Give, as far as possible, the technical name of the logical relation between each of the above propositions and each other.

23. 'Some small sects are said to have no discreditable members, because they do not receive such, and extrude all who begin to verge upon the character.' Point out how

this statement illustrates logical conversion.

24. Can we logically infer that because heat expands bodies, therefore cold contracts them?

25. Does it follow that because every city contains a cathedral, therefore the creation of a city involves the creation of a cathedral, or the creation of a cathedral involves the creation of a city?

26. All English Dukes are members of the House of Lords. Does it follow by immediate inference by complex conception that the creation of an English Duke is the creation of a member of the House of Lords?

27. Give every possible converse of the following propositions—

(1) Two straight lines cannot enclose space. (2) All trade-winds depend on heat.

(3) Some students do not fail in anything.

[M.]

28. Give the logical opposites, converse and contrapositive, of Euclid's (so-called) twelfth axiom—

If a straight line meet two straight lines, so as to make the interior angles on the same side of it taken together less than two right angles, those straight lines being continually produced shall at length meet upon that side on which are the angles which are less than two right angles.

29. How is the above proposition related to this other :If a straight line fall upon two parallel straight lines, it

makes the two interior angles upon the same side together equal to two right angles?

[R.]

30. From 'Some members of Parliament are all the ministers' (Elementary Lessons, p. 325, No. 3 [4]), can we infer that 'some place-seeking prejudiced and incapable members of Parliament are all the place-seeking prejudiced and incapable ministers'?

31. Is it perfectly logical to argue that because two subcontrary propositions may both be true at the same time, therefore their contradictories, which are contrary to each other, may both be false?

32. Is it perfectly logical to argue thus ?—If contrary propositions are both false, their respective contradictories, which are sub-contraries to each other, are both true. Now as this result is possible, it is therefore possible that the contraries may both be false.

33. What is the logical relation, if any, between the two assertions in Proverbs, chap. xi. 1, 'A false balance is abomination to the Lord: but a just weight is his delight'?

34. Examine the verses of Proverbs, chap. x. to xv. and assign the relation between the two opposed assertions which make nearly all the verses.

35. What is the nature of the step from 'anger is a short madness' to 'madness is a long passion'?

[R.]

36. The angles at the base of an isosceles triangle are equal.' What can be inferred from this proposition by obversion, conversion, and contraposition, without any appeal to geometrical proof?

37. From the assertion 'The improbable is not impossible,' what can we learn, if anything, about (1) the possible, (2) the probable, (3) the not-improbable, (4) the impossible, (5) the not-impossible?

38. How would a logician express the relations between the following statements of four interlocutors?

(1) None but traitors would do so base a deed.

(2) And not all traitors.

(3) Some would.

(4) No; not even traitors.

[College Moral Science Examination, Cambridge.]

39. What difficulties or absurdities do you meet in converting the following propositions?

(1) Some books are dictionaries.

(2) No triangle has one side equal to the sum of the other two.

(3) Every one is the best judge of his own interests.

(4) A few men are both scientific discoverers and men of business.

(5) Whatever is, is right.

(6) Some men are wise in their own conceit.

(7) "The eye sees not itself,

But by reflection, by some other things."

CHAPTER VII.

DEFINITION AND DIVISION.

1. ALMOST all text-books of Deductive Logic give rules for judging of the correctness of definitions, and for dividing up notions into subaltern genera and species. On attempting, however, to treat these parts of logic in the manner of this work, it has come home to me very strongly that they are beyond the sphere of Formal Logic, and involve the matter of thought. In form there is nothing peculiar to a definition; in fact the very same proposition may be a definition to one person and a theorem to another. It is open to us for instance to define the number 9 as 9 = 3 × 3; or, 9 = 8 + 1; or, 9 = 7 + 2, &c.; but having selected at will any one of these equations as a definition, the other equations follow as theorems. The perplexity in which the theory of parallel lines is involved partly arises from the fact that there is choice of definitions, some mathematicians choosing one way and some the other. It is quite apparent, too, that the same proposition may afford different knowledge to different people. For instance, 'John Herschel was the only son of William Herschel' would serve as a definition of John Herschel to any one who knew only William Herschel, and of William Herschel to one who only knew John. To one who knew both it might be a theorem. Similar remarks might be made concerning the distinction between ampliative and explicative propositions.

2. These in addition to other considerations convince me that any attempt to treat definition as a part of Formal Logic must be theoretically unsound and practically unsatisfactory. The case is somewhat similar with Logical Division, which, so far as it belongs to Formal Logic, can be nothing more than that method of Dichotomous Division fully developed in the later chapters on Equational Logic. Anything more than this must involve material knowledge, and should be treated in a different work and in a different On these grounds I have decided not to attempt any explication of Definition and Division here, but to confine this chapter to a collection of questions, such as are to be commonly found in examination papers. The student may be referred for the current doctrines to the Elementary Lessons, No. XII. and XIII.; Fowler's Deductive Logic, Chapters VII. and VIII.; Duncan's Logic, Chapter VI. etc. 3. Examine the following definitions

manner.

(1) Conversion is the changing of terms in a proposition.

(2) Opposed propositions are those which differ in quantity and quality.

(3) Contradictory opposition is the opposition of contradictories.

[R.]

4. Define any of the following terms, notions, or classes