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.
[419] Compare pages [149] to 151.
[420] Ordinary formal logic is included under the logic of relatives interpreted in the widest sense, but only in a more generalised form than that in which it is customarily treated.
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.]
389 335. 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.]