323 Compare with the above the following syllogism which has two singular premisses:—The Lord Chancellor receives a higher salary than the Prime Minister; Lord Herschell is the Lord Chancellor; therefore, Lord Herschell receives a higher salary than the Prime Minister. These premisses would presumably be compounded by Bain into the single proposition, “The Lord Chancellor, Lord Herschell, receives a higher salary than the Prime Minister.”
Do not Bain’s criticisms apply to these syllogisms as much as to the syllogism with two singular premisses? The method of treatment adopted is indeed particularly applicable to syllogisms in which the middle term is subject in both premisses. But we may always combine the two premisses of a syllogism in a single statement, and it is always true that the conclusion of a syllogism contains a part of, and only a part of, the information contained in the two premisses taken together; hence we may always get Bain’s result.324 In other words, in the conclusion of every syllogism “we repeat less than we are entitled to say,” or, if we care to put it so, “drop from a complex statement some portion not desired at the moment.”
324 It may be pointed out that the general method adopted by Boole in his Laws of Thought is to sum up all his given propositions in a single proposition, and then eliminate the terms that are not required. Compare also the methods employed in Appendix C of the present work.
300 208. Charge of incompleteness brought against the ordinary syllogistic conclusion.—This charge (a consideration of which will appropriately supplement the discussion contained in the preceding section) is brought by Jevons (Principles of Science, 4, § 8) against the ordinary syllogistic conclusion. The premisses Potassium floats on water, Potassium is a metal yield, according to him, the conclusion Potassium metal is potassium floating on water. But “Aristotle would have inferred that some metals float on water. Hence Aristotle’s conclusion simply leaves out some of the information afforded in the premisses ; it even leaves us open to interpret the some metals in a wider sense than we are warranted in doing.”
In reply to this it may be remarked: first, that the Aristotelian conclusion does not profess to sum up the whole of the information contained in the premisses of the syllogism; secondly, that some must here be interpreted to mean merely “not none,” “one at least.” The conclusion of the above syllogism might perhaps better be written “some metal floats on water,” or “some metal or metals &c.” Lotze remarks in criticism of Jevons: “His whole procedure is simply a repetition or at the outside an addition of his two premisses; thus it merely adheres to the given facts, and such a process has never been taken for a Syllogism, which always means a movement of thought that uses what is given for the purpose of advancing beyond it…… The meaning of the Syllogism, as Aristotle framed it, would in this case be that the occurrence of a floating metal Potassium proves that the property of being so light is not incompatible with the character of metal in general” (Logic, II. 3, note). This criticism is perhaps pushed a little too far. It is hardly a fair description of Jevons’s conclusion to say that it is the mere sum of the premisses; for it brings out a relation between two terms which was not immediately apparent in the premisses as they originally stood. Still there can be no doubt that the elimination of the middle term is the very gist of syllogistic reasoning as ordinarily understood.
It may be added, as an argumentum ad hominem against Jevons, that his own conclusion also leaves out some of the information afforded in the premisses. For we cannot pass 301 back from the proposition Potassium metal is potassium floating on water to either of the original premisses.
209. The connexion between the Dictum de omni et nullo and the ordinary Rules of the Syllogism.—The dictum de omni et nullo was given by Aristotle as the axiom on which all syllogistic inference is based. It applies directly, however, to those syllogisms only in which the major term is predicate in the major premiss, and the minor term subject in the minor premiss (i.e., to what are called syllogisms in figure 1). The rules of the syllogism, on the other hand, apply independently of the position of the terms in the premisses. Nevertheless, it is interesting to trace the connexion between them. It will be found that all the rules are involved in the dictum, but some of them in a less general form, in consequence of the distinction just pointed out.
The dictum may be stated as follows:—“Whatever is predicated, whether affirmatively or negatively, of a term distributed may be predicated in like manner of everything contained under it.”
(1) The dictum provides for three and only three terms; namely, (i) a certain term which must be distributed, (ii) something predicated of this term, (iii) something contained under it. These terms are respectively the middle, major, and minor. We may consider the rule relating to the ambiguity of terms to be also contained here, since if any term is ambiguous we have practically more than three terms.
(2) The dictum provides for three and only three propositions; namely, (i) a proposition predicating something of a term distributed, (ii) a proposition declaring something to be contained under this term, (iii) a proposition making the original predication of the contained term. These propositions constitute respectively the major premiss, the minor premiss, and the conclusion, of the syllogism.
