CHAPTER III.
THE REDUCTION OF SYLLOGISMS.
256. The Problem of Reduction.—By reduction is meant a process whereby the reasoning contained in a given syllogism is expressed in some other mood or figure. Unless an explicit statement is made to the contrary, reduction is supposed to be to figure 1.
The following syllogism in figure 3 may be taken as an example:
| All M is P, | |
| Some M is S, | |
| therefore, | Some S is P. |
It will be seen that by simply converting the minor premiss, we have precisely the same reasoning in figure 1.
This is an example of direct or ostensive reduction.
257. Indirect Reduction.—A proposition is established indirectly when its contradictory is proved false; and this is effected if it can be shewn that a consequence of the truth of its contradictory would be self-contradiction.
The method of indirect proof is in several cases adopted by Euclid; and it may be employed in the reduction of syllogisms from one mood to another. Thus, AOO in figure 2 is usually reduced in this manner. The argument may be stated as follows:—
From the premisses,—
| All P is M, | |
| Some S is not M, | |
| it follows that | Some S is not P ; |
for if this conclusion is not true, then, by the law of excluded 319 middle, its contradictory (namely, All S is P) must be so; and, the premisses being given true, the three following propositions must all be true, namely,
| All P is M, |
| Some S is not M, |
| All S is P. |
But combining the first and the third of these we have a syllogism in figure 1, namely,
| All P is M, | |
| All S is P, | |
| yielding the conclusion | All S is M. |
Some S is not M and All S is M are, therefore, true together; but, by the law of contradiction, this is absurd, since they are contradictories.
Hence it has been shewn that the consequence of supposing Some S is not P false is a self-contradiction; and we may accordingly infer that it is true.
It will be observed that the only syllogism made use of in the above argument is in figure 1; and the process may, therefore, be regarded as a reduction of the reasoning to figure 1.
This method of reduction is called Reductio ad impossibile, or Reductio per impossibile,[341] or Deductio ad impossibile, or Deductio ad absurdum. It is the only way of reducing AOO in figure 2 or OAO in figure 3 to figure 1, unless negative terms are used (as in obversion and contraposition); and it was adopted by the old writers in consequence of their objection to negative terms.
[341] Compare Mansel’s Aldrich, pp. 88, 89.
It will be shewn [later on] in this chapter that by employing the method of indirect reduction systematically we can bring out with great clearness the relation between the different moods and figures of the syllogism.
258. The mnemonic lines Barbara, Celarent, &c.—The mnemonic hexameter verses (which are spoken of by De Morgan as “the magic words by which the different moods have been denoted for many centuries, words which I take to be more full of meaning than any that ever were made”) are usually given as follows: 320
Barbără, Cēlārent, Dărĭi, Fĕrĭōque prioris:
Cēsărĕ, Cāmēstres, Festīnŏ, Bărōcŏ, secundae:
Tertia, Dāraptī, Dĭsămis, Dātīsĭ, Fĕlapton,
Bōcardō, Fērīsŏn, habet: Quarta insuper addit
Brāmantip, Cămĕnes, Dĭmăris, Fēsāpŏ, Frĕsīson.
Each valid mood in every figure, unless it be a subaltern mood, is here represented by a separate word; and in the case of a mood in any of the so-called imperfect figures (i.e., figures 2, 3, 4), the mnemonic gives full information for its reduction to figure 1, the so-called perfect figure.
The only meaningless letters are b (not initial), d (not initial), l, n, r, t ; the signification of the remainder is as follows:—
The vowels give the quality and quantity of the propositions of which the syllogism is composed; and, therefore, really give the syllogism itself, if the figure is also known. Thus, Camenes in figure 4 represents the syllogism—
| All P is M, | |
| No M is S, | |
| therefore, | No S is P. |
The initial letters in the case of figures 2, 3, 4 shew to which of the moods of figure 1 the given mood is to be reduced, namely, to that which has the same initial letter. The letters B, C, D, F were chosen for the moods of figure 1 as being the first four consonants in the alphabet.
