the premisses are, by supposition, affirmative; therefore the conclusion must be affirmative. Conversely, it can be shown that to prove an affirmative conclusion, both the premisses must be affirmative. For, if one of the premisses be negative, the conclusion will, by Rule 6, be negative; therefore both the premisses must be affirmative.

8. If both the premisses be particular, nothing can be inferred. The two particular premisses are either II, IO, or 00 in any order. In the first combination the middle term is not distributed in either of the premisses. In the second, it may be distributed by being the predicate in O, but as the conclusion must be negative, a term will be distributed, also, in the conclusion, which was not distributed in the premisses; hence there will be an illicit process either of the subject or of the predicate in the conclusion. No conclusion follows from the last combination, both the premisses being negative. Hence it is true universally that nothing can be inferred if both the premisses be particular.

9. If one of the premisses be particular, the conclusion must be particular. If one premiss be particular, the other must be universal, for from two particular premisses nothing can be inferred.

Hence, the two premisses are either IA, or IE, or OA, or OE in any order. The conclusion of IA or AI must be particular, because in the premisses only one term (the subject in A) is distributed, and that, therefore, must be the middle term; and if the conclusion were universal, a term would be distributed in it which was not distributed in the premisses; hence there would be an illicit process. The conclusion of IE or EI must be particular, for if it were universal, there would be, as in the preceding case, an illicit process. In the premisses two terms only are distributed; of these one must be the middle term, and the other one only, therefore, can be distributed in the conclusion. But the conclusion must be negative, as one of the premisses is negative, and if it were, also, universal, both its subject and predicate would be distributed; and hence there would be a

term distributed in the conclusion, which was not distributed in the premisses. Similarly, the conclusion from OA or AO must be particular; only two terms are distributed in the premisses; of these one must be the middle term, and the other the predicate of the conclusion, which will be negative, and have, therefore, the predicate distributed. Hence the subject of the conclusion must be undistributed, that is, the conclusion must be particular; otherwise there would be an illicit process. No conclusion follows from OE, as both the premisses are negative.

This rule can also be proved from the diagrams. Take the combination IA. From the 3rd and 2nd diagrams follows a

particular conclusion, 'Some C is A,' and from the 1st and 2nd follows a particular conclusion, 'Some C is A.' In some cases, as in the 2nd and 2nd, a universal may follow; but as this does not follow in the other cases, it is inadmissible.

From this rule, it is evident that if the conclusion is universal, both the premisses must be universal. For, if one of the premisses be particular, the conclusion will be particular. Therefore both the premisses must be universal.

The last three rules, viz., the 7th, 8th, and 9th, are merely consequences of the other rules. A violation of any of those three rules is a result of the violation of some of the other rules. If the other rules are carefully observed, the last three must be observed along with them, and can not be violated.

§ 5. Division of Categorical Syllogisms into Figures.

Every valid categorical syllogism must conform to the nine rules, or conditions laid down and proved above. By the help of those rules, we can easily distinguish a valid from an invalid

categorical syllogism. Given any combination of two premisses and a conclusion, we can, by the aid of the rules, determine whether the conclusion follows from the premisses or not. When only two premisses are given, we can determine whether they lead to any conclusion, and if so, to what conclusion.

In every categorical syllogism there must be two premisses and a conclusion determined by the premisses. Given the premisses, the nature of the legitimate conclusion is given along with them. In the premisses, the middle term may have different positions in different syllogisms, and the primary division of categorical syllogisms is founded on the difference in position of the middle term in relation to the extremes in the premisses. The division is into three classes, technically called Figures, and is as follows:

(1) The middle term is the subject in one premiss, and predicate in the other.

(2) The middle term is the predicate in both the premisses. (3) The middle term is the subject in both the premisses. Taking B to be the middle term and A and C the extremes, the three classes may be thus symbolically expressed :

The conclusion expresses a relation between C and A, and is represented by a proposition whose subject and predicate are either A and C or C and A respectively.

If we always take C as the subject and A as the predicate in the conclusion, and call them the minor and the major term, and the two premisses in which they occur the minor and the major premiss respectively1, we get four classes or Figures as follows:

1 It should be observed that the distinction between the major and the minor term is purely conventional. There is no reason why the subject of the conclusion should be called the minor and the predicate the major term. It is due to usage that the two names 'minor term'

(1) In the 1st figure the middle term is the subject in the major premiss, and predicate in the minor premiss.

(2) In the 2nd, the middle term is the predicate in both the premisses.

(3) In the 3rd, the middle term is the subject in both the premisses.

(4) In the 4th, the middle term is the predicate in the major premiss and subject in the minor.

The conclusion is always a proposition, having C and A respectively for its subject and predicate.

The first classification or division is founded on the difference in position of the middle term in the premisses. The second is founded on this difference and on the distinction between the predicate and the subject in the conclusion, or between the major and the minor term, and the consequent distinction between the major and the minor premiss.

On the first method of classification of syllogisms there are three Figures, and on the second method there are four. On the first method the conclusion is of the form C A or of the form A C; and, on the second method, it is always of the form CA. As best adapted for teaching and as sanctioned by high authorities, we shall adopt here the four-fold classification, and take the conclusion to be always of the form C A1.

and 'major term' are applied to the subject and the predicate, respectively, in the conclusion. The definition of the minor term is that it is the subject, and the definition of the major term is that it is the predicate, in the conclusion; in other words, the term that is the subject in the conclusion is defined as the minor term, and the term that is the predicate as the major term of a syllogism.

1 Some logicians obtain the four figures by a double division. Ueberweg, for example, first divides all categorical syllogisms into

§ 6. Subdivision of Categorical Syllogisms in each Figure into Moods.

A syllogism may differ from another not only in the position of the middle term in the premisses, but also in the quantity and quality of the two premisses themselves. Each of the two premisses of a syllogism in each figure may consist of any one of the four propositional forms A, E, I, and O. The major premiss may be any one of these four forms, and the minor, again, may be any one of them. Thus there may be sixteen possible combinations of premisses in each figure, the first letter in each combination representing the major premiss, and the second letter the minor premiss, of a possible syllogism:

Theoretically there can not be any other combination of premisses. All possible ones are enumerated in the list above. Of course each of these combinations does not lead to a valid conclusion, and does not, therefore, constitute a valid syllogism. By the rules given above, and by the method of the comparison of the diagrams, we shall now test these combinations, and find out which of them yield valid forms of syllogism, technically called Moods, and which do not, in each figure.

Of the sixteen combinations we may at once reject EE, EO, OE, and OO as invalid in all figures, because no conclusion three chief classes, called Figures in the more comprehensive sense (the three-fold classification given above), and then subdivides the first of these three classes into two according as the middle term is the subject in the major premiss and predicate in the minor, or the predicate in the major premiss and subject in the minor, the former subdivision corresponding to the first, and the latter to the fourth of the four-fold classification given above. The second and third primary classes do not give rise to any subdivisions. The four classes thus obtained by a double division are called by him Figures in the narrower

sense.