(3) The dictum prescribes not merely that the middle term shall be distributed once at least in the premisses, but more definitely that it shall be distributed in the major premiss,—“Whatever is predicated of a term distributed.”325
325 This is another form of what will be found to be a special rule of figure 1, namely, that the major premiss must be universal. Compare section 244.
302 (4) Illicit process of the major is provided against indirectly. This fallacy can be committed only when the conclusion is negative; but the words “in like manner” declare that if there is a negative conclusion, the major premiss must also be negative; and since in any syllogism to which the dictum directly applies, the major term is predicate of this premiss, it will be distributed in its premiss as well as in the conclusion. Illicit process of the minor is provided against inasmuch as the dictum warrants us in making our predication in the conclusion only of what has been shewn in the minor premiss to be contained under the middle term.
(5) The proposition declaring that something is contained under the term distributed must necessarily be an affirmative proposition. The dictum provides, therefore, that the premisses shall not both be negative.326
326 It really provides that the minor premiss shall be affirmative, which again is one of the special rules of figure 1.
(6) The words “in like manner” clearly provide against a breach of the rule that if one premiss is negative, the conclusion must be negative, and vice versâ.
EXERCISES.327
327 The following exercises may be solved without any knowledge beyond what is contained in the preceding chapter, the assumption however being made that if no rule of the syllogism as given in section 199 or section 201 is broken, then the syllogism is valid.
210. If P is a mark of the presence of Q, and R of that of S, and if P and R are never found together, am I right in inferring that Q and S sometimes exist separately? [V.]
The premisses may be stated as follows:
All P is Q,
All R is S,
No P is R ;
and in order to establish the desired conclusion we must be able to
infer at least one of the following,—Some Q is not S,
Some S is not Q.
But neither of these propositions can be inferred; for they
distribute respectively S and Q, and neither of these
terms is distributed in the given premisses. The question is,
therefore, to be answered in the negative.
303 211. If it be known concerning a syllogism in the Aristotelian system that the middle term is distributed in both premisses, what can we infer as to the conclusion? [C.]
If both premisses are affirmative, they can between them distribute only two terms, and by hypothesis the middle term is
distributed twice in the premisses; hence the minor term cannot be
distributed in the premisses, and it follows that the conclusion must
be particular.
If one of the premisses is negative, there may be three distributed
terms in the premisses; these must, however, be the middle term twice
(by hypothesis) and the major term (since the conclusion must now be
negative and will therefore distribute the major term); hence the
minor term cannot be distributed in the premisses, and it again
follows that the conclusion must be particular.
But either both premisses will be affirmative, or one affirmative
and the other negative; in any case, therefore, we can infer that the
conclusion will be particular.
212. Shew directly in how many ways it is possible to prove the conclusions SaP, SeP ; point out those that conform immediately to the Dictum de omni et nullo ; and exhibit the equivalence between these and the remainder. [W.]
(1) To prove All S is P.
Both premisses must be affirmative, and both must be universal.
S being distributed in the conclusion must be distributed in
the minor premiss, which must therefore be All S is M.
M not being distributed in the minor must be distributed in
the major, which must therefore be All M is P.
SaP can therefore be proved in only one way, namely,
| All M is P, | |
| All S is M, | |
| therefore, | All S is P ; |
and this
syllogism conforms immediately to the Dictum.
(2) To prove No S is P.
Both premisses must be universal, and one must be negative while
the other is affirmative; i.e., one premiss must be E
and the other A.
First, let the major be E, i.e., either No
M is P or No P is M. In each case the minor must be
affirmative and must distribute S ; therefore, it will be All
S is M.
304 Secondly, let
the minor be E, i.e., either No S is M or No M
is S. In each case the major must be affirmative and must
distribute P ; therefore, it will be All P is M.
We can then prove SeP in four ways, thus,—
| (i) | MeP, | (ii) | PeM, | (iii) | PaM, | (iv) | PaM, |
| SaM, | SaM, | SeM, | MeS, | ||||
| ⎯⎯ | ⎯⎯ | ⎯⎯ | ⎯⎯ | ||||
| SeP. | SeP. | SeP. | SeP. |
Of these, (i) only conforms immediately to the dictum, and
we have to shew the equivalence between it and the others.
The only difference between (i) and (ii) is that the major premiss
of the one is the simple converse of the major premiss of the other;
they are, therefore, equivalent. Similarly the only difference between
(iii) and (iv) is that the minor premiss of the one is the simple
converse of the minor premiss of the other; they are, therefore,
equivalent.
