Deductive Logic Part 22

-- 545. The other premiss, in which the minor term is compared with the middle, is called the Minor Premiss.

-- 546. The order in which the premisses occur in a syllogism is indifferent, but it is usual, for convenience, to place the major premiss first.

-- 547. The following will serve as a typical instance of a syllogism--

Middle term Major term Major Premiss. All mammals are warm-blooded | Antecedent > or Minor term Middle term | Premisses Minor Premiss. All whales are mammals /

Minor term Major term Consequent or .'. All whales are warm-blooded > Conclusion.

-- 548. The reason why the names 'major, 'middle' and 'minor' terms were originally employed is that in an affirmative syllogism such as the above, which was regarded as the perfect type of syllogism, these names express the relative quant.i.ty in extension of the three terms.

[Ill.u.s.tration]

-- 549. It must be noticed however that, though the middle term cannot be of larger extent than the major nor of smaller extent than the minor, if the latter be distributed, there is nothing to prevent all three, or any two of them, from being coextensive.

-- 550. Further, when the minor term is undistributed, we either have a case of the intersection of two cla.s.ses, from which it cannot be told which of them is the larger, or the minor term is actually larger than the middle, when it stands to it in the relation of genus to species, as in the following syllogism--

All Negroes have woolly hair.

Some Africans are Negroes.

.'. Some Africans have woolly hair.

[Ill.u.s.tration]

-- 551. Hence the names are not applied with strict accuracy even in the case of the affirmative syllogism; and when the syllogism is negative, they are not applicable at all: since in negative propositions we have no means of comparing the relative extension of the terms employed. Had we said in the major premiss of our typical syllogism, 'No mammals are cold-blooded,' and drawn the conclusion 'No whales are cold-blooded,' we could not have compared the relative extent of the terms 'mammal' and 'cold-blooded,' since one has been simply excluded from the other.

[Ill.u.s.tration]

-- 552. So far we have rather described than defined the syllogism. All the products of thought, it will be remembered, are the results of comparison. The syllogism, which is one of them, may be so regarded in two ways--

(1) As the comparison of two propositions by means of a third.

(2) As the comparison of two terms by means of a third or middle term.

-- 553. The two propositions which are compared with one another are the major premiss and the conclusion, which are brought into connection by means of the minor premiss. Thus in the syllogism above given we compare the conclusion 'All whales are warm-blooded' with the major premiss 'All mammals are warm-blooded,' and find that the former is contained under the latter, as soon as we become acquainted with the intermediate proposition 'All whales are mammals.'

-- 554. The two terms which are compared with one another are of course the major and minor.

-- 555. The syllogism is merely a form into which our deductive inferences may be thrown for the sake of exhibiting their conclusiveness. It is not the form which they naturally a.s.sume in speech or writing. Practically the conclusion is generally stated first and the premisses introduced by some causative particle as 'because,' 'since,' 'for,' &c. We start with our conclusion, and then give the reason for it by supplying the premisses.

-- 556. The conclusion, as thus stated first, was called by logicians the Problema or Quaestio, being regarded as a problem or question, to which a solution or answer was to be found by supplying the premisses.

-- 557. In common discourse and writing the syllogism is usually stated defectively, one of the premisses or, in some cases, the conclusion itself being omitted. Thus instead of arguing at full length

All men are fallible, The Pope is a man, .'. The Pope is fallible,

we content ourselves with saying 'The Pope is fallible, for he is a man,' or 'The Pope is fallible, because all men are so'; or perhaps we should merely say 'All men are fallible, and the Pope is a man,'

leaving it to the sagacity of our hearers to supply the desired conclusion. A syllogism, as thus elliptically stated, is commonly, though incorrectly, called an Enthymeme. When the major premiss is omitted, it is called an Enthymeme of the First Order; when the minor is omitted, an Enthymeme of the Second Order; and when the conclusion is omitted an Enthymeme of the Third Order.

CHAPTER IX.

_Of Mood and Figure._

-- 558. Syllogisms may differ in two ways--

(1) in Mood;

(2) in Figure.

-- 559. Mood depends upon the kind of propositions employed. Thus a syllogism consisting of three universal affirmatives, AAA, would be said to differ in mood from one consisting of such propositions as EIO or any other combination that might be made. The syllogism previously given to prove the fallibility of the Pope belongs to the mood AAA. Had we drawn only a particular conclusion, 'Some Popes are fallible,' it would have fallen into the mood AAI.

-- 560. Figure depends upon the arrangement of the terms in the propositions. Thus a difference of figure is internal to a difference of mood, that is to say, the same mood can be in any figure.

-- 561. We will now show how many possible varieties there are of mood and figure, irrespective of their logical validity.

-- 562. And first as to mood.

Since every syllogism consists of three propositions, and each of these propositions may be either A, E, I, or O, it is clear that there will be as many possible moods as there can be combinations of four things, taken three together, with no restrictions as to repet.i.tion. It will be seen that there are just sixty-four of such combinations. For A may be followed either by itself or by E, I, or O. Let us suppose it to be followed by itself. Then this pair of premisses, AA, may have for its conclusion either A, E, I, or O, thus giving four combinations which commence with AA. In like manner there will be four commencing with AE, four with AI, and four with AO, giving a total of sixteen combinations which commence with A. Similarly there will be sixteen commencing with E, sixteen with I, sixteen with O--in all sixty-four. It is very few, however, of these possible combinations that will be found legitimate, when tested by the rules of syllogism.

-- 563. Next as to figure.

There are four possible varieties of figure in a syllogism, as may be seen by considering the positions that can be occupied by the middle term in the premisses. For as there are only two terms in each premiss, the position occupied by the middle term necessarily determines that of the others. It is clear that the middle term must either occupy the same position in both premisses or not, that is, it must either be subject in both or predicate in both, or else subject in one and predicate in the other. Now, if we are not acquainted with the conclusion of our syllogism, we do not know which is the major and which the minor term, and have therefore no means of distinguishing between one premiss and another; consequently we must Stop here, and say that there are only three different arrangements possible. But, if the Conclusion also be a.s.sumed as known, then we are able to distinguish one premiss as the major and the other as the minor; and so we can go further, and lay down that, if the middle term does not hold the same position in both premisses, it must either be subject in the major and predicate in the minor, or else predicate in the major and subject in the minor.

-- 564. Hence there result

_The Four Figures._

When the middle term is subject in the major and predicate in the minor, we are said to have the First Figure.

When the middle term is predicate in both premisses, we are said to have the Second Figure.

When the middle term is subject in both premisses, we are said to have the Third Figure.

When the middle term is predicate in the major premiss and subject in the minor, we are said to have the Fourth Figure.

-- 565. Let A be the major term; B the middle. C the minor.

Figure I. Figure II. Figure III. Figure IV.

B--A A--B B--A A--B C--B C--B B--C B--C C--A C--A C--A C--A

All these figures are legitimate, though the fourth is comparatively valueless.

-- 566. It will be well to explain by an instance the meaning of the a.s.sertion previously made, that a difference of figure is internal to a difference of mood. We will take the mood EIO, and by varying the position of the terms, construct a syllogism in it in each of the four figures.

I.

E No wicked man is happy.