| All A is B, | |
| No C is B, | |
| [therefore, | No A is C], |
| All D is C, | |
| [therefore, | No A is D], |
| All E is D, | |
| therefore, | No A is E. |
It has, to begin with, been objected that the above is Goclenian, and not Aristotelian, in form, “the subject of each premiss after the first being the predicate of the succeeding one.” This overlooks the more fundamental characteristic of the Aristotelian sorites, that the first premiss and the suppressed conclusions are all minors in their respective syllogisms. It has further been objected that the following analysis might serve in lieu of the one given above:—AaB, CeB, [∴ CeA,] DaC, [∴ DeA], EaD, ∴ AeE. No doubt this analysis is a possible one, but the objection to it is its heterogeneous character. The first premiss and the first suppressed conclusion are majors, while the last suppressed conclusion is a minor. Again, the first syllogism is in figure 2, the second in figure 1, and the third in figure 4. It must be granted that what has been above called a heterogeneous analysis is in some cases the only one available, but it is better to adopt something more homogeneous where possible. If the first premiss of a sorites contains the subject, and the last the predicate, of the conclusion, then the last premiss is necessarily the major of the final syllogism; and hence the rule may be laid down that we can work out such a sorites homogeneously only by treating the first premiss and all the suppressed conclusions as minors, and all the remaining premisses as majors, in their respective syllogisms. A corresponding rule may be laid down if the first premiss contains the predicate, and the last the subject, of the conclusion.
It will be found that a sorites in figure 4 cannot have more than a limited number of premisses. This point is raised in section [335].
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:—
| 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 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]).
[409] Using other letters, this is the example given by Mill, Logic, ii. 2, § 1, note, and quoted by Herbert Spencer, Principles of Psychology, II. p. 88. The more general problem of which the above is a special instance is as follows: Given that there are n M’s in existence, and that a M’s are S while b M’s are P, to determine what is the least number of S’s that are also P’s. It is clear that we have no conclusion at all unless a + b > n, i.e., unless there is ultra-total distribution of the middle term. If this condition is satisfied, then supposing the (n − b) M’s which are not-P are all of them found amongst the MS’s, there will still be some MS’s left which are P’s, namely, a − (n − b). Hence the least number of S’s that are also P’s must be a + b − n.
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.