Finally, we may shew that (iv) is equivalent to (i) by transposing
the premisses and converting the conclusion.
213. Given that the major term is distributed in the premisses and undistributed in the conclusion of a valid syllogism, determine the syllogism. [C.]
Since the major term is undistributed in the conclusion, the conclusion—and, therefore, both premisses—must be affirmative. Hence, in order to distribute P, the major premiss must be PaM ; and in order to distribute M (which is not distributed in the major premiss), the minor premiss must be MaS. It follows that the syllogism must be
| All P is M, | |
| All M is S, | |
| therefore, | Some S is P. |
214. Prove that if three
propositions involving three terms (each of which occurs in two of the
propositions) are together incompatible, then (a) each term is
distributed at least once, and (b) one and only one of the
propositions is negative.
Shew that these rules are equivalent to the rules of the syllogism.
[J.]
No two of the propositions can be formally incompatible with one another, since they do not contain the same terms. But each pair must be incompatible with the third, i.e., the contradictory of any one must be deducible from the other two. It follows that 305 we shall have three valid syllogisms, in which the given propositions taken in pairs are the premisses, whilst the contradictory of the third proposition is in each case the conclusion.328
Then (a) each term must be distributed once at
least. For if any one of the terms failed to be distributed at
least once, we should obviously have undistributed middle in one of
our syllogisms; and (since a term undistributed in a proposition is
distributed in its contradictory) illicit major or minor in the two
others. If, however, the above condition is fulfilled, it is clear
that we cannot have either undistributed middle, or illicit major or
minor. Hence rule (a) is equivalent to the syllogistic rules
relating to the distribution of terms.
Again, (b) one of the propositions must be negative, but
not more than one of them can be negative. For if all three were
affirmative, then (since the contradictory of an affirmative is
negative) we should in each of our syllogisms infer a negative from
two affirmatives; and if two were negative, we should have two
negative premisses in one of our syllogisms, and (since the
contradictory of a negative is affirmative) an affirmative conclusion
with a negative premiss in each of the others. If, however, the above
condition is fulfilled, it is clear that we cannot have either two
negative premisses, or two affirmative premisses with a negative
conclusion, or a negative premiss with an affirmative conclusion.
Hence rule (b) is equivalent to the syllogistic rules relating
to quality.
328 Every syllogism involves two others, in each of which one of the original premisses combined with the contradictory of the conclusion proves the contradictory of the other original premiss. Hence the three syllogisms referred to in the text mutually involve one another. Compare sections 264, 265.
215. Explain what is meant by a syllogism ; and put the following argument into syllogistic form:—"We have no right to treat heat as a substance, for it may be transformed into something which is not heat, and is certainly not a substance at all, namely, mechanical work.” [N.]
216. Put the following argument into syllogistic form:—How can anyone maintain that pain is always an evil, who admits that remorse involves pain, and yet may sometimes be a real good? [V.]
306 217. It has been pointed out by Ohm that reasoning to the following effect occurs in some works on mathematics:—“A magnitude required for the solution of a problem must satisfy a particular equation, and as the magnitude x satisfies this equation, it is therefore the magnitude required.” Examine the logical validity of this argument. [C.]
218. Obtain a conclusion from the two negative premisses,—No P is M, No S is M. [K.]
219. If it is false that the attribute B is ever found coexisting with A, and not less false that the attribute C is sometimes found absent from A, can you assert anything about B in terms of C? [C.]
220. Give examples (in symbols—taking S, M, P, as minor, middle, and major terms, respectively) in which, attempting to infer a universal conclusion where we have a particular premiss, we commit respectively one but one only of the following fallacies,—(a) undistributed middle, (b) illicit major, (c) illicit minor. Give also an example in which, making the same attempt, we commit none of the above fallacies. [K.]
221. Can an apparent syllogism break directly all the rules of the syllogism at once? [K.]
222. Can you give an instance of an invalid syllogism in which the major premiss is universal negative, the minor premiss affirmative, and the conclusion particular negative? If not, why not? [K.]
223. Shew that
(i) If both premisses of a syllogism are affirmative, and one but
only one of them universal, they will between them distribute only one
term;
(ii) If both premisses are affirmative and both universal, they
will between them distribute two terms;
(iii) If one but only one premiss is negative, and one but only one
premiss universal, they will between them distribute two terms;
(iv) If one but only one premiss is negative, and both premisses
are universal, they will between them distribute three terms. [K.]
