Part 3 — CHAPTER VII.

IRREGULAR AND COMPOUND SYLLOGISMS.

322. The Enthymeme.—By the enthymeme, Aristotle meant what has been called the “rhetorical syllogism” as opposed to the apodeictic, demonstrative, theoretical syllogism. The following is from Mansel’s notes to Aldrich (pp. 209 to 211): “The enthymeme is defined by Aristotle, συλλογισμὸς ἐξ εἰκότων ἤ σημείων. The εἰκὸς and σημεῖων themselves are propositions; the former stating a general probability, the latter a fact, which is known to be an indication, more or less certain, of the truth of some further statement, whether of a single fact or of a general belief. The former is a proposition nearly, though not quite, universal ; as ‘Most men who envy hate’: the latter is a singular proposition, which however is not regarded as a sign, except relatively to some other proposition, which it is supposed may be inferred from it. The εἰκός, when employed in an enthymeme, will form the major premiss of a syllogism such as the following:

Most men who envy hate,

This man envies,

therefore, This man (probably) hates.

“The reasoning is logically faulty; for, the major premiss not being absolutely universal, the middle term is not distributed.

“The σημεῖων will form one premiss of a syllogism which may be in any of the three figures, as in the following examples:

Figure 1. All ambitious men are liberal,

Pittacus is ambitious,

therefore, Pittacus is liberal.

Figure 2. All ambitions men are liberal,

Pittacus is liberal,

therefore, Pittacus is ambitious.

Figure 3. Pittacus is liberal,

Pittacus is ambitious,

therefore, All ambitious men are liberal.

“The syllogism in the first figure alone is logically valid. In the second, there is an undistributed middle term; in the third, an illicit process of the minor.”400

An enthymeme is now usually defined as a syllogism incompletely stated, one of the premisses or the conclusion being understood but not expressed.401 The arguments of everyday life are to a large extent enthymematic in this sense; and the same may be said of fallacious arguments, which are seldom completely stated, or their want of cogency would be more quickly recognised.

An enthymeme is said to be of the first order when the major premiss is suppressed; of the second order when the minor premiss is suppressed; and of the third order when the conclusion is suppressed.

Thus, “Balbus is avaricious, and therefore, he is unhappy,” is an enthymeme of the first order; “All avaricious persons are unhappy, and therefore, Balbus is unhappy,” is an enthymeme of the second order; “All avaricious persons are unhappy, and Balbus is avaricious,” is an enthymeme of the third order.

323. The Polysyllogism and the Epicheirema.—A chain of syllogisms, that is, a series of syllogisms so linked together that the conclusion of one becomes a premiss of another, is called a polysyllogism. In a polysyllogism, any individual syllogism the conclusion of which becomes the premiss of a succeeding one is called a prosyllogism, any individual syllogism one of the premisses of which is the conclusion of a preceding syllogism is called an episyllogism. Thus,—

All C is B,⎫

All B is C,⎬ prosyllogism,

therefore, All B is D,⎭⎫

but All A is B,⎬ episyllogism.

therefore, All A is D,⎭

The same syllogism may of course be both an episyllogism and a prosyllogism, as would be the case with the above episyllogism if the chain were continued further.

A chain of reasoning402 is said to be progressive (or synthetic or episyllogistic) when the progress is from prosyllogism to episyllogism. Here the premisses are first given, and we pass on by successive steps of inference to the ultimate conclusion which they yield. A chain of reasoning is, on the other hand, said to be regressive (or analytic or prosyllogistic) when the progress is from episyllogism to prosyllogism. Here the ultimate conclusion is first given and we pass back by successive steps of proof to the premisses on which it may be based.403

An epicheirema is a polysyllogism with one or more prosyllogisms briefly indicated only. That is, one or more of the syllogisms of which the polysyllogism is composed are enthymematic. The following is an example:

All B is D, because it is C,

All A is B,

therefore, All A is D.404

324. The Sorites.—A sorites is a polysyllogism in which all the conclusions are omitted except the final one, the premisses being given in such an order that any two successive propositions contain a common term. Two forms of sorites are usually recognised, namely, the so-called Aristotelian sorites and the Goclenian sorites. In the former, the premiss stated first contains the subject of the conclusion, while the term common to any two successive premisses occurs first as predicate and then as subject; in the latter, the premiss stated first contains the predicate of the conclusion, while the term common to any two successive premisses occurs first as subject and then as predicate. The following are examples:

Aristotelian Sorites,—All A is B,

All B is C,

All C is D,

All D is E,

therefore, All A is E.