Thus, Camestres is reduced to Celarent,—
| All P is M, | ⟍ ⟋ | No M is S, | |
| No S is M, | ⟋ ⟍ | All P is M, | |
| therefore, | No S is P. | therefore, | No P is S, |
| therefore, | No S is P.[ 342] |
[342] The order of inference in this and in other reductions might be made clear by the use of arrows, representing inference, as follows:
| All P is M, | ⟍ ↗ | No M is S, |
| No S is M, | ⟋ ↘ | All P is M, |
| ↓ | ||
| No S is P. | ← | No P is S, |
s (in the middle of a word) indicates that in the process of reduction the preceding proposition is to be simply converted. 321 Thus, in reducing Camestres to Celarent, as shewn above, the minor premiss is simply converted.
s (at the end of a word) shews that the conclusion of the new syllogism has to be simply converted in order that the given conclusion may be obtained. This again is illustrated in the reduction of Camestres. The final s does not affect the conclusion of Camestres itself, but the conclusion of Celarent to which it is reduced.[343]
[343] This peculiarity in the signification of s and p when they are final letters is sometimes overlooked. The point to be noted is that the conclusion of the syllogism originally given is not, like the original premisses, a datum from which we set out, but a result that we have to reach. It follows that the conclusion to be manipulated, if any, must be the conclusion of the syllogism obtained by reduction, not the conclusion of the original syllogism. This is clearly shewn in the case of Camestres by the method adopted in the last preceding [note] to illustrate the reduction of Camestres to Celarent. The reduction of Disamis, Bramantip, Camenes, Dimaris to figure 1 might be illustrated similarly.
p (in the middle of a word) signifies that the preceding proposition is to be converted per accidens ; as, for example, in the reduction of Darapti to Darii,—
| All M is P, | All M is P, | ||
| All M is S, | Some S is M, | ||
| therefore, | Some S is P. | therefore, | Some S is P. |
p (at the end of a word[344]) implies that the conclusion obtained by reduction is to be converted per accidens. Thus, in Bramantip, the p does not relate to the I conclusion of the mood itself;[345] it really relates to the A conclusion of the syllogism in Barbara which is given by reduction. Thus,—
| All P is M, | ⟍ ⟋ | All M is S, | |
| All M is S, | ⟋ ⟍ | All P is M, | |
| therefore, | Some S is P. | therefore, | All P is S, |
| therefore, | Some S is P. |
[344] See the last preceding [note].
[345] Compare, however, Hamilton, Logic, I. p. 264, and Spalding, Logic, pp. 230, 1.
m indicates that in reduction the premisses have to be transposed (metathesis praemissarum); as just shewn in the case of Bramantip, and also in the case of Camestres.
c signifies that the mood is to be reduced indirectly (i.e., by 322 reductio per impossibile in the manner shewn in the preceding [section]); and the position of the letter indicates that in this process of indirect reduction the first step is to omit the premiss preceding it, i.e., the other premiss is to be combined with the contradictory of the conclusion (conversio syllogismi, or ductio per contradictoriam propositionem sive per impossibile), The letter c is by some writers replaced by k, Baroko and Bokardo being given as the mnemonics, instead of Baroco and Bocardo.
The following lines are sometimes added to the verses given above, in order to meet the case of the subaltern moods:—
Quinque Subalterni, totidem Generalibus orti,
Nomen habent nullum, nec, si bene colligis, usum.[346]
[346] The mnemonics have been written in various forms. Those given above are from Aldrich, and they are the ones that are in general use in England. Wallis in his Institutio Logicae (1687) gives for the fourth figure, Balani, Cadere, Digami, Fegano, Fedibo. P. van Musschenbroek in his Institutiones Logicae (1748) gives Barbari, Calentes, Dibatis, Fespamo, Fresisom. This variety of forms for the moods of figure 4 is no doubt due to the fact that the recognition of this figure at all was quite exceptional until comparatively recently. Compare sections [262], [263].
According to Ueberweg (Logic, § 118) the mnemonics run,—
Barbara, Celarent primae, Darii Ferioque.
Cesare, Camestres, Festino, Baroco secundae.
Tertia grande sonans recitat Darapti, Felapton,
Disamis, Datisi, Bocardo, Ferison. Quartae
Sunt Bamalip, Calemes, Dimatis, Fesapo, Fresison.
Ueberweg gives Camestros and Calemos for the weakened moods of Camestres and Calemes. This is not, however, quite accurate. The mnemonics should be Camestrop and Calemop.
