Studies and Exercises in Formal Logic

327. Ultra-total Distribution of the Middle Term.—The ordinary syllogistic rule relating to the distribution of the 377 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:—

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 378 (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:—

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. 379

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 380 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:—

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

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

(5) AUA in figure 2 runs as follows,—

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 381 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.

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,

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. 382

(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 383 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.

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

330. Formal Inferences not reducible to ordinary Syllogisms.415—The following is an example of what is usually called the argument à fortiori: 385

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 386 greater than a second thing which is greater than a third thing is itself greater than that third thing.

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 387 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 388 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.

(1) Let every proposition be interpreted as implying the existence both of its subject and of its predicate.421 In this case, the existence of the major, middle, and minor terms is in every case guaranteed by the premisses, and therefore no further assumption with regard to existence is required in order that the conclusion may be legitimately obtained.422 We may regard the above supposition as that which is tacitly made in the ordinary doctrine of the syllogism.

(2) Let every proposition be interpreted as implying the existence of its subject. Under this supposition an affirmative proposition ensures the existence of its predicate also; but not so a negative proposition. It follows that any mood will be valid unless the minor term is in its premiss the predicate of a negative proposition. This cannot happen either in figure 1 or in figure 2, since in these figures the minor is always subject in its premiss; nor in figure 3, since in this figure the minor 391 premiss is always affirmative. In figure 4, the only moods with a negative minor are Camenes and its weakened form AEO. Our conclusion then is that on the given supposition every ordinarily recognised mood is valid except these two.423

(3) Let no proposition be interpreted as implying the existence either of its subject or of its predicate. Taking S, M, P, as the minor, middle, and major terms respectively, the conclusion will imply that if there is any S there is some P or not-P (according as it is affirmative or negative). Will the premisses also imply this? If so, then the syllogism is valid; but not otherwise.

It has been shewn in section 212 that a universal affirmative conclusion, All S is P, can be proved only by means of the premisses, All M is P, All S is M ; and it is clear that these premisses themselves imply that if there is any S there is some P. On our present supposition, then, a syllogism is valid if its conclusion is universal affirmative.

Again, as shewn in section 212, a universal negative conclusion, No S is P, can be proved only in the following ways:—

392 In (i) the minor premiss implies that if S exists then M exists, and the major premiss that if M exists then not-P exists. In (ii) the minor premiss implies that if S exists then not-M exists, and the major premiss that if not-M exists then not-P exists (as shewn in section 158). Hence a syllogism is valid if its conclusion is universal negative.

Next, let the conclusion be particular. In figure 1, the implication of the conclusion with regard to existence is contained in the premisses themselves, since the minor term is the subject of an affirmative minor premiss, and the middle term the subject of the major premiss. In figure 2, we may consider the weakened moods disposed of in what has been already said with regard to universal conclusions; for under our present supposition subalternation is a valid process. The remaining moods with particular conclusions in this figure are Festino and Baroco. In the former, the minor premiss implies that if S exists then M exists, and the major that if M exists then not-P exists; in the latter, the minor premiss implies that if S exists then not-M exists, and the major that if not-M exists then not-P exists.

All the ordinarily recognised moods, then, of figures 1 and 2 are valid. But it is otherwise with moods yielding a particular conclusion in figures 3 and 4, with the single exception of the weakened form of Camenes (which is itself the only mood with a universal conclusion in these figures). Subalternation being a valid process, the legitimacy of the latter follows from the legitimacy of Camenes itself. But in all other cases in figures 3 and 4, the minor term is the predicate of an affirmative minor premiss. Its existence, therefore, carries no further implication of existence with it in the premisses. It does so in the conclusion. Hence all the moods of figures 3 and 4, with the exception of AEE and AEO in the latter figure, are invalid. Take, as an example, a syllogism in Darapti,—

The conclusion implies that if S exists P exists; but 393 consistently with the premisses, S may be existent while M and P are both non-existent. An implication is, therefore, contained in the conclusion which is not justified by the premisses.

Hence on the supposition that no proposition implies the existence either of its subject or of its predicate all the ordinarily recognised moods of figures 1 and 2 are valid, but none of those of figures 3 and 4 excepting Camenes and the weakened form of Camenes.424