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
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.
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.]
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 word344) 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. |
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.
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
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.