Professor Carveth Read (Logic, pp. 126, 7) suggests an ingenious modification of the verses, so as to make each mnemonic immediately suggest the figure to which the corresponding mood belongs, at the same time abolishing all the unmeaning letters. He takes l as the sign of the first figure, n of the second, r of the third, and t of the fourth. The lines (to be scanned, says Professor Read, discreetly) then run
Ballala, Celallel, Dalii, Felioque prioris.
Cesane, Camesnes, Fesinon, Banoco secundae.
Tertia Darapri, Drisamis, Darisi, Ferapro,
Bocaro, Ferisor habet. Quanta insuper addit
Bamatip, Cametes, Dimatis, Fesapto, Fesistot.
Professor Mackenzie suggests that, if this plan is adopted, it would be better to take r for the first figure (figura recta, the straightforward figure), n for the second figure (figura negativa), t for the third figure (figura tertia or particularis), and l for the fourth figure (figura laeva, the left-handed figure). Compare also Mrs Ladd Franklin, Studies in Logic, Johns Hopkins University, p. 40.
323 259. The direct reduction of Baroco and Bocardo.—These moods may be reduced directly to the first figure by the aid of obversion and contraposition as follows.[347]
Baroco:—
| All P is M, | |
| Some S is not M, | |
| therefore, | Some S is not P, |
is reducible to Ferio by the contraposition of the major premiss and the obversion of the minor, thus,—
| No not-M is P, | |
| Some S is not-M, | |
| therefore, | Some S is not P. |
[347] Another method is to reduce Baroco and Bocardo by the process of ἔκθεσις to other moods of figures 2 and 3, and thence to figure 1. Ueberweg writes, “Baroco may also be referred to Camestres when those (some) S of which the minor premiss is true are placed under a special notion and denoted by Sʹ. Then the conclusion must hold good universally of Sʹ, and consequently particularly of S. Aristotle calls such a procedure ἔκθεσις” (Logic, § 113). As regards Bocardo, “Aristotle remarks that this mood may be proved without apagogical procedure (reductio ad impossibile) by the ἐκθέσθαι or λαμβάνειν of that part of the middle notion which is true of the major premiss. If we denote this part by N, then we get the premisses; NeP ; NaS: from which follows (in Felapton) SoP ; which was to be proved” (§ 115). The procedure is, however, rather more complicated than appears in the above statements. In the case of Baroco (PaM, SoM, ∴ SoP), let the S’s which are not M (of which by hypothesis there are some) be denoted by X ; then we have PaM, XeM, ∴ XeP (Camestres); but XaS, and hence we have further XeP, XaS, ∴ SoP (Felapton). In the case of Bocardo (MoP, MaS, ∴ SoP), let the M’s which are not P (of which by hypothesis there are some) be denoted by N ; then we have MaS, NaM, ∴ NaS (Barbara); and hence NeP, NaS, ∴ SoP (Felapton). The argument in both cases suggests questions connected with the existential import of propositions; but the consideration of such questions must for the present be deferred.
Faksoko has been suggested as a mnemonic for this method of reduction, k denoting obversion, so that ks demotes obversion followed by conversion (i.e., contraposition).
Whately’s mnemonic Fakoro (Elements of Logic, p. 97) does not indicate the obversion of the minor premiss (r being with him an unmeaning letter).
| Some M is not P, | |
| All M is S, | |
| therefore, | Some S is not P, |
is reducible to Darii by the contraposition of the major premiss and the transposition of the premisses, thus,—
| All M is S, | |
| Some not-P is M, | |
| therefore, | Some not-P is S. |
Some not-P is S is not indeed our original conclusion, but the latter can be obtained from it by conversion followed by obversion. This method of reduction may be indicated by Doksamosk (which again is obviously preferable to Dokamo, suggested by Whately, since the latter would make it appear as if we immediately obtained the original conclusion in Darii.)
260. Extension of the Doctrine of Reduction.—The doctrine of reduction may be extended, and it can be shewn not merely that any syllogism may be reduced to figure 1, but also that it may be reduced to any given mood (not being a subaltern mood) of that figure.[348] This position will obviously be established if we can shew that Barbara, Celarent, Darii, and Ferio are mutually reducible to one another.
[348 ] Compare, further, sections [284], [285].
