CHAPTER II.
THE FIGURES AND MOODS OF THE SYLLOGISM.
243. Figure and Mood.—By the figure of a syllogism is meant the position of the terms in the premisses. Denoting the major, middle, and minor terms by the letters P, M, S respectively, and stating the major premiss first, we have four figures of the syllogism as shewn in the following table:—
| Fig. 1. | Fig. 2. | Fig. 3. | Fig. 4. |
| M – P | P – M | M – P | P – M |
| S – M | S – M | M – S | M – S |
| ⎯⎯ | ⎯⎯ | ⎯⎯ | ⎯⎯ |
| S – P | S – P | S – P | S – P |
By the mood of a syllogism is meant the quantity and quality of the premisses and conclusion. For example, AAA is a mood in which both the premisses and also the conclusion are universal affirmatives; EIO is a mood in which the major is a universal negative, the minor a particular affirmative, and the conclusion a particular negative. It is clear that if figure and mood are both given, the syllogism is given.
244. The Special Rules of the Figures; and the Determination of the Legitimate Moods in each Figure.[329]—It may first of all be shewn that certain combinations of premisses are incapable of yielding a valid conclusion in any figure. A priori, there are possible the following sixteen different combinations of premisses, the major premiss being always stated first:—AA, AI, AE, AO, IA, II, IE, IO, EA, EI, EE, EO, OA, OI, OE, OO. Referring back, however, to the syllogistic rules and corollaries (as given in sections [199], [200]), we find that EE, 310 EO, OE, OO (being combinations of negative premisses) yield no conclusion by rule 5; that II, IO, OI (being combinations of particular premisses) are excluded by corollary i.; and that IE is excluded by corollary iii., which tells us that nothing follows from a particular major and a negative minor.
[329] The method of determination here adopted is only one amongst several possible methods. Another is suggested, for example, in sections [212], [233].
We are left then with the following eight possible combinations:—AA, AI, AE, AO, IA, EA, EI, OA ; and we may go on to enquire in which figures these will yield conclusions. In pursuing this enquiry, special rules of the various figures may be determined, which, taken together with the three corollaries established in section [200], replace the general rules of distribution. These special rules, supplemented by the general rules of quality and the corollaries,[330] will enable the validity of the different moods to be tested by a mere inspection of the form of the propositions of which they consist.
[330] The general rules of quality and the corollaries can be directly applied without reference to the position of the terms in the premisses of a syllogism. This is not the case with the general rules of distribution. The object of the special rules is, in the case of each particular figure, to substitute for the general rules of distribution special rules of quantity and quality.
The special rules[331] and the legitimate moods of Figure 1.
[331] As indicated in section [209], the special rules of figure 1 follow immediately from the dictum de omni et nullo.
The position of the terms in figure 1 is shewn thus,—
M – P
S – M
⎯⎯
S – P
and it can be deduced from the general rules of the syllogism that in this figure:—
(1) The minor premiss must be affirmative. For if it were negative, the major premiss would have to be affirmative by rule 5, and the conclusion negative by rule 6. The major term would therefore be distributed in the conclusion, and undistributed in its premiss; and the syllogism would be invalid by rule 4.
(2) The major premiss must be universal. For the middle term, being undistributed in the affirmative minor premiss, must be distributed in the major premiss.
311 Rule (1) shews that AE and AO and rule (2) that IA and OA, yield no conclusions in this figure. We are accordingly left with only four combinations, namely, AA, AI, EA, EI From the rules that a particular premiss cannot yield a universal conclusion or a negative premiss an affirmative conclusion, while conversely a negative conclusion requires a negative premiss, it follows further that AA will justify either of the conclusions A or I, EA either E or O, AI only I, EI only O. There are then six moods in figure 1 which do not offend against any of the rules of the syllogism,[332] namely, AAA, AAI, AII, EAE, EAO, EIO.