Goclenian Sorites,—All D is E,

All C is D,

All B is C,

All A is B,

therefore, All A is E.

It will be found that, in the case of the Aristotelian sorites, if the argument is drawn out in full, the first premiss and the suppressed conclusions all appear as minor premisses in successive syllogisms. Thus, the Aristotelian sorites given above may be analysed into the three following syllogisms,—

(1) All B is C,

All A is B,

therefore, All A is C ;

(2) All C is D,

All A is C,

therefore, All A is D ;

(3) All D is E,

All A is D,

therefore, All A is E.

Here the premiss originally stated first is the minor premiss of (1), the conclusion of (1) is the minor premiss of (2), that of (2) the minor premiss of (3); and so it would go on if the number of propositions constituting the sorites were increased.

In the Goclenian sorites, the premisses are the same, but their order is reversed, and the result of this is that the premiss originally stated first and the suppressed conclusions become major premisses in successive syllogisms. Thus, the Goclenian sorites given above may be analysed into the three following syllogisms,—

(1) All D is E,

All C is D,

therefore, All C is E ;

(2) All C is E,

All B is C,

therefore, All B is E ;

(3) All B is E,

All A is B,

therefore, All A is E.

Here the premiss originally stated first is the major premiss of (1), the conclusion of (1) is the major premiss of (2); and so on.

The so-called Aristotelian sorites405 is that to which the greater prominence is usually given; but it will be observed that the order of premisses in the Goclenian form is that which corresponds to the customary order of premisses in a simple syllogism.406

A sorites may of course consist of conditional or hypothetical propositions; and it is not at all unusual to find propositions of these kinds combined in this manner. Theoretically a sorites might also consist of alternative propositions; but it is not likely that this combination would ever occur naturally.

325. The Special Rules of the Sorites.—The following special rules may be given for the ordinary Aristotelian sorites, as defined in the preceding section:— (1) Only one premiss can be negative; and if one is negative, it must be the last. (2) Only one premiss can be particular; and if one is particular, it must be the first.

Any Aristotelian sorites may be represented in skeleton form, the quantity and quality of the premisses being left undetermined, as follows:—

SM₁

M₁, M₂

M₂, M₃

………………

………………

M_n−2, M_n−1

M_n−1, M_n

M_n, P

⎯⎯⎯⎯⎯⎯

S P

(1) There cannot be more than one negative premiss, for if there were—since a negative premiss in any syllogism necessitates a negative conclusion—we should in analysing the sorites somewhere come upon a syllogism containing two negative premisses.

Again, if one premiss is negative, the final conclusion must be negative. Hence P must be distributed in the final conclusion. Therefore, it must be distributed in its premiss, i.e., the last premiss, which must accordingly be negative. If any premiss then is negative, this is the one.

(2) Since it has been shewn that all the premisses, except the last, must be affirmative, it is clear that if any, except the first, were particular, we should somewhere commit the fallacy of undistributed middle.

The special rules of the Goclenian sorites, as defined in the preceding section, may be obtained by transposing “first” and “last” in the above.

326. The possibility of a Sorites in a Figure other than the First.—It will have been noticed that in our analysis both of the Aristotelian and of the Goclenian sorites all the resulting syllogisms are in figure 1. Such sorites may accordingly be said to be themselves in figure 1. The question arises whether a sorites is possible in any other figure.

The usual answer to this question is that the first or the last syllogism of a sorites may be in figure 2 or 3 (e.g., in figure 2 we may have A is B, B is C, C is D, D is E, F is not E, therefore, A is not F) but that it is impossible that all the steps should be in either of these figures.407 “Every one,” says Mill, “who understands the laws of the second and third figures (or even the general laws of the syllogism) can see that no more than one step in either of them is admissible in a sorites, and that it must either be the first or the last” (Examination of Hamilton, pp. 514, 5).

This treatment of the question seems, however, open to refutation by the simple method of constructing examples. Take, for instance, the following sorites:—

(i) Some S is not M₁,

All M₂ is M₁,

All M₃ is M₂,

All M₄ is M₃,

All P is M₄,

therefore, Some S is not P.