Barbara may be reduced to Celarent by the obversion of the major premiss and also of the new conclusion thereby obtained. Thus, using arrows, as in the [note] on page 320,
| All M is P, | → | No M is not-P, |
| All S is M, | → | All S is M, |
| ↓ | ||
| All S is P. | ← | No S is not-P. |
Conversely, Celarent is reducible to Barbara ; and in a similar manner, by obversion of major premiss and conclusion, Darii and Ferio are reducible to one another.
It will now suffice if we can shew that Barbara and Darii are mutually reducible to one another. Clearly the only method possible here is the indirect method.
Take Barbara,
| MaP, | |
| SaM, | |
| ⎯⎯ | |
| ∴ | SaP ; |
325 for, if not, then we have SoP ; and MaP, SaM, SoP must be true together. From SoP by first obverting and then converting (and denoting not-P by Pʹ) we get PʹiS, and combining this with SaM we have the following syllogism in Darii,—
| SaM, | |
| PʹiS, | |
| ⎯⎯ | |
| ∴ | PʹiM. |
PʹiM by conversion and obversion becomes MoP ; and therefore MaP and MoP are true together; but this is impossible, since they are contradictories. Therefore, SoP cannot be true, i.e., the truth of SaP is established.
Similarly, Darii may be indirectly reduced to Barbara.[349]
| MaP, | (i) | |
| SiM, | (ii) | |
| ⎯⎯ | ||
| ∴ | SiP. | (iii) |
The contradictory of (iii) is SeP, from which we obtain PaSʹ. Combining with (i), we have—
| PaSʹ, | ||
| MaP, | ||
| ⎯⎯ | ||
| ∴ | MaSʹ | in Barbara. |
But from this conclusion we may obtain SeM, which is the contradictory of (ii).
[349] It has been maintained, that this reduction is unnecessary, and that, to all intents and purposes, Darii is Barbara, since the “some S” in the minor is, and is known to be, the same some as in the conclusion. Compare section [269].
261. Is Reduction an essential part of the Doctrine of the Syllogism?—According to the original theory of reduction, the object of the process is to be sure that the conclusion is a valid inference from the premisses. The validity of a syllogism in figure 1 may be directly tested by reference to the dictum de omni et nullo: but this dictum has no direct application to syllogisms in the remaining three figures. Thus, Whately says, “As it is on the dictum de omni et nullo that all reasoning ultimately depends, so all arguments may be in one way or other brought into some one of the four moods in the first figure: and a syllogism is, in that case, said to be reduced” (Elements of Logic, p. 93). Professor Fowler puts the same position somewhat more guardedly, “As we have adopted no canon for the 2nd, 3rd, and 4th figures, we have as yet 326 no positive proof that the six moods remaining in each of those figures are valid: we merely know that they do not offend against any of the syllogistic rules. But if we can reduce them, i.e., bring them back to the first figure, by shewing that they are only different statements of its moods, or in other words, that precisely the same conclusions can be obtained from equivalent premisses in the first figure, their validity will be proved beyond question” (Deductive Logic, p. 97).
Reduction is, on the other hand, regarded by some logicians as both unnecessary and unnatural. It is, in the first place, said to be unnecessary, on the ground that the dictum de omni et nullo has no claim to be regarded as the paramount law for all valid inference.[350] In sections [270] to 272 it will be shewn that dicta can be formulated for the other figures, which may be regarded as making them independent of the first, and putting them on a level with it. It may also be maintained that in any mood the validity of a particular syllogism is as self-evident as that of the dictum de omni et nullo itself; and that, therefore, although axioms of syllogism are useful as generalisations of the syllogistic process, they are needless in order to establish the validity of any given syllogism. This view is indicated by Ueberweg.
[350] Compare Thomson, Laws of Thought, p. 172.
Reduction is, in the second place, said to be unnatural, inasmuch as it often involves the substitution of an unnatural and indirect for a natural and direct predication. Figures 2 and 3 at any rate have their special uses, and certain reasonings fall naturally into these figures rather than into the first figure.[351]
[351] Compare a quotation from Lambert (Neues Organon, §§ 230, 231) given by Sir W. Hamilton (Logic, II. p. 438).
The following example is given by Thomson (Laws of Thought, p. 174): “Thus, when it was desirable to shew by an example that zeal and activity did not always proceed from selfish motives, the natural course would be some such syllogism as the following. The Apostles sought no earthly reward, the Apostles were zealous in their work; therefore, 327 some zealous persons seek not earthly reward.” In reducing this syllogism to figure 1, we have to convert our minor into “Some zealous persons were Apostles,” which is awkward and unnatural.
