Any Aristotelian sorites may be represented in skeleton form, the quantity and quality of the premisses being left undetermined, as follows:—
| S | M1 |
| M1, | M2 |
| M2, | M3 |
| ……… | ……… |
| ……… | ……… |
| Mn−2, | Mn−1 |
| Mn−1, | Mn |
| Mn, | P |
| ⎯⎯⎯ | ⎯⎯⎯ |
| S | P |
373 (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 374 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).
[407] Sir William Hamilton indeed professes to give sorites in the second and third figures, which have, he says, been overlooked by other logicians (Logic, II. p. 403). It appears, however, that by a sorites in the second figure he means such a reasoning as the following,—No B is A, No C is A, No D is A, No E is A, All F is A, therefore, No B, or C, or D, or E, is F ; and by a sorites in the third figure such as the following,—A is B, A is C, A is D, A is E, A is F, therefore, Some B, and C, and D, and E, are F. He does not himself give these examples; but that they are of the kind which he intends may be deduced from his not very lucid statement, “In second and third figures, there being no subordination of terms, the only sorites competent is that by repetition of the same middle. In first figure, there is a new middle term for every new progress of the sorites; in second and third, only one middle term for any number of extremes. In first figure, a syllogism only between every second term of the sorites, the intermediate term constituting the middle term. In the others, every two propositions of the common middle term form a syllogism.” But it is clear that in the accepted sense of the term these are not sorites at all. In each case the conclusion is a mere summation of the conclusions of a number of syllogisms having a common premiss; in neither case is there any chain argument. Hamilton’s own definition of the sorites, involved as it is, might have saved him from this error. He gives for his definition, “When, on the common principle of all reasoning,—that the part of a part is a part of the whole,—we do not stop at the second gradation, or at the part of the highest part, and conclude that part of the whole, but proceed to some indefinitely remoter part, as D, E, F, G, H, &c., which, on the general principle, we connect in the conclusion with its remotest whole,—this complex reasoning is called a Chain-Syllogism or Sorites” (Logic, I. p. 366). In connexion with Hamilton’s treatment of this question, Mill very justly remarks, “If Sir W. Hamilton had found in any other writer such a misuse of logical language as he is here guilty of, he would have roundly accused him of total ignorance of logical writers” (Examination of Hamilton, p. 515).
This treatment of the question seems, however, open to refutation by the simple method of constructing examples. Take, for instance, the following sorites:—