(ii) Some M₄ is not P,

All M₄ is M₃,

All M₃ is M₂,

All M₂ is M₁,

All M₁ is S,

therefore, Some S is not P.

Analysing the first of the above, and inserting the suppressed conclusions in square brackets, we have—

Some S is not M₁,

All M₂ is M₁,

[therefore, Some S is not M₂,]

All M₃ is M₂,

[therefore, Some S is not M₃,]

All M₄ is M₃,

[therefore, Some S is not M₄,]

All P is M₄,

therefore, Some S is not P.

This is the only resolution of the sorites possible unless the order of the premisses is transposed, and it will be seen that all the resulting syllogisms are in figure 2 and in the mood Baroco. The sorites may accordingly be said to be in the same mood and figure. It is analogous to the Aristotelian sorites, the subject of the conclusion appearing in the premiss stated first, and the suppressed premisses being all minors in their respective syllogisms.

The corresponding analysis of (ii) yields the following:—

Some M₄ is not P,

All M₄ is M₃,

[therefore, Some M₃ is not P,]

All M₃ is M₂,

[therefore, Some M₂ is not P,]

All M₂ is M₁,

[therefore, Some M₁ is not P,]

All M₁ is S,

therefore, Some S is not P.

These syllogisms are all in figure 3 and in the mood Bocardo ; and the sorites itself may be said to be in the same mood and figure. It is analogous to the Goclenian sorites, the predicate of the conclusion appearing in the premiss stated first, and the suppressed premisses being majors in their respective syllogisms.

It will be observed that the rules given in the preceding section have not been satisfied in either of the above sorites, the reason being that the rules in question correspond to the special rules of figure 1, and do not apply unless the sorites is in that figure. For such sorites as are possible in figures 2, 3, and 4, other rules might be framed corresponding to the special rules of these figures in the case of the simple syllogism.

It is not maintained that sorites in other figures than the first are likely to be met with in common use, but their construction is of some theoretical interest.408

327. Ultra-total Distribution of the Middle Term.—The ordinary syllogistic rule relating to the distribution of the middle term does not contemplate the recognition of any signs of quantity other than all and some ; and if other signs are recognised, the rule must be modified. For example, the admission of the sign most yields the following valid reasoning, although the middle term is not distributed in either of the premisses:—

Most M is P,

Most M is S,

therefore, Some S is P.

Interpreting most in the sense of more than half, it clearly follows from the above premisses that there must be some M which is both S and P. But we cannot say that in either premiss the term M is distributed.

In order to meet cases of this kind, Hamilton (Logic, II. p. 362) gives the following modification of the rule relating to the distribution of the middle term: “The quantifications of the middle term, whether as subject or predicate, taken together, must exceed the quantity of that term taken in its whole extent”; in other words, we must have an ultra-total distribution of the middle term in the two premisses taken together.

De Morgan (Formal Logic, p. 127) writes as follows: “It is said that in every syllogism the middle term must be universal in one of the premisses, in order that we may be sure that the affirmation or denial in the other premiss may be made of some or all of the things about which affirmation or denial has been made in the first. This law, as we shall see, is only a particular case of the truth: it is enough that the two premisses together affirm or deny of more than all the instances of the middle term. If there be a hundred boxes, into which a hundred and one articles of two different kinds are to be put, not more than one of each kind into any one box, some one box, if not more, will have two articles, one of each kind, put into it. The common doctrine has it, that an article of one particular kind must be put into every box, and then some one or more of another kind into one or more of the boxes, before it may be affirmed that one or more of different kinds are found together.” De Morgan himself works the question out in detail in his treatment of the numerically definite syllogism (Formal Logic, pp. 141 to 170). The following may be taken as an example of numerically definite reasoning:—If 70 per cent. of M are P, and 60 per cent. are S, then at least 30 per cent. are both S and P.409 The argument may be put as follows: On the average, of 100 M’s 70 are P and 60 are S ; suppose that the 30 M’s which are not P are S, still 30 S’s are to be found in the remaining 70 M’s which are P’s; and this is the desired conclusion. Problems of this kind constitute a borderland between formal logic and algebra. Some further examples will be given in chapter 8 (section 345).