Take again this syllogism, “Every reasonable man wishes the Reform Bill to pass, I don’t, therefore, I am not a reasonable man.” Reduced in the regular way to Celarent, the major premiss becomes, “No person wishing the Reform Bill to pass is I,” yielding the conclusion, “No reasonable man is I.”
Further illustrations of this point will be found if we reduce to figure 1, syllogisms with such premisses as the following:—All orchids have opposite leaves, This plant has not opposite leaves; Socrates is poor, Socrates is wise.
The above arguments justify the position that reduction is not a necessary part of the doctrine of the syllogism, so far as the establishment of the validity of the different moods is concerned.[352]
[352] Hamilton (Logic, I. p. 433) takes a curious position in regard to the doctrine of reduction. “The last three figures,” he says, “are virtually identical with the first.” This has been recognised by logicians, and hence “the tedious and disgusting rules of their reduction.” But he himself goes further, and extinguishes these figures altogether, as being merely “accidental modifications of the first,” and “the mutilated expressions of a complex mental process.” A somewhat similar position is taken by Kant in his essay On the Mistaken Subtilty of the Four Figures. Kant’s argument is virtually based on the two following propositions: (1) Reasonings in figures 2, 3, 4 require to be implicitly, if not explicitly, reduced to figure 1, in order that their validity may be apparent; for example, in Cesare we must have covertly performed the conversion of the major premiss in thought, since otherwise our premisses would not be conclusive; (2) No reasonings ever fall naturally into any of the moods of figures 2, 3, 4, which are, therefore, a mere useless invention of logicians. On grounds already indicated, both these propositions must be regarded as erroneous. A further error seems to be involved in the following passage from the same essay of Kant’s: “It cannot be denied that we can draw conclusions legitimately in all these figures. But it is incontestable that all except the first determine the conclusion only by a roundabout way, and by interpolated inferences, and that the very same conclusion would follow from the same middle term in the first figure by pure and unmixed reasoning.” The latter part of this statement cannot be justified in such a case as that of Baroco.
At the same time, no treatment of the syllogism can be 328 regarded as scientific or complete until the equivalence between the moods in the different figures has been shewn; and for this purpose, as well as for its utility as a logical exercise, a full treatment of the problem of reduction should be retained.[353]
[353] See, further, sections [266], [268].
262. The Fourth Figure.—Figure 4 was not as such recognised by Aristotle; and its introduction having been attributed by Averroës to Galen, it is frequently spoken of as the Galenian Figure. It does not usually appear in works on Logic before the beginning of the eighteenth century, and even by modern logicians its use is sometimes condemned. Thus Bowen (Logic, p. 192) holds that “what is called the fourth figure is only the first with a converted conclusion; that is we do not actually reason in the fourth, but only in the first, and then if occasion requires, convert the conclusion of the first.” This account of figure 4 cannot, however, be accepted, since it will not apply to Fesapo or Fresison. For example, from the premisses of Fesapo (No P is M and All M is S) no conclusion whatever is obtainable in figure 1.[354]
[354] For the most part the critics of the fourth figure seem to identify it altogether with Bramantip. The following extract from Father Clarke’s Logic (p. 337) will serve to illustrate the contumely to which this poor figure is sometimes subjected: “Ought we to retain it? If we do, it should be as a sort of syllogistic Helot, to shew how low the syllogism can fall when it neglects the laws on which all true reasoning is founded, and to exhibit it in the most degraded form which it can assume without being positively vicious. Is it capable of reformation? Not of reformation, but of extinction…… Where the same premisses in the first figure would prove a universal affirmative, this feeble caricature of it is content with a particular; where the first figure draws its conclusion naturally and in accordance with the forms into which human thought instinctively shapes itself, this perverted abortion forces the mind to an awkward and clumsy process which rightly deserves to be called ‘inordinate and violent.’” Father Clarke’s own violence appears to be attributable mainly to the fact that figure 4 was not, as such, recognised by Aristotle.
