Studies and Exercises in Formal Logic

It is indeed impossible to treat the syllogism scientifically and completely without admitting in some form or other the moods of figure 4. In an à priori separation of figures according to the position of the major and minor terms in the premisses, this figure necessarily appears, and it yields conclusions which are not directly obtainable from the same premisses in any other figure. It is not actually in frequent use, but reasonings may sometimes not unnaturally fall into it; for example, None of the Apostles were Greeks, Some Greeks are worthy of all honour, therefore, Some worthy of all honour are not Apostles.

263. Indirect Moods.—The earliest form in which the mnemonic verses appeared was as follows:—

Aristotle recognised only three figures: the first figure, which he considered the type of all syllogisms and which he 330 called the perfect figure, the dictum de omni et nullo being directly applicable to it alone; and the second and third figures, which he called imperfect figures, since it was necessary to reduce them to the first figure, in order to obtain a test of their validity.

Before the fourth figure, however, was commonly recognised as such, its moods were recognised in another form, namely, as indirect moods of the first figure; and the above mnemonics—Baralipton, Celantes, Dabitis, Fapesmo, Frisesomorum—represent these moods so regarded.357

The conception of indirect moods may be best explained by starting from a definition of figure, which contains no reference to the distinction between major and minor terms, and which accordingly yields only three figures instead of four, namely: Figure 1, in which the middle term is subject in one of the premisses and predicate in the other; Figure 2, in which the middle term is predicate in both premisses; Figure 3, in which the middle term is subject in both premisses. The moods of figure 1 may then be distinguished as direct or indirect according as the position of the terms in the conclusion is the same as their position in the premisses or the reverse.358 Thus, with 331 the premisses MaP, SaM, we have a direct conclusion SaP, and an indirect conclusion PiS. These are respectively Barbara and Baralipton. Similarly, Celantes corresponds to Celarent, and Dabitis to Darii. With the premisses MeP, SiM, we obtain the direct conclusion SoP, but nothing can be inferred of P in terms of S. There is, therefore, no indirect mood corresponding to Ferio. On the other hand, Fapesmo and Frisesomorum (the Fesapo and Fresison of the fourth figure) have no corresponding direct moods.

Clearly it is no more than a formal difference whether the five moods in question are recognised in the manner just indicated, or as constituting a distinct figure; but, on the whole, the latter alternative seems less likely to give rise to confusion.

The distinction between direct and indirect moods as above expressed is for obvious reasons confined to the first figure. It will be observed, however, that in the traditional names of the indirect moods of the first figure the minor premiss precedes the major, and if we seek to apply a distinction between direct and indirect moods in the case of the second and third figures, it can only be with reference to the conventional order of the premisses. Thus, in the second figure, taking the premisses PeM, SaM, we may infer either SeP or PeS, and if we call a syllogism direct or indirect according as the major premiss precedes the minor, or vice versâ, then PeM, SaM, SeP will be a direct mood, and PeM, SaM, PeS an indirect mood. The former of these syllogisms is Cesare, and the latter is Camestres with the premisses transposed.359 Hence the latter will immediately become a direct mood by merely changing the order of the premisses; and the artificiality of the distinction is at once apparent. The result will be found to be similar in other cases, and the distinction may, therefore, be rejected so far as figures 2 and 3 are concerned.

264. Further discussion of the process of Indirect Reduction.—The discussion of the problem of reduction in the preceding pages has in the main followed the traditional lines. It 332 is, however, desirable to treat the process of indirect reduction in a rather more independent and systematic manner. By doing so, we shall find that the process enables us to exhibit very clearly and symmetrically the relations between the first three figures, and also the distinctive functions of these figures.

The argument on which indirect reduction is based is one of which we have several times made use (e.g., in the proof of the second corollary adopted from De Morgan in section 200, and in certain of the proofs contained in section 202), namely, that if X and Y together prove Z, then X and the denial of Z must prove the denial of Y, and vice versâ.

The process may conveniently be exhibited as the contraposition of a hypothetical. Thus, from the proposition X being given, if Y then Z we may infer by contraposition X being given, if not Z then not Y ; and we can equally pass back from the contrapositive to the original proposition.