[332] Rule (2) provides against undistributed middle, and rule (1) against illicit major. We cannot have illicit minor, unless we have a universal conclusion with a particular premiss, and this also has been provided against.
Mr Johnson points out that the following symmetrical rules may be laid down for the correct distribution of terms in the different figures; and that these rules (three in each figure) taken together with the rules of quality are sufficient to ensure that no syllogistic rule is broken.
(i) To avoid undistributed middle: in figure 1, If the minor is affirmative, the major must be universal; in figure 4, If the major is affirmative, the minor must be universal; in figure 2, One premiss must be negative; in figure 3, One premiss must be universal. (The last of these rules is of course superfluous if the corollaries contained in section [200] are supposed given.)
(ii) To avoid illicit major: in figures 1 and 3, If the conclusion is negative, the major must be negative and, therefore, the minor affirmative; in figures 2 and 4, If the conclusion is negative, the major must be universal.
(iii) To avoid illicit minor: in figures 1 and 2, If the minor is particular, the conclusion must be particular; in figures 3 and 4, If the minor is affirmative, the conclusion must be particular. (The first of these two rules is again superfluous as a special rule if the corollaries are supposed given.)
The above rules are substantially identical with those given in the text.
The actual validity of these moods may be established by shewing that the axiom of the syllogism, the dictum de omni et nullo, applies to them; or by taking them severally and shewing that in each case the cogency of the reasoning is self-evident.
The special rules and the legitimate moods of Figure 2.
The position of the terms in figure 2 is shewn thus,—
P – M
S – M
⎯⎯
S – P ;
312 and its special rules (which the reader is recommended to deduce from the general rules of the syllogism for himself) are,—
(1) One premiss must be negative ;
(2) The major premiss must be universal.
The application of these rules again leaves six moods, namely, AEE, AEO, AOO, EAE, EAO, EIO.
Recourse cannot now he had directly to the dictum de omni et nullo in order to shew positively that these moods are legitimate. It may, however, be shewn in each case that the cogency of the reasoning is self-evident. The older logicians did not adopt this course; their method was to shew that, by the aid of immediate inferences, each mood could be reduced to such a form that the dictum did apply directly to it. The doctrine of reduction resulting from the adoption of this method will be discussed in the following [chapter].
The special rules and the legitimate moods of Figure 3.
The position of the terms in this figure is shewn thus,—
M – P
M – S
⎯⎯
S – P ;
and its special rules are,—
(1) The minor must be affirmative ;
(2) The conclusion must be particular.
Proceeding as before, we are left with six valid moods, namely, AAI, AII, EAO, EIO, IAI, OAO.
The special rules and the legitimate moods of Figure 4.
The position of the terms in this figure is shewn thus,—
P – M
M – S
⎯⎯
S – P ;
and the following may be given as its special rules,—
(1) If the major is affirmative, the minor must be universal ;
(2) If either premiss is negative, the major must be universal ; 313
(3) If the minor is affirmative, the conclusion must be particular.[333]
[333] The special rules of the fourth figure are variously stated. They are given in the above form in the Port Royal Logic, pp. 202, 203. See, also, section [255].
The result of the application of these rules is again six valid moods, namely, AAI, AEE, AEO, EAO, EIO, IAI.
Our final conclusion then is that there are 24 valid moods, namely, six in each figure.
In Figure 1, AAA, AAI, EAE, EAO, AII, EIO.
In Figure 2, EAE, EAO, AEE, AEO, EIO, AOO.
In Figure 3, AAI, IAI, AII, EAO, OAO, EIO.
In Figure 4, AAI, AEE, AEO, EAO, IAI, EIO.
245. Weakened Conclusions and Subaltern Moods.—When from premisses that would have justified a universal conclusion we content ourselves with inferring a particular (as, for example, in the syllogism All M is P, All S is M, therefore, Some S is P), we are said to have a weakened conclusion, and the syllogism is said to be a weakened syllogism or to be in a subaltern mood (because the conclusion might be obtained by subaltern inference[334] from the conclusion of the corresponding unweakened mood).