Thomson’s ground of rejection is that in the fourth figure the order of thought is wholly inverted, the subject of the conclusion having been a predicate in the premisses, and the predicate a subject. “Against this the mind rebels; and we can ascertain that the conclusion is only the converse of the real one, by proposing to ourselves similar sets of premisses, to 329 which we shall always find ourselves supplying a conclusion so arranged that the syllogism is in the first figure, with the second premiss first” (Laws of Thought, p. 178). As regards the first part of this argument, Thomson himself points out that the same objection applies partially to figures 2 and 3. It no doubt helps to explain why as a matter of fact reasonings in figure 4 are not often met with;[355] but it affords no sufficient ground for altogether refusing to recognise this figure. The second part of Thomson’s argument is, for a reason already stated, unsound. The conclusion, for example, of Fresison cannot be “the converse of the real conclusion,” since (being an O proposition) it is not the converse of any other proposition whatsoever.
[355] The reasons why figure 4, “with its premisses looking one way, and its conclusion another,” is seldom used, are elaborated by Karslake, Aids to the Study of Logic, I. pp. 74, 5.
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:—
Barbara, Celarent, Darii, Ferio, Baralipton,
Celantes, Dabitis, Fapesmo, Frisesomorum,
Cesare, Camestres, Festino, Baroco, Darapti,
Felapton, Disamis, Datisi, Bocardo, Ferison.[356]
[356] First published in the Summulae Logicales of Petrus Hispanus, afterwards Pope John XXI., who died in 1277. The mnemonics occur in an earlier unpublished work of William Shyreswood, who died as Chancellor of Lincoln in 1249.
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]
[357] From the 14th to the 17th century the mnemonics found in works on Logic usually give the moods of the fourth figure in this form, or else omit them altogether. Wallis (1687) recognises them in both forms, giving two sets of mnemonics.
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.
[358] It follows that if we compare the conclusion of an indirect mood with the conclusion of the corresponding direct mood (where such correspondence exists), we shall find that the terms have changed places. Mansel’s definition of an indirect mood as “one in which we do not infer the immediate conclusion, but its converse” (Aldrich, p. 78) must, however, be rejected for the reason that it cannot be applied to Fapesmo and Frisesomorum, which are indirect moods having no corresponding valid direct moods at all. In these we cannot be said to infer “the converse of the immediate conclusion,” for there is no immediate conclusion. Mansel deals with these two moods very awkwardly. “Fapesmo and Frisesomorum,” he remarks, “have negative minor premisses, and thus offend against a special rule of the first figure; but this is checked by a counterbalancing transgression. For by simply converting O, we alter the distribution of the terms, so as to avoid an illicit process.” But the notion that we can counterbalance one violation of law by committing a second cannot be allowed. The truth of course is that, in the first place, the special rules of the first figure as ordinarily given do not apply to the indirect moods; and in the second place, the conclusion O is not obtained by conversion at all.
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.
[359] Take, again, the premisses MaP, MoS. Here there is no direct conclusion, but only an indirect conclusion PoS. This, however, is merely Bocardo with the premisses transposed.
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.”
[360] See Baldwin’s Dictionary of Philosophy, art. Symbolic Logic. It is shewn in this article that the whole of syllogistic reasoning may be summed up in the following antilogism, the symbolism of section [138] being made use of,—
[(AB = 0)(bC = 0)(AC > 0)] = 0.
The fifteen moods containing neither a strengthened premiss nor a weakened conclusion may, by the aid of conversions and obversions, be obtained from this antilogism according as the contradictory of one or other of the three incompatibles is taken as the conclusion.
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
| S ⎯ M, M ⎯ P, ∴ (S ⎯ P)ʹ. | (1) |
The second syllogism becomes
| M ⎯ P, S ⎯ P, ∴ (S ⎯ M)ʹ; | (2) |
and the third is
| S ⎯ P, S ⎯ M, ∴ (M ⎯ P)ʹ. | (3) |
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,—
| P ⎯ M, S ⎯ M, ∴ (S ⎯ P)ʹ; | (1) |
| S ⎯ M, S ⎯ P, ∴ (P ⎯ M)ʹ; | (2) |
| S ⎯ P, P ⎯ M, ∴ (S ⎯ M)ʹ. | (3) |
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
| M ⎯ P, M ⎯ S, ∴ (S ⎯ P)ʹ; | (1) |
| M ⎯ S, S ⎯ P, ∴ (M ⎯ P)ʹ; | (2) |
| S ⎯ P, M ⎯ P, ∴ (M ⎯ S)ʹ. | (3) |
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]
[361] If we were to start with a syllogism in figure 1, the major premiss being stated first, then the cyclic order of figures would be 1, 3, 2, and in figures 2 and 3 the minor premiss would precede the major.
