Studies and Exercises in Formal Logic

³⁹⁰ When the alternative major premiss is equivalent not to a true hypothetical but to a conditional (as in the second of the above examples), the syllogism may be reduced to pure categorical form (unless the categorical and conditional forms of proposition are in some way differentiated from one another). Thus,

³⁹¹ It will be observed that, interpreting the alternants as not necessarily exclusive of one another, the modus ponendo tollens in the above form is equivalent to one of the fallacies in the mixed hypothetical syllogism mentioned in section 306.

³⁹² This is in the stricter sense a disjunctive syllogism, the modus tollendo ponens being an alternative syllogism. The reader must, however, be careful to remember that the latter is what is ordinarily meant by the disjunctive syllogism in logical text-books.

³⁹³ In the strict use of the term, a dilemma implies only two alternants in the alternative premiss; if there are more than two alternants we have a trilemma, or a tetralemma, or a polylemma, as the case may be.

³⁹⁴ This application of the terms major and minor is somewhat arbitrary. The dilemmatic force of the argument is indeed made more apparent by stating the alternative premiss (i.e., the so-called minor premiss) first.

³⁹⁵ A further form of argument may be distinguished in which the alternation contained in the so-called minor premiss is affirmed only hypothetically, and in which, therefore, the conclusion also is hypothetical. For example,

This might be called the hypothetical dilemma. It admits of varieties corresponding to the varieties of the ordinary dilemma; but no detailed treatment of it seems called for.

³⁹⁶ It will usually be a simple categorical; but see the following note.

³⁹⁷ The following is a simple, not a complex, constructive dilemma:

If A is B, E is F or G is H ; and if C is D, E is F or G is H ;
but Either A is B or C is D ;
therefore, Either E is F or G is H.

The hypotheticals which here constitute the major premiss have a common consequent; but since this is itself alternative, the conclusion appears in the alternative form. This case is analogous to the following,—All M is P or Q, All S is M, therefore, All S is P or Q,—where the conclusion of an intrinsically categorical syllogism also appears in the alternative form. Compare the note on page 359.

³⁹⁸ The following is a simple, not a complex, destructive dilemma:

If both P and Q are true then X is true, and under the same hypothesis Y is true ;
but Either X or Y is not true ;
therefore, Either P or Q is not true.

³⁹⁹ Compare Ueberweg, Logic, § 123.

⁴⁰⁰ On this subject the student may be referred to the remainder of the note from which the above extract is taken, and to Hamilton, Discussions, pp. 152 to 156. Compare also Karslake, Aids to the Study of Logic, Book II.

⁴⁰¹ This account of the enthymeme appears to have been originally based on the erroneous idea that the name signified the retention of one premiss in the mind, ἐν θυμῷ. Thus, in the Port Royal Logic, an enthymeme is described as “a syllogism perfect in the mind, but imperfect in the expression, since some one of the propositions is suppressed as too clear and too well known, and as being easily supplied by the mind of those to whom we speak” (p. 229). As regards the true origin of the name enthymeme, see Mansel’s Aldrich, p. 218.

⁴⁰² The distinction which follows is ordinarily applied to chains of reasoning only; but the reader will observe that it admits of application to the case of the simple syllogism also.

⁴⁰³ On the distinction between progressive and regressive arguments, see Ueberweg, Logic, § 124.

⁴⁰⁴ A distinction has been drawn between single and double epicheiremas according as reasons are enthymematically given in support of one or both of the premisses of the ultimate syllogism. The example given in the text is a single epicheirema; the following is an example of a double epicheirema:

The epicheirema is sometimes defined as if it were essentially a regressive chain of reasoning. But this is hardly correct, if, as is usually the case, examples such as the above are given; for it is clear that in these examples the argument is only partly regressive.

⁴⁰⁵ This form of sorites ought not properly to be called Aristotelian; but it is generally so described in logical text-books. The name sorites is not to be found in any logical treatise of Aristotle, though in one place he refers vaguely to the form of reasoning which the name is now employed to express. The distinct exposition of this form of reasoning is attributed to the Stoics, and it is designated sorites by Cicero; but it was not till much later that the name came into general use amongst logicians in this sense. The form of sorites called the Goclenian was first given by Professor Rudolf Goclenius of Marburg (1547 to 1628) in his Isagoge in Organum Aristotelis, 1598. Compare Hamilton, Logic, I. p. 375; and Ueberweg, Logic, § 125. It may be added that the term sorites (which is derived from σωρὸς, a heap) was used by ancient writers in a different sense, namely, to designate a particular sophism, based on the difficulty which is sometimes found in assigning an exact limit to a notion. “It was asked,—was a man bald who had so many thousand hairs; you answer, No: the antagonist goes on diminishing and diminishing the number, till either you admit that he who was not bald with a certain number of hairs, becomes bald when that complement is diminished by a single hair; or you go on denying him to be bald, until his head be hypothetically denuded.” A similar puzzle is involved in the question,—On what day does a lamb become a sheep? Sorites in this sense is also called sophisma polyzeteseos or fallacy of continuous questioning. See Hamilton, Logic, i. p. 464.

⁴⁰⁶ The mistake is sometimes made of speaking of the Goclenian sorites as a regressive form of argument. It is clear, however, that in both forms of sorites we pass continuously from premisses to conclusions, not from conclusions to premisses.

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).