Since the contradictory of the conclusion of a syllogism may be combined with either of the original premisses, it follows that every valid syllogism carries with it the validity of two other syllogisms. Hence all valid syllogisms must be capable of being arranged in sets of three which are mutually equivalent.

The three equivalent syllogisms may be symmetrically expressed as follows (where P and Pʹ, Q and Qʹ, R and Rʹ are respectively contradictories):

(i) premisses, P and Q ; conclusion Rʹ ;
(ii) premisses, Q and R ; conclusion Pʹ ;
(iii) premisses, R and P ; conclusion Qʹ.

It must be understood that the order of the premisses in these syllogisms is not intended to indicate which is major and which minor.

265. The Antilogism.—Each of the three equivalent syllogisms just given involves further the formal incompatibility of the three propositions P, Q, R (compare section 214). Three propositions, containing three and only three terms, which are thus formally incompatible with one another, constitute what has been called by Mrs Ladd Franklin an antilogism.360 Thus, 333 the syllogism, “MaP, SaM, therefore, SaP,” has for its equivalent antilogism, “MaP, SaM, SoP are three propositions that are formally incompatible with one another.”

266. Equivalence of the Moods of the first three Figures shewn by the Method of Indirect Reduction.—If one of our three equivalent syllogisms is in one of the first three figures, then it can be shewn that the two others will be in the remaining two of these figures.

Thus, let P, Q, ∴ Rʹ be in figure 1, the minor premiss being stated first. It may then be written

The second syllogism becomes

and the third is

It will be seen that (2) is in figure 2, and (3) in figure 3.

Next, let P, Q, ∴ Rʹ be in figure 2, the major premiss being stated first. We then have for our three syllogisms,—

Here (2) is in figure 3, (3) in figure 1.

Finally, let P, Q, ∴ Rʹ be in figure 3, the major premiss being stated first. We have

Here (2) is in figure 1, (3) in figure 2.

Hence we see that, starting with a syllogism in any one of the first three figures (the minor premiss preceding the major in figure 1, but following it in figures 2 and 3), and taking the 334 propositions in the above cyclic order, then the figures will always recur in the cyclic order 1, 2, 3.361

It follows that (as we already know to be the case) there must be an equal number of valid syllogisms in each of the first three figures, and that they may be arranged in sets of equivalent trios. These equivalent trios will be found to be as follows (sets containing strengthened premisses or weakened conclusions being enclosed in square brackets);

Barbara, Baroco, Bocardo;

[AAI, AEO, Felapton;]

Celarent, Festino, Disamis;

[EAO, EAO, Darapti;]

Darii, Camestres, Ferison;

Ferio, Cesare, Datisi.

The corresponding antilogisms are AAO, [AAE,] EAI, [EAA,] AIE, EIA.362

267. The Moods of Figure 4 in their relation to one another.—We have seen that in the equivalent trios of syllogisms yielded by the process of indirect reduction we never have in any one trio more than one syllogism in figure 1, or in figure 2, or in figure 3. Figure 4 is, however, self-contained in the sense that if we start with a syllogism in this figure, both the other syllogisms will be in the same figure. Proceeding as in the last section, we may shew this as follows, the major premiss being stated first:363

It follows that in figure 4 the number of valid syllogisms must be some multiple of three. The number is, as we know, six. There are, therefore, two equivalent trios; and they will be found to be as follows: 335

The equivalent antilogisms are [AAE,] AEI. Comparing this result with that obtained in the preceding section, we see that the only valid antilogistic combinations are AAO and AEI, with the addition of AAE (in which one of the three propositions is unnecessarily strengthened).364

268. Equivalence of the Special Rules of the First Three Figures.—Let the following be a valid syllogism in figure 1,—

Then the corresponding valid syllogism in figure 2 will be

and the corresponding valid syllogism in figure 3 will be

The special rules of figure 1 are

that is, (1) must be affirmative, (2) must be universal.