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]
[362] The position of the terms in these antilogisms corresponds to that of figure 1, the major premiss being stated first.
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]
| P ⎯ M, M ⎯ S, ∴ (S ⎯ P)ʹ; | (1) |
| M ⎯ S, S ⎯ P, ∴ (P ⎯ M)ʹ; | (2) |
| S ⎯ P, P ⎯ M, ∴ (M ⎯ S)ʹ. | (3) |
[363] It will be found that it comes to just the same thing if the minor premiss is stated first.
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
[Bramantip, AEO, Fesapo;]
Camenes, Fresison, Dimaris.
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]
[364] This result might be inferred from the rules given in section [214].
268. Equivalence of the Special Rules of the First Three Figures.—Let the following be a valid syllogism in figure 1,—
| (minor) | S ⎯ M, | (1) | |
| (major) | M ⎯ P, | (2) | |
| (conclusion) | ∴ | (S ⎯ P)ʹ. | (3) |
Then the corresponding valid syllogism in figure 2 will be
| (major) | M ⎯ P, | (2) | |
| (minor) | S ⎯ P, | contradictory of (3) | |
| (conclusion) | ∴ | (S ⎯ M)ʹ; | contradictory of (1) |
and the corresponding valid syllogism in figure 3 will be
| (major) | S ⎯ P, | contradictory of (3) | |
| (minor) | S ⎯ M, | (1) | |
| (conclusion) | ∴ | (M ⎯ P)ʹ. | contradictory of (2) |
The special rules of figure 1 are
| minor | affirmative, |
| major | universal, |
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,—
| major | universal, |
| conclusion | negative [and hence one premiss negative]. |
In figure 3, (1) is the minor, and the contradictory of (2) is the conclusion. Therefore, in figure 3 we must have the rules,—
| minor | affirmative, |
| conclusion | particular. |
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]
[365] The complete rules for the antilogisms of the first three figures, as given at the end of section [266], are (a) first proposition universal, (b) second proposition affirmative, (c) third proposition opposite in quality to the first, and (unless it is strengthened) opposite in quantity to the second. These rules replace all general rules.
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]
| All B is C (or is not C), | (Rule) | |
| All (or some) A is B, | (Case) | |
| therefore, | All (or some) A is C (or is not C). | (Result) |
[366] Compare C. S. Peirce in the Johns Hopkins Studies in Logic, p. 148, and Sigwart, Logic, i. p. 354. Sigwart gives the following formula:
| If anything is M it is P (or is not P), | |
| Certain subjects S are M, | |
| therefore, | They are P (or are not P). |
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
| All B is C (or is not C), | (Rule) | |
| Some (or all) A is not C (or is C), | (Denial of Result) | |
| therefore, | Some (or all) A is not B. | (Denial of Case) |
[367] Sigwart’s way of putting it (Logic, i. p. 354) is that in figure 2, instead of inferring from ground to consequence, we infer from invalidity of consequence to invalidity of ground; and he gives the following scheme:
| If anything is P it is M (or is not M), | |
| Certain subjects S are not M (or are M), | |
| therefore, | They are not P. |
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]
[368] The dictum for figure 2, sometimes called the dictum de diverso, is expressed in the above form by Mansel (Aldrich, p. 86). It was given by Lambert in the form, “If one term is contained in, and another excluded from, a third term, they are mutually excluded.” This is at least expressed loosely, since it would appear to warrant a universal conclusion, if any conclusion at all, in Festino and Baroco. Bailey (Theory of Reasoning, p. 71) gives the following pair of maxims for figure 2,—“When the whole of a class possess a certain attribute, whatever does not possess the attribute does not belong to the class. When the whole of a class is excluded from the possession of an attribute, whatever possesses the attribute does not belong to the class.”
[369] Lambert is usually regarded as the originator of the idea of framing dicta that shall be directly applicable to figures other than the first. Thomson, however, points out that it is an error to suppose that Lambert was the first to invent such dicta. “More than a century earlier, Keckermann saw that each figure had its own law and its own peculiar use, and stated them as accurately, if less concisely, than Lambert” (Laws of Thought, p. 173, note). Distinct principles for the second and third figures are laid down also in the Port Royal Logic, which was published in 1662.