[334] In treating the syllogism on the traditional lines it is assumed that S, M, P all represent existing classes. Subaltern inference is, therefore, a valid process.
In the [preceding] section it has been shewn that in each figure there are six moods which do not offend against any of the syllogistic rules: so that in all there are 24 distinct valid moods. Five of these, however, have weakened conclusions; and, since we are not likely to be satisfied with a particular conclusion when the corresponding universal can be obtained from the same premisses, these moods are of no practical importance. Accordingly when the moods of the various figures are enumerated (as in the mnemonic verses) they are usually omitted. Still, their recognition gives a completeness to the theory of the syllogism, which it cannot otherwise possess. There is also a symmetry in the result of 314 their recognition as yielding exactly six legitimate moods in each figure.[335]
[335] It has been remarked that 19 being a prime number at once suggests incompleteness or artificiality in the common enumeration.
The subaltern moods are,—
In Figure 1, AAI, EAO ;
In Figure 2, EAO, AEO ;
In Figure 4, AEO.
It is obvious that there can be no weakened conclusion in Figure 3, since in no case is it possible to infer more than a particular conclusion in this figure.
AAI in Figure 4 is sometimes spoken of as a subaltern mood. But this is a mistake. With the premisses All P is M, All M is S, the conclusion Some S is P is certainly in one sense weaker than the premisses would warrant since the universal conclusion All P is S might have been inferred. But All P is S is not the universal corresponding to Some S is P. The subjects of these two propositions are different; and we infer all that we possibly can about S when we say that some S is P. In other words, regarded as a mood of figure 4, this mood is not a subaltern. AAI in figure 4 is thus differentiated from AAI in figure 1, and its inclusion in the mnemonic verses justified.
246. Strengthened Syllogisms.—If in a syllogism the same conclusion can still be obtained although for one of the premisses we substitute its subaltern, the syllogism is said to be a strengthened syllogism. A strengthened syllogism is thus a syllogism with an unnecessarily strengthened premiss.[336]
[336] Compare De Morgan, Formal Logic, pp. 91, 130. De Morgan calls a syllogism fundamental, when neither of its premisses is stronger than is necessary to produce the conclusion (Formal Logic, p. 77).
For example, the conclusion of the syllogism—
| All M is P, | |
| All M is S, | |
| therefore, | Some S is P, |
could equally be obtained from the premisses All M is P, Some M is S ; or from the premisses Some M is P, All M is S.
By trial we may find that every syllogism in which there 315 are two universal premisses with a particular conclusion is a strengthened syllogism, with the single exception of AEO in the fourth figure.[337]
[337] A general proof of this proposition will be given in section [351].
In a full enumeration there are two strengthened syllogisms in each figure:—
In Figure 1, AAI, EAO ;
In Figure 2, EAO, AEO ;
In Figure 3, AAI, EAO ;
In Figure 4, AAI, EAO.
It will be observed that in figures 1 and 2, a syllogism having a strengthened premiss may also be regarded as a syllogism having a weakened conclusion, and vice versâ ; but that in figures 3 and 4, the contrary holds in both cases. The only syllogism with a weakened conclusion in either of these figures is AEO in figure 4; and in this mood no conclusion is obtainable if either of the premisses is replaced by its subaltern.
If syllogisms containing either a strengthened premiss or a weakened conclusion are omitted, we are left with 15 valid moods, namely, 4 in each of the first three figures and 3 in figure 4.
247. The peculiarities and uses of each of the four figures of the syllogism.[338]—Figure 1. In this figure it is possible to prove conclusions of all the forms A, E, I, O; and it is the only figure in which a universal affirmative conclusion can be proved. This alone makes it by far the most useful and important of the syllogistic figures. All deductive science, the object of which is to establish universal affirmatives, tends to work in AAA in this figure.