328. The Quantification of the Predicate and the Syllogism.—It will be convenient to consider briefly in this chapter the application of the doctrine of the quantification of the predicate to the syllogism; the result is the reverse of simplification.410 The most important points that arise may be brought out by considering the validity of the following syllogisms: in figure 1, UUU, IUη, AYI; in figure 2, ηUO, AUA; in figure 3, YAI. In the next section we will proceed more systematically, U and ω being left out of account.

(1) UUU in figure 1 is valid:—

All M is all P,

All S is all M,

therefore, All S is all P.

It will be observed that whenever one of the premisses is U, the conclusion may be obtained by substituting S or P (as the case may be) for M in the other premiss.

Without the use of quantified predicates, the above reasoning may be expressed by means of the two following syllogisms:

All M is P,All M is S,

All S is M,All P is M,

therefore, All S is P ;therefore, All P is S.

(2) IUη in figure 1 is invalid, if some is used in its ordinary logical sense. The premisses are Some M is some P and All S is all M. We may, therefore, obtain the legitimate conclusion by substituting S for M in the major premiss. This yields Some S is some P.

If, however, some is here used in the sense of some only, No S is some P follows from Some S is some P, and the original syllogism is valid, although a negative conclusion is obtained from two affirmative premisses.

This syllogism is given as valid by Thomson (Laws of Thought, § 103); but apparently only through a misprint for IEη. In his scheme of valid syllogisms (thirty-six in each figure), Thomson seems consistently to interpret some in its ordinary logical sense. Using the word in the sense of some only, several other syllogisms would be valid that he does not give as such.411

(3) AYI in figure 1, some being used in its ordinary logical sense, is equivalent to AAI in figure 3 in the ordinary syllogistic scheme, and is valid. But it is invalid if some is used in the sense of some only, for the conclusion now implies that S and P are partially excluded from each other as well as partially coincident, whereas this is not implied by the premisses. With this use of some, the correct conclusion can be expressed only by stating an alternative between SuP, SaP, SyP, and SiP. This case may serve to illustrate the complexities in which we should be involved if we were to attempt to use some consistently in the sense of some only.412

(4) ηUO in figure 2 is valid:—

No P is some M,

All S is all M,

therefore, Some S is not any P.

Without the use of quantified predicates, we can obtain the same conclusion in Bocardo, thus,—

Some M is not P,

All M is S,

therefore, Some S is not P.

It will be observed that both (3) and (4) are strengthened syllogisms.

(5) AUA in figure 2 runs as follows,—

All P is some M,

All S is all M,

therefore, All S is some P.

Here we have neither undistributed middle nor illicit process of major or minor, nor is any rule of quality broken, and yet the syllogism is invalid.413 Applying the rule given above that “whenever one of the premisses is U, the conclusion may be obtained by substituting S or P (as the case may be) for M in the other premiss,” we find that the valid conclusion is Some S is all P. More generally, it follows from this rule of substitution that if one premiss is U while in the other premiss the middle term is undistributed, then the term combined with the middle term in the U premiss must be undistributed in the conclusion. This appears to be the one additional syllogistic rule required if we recognise U propositions in syllogistic reasonings.

All danger of fallacy is avoided by breaking up the U proposition into two A propositions. In the case before us we have,—All P is M, All M is S ; All P is M, All S is M. From the first of these pairs of premisses we get the conclusion All P is S ; in the second pair the middle term is undistributed, and therefore no conclusion is yielded at all.

(6) YAI in figure 3 is valid:—

Some M is all P,

All M is some S,

therefore, Some S is some P.

The conclusion is however weakened, since from the given premisses we might infer Some S is all P.414 It will be observed that when we quantify the predicate, the conclusion of a syllogism may be weakened in respect of its predicate as well as in respect of its subject. In the ordinary doctrine of the syllogism this is for obvious reasons not possible.

Without quantification of the predicate the above reasoning may be expressed in Bramantip, thus,

All P is M,

All M is S,

therefore, Some S is P.

We could get the full conclusion, All P is S, in Barbara.