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:
| Some (or all) A is not C (or is C), | (Denial of Result) | |
| All (or some) A is B, | (Case) | |
| therefore, | Some B is not C (or is C). | (Denial of Rule) |
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]
[370] Lambert gave the following dictum de exemplo for figure 3:—“Two terms which contain a common part partly agree, or if one contains a part which the other does not, they partly differ.” This maxim is open to exception. The proposition “If one term contains a part which another does not, they partly differ” applied to MeP, MaS, would appear to justify PoS just as much as SoP, or else to yield an alternative between these two. Mr Johnson gives a single formula for figure 3, namely, “A statement may be applied to part of a class, if it applies wholly [or at least partly] to a set of objects that are at least partly [or wholly] included in that class.” This is correct, but perhaps not very easy to grasp.
[371] These dicta (or dicta corresponding to them) are sometimes called respectively the dictum de exemplo and the dictum de excepto.
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]
EXERCISES.
273. Reduce Barbara to Bocardo, Bocardo to Baroco, Baroco to Barbara. [K.]
274. Reduce Ferio to figure 2, Festino to figure 3, Felapton to figure 4. [K.]
275. Reduce Camestres to Datisi. Why cannot Camestres be reduced either directly or indirectly to Felapton? Can Felapton be reduced to Camestres? [K.]
276. Assuming that in the first figure the major must be universal and the minor affirmative, shew by reductio ad absurdum that the conclusion in the second figure must be negative and in the third particular. [J.]
277. State the following argument in a syllogism of the third figure, and reduce it, both directly and indirectly, to the first:—Some things worthy of being known are not directly useful, for every truth is worthy of being known, while not every truth is directly useful. [M.]
278. State the figure and mood of the following syllogism; reduce it to the first figure; and examine whether there is anything unnatural in the argument as it stands:—
None who dishonour the king can be true patriots; for a true patriot must respect the law, and none who respect the law would dishonour the king. [J.]
279. “Rejecting the fourth figure and the subaltern moods, we may say with Aristotle: A is proved only in one figure and one mood, E in two figures and three moods, I in two figures and four moods, O in three figures and six moods. For this reason, A is declared by Aristotle to be the most difficult proposition to establish, and the easiest to overthrow; O, the reverse.” Discuss the fitness of these data to establish the conclusion. [K.]
280. Prove, from the general rules of the syllogism, that the number of possible moods, irrespective of difference of figure, is 11.
In the 19 moods of the mnemonic verses, only 10 out of the possible 11 moods are represented. Find the missing mood, and account for its absence from the verses. [L.]
281. Given
(1) the conclusion of a syllogism in the first figure,
(2) the minor premiss of a syllogism in the second figure,
(3) the major premiss of a syllogism in the third figure,
340 examine in each case how far the quality and quantity of the two remaining propositions of the syllogism can be determined (it being given that the syllogism does not contain a strengthened premiss or a weakened conclusion).
Express the result, as far as possible, in general terms in each figure. [J.]
282. Find out in which of the valid syllogistic moods the combination of one premiss with the subcontrary of the conclusion would establish the subcontrary of the other premiss. [L.]
283. Construct a syllogism in accordance with each of the following two dicta:—
(1) Any object that is found to lack a property known to belong to all members of a class must be excluded from that class;
(2) If any objects that have been included in a class are found to lack a certain property, then that property cannot be predicated of all members of the class.
Assign the mood and figure of each argument, and shew the relations between the above dicta and the dictum de omni et nullo. [L.]
284. Shew that any given mood may be directly reduced to any other mood, provided (1) that the latter contains neither a strengthened premiss nor a weakened conclusion, and (2) that if the conclusion of the former is universal, the conclusion of the latter is also universal [K.]
285. Shew that any given mood may be directly or indirectly reduced to any other mood, provided that the latter has not either a strengthened premiss or a weakened conclusion, unless the same is true of the former also. [K.]
286. Examine the following statement of De Morgan’s:—“There are but six distinct syllogisms. All others are made from them by strengthening one of the premisses, or converting one or both of the premisses, where such conversion is allowable; or else by first making the conversion, and then strengthening one of the premisses.” [K.]
287. Shew, by the aid of the process of indirect reduction, that the special rules for Figure 4 given in section [244] are mutually deducible from one another. [RR.]