[338] On the distinctive characteristics of the different figures, see also sections [269] to 271.
Another point to notice is that only in this figure is it the case that both the subject of the conclusion is subject in the premisses, and the predicate of the conclusion predicate in the premisses; in figure 2 the predicate of the conclusion is subject in the major premiss; in figure 3 the subject of the conclusion is predicate in the minor premiss; and in figure 4 there is a double inversion.[339] This no doubt partly 316 accounts for the fact that a reasoning expressed in figure 1 so often seems more natural than the same reasoning expressed in any other figure.[340]
[339] The double inversion in figure 4 is one of the reasons given by Thomson for rejecting that figure altogether. Compare section [262].
[340] Compare Solly, Syllabus of Logic, pp. 130 to 132.
Figure 2. In this figure, only negatives can be proved; and therefore it is chiefly used for purposes of disproof. For example, Every real natural poem is naïve ; those poems of Ossian which Macpherson pretended to discover are not naïve (but sentimental); hence they are not real natural poems (Ueberweg, System of Logic, § 113). It has been called the exclusive figure; because by means of it we may go on excluding various suppositions as to the nature of something under investigation, whose real character we wish to ascertain (a process called abscissio infiniti). For example, Such and such an order has such and such properties, This plant has not those properties ; therefore, It does not belong to that order. A syllogism of this kind may be repeated with a number of different orders till the enquiry is so narrowed down that the place of the plant is easily determined. Whately (Elements of Logic, p. 92) gives an example from the diagnosis of a disease.
Figure 3. In this figure, only particulars can be proved. It is frequently useful when we wish to take objection to a universal proposition laid down by an opponent by establishing an instance in which such universal proposition does not hold good.
It is the natural figure when the middle term is a singular term, especially if the other terms are general. It has been already shewn that if one and only one term of an affirmative proposition is singular, that term is almost necessarily the subject. For example, such a reasoning as Socrates is wise, Socrates is a philosopher, therefore, Some philosophers are wise, can only with great awkwardness be expressed in any figure other than figure 3.
Figure 4. This figure is seldom used, and some logicians have altogether refused to recognise it. We shall return to a discussion of it subsequently. See section [262].
Lambert in his Neues Organon expresses the uses of the different syllogistic figures as follows: “The first figure is suited to the discovery or proof of the properties of a thing; 317 the second to the discovery or proof of the distinctions between things; the third to the discovery or proof of instances and exceptions; the fourth to the discovery or exclusion of the different species of a genus.”
EXERCISES.
248. Why is IE an inadmissible, while EI is an admissible, mood in every figure of the syllogism? [L.]
249. What moods are good in the first figure and faulty in the second, and vice versâ? Why are they excluded in one figure and not in the other? [O.]
250. (i) Shew that O cannot stand as premiss in figure 1, as major in figure 2, as minor in figure 3, as premiss in figure 4.
(ii) Shew that it is impossible to have the conclusion in A in any figure but the first. What fallacies would be committed if there were such a conclusion to a reasoning in any other figure? [C.]
251. Two valid syllogisms in the same figure have the same major, middle, and minor terms, and their major premisses are subcontraries; determine—without reference to the mnemonic verses—what the syllogisms must be. [K.]
252. Prove, by general reasoning, that any mood valid both in figure 2 and in figure 3 is valid also in figure 1 and in figure 4. [C.]
253. Shew, without individual reference to the different figures, that EAO is a strengthened syllogism in every figure, and that AAI is a strengthened syllogism whenever it is valid. [K.]
254. Shew, by general reasoning, that every valid syllogism in which the middle term is twice distributed contains a strengthened premiss. Does it follow that it must have also a weakened conclusion? [K.]
255. Shew that the following two rules would suffice as the special rules for the fourth figure: (i) The conclusion and major cannot have the same form unless it be particular affirmative; (ii) The conclusion and minor cannot have the same form unless it be universal negative. [J.]