329. Table of valid moods resulting from the recognition of Y and η in addition to A, E, I, O.—If we adopt the sixfold schedule of propositions obtained by adding Only S is P (Y) and Not only S is P (η) to the ordinary fourfold schedule, as in section 150, every proposition is simply convertible, and, therefore, a valid mood in any figure is reducible to any other figure by the simple conversion of one or both of the premisses. Hence if the valid moods of any one figure are determined, those of the remaining figures may be immediately deduced therefrom.

It will be found that in each figure there are twelve valid moods, which are neither strengthened nor weakened. This result may be established by either of the two alternative methods which follow.

I. We may enquire what various combinations of premisses will yield conclusions of the forms A, Y, E, I, O, η, respectively.

It will suffice, as we have already seen, to consider some one figure. We may, therefore, take figure 1, so that the position of the terms will be—

MP

SM

⎯⎯⎯⎯

SP

(i) To prove SaP, both premisses must be affirmative; and, in order to avoid illicit minor, the minor premiss must be SaM. It follows that the major must be MaP or there would be undistributed middle. Hence AAA is the only valid mood yielding an A conclusion.

(ii) To prove SyP, both premisses must be affirmative; and, in order to avoid illicit major, the major premiss must be MyP. It follows that the minor must be SyM, in order to avoid undistributed middle. Hence YYY is the only valid mood yielding a Y conclusion.

(iii) To prove SeP, the major must be (1) MeP or (2) MyP or (3) MoP in order to avoid illicit major. If (1), the minor must be SaM or there would be either two negative premisses or illicit minor; if (2), it must be SeM or there would be undistributed middle or illicit minor; if (3), it must be affirmative and distribute both S and M, which is impossible. Hence EAE and YEE are the only valid moods yielding an E conclusion.

(iv) To prove SiP, both premisses must be affirmative, and since SaM would necessarily be a strengthened premiss, the minor must be (1) SiM or (2) SyM. If (1), the major must be MaP or there would be undistributed middle; and if (2), it must be MiP or there would be a strengthened premiss. Hence AII and IYI are the only valid (unstrengthened and unweakened) moods yielding an I conclusion.

(v) To prove SoP, the major must be (1) MeP or (2) MyP or (3) MoP or there would be illicit major. If (1), the minor must be SiM or there would be a strengthened premiss; if (2), it must be SoM or there would be either two affirmative premisses with a negative conclusion or undistributed middle or a strengthened premiss; and if (3), it must be SyM or there would be two negative premisses or undistributed middle. Hence EIO, YOO, OYO are the only valid (unstrengthened and unweakened) moods yielding an O conclusion.

(vi) To prove SηP, the minor must be (1) SeM or (2) SaM or (3) SηM or there would be illicit minor. If (1), the major must be MiP or there would be a strengthened premiss; if (2), the major must be MηP or there would be undistributed middle or two affirmative premisses with a negative conclusion or a strengthened premiss; and if (3), the major must be MaP or there would be undistributed middle or two negative premisses. Hence IEη, ηAη, Aηη are the only valid (unstrengthened and unweakened) moods yielding an η conclusion.

By converting one or both of the premisses we may at once deduce from the above a table of valid (unstrengthened and unweakened) moods for all four figures as follows:—

Fig. 1. Fig. 2. Fig. 3. Fig. 4.

AAAYAAAYAYYA

YYYAYYYAYAAY

EAEEAEEYEEYE

YEEAEEYEEAEE

AIIYIIAIIYII

IYIIYIIAIIAI

EIOEIOEIOEIO

YOOAOOYηOAηO

OYOηYOOAOηAO

IEηIEηIEηIEη

ηAηOAηηYηOYη

AηηYηηAOηYOη

II. The above table may also be obtained by (1) taking all the combinations of premisses that are à priori possible, (2) establishing special rules for the particular figure selected, which (taken together with the rules of quality) will enable us to exclude the combinations of premisses which are either invalid or strengthened whatever the conclusion may be, (3) assigning the valid unweakened conclusion in the remaining cases.

The following are all possible combinations of premisses, valid and invalid:

AA (b)YAIAEA (b)OAηA(b) (c)

AYYY (a)IY (a) EYOY (a)ηY

AIYI (a)II (a) EIOI (a)ηI (c)

AE (b)YEIE[EE] (b) [OE][ηE] (b)

AOYO (a)IO (a) [EO][OO] (a)[ηO]

Aη (b) (c)YηIη (c)[Eη] (b)[Oη][ηη] (b) (c)

The combinations in square brackets are excluded by the rule that from two negative premisses nothing follows.

