(I). Some C is A

All men are rational,

Some imperfect beings are men;

.. Some imperfect beings are rational.

This is in the perfect mood Darii. Its conclusion is the same

as that of the original syllogism.

IV. Take the mood Felapton of the 3rd figure

(E)

No B is A

(A)

All B is C

(0) .. Some C is not A

No men are perfect,

All men are rational;

.. Some rational beings are not perfect.

By converting per accidens the minor premiss we get the following:

(E) No B is A

No men are perfect,
Some rational beings are men;

(0).. Some C is not A .. Some rational beings are not perfect.

This is in the perfect mood Ferio. Its conclusion is the same as that of the original syllogism.

V. Take the mood Bramantip of the 4th figure—

By transposing the premisses we get the following :—

This is a syllogism in the perfect mood Barbara. The converse of its conclusion is the same as the conclusion of the original syllogism.

VI. Take the mood Dimaris of the 4th figure

This is a syllogism in the perfect mood Darii. The converse of its conclusion is the same as the conclusion of the original syllogism.

VII. Take the mood Fresison of the 4th figure

This is in the perfect mood Ferio. The conclusion is the same as the original conclusion.

The directions given above for reduction are not sufficient for the two imperfect moods Baroko and Bokardo. The older logicians reduced them by the method to be next described, namely, Indirect Reduction. They may be, however, reduced to the first figure, by the method of Direct Reduction, thus :—

(0) Some C is not B

All men are mortal,

Some beings are not mortal;

(0).. Some C is not A .. Some beings are not men.

By contraposing the major premiss, and obverting the minor premiss, we get the following syllogism :—

(E) No not-B is A Some C is not-B (0).. Some C is not A

(I)

· No immortal being is man,

Some beings are immortal;

.. Some beings are not men.

This is a syllogism in the perfect mood Ferio, of which 'A' and 'C' are the major and minor terms, and 'not-B' the middle term.

IX. Bokardo of the 3rd figure

This is a syllogism in the perfect mood Darii, of which ‘C' and 'not-A' are the major and minor terms, and 'B' the middle term. By converting simply the conclusion of the new syllogism and then obverting the converse, we can easily obtain the conclusion of the original syllogism.

The processes employed for reducing them will be sufficiently indicated if Baroko and Bokardo be called Facoko and Doclamosk respectively, c signifying that the proposition signified by the vowel before it is to be contraposed, k that the proposition is to be obverted, and s as usual, that the proposition is to be simply converted.

§ 6. Indirect Reduction, or, Reductio per deductionem ad impossibile.