224. Ascertain how many distributed terms there may be in the premisses of a syllogism more than in the conclusion. [L.]
225. If the minor premiss of a syllogism is negative, what do you know about the position of the terms in the major? [O’S.]
307 226. If the major term of a syllogism is the predicate of the major premiss, what do you know about the minor premiss? [L.]
227. How much can you tell
about a valid syllogism if you know (1) that only the middle term is
distributed;
(2) that only the middle and minor terms are distributed;
(3) that all three terms are distributed? [W.]
228. What can be determined respecting a valid syllogism under each of the following conditions: (1) that only one term is distributed, and that only once; (2) that only one term is distributed, and that twice; (3) that two terms only are distributed, each only once; (4) that two terms only are distributed, each twice? [L.]
229. Two propositions are given having a term in common. If they are I and A, shew that either no conclusion or two can be deduced; but if I and E, always and only one. [T.]
230. Find out, from the rules of the syllogism, what are the valid forms of syllogism in which the major premiss is particular affirmative. [J.]
231. Given (a) that the major premiss, (b) that the minor premiss, of a valid syllogism is particular negative, determine in each case the syllogism. [K.]
232. Given that the major premiss of a valid syllogism is affirmative, and that the major term is distributed both in premisses and conclusion, while the minor term is undistributed in both, determine the syllogism. [N.]
233. Shew directly in how many ways it is possible to prove the conclusions SiP, SoP. [W.]
234. Shew that if the rule that a negative conclusion requires a negative premiss be omitted from the general rules of the syllogism, the only invalid syllogism thereby admitted is such that, if its conclusion be false whilst its premisses are true, the three terms of the syllogism must be absolutely coextensive. [O’S.]
235. Find, by direct application of the fundamental rules of syllogism, what are the valid forms of syllogism in which neither of the premisses is a universal proposition having the same quality as the conclusion. [J.]
308 236. In what
cases will contradictory major premisses both yield conclusions when
combined with the same minor?
How are the conclusions related?
Shew that in no case will contradictory minor premisses both yield
conclusions when combined with the same major. [O’S.]
237. (a) All just
actions are praiseworthy; (b) No unjust actions are expedient;
(c) Some inexpedient actions are not praiseworthy; (d)
Not all praiseworthy actions are inexpedient.
Do (c) and (d) follow from (a) and (b)?
[K.]
238. Reduce the following
arguments to ordinary syllogistic form:
(i) No M is S, Whatever is not M is P, therefore,
All S is P ;
(ii) It cannot be that no not-S is P, for some M is P and no M is S ;
(iii) It is impossible for the three propositions, All M is
P, Anything that is not M is not S, Some things that are
not P are S, all to be true together;
(iv) Everything is M or P, Nothing is both S and M,
therefore, All S is P. [K.]
239. Shew that the following
syllogisms break directly or indirectly all the rules of the
syllogism:
(1) All P is M, All S is M, therefore, Some S is
not P ;
(2) All M is P, All M is S, therefore, No S is
P. [K.]
[The so-called rules that every syllogism contains three and only three terms, and that every syllogism consists of three and only three propositions, are not here included under the rules of the syllogism.]
240. In a circular argument involving two valid syllogisms, Q and U are used as premisses to prove R, while R and V are used as premisses to prove Q ; shew that U and V must be a pair of complementary propositions, i.e., of the forms All M is N and All N is M respectively. [J.]
241. Shew that if two valid syllogisms have a common premiss while the other premisses are contradictories, both the conclusions must be particular. [K.]
242. Given the premisses of a valid syllogism, examine in what cases it is (a) possible, (b) impossible, to determine which is the minor term and which the major term. [J.]
CHAPTER II.
THE FIGURES AND MOODS OF THE SYLLOGISM.
243. Figure and Mood.—By the figure of a syllogism is meant the position of the terms in the premisses. Denoting the major, middle, and minor terms by the letters P, M, S respectively, and stating the major premiss first, we have four figures of the syllogism as shewn in the following table:—
| Fig. 1. | Fig. 2. | Fig. 3. | Fig. 4. |
| M – P | P – M | M – P | P – M |
| S – M | S – M | M – S | M – S |
| ⎯⎯ | ⎯⎯ | ⎯⎯ | ⎯⎯ |
| S – P | S – P | S – P | S – P |
By the mood of a syllogism is meant the quantity and quality of the premisses and conclusion. For example, AAA is a mood in which both the premisses and also the conclusion are universal affirmatives; EIO is a mood in which the major is a universal negative, the minor a particular affirmative, and the conclusion a particular negative. It is clear that if figure and mood are both given, the syllogism is given.