In figure 2, (2) is the major, and the contradictory of (1) is the conclusion. Therefore, in figure 2 we must have the rules,—

In figure 3, (1) is the minor, and the contradictory of (2) is the conclusion. Therefore, in figure 3 we must have the rules,—

Thus the special rules of figures 2 and 3 are shewn to be deducible from the special rules of figure 1. We might equally 336 well start from the special rules of figure 2 or of figure 3 and deduce the rules of the two other figures.365

269. Scheme of the Valid Moods of Figure l.—So far as the nature of the reasoning involved is concerned, there is practically no distinction between Barbara and Darii, or between Celarent and Ferio. For in each case, if S is the minor term, the S’s referred to in the conclusion are precisely the same S’s as those referred to in the minor premiss.

Again, the only difference between Barbara and Celarent, or between Darii and Ferio, is that the universal rule which the minor premiss enables us to apply to a particular case is in Barbara and Darii a universal affirmation, while in Celarent and Ferio it is a universal denial.

We may, therefore, sum up all four moods in the following scheme:366

This way of setting out the valid moods of figure 1 shews clearly how they are all included under the dictum de omni et nullo.

270. Scheme of the Valid Moods of Figure 2.—Applying the principle of indirect reduction, we may immediately obtain from the scheme given in the last preceding section the following scheme, summing up the valid moods of figure 2:367 337

This scheme may be expressed in the following dictum,—“If a certain attribute can be predicated, affirmatively or negatively, of every member of a class, any subject of which it cannot be so predicated does not belong to the class.”368 This dictum may, like the dictum de omni et nullo, claim to be axiomatic, and it is related to the valid syllogisms of figure 2 just as the dictum de omni et nullo is related to the valid syllogisms of figure 1.369

271. Scheme of the Valid Moods of Figure 3.—Dealing with figure 3 in the same way as we have done with figure 2, we get the following scheme, summing up the valid moods of that figure:

It is not easy to express this scheme in a single self-evident maxim.370 Separate dicta of an axiomatic character may, 338 however, be formulated for the affirmative and negative moods respectively of figure 3, namely, “If two attributes can both be affirmed of a class, and one at least of them universally so, then these two attributes sometimes accompany each other,” “If one attribute can be affirmed while another is denied of a class, either the affirmation or the denial being universal, then the former attribute is not always accompanied by the latter.”371

272. Dictum for Figure 4.—The following dictum, called the dictum de reciproco, was formulated by Lambert for figure 4:—“If no M is B, no B is this or that M ; if C is (or is not) this or that B, there are B’s which are (or are not) C.” The first part of this dictum is intended to apply to Camenes, and the second part to the remaining moods of the fourth figure; but the application can hardly in either case be regarded as self-evident. Several other axioms have been constructed for figure 4; but they are, as a rule, little more than a bare enumeration of the valid moods of that figure, whilst at the same time they are less self-evident than these moods considered individually. The following axiom, however, suggested by Mr Johnson, is not open to these criticisms: “Three classes cannot be so related, that the first is wholly included in the second, the second wholly excluded from the third, and the third partly or wholly included in the first.” This dictum affirms the validity of two antilogisms; in other words, it declares the mutual incompatibility of each of the following trios of propositions: XaY, YeZ, ZiX ; XaY, YeZ, ZaX ; and it will be found that these incompatibles yield the six valid moods of the fourth figure.372

339  

To obtain the conclusion, each of the cases yielded by the major premiss must now be combined with each of those yielded by the minor. This gives four combinations,373 and whatever is true of S in terms of P in all of them is the conclusion required.

342

343 Taking them together we have six combinations, some of which themselves yield more than one case:—

344 So far as S and P are concerned (M being left out of account) these nine cases are reducible to the following three:

It must be admitted that this is very complex, and that it would be a serious matter if in the first instance we had to work through all the different moods in this manner.374 Still, for purposes of illustration, this very complexity has a certain advantage. It shews how many relations between three terms in respect of extension are left to us, even with two premisses given.

289. The application of Lambert’s diagrammatic scheme to syllogistic reasonings.—As applied to syllogisms, Lambert’s lines are much less cumbrous than Euler’s circles. The main point to notice is that it is in general necessary that the line standing for the middle term should not be dotted over any part of its extent.375 This condition can be satisfied by selecting the appropriate alternative form in the case of A, I, and O propositions, as given in section 127. As examples we may represent Barbara, Baroco, Datisi, and Fresison by Lambert’s method.