Taking the third figure, in which the middle term is subject in each premiss, and remembering that the subject is distributed in A, E, η and in these only, while the predicate is distributed in Y, E, O and in these only, the following special rules are obtainable:

(a) One premiss must be A, E, or η, or the middle term would not be distributed in either premiss;

(b) One premiss must be Y, I, or O, or the middle term would be distributed in both premisses, and there would hence be a strengthened premiss;

(c) If either premiss is negative, one of the premisses must be Y, E, or O, for otherwise (since the conclusion must be negative, distributing one of its terms) there would be illicit process either of major or minor.

These rules exclude the combinations of premisses marked respectively (a), (b), (c) above.

Assigning the valid unweakened conclusion in the case of each of the twelve combinations which remain, we have the following; AYA, AII, AOη, YAY, YEE, YηO, IAI, IEη, EYE, EIO, OAO, ηYη. From this, the table of valid (unstrengthened and unweakened) moods for all four figures may be expanded as before.

330. Formal Inferences not reducible to ordinary Syllogisms.415—The following is an example of what is usually called the argument à fortiori:

B is greater than C,

A is greater than B,

therefore, A is greater than C.

As this stands, it is clearly not in the ordinary syllogistic form since it contains four terms; an attempt is, however, sometimes made to reduce it to ordinary syllogistic form as follows:

B is greater than C,

therefore, Whatever is greater than B is greater than C,

but A is greater than B,

therefore, A is greater than C.

With De Morgan, we may treat this as a mere evasion, or as a petitio principii. The principle of the argument à fortiori is really assumed in passing from B is greater than C to Whatever is greater than B is greater than C. It may indeed be admitted that by the above reduction the argument à fortiori is resolved into a syllogism together with an immediate inference. But this immediate inference is not one that can be justified so long as we recognise only such relations between terms or classes as are implied by the ordinary copula; and if anyone declined to admit the validity of the argument à fortiori he would decline to admit the validity of the step represented by the immediate inference.

The following attempted resolution416 must be disposed of similarly:

Whatever is greater than a greater than C is greater than C, A is greater than a greater than C, therefore, A is greater than C.

At any rate, it is clear that this cannot be the whole of the reasoning, since B no longer appears in the premisses at all.

The point at issue may perhaps be most clearly indicated by saying that whilst the ordinary syllogism may be based upon the dictum de omni et nullo, the argument à fortiori cannot be made to rest entirely upon this axiom. A new principle is required and one which must be placed on a par with the dictum de omni et nullo, not in subordination to it. This new principle may be expressed in the form, Whatever is greater than a second thing which is greater than a third thing is itself greater than that third thing.

Mansel (Aldrich, pp. 199, 200) treats the argument à fortiori as an example of a material consequence on the ground that it depends upon “some understood proposition or propositions, connecting the terms, by the addition of which the mind is enabled to reduce the consequence to logical form.” He would effect the reduction in one of the ways already referred to. This, however, begs the question that the syllogistic is the only logical form. As a matter of fact the cogency of the argument à fortiori is just as intuitively evident as that of a syllogism in Barbara itself. Why should no relation be regarded as formal unless it can be expressed by the word is? Touching on this case, De Morgan remarks that the formal logician has a right to confine himself to any part of his subject that he pleases; “but he has no right except the right of fallacy to call that part the whole” (Syllabus, p. 42).

There are an indefinite number of other arguments which for similar reasons cannot be reduced to syllogistic form. For example,—A equals B, B equals C, therefore, A equals C ;417 X is a contemporary of Y, and Y of Z, therefore, X is a contemporary of Z ; A is a brother of B, B is a brother of C, therefore, A is a brother of C ; A is to the right of B, B is to the right of C, therefore, A is to the right of C ; A is in tune with B, and B with C, therefore, A is in tune with C. All these arguments depend upon principles which may be placed on a par with the dictum de omni et nullo, and which are equally axiomatic in the particular systems to which they belong.

The claims that have been put forward on behalf of the syllogism as the exclusive form of all deductive reasoning must accordingly be rejected.