244. The Special Rules of the Figures; and the Determination of the Legitimate Moods in each Figure.329—It may first of all be shewn that certain combinations of premisses are incapable of yielding a valid conclusion in any figure. A priori, there are possible the following sixteen different combinations of premisses, the major premiss being always stated first:—AA, AI, AE, AO, IA, II, IE, IO, EA, EI, EE, EO, OA, OI, OE, OO. Referring back, however, to the syllogistic rules and corollaries (as given in sections 199, 200), we find that EE, 310 EO, OE, OO (being combinations of negative premisses) yield no conclusion by rule 5; that II, IO, OI (being combinations of particular premisses) are excluded by corollary i.; and that IE is excluded by corollary iii., which tells us that nothing follows from a particular major and a negative minor.
329 The method of determination here adopted is only one amongst several possible methods. Another is suggested, for example, in sections 212, 233.
We are left then with the following eight possible combinations:—AA, AI, AE, AO, IA, EA, EI, OA ; and we may go on to enquire in which figures these will yield conclusions. In pursuing this enquiry, special rules of the various figures may be determined, which, taken together with the three corollaries established in section 200, replace the general rules of distribution. These special rules, supplemented by the general rules of quality and the corollaries,330 will enable the validity of the different moods to be tested by a mere inspection of the form of the propositions of which they consist.
330 The general rules of quality and the corollaries can be directly applied without reference to the position of the terms in the premisses of a syllogism. This is not the case with the general rules of distribution. The object of the special rules is, in the case of each particular figure, to substitute for the general rules of distribution special rules of quantity and quality.
The special rules331 and the legitimate moods of Figure 1.
331 As indicated in section 209, the special rules of figure 1 follow immediately from the dictum de omni et nullo.
The position of the terms in figure 1 is shewn thus,—
M – P
S – M
⎯⎯
S – P
and it can
be deduced from the general rules of the syllogism that in this
figure:—
(1) The minor premiss must be affirmative. For if it were
negative, the major premiss would have to be affirmative by rule 5,
and the conclusion negative by rule 6. The major term would therefore
be distributed in the conclusion, and undistributed in its premiss;
and the syllogism would be invalid by rule 4.
(2) The major premiss must be universal. For the middle
term, being undistributed in the affirmative minor premiss, must be
distributed in the major premiss.
311 Rule (1) shews that AE and AO and rule (2) that IA and OA, yield no conclusions in this figure. We are accordingly left with only four combinations, namely, AA, AI, EA, EI From the rules that a particular premiss cannot yield a universal conclusion or a negative premiss an affirmative conclusion, while conversely a negative conclusion requires a negative premiss, it follows further that AA will justify either of the conclusions A or I, EA either E or O, AI only I, EI only O. There are then six moods in figure 1 which do not offend against any of the rules of the syllogism,332 namely, AAA, AAI, AII, EAE, EAO, EIO.
332 Rule (2) provides against undistributed middle, and rule (1) against illicit major. We cannot have illicit minor, unless we have a universal conclusion with a particular premiss, and this also has been provided against.
Mr Johnson points out that the following symmetrical rules may be laid down for the correct distribution of terms in the different figures; and that these rules (three in each figure) taken together with the rules of quality are sufficient to ensure that no syllogistic rule is broken.
(i) To avoid undistributed middle: in figure 1, If the minor is affirmative, the major must be universal; in figure 4, If the major is affirmative, the minor must be universal; in figure 2, One premiss must be negative; in figure 3, One premiss must be universal. (The last of these rules is of course superfluous if the corollaries contained in section 200 are supposed given.)
(ii) To avoid illicit major: in figures 1 and 3, If the conclusion is negative, the major must be negative and, therefore, the minor affirmative; in figures 2 and 4, If the conclusion is negative, the major must be universal.
(iii) To avoid illicit minor: in figures 1 and 2, If the minor is particular, the conclusion must be particular; in figures 3 and 4, If the minor is affirmative, the conclusion must be particular. (The first of these two rules is again superfluous as a special rule if the corollaries are supposed given.)
The above rules are substantially identical with those given in the text.
The actual validity of these moods may be established by shewing that the axiom of the syllogism, the dictum de omni et nullo, applies to them; or by taking them severally and shewing that in each case the cogency of the reasoning is self-evident.
The special rules and the legitimate moods of Figure 2.
