I. Baroko of the second figure may be thus reduced by this method :

The conclusion of this syllogism is true if the premisses be true. If the conclusion 'Some C is not A' be not true, then its contradictory 'All C is A' must be true by Opposition, because of two contradictory propositions one must be true. Then combining this with the major premiss of the given syllogism, we have the following new syllogism in the perfect mood Barbara :

If the conclusion of this syllogism be true, its contradictory 'Some C is not B' must be false by Opposition; because of two contradictory propositions one must be false. But the latter is the minor premiss of the original syllogism, and is therefore true by supposition. Hence its contradictory, the conclusion of the new syllogism, must be false; and the falsity must be due either to the process of reasoning or to the premisses. The falsity can not be due to the process of reasoning, for the new syllogism is in the perfect mood Barbara; it must therefore be due to the premisses. It can not be due to the major premiss, which is also the major premiss of the original syllogism, and is therefore true by supposition: hence it must be due to the minor premiss‘All C is A,' that is, this premiss must be false, and its contradictory 'Some C is not A,' the conclusion of the original syllogism, is therefore true.

II. Bokardo of the 3rd figure may be thus reduced by this method:

(0)

(A)

Some B is not A,
All B is C;

(0) .. Some C is not A.

The conclusion of this syllogism is true, if the premisses be true. If the conclusion be not true, its contradictory 'All C is A' must be true by Opposition. Then taking this as a major premiss, and the minor premiss of the original syllogism as a minor premiss, we can form the following new syllogism in the perfect mood Barbara :

If the conclusion 'All B is A' be true, then its contradictory 'Some B is not A' must be false by Opposition; but this is not possible, as the latter is the major premiss of the original syllogism, and therefore true by supposition. Hence the former 'All B is A' must be false; and the falsity not being due to the reasoning process which is in the perfect mood Barbara, nor to the minor premiss 'All B is C' of the new syllogism, which is also the minor premiss of the original syllogism, and therefore true by supposition, it must be due to the falsity of the major premiss 'All C is A.' This proposition being false, its contradictory 'Some C is not A,' the conclusion of the original syllogism, is true.

The initial letter B of these two moods signifies that the new syllogism which arises in the process of reduction is in the mood Barbara, and the letter & indicates that the older logicians reduced them by the Indirect method.

The Indirect method of Reduction is also applicable to the other imperfect moods.

III. Take, for example, Cesare of the 2nd figure-

If this conclusion be not true, its contradictory 'Some C is A' must be true by Opposition. We can now form the following new syllogism in the perfect mood Ferio

If this conclusion be true, its contradictory 'All C is B' must be false. But this is not possible, as the proposition 'All C is B' is the minor premiss of the original syllogism, and therefore true by supposition. Hence the conclusion of the new syllogism is not true; and its falsity not being due to the reasoning process, nor to the major premiss of the syllogism, must be due to the falsity of the minor premiss 'Some C is A.' Hence this proposition is false, and its contradictory 'No C is A,' the conclusion of the original syllogism, is true.

IV. Take the mood Darapti of the 3rd figure

If this conclusion be not true, its contradictory 'No C is A' must be true. With this as a major premiss, and the minor premiss of the original syllogism as a minor premiss, we can form the following new syllogism in the perfect mood Celarent—

If this conclusion be true, its contrary 'All B is A' must be false by Opposition, because two contrary propositions can not both be true, and one must be false. But 'All B is A' being the major premiss of the original syllogism can not be false; hence No B is A,' the conclusion of the new syllogism, can not be true and must be false, the falsity being due, as in the preceding cases, to the major premiss 'No C is A' being false. This proposition being false, its contradictory 'Some C is A,' the conclusion of the original syllogism, must be true. § 7. Exercises.

1. What is Reduction? Is it necessary? Define Direct and Indirect Reduction, and distinguish them from each other.

2. Reduce by the Direct method the following moods:-Cesare, Disamis, Datisi, Ferison, Bramantip, Camenes, and Fesapo.

3. Reduce the following moods by the Indirect method:-Camestres, Felapton, Bramantip, Festino, Camenes, Dimaris, and Disamis. 4. Reduce both by the Direct and by the Indirect method the two moods Baroko and Bokardo.

5. Show by the Aristotelian method that the moods AAA, EAA, AII, and AEA are invalid in the second figure.

6. Find by the same method the conclusion, if any, to which the following combinations lead in the imperfect figures:-AA, AE, EA, OA, AO, and EI.

7. Show by the same method that the moods AAA, EAE, AEE are invalid in the third figure.

8. Determine by the same method the valid moods in the second figure.

9. Give concrete examples of the following moods, and reduce them both by the Direct and by the Indirect method:-Bramantip, Disamis, Baroko, Fesapo, and Bokardo.

10. Reduce the following pairs of premisses to the first figure and draw the conclusion, if any, which follows from each pair:

(i) No X is Y, all Y is Z, (ii) No X is Y, all Z is Y.

(iii) All Y is X, all Y is Z.

(iv) No Y is X, all Y is Z.

11. Test the following inferences by the method of Diagrams and also by the Aristotelian and scholastic methods.

(i) No A is B; no C is not-B; therefore all C is not-A.

(ii) All A is B; all C is not-B; therefore no C is A.

(iii) No not-B is C; all not-B is A; therefore some C is not-A.
(iv) None but material bodies gravitate; air gravitates: therefore
air is a material body.

(v) Plants alone have flowers; zoophytes have no flowers; there-
fore they are not plants.

CHAPTER V.

THE VARIOUS KINDS OF SYLLOGISMS.

§ 1. A Syllogism consists of two premisses and the conclusion which follows from them. It is evident that the two premisses of a syllogism may differ in Quality, Quantity, Relation, or Modality. The various kinds or divisions of syllogisms are founded upon the modifications of these general characters of their premisses. We have seen in a previous chapter that the division into Moods is founded upon the difference in Quantity and Quality of the two premisses. The division of syllogisms into Pure and Mixed is founded upon the difference in Relation of the premisses. The division into (1) Necessary, (2) Assertory, and (3) Probable is founded upon the difference in Modality of the premisses. The various kinds or divisions may be shown thus in a tabular view :

The two classes of Pure and Mixed syllogisms, founded on the difference in Relation of the premisses, are thus subdivided.