Such claims have been made, for example, by Whately. Syllogism, he says, is “the form to which all correct reasoning may be ultimately reduced” (Logic, p. 12). Again, he remarks, “An argument thus stated regularly and at full length is called a Syllogism; which, therefore, is evidently not a peculiar kind of argument, but only a peculiar form of expression, in which every argument may be stated” (Logic, p. 26).418

Spalding seems to have the same thing in view when he says,—“An inference, whose antecedent is constituted by more propositions than one, is a mediate inference. The simplest case, that in which the antecedent propositions are two, is the syllogism. The syllogism is the norm of all inferences whose antecedent is more complex; and all such inferences may, by those who think it worth while, be resolved into a series of syllogisms” (Logic, p. 158).

J. S. Mill endorses these claims. “All valid ratiocination,” he observes, “all reasoning by which from general propositions previously admitted, other propositions equally or less general are inferred, may be exhibited in some of the above forms,” i.e., the syllogistic moods (Logic, II. 2, § 1).

What is required in order to fill the logical gap created by the admission that the syllogism is not the norm of all valid formal inference has been called the logic of relatives.419 The function of the logic of relatives is to take account of relations generally, and not “those merely which are indicated by the ordinary logical copula is” (Venn, Symbolic Logic, p. 400).420 The line which this branch of logic may take, if it is ever fully worked out, is indicated by the following passage from De Morgan (Syllabus, pp. 30, 31):—“A convertible copula is one in which the copular relation exists between two names both ways: thus ‘is fastened to,’ ‘is joined by a road with,’ ‘is equal to,’ &c. are convertible copulae. If ‘X is equal to Y’ then ‘Y is equal to X,’ &c. A transitive copula is one in which the copular relation joins X with Z whenever it joins X with Y and Y with Z. Thus ‘is fastened to’ is usually understood as a transitive copula: ‘X is fastened to Y’ and ‘Y is fastened to Z’ give ‘X is fastened to Z.’” The student may further be referred to Venn, Symbolic Logic, pp. 399 to 404; and also to Mr Johnson’s articles on the Logical Calculus in Mind, 1892, especially pp. 26 to 28 and 244 to 250.

EXERCISES.

331. Shew that if either of two given propositions will suffice to expand a given enthymeme of the first or second order into a valid syllogism, then the two propositions will be equivalent to each other, provided that neither of them constitutes a strengthened premiss. [J.]

332. Given one premiss and the conclusion of a valid syllogism within what limits may the other premiss be determined? Shew that the problem is equally determinate with that in which we are given both the premisses and have to find the conclusion. In what cases is it absolutely determinate? [K.]

333. Construct a valid sorites consisting of five propositions and having Some A is not B as its first premiss. Point out the mood and figure of each of the distinct syllogisms into which the sorites may be resolved. [K.]

334. Discuss the character of the following sorites, in each case indicating how far more than one analysis is possible: (i) Some D is E, All D is C, All C is B, All B is A, therefore, Some A is E ; (ii) Some A is B, No C is B, All D is C, All E is D, therefore, Some A is not E ; (iii) All E is D, All D is C, All C is B, All B is A, therefore, Some A is E ; (iv) No D is E, Some D is C, All C is B, All B is A, therefore, Some A is not E. [K.]

335. Discuss the possibility of a sorites which is capable of being analysed so as to yield valid syllogisms all of which are in figure 4. Determine the maximum number of propositions of which such a sorites can consist. [K.]

336. Examine the validity of the following moods: in figure 1, UAU, YOO, EYO; in figure 2, AAA, AYY, UOω; in figure 3, YEE, OYO, AωO. [C.]

337. Enquire in what figures, if any, the following moods are valid, noting cases in which the conclusion is weakened:—AUI; YAY; UOη; IUη; UEO. [L.]

338. If some is used in the sense of “some, but not all,” what can be inferred from the propositions All M is some P, All M is some S? [K.]

339. Giving to some its ordinary logical meaning, shew that, in any syllogism expressed with quantified predicates, a premiss of the form U may always be regarded as a strengthened premiss unless the conclusion is also of the form U. [K.]

340. Is it possible that there should be three propositions such that each in turn is deducible from the other two? [V.]

341. Determine special rules for figures 1, 2, and 4, corresponding to the special rules for figure 3 given in section 329. [K.]