The position of the terms in figure 2 is shewn thus,—
P – M
S – M
⎯⎯
S – P ;
312 and its special rules (which the reader is recommended to deduce from the general rules of
the syllogism for himself) are,—
(1) One premiss must be negative ;
(2) The major premiss
must be universal.
The application of these rules again leaves six moods, namely, AEE, AEO, AOO, EAE, EAO, EIO.
Recourse cannot now he had directly to the dictum de omni et nullo in order to shew positively that these moods are legitimate. It may, however, be shewn in each case that the cogency of the reasoning is self-evident. The older logicians did not adopt this course; their method was to shew that, by the aid of immediate inferences, each mood could be reduced to such a form that the dictum did apply directly to it. The doctrine of reduction resulting from the adoption of this method will be discussed in the following chapter.
The special rules and the legitimate moods of Figure 3.
The position of the terms in this figure is shewn thus,—
M – P
M – S
⎯⎯
S – P ;
and its special rules are,—
(1) The minor must be affirmative ;
(2) The conclusion must be particular.
Proceeding as before, we are left with six valid moods, namely, AAI, AII, EAO, EIO, IAI, OAO.
The special rules and the legitimate moods of Figure 4.
The position of the terms in this figure is shewn thus,—
P – M
M – S
⎯⎯
S – P ;
and the following may be given as its special rules,—
(1) If the major is affirmative, the minor must be
universal ;
(2) If either premiss is negative, the major must be
universal ; 313
(3) If the minor is affirmative, the conclusion must be
particular.333
333 The special rules of the fourth figure are variously stated. They are given in the above form in the Port Royal Logic, pp. 202, 203. See, also, section 255.
The result of the application of these rules is again six valid moods, namely, AAI, AEE, AEO, EAO, EIO, IAI.
Our final conclusion then is that there are 24 valid moods, namely, six in each figure.
In Figure 1, AAA, AAI, EAE, EAO,
AII, EIO.
In Figure 2, EAE, EAO,
AEE, AEO, EIO, AOO.
In Figure 3,
AAI, IAI, AII, EAO, OAO,
EIO.
In Figure 4, AAI, AEE, AEO,
EAO, IAI, EIO.
245. Weakened Conclusions and Subaltern Moods.—When from premisses that would have justified a universal conclusion we content ourselves with inferring a particular (as, for example, in the syllogism All M is P, All S is M, therefore, Some S is P), we are said to have a weakened conclusion, and the syllogism is said to be a weakened syllogism or to be in a subaltern mood (because the conclusion might be obtained by subaltern inference334 from the conclusion of the corresponding unweakened mood).
334 In treating the syllogism on the traditional lines it is assumed that S, M, P all represent existing classes. Subaltern inference is, therefore, a valid process.
In the preceding section it has been shewn that in each figure there are six moods which do not offend against any of the syllogistic rules: so that in all there are 24 distinct valid moods. Five of these, however, have weakened conclusions; and, since we are not likely to be satisfied with a particular conclusion when the corresponding universal can be obtained from the same premisses, these moods are of no practical importance. Accordingly when the moods of the various figures are enumerated (as in the mnemonic verses) they are usually omitted. Still, their recognition gives a completeness to the theory of the syllogism, which it cannot otherwise possess. There is also a symmetry in the result of 314 their recognition as yielding exactly six legitimate moods in each figure.335
335 It has been remarked that 19 being a prime number at once suggests incompleteness or artificiality in the common enumeration.
The subaltern moods are,—
In Figure 1, AAI, EAO ;
In Figure 2, EAO, AEO ;
In Figure 4, AEO.
It is obvious that there can be no weakened conclusion in Figure 3, since in no case is it possible to infer more than a particular conclusion in this figure.
AAI in Figure 4 is sometimes spoken of as a subaltern mood. But this is a mistake. With the premisses All P is M, All M is S, the conclusion Some S is P is certainly in one sense weaker than the premisses would warrant since the universal conclusion All P is S might have been inferred. But All P is S is not the universal corresponding to Some S is P. The subjects of these two propositions are different; and we infer all that we possibly can about S when we say that some S is P. In other words, regarded as a mood of figure 4, this mood is not a subaltern. AAI in figure 4 is thus differentiated from AAI in figure 1, and its inclusion in the mnemonic verses justified.
246. Strengthened Syllogisms.—If in a syllogism the same conclusion can still be obtained although for one of the premisses we substitute its subaltern, the syllogism is said to be a strengthened syllogism. A strengthened syllogism is thus a syllogism with an unnecessarily strengthened premiss.336