(8). Show that we cannot tell what percentage of B or of C the same individuals make up unless we know how much of B or of C is not A. (7). In the easy case in which all B is A, and all C is A, find what percentage of B or of C must be made up by the individuals which are both B and C at once. (8). Mill asserts that these numerical formulæ of De Morgan's are useless for the ordinary purposes of logic; why so? (e). May they be of use in any other sciences ? 5. The first advantage claimed by Sir William Hamilton from the quantification of the predicate is, that "it recals the two terms of the proposition to their true relation." (a). On what grounds is this challenged by Mill? (B). In a subsequent chapter Mill advocates a change in the form of the universal affirmative proposition so as to bring out more clearly what he holds to be the true relation of the terms? (7). What effects has the quantification of the predicate (1) On conversion, (2) on the six general rules of syllogism, and (3) on the importance of mere variation of figure ? 6. What hypotheses, now discarded, gave some meaning and reality to (a). The dictum de omni; and (B). To the axiom "whatever is, is," respectively? (7). Show that Condillac's absurd definition of Science was the necessary result of the Nominalist theory of the import of propositions combined with the kindred theory of syllogistic reasoning. 7. Mill contends that "in every syllogism, considered as an argument to prove the conclusion, there is a petitio principii," and all the examples he gives represent the major premiss as begging the question. Give an example or two in which the minor premiss begs the question, and some other example in which the conclusion is contained, not in either premiss, but in the combination of the two? 8. Those who regard syllogism as a means of proving propositions overlook the distinction between two parts of the process of philoso phizing, and mistake the nature of generalisation? 9. (a). Give Mill's summary of his doctrine of the function of the syllogism. (8). His summary concludes with the words "But a record remains, not, indeed, descriptive of the facts themselves, but showing how those cases may be distinguished, respecting which those facts, when known, were considered to warrant a given inference." Clear up this obscure passage. by taking as an example the "record ” or major premise, "all mammals are warm-blooded," and some suitable minor; and then showing what the cases, the facts, and the inference would be. 1. It appears from the quotation prefixed to B. II. that "syllogism" is not the exact translation of Aristotle's Evλλoyiqμós? 2. Brown contended that the minor premiss in a syllogism was superfluous. (a). Give his argument, and (B). Expose its errors. 3. (a). What function is assigned by Whately to the major premiss? (8). Mill illustrates in various ways the irrelevance of this statement, considered as an objection to his view of the syllogism? 4. "It cannot be a necessary condition of reasoning," says Mill, "that we should begin by making an assertion which is afterwards to be employed in proving itself." (a). To what objection, urged by De Morgan, and an Edinburgh Reviewer, is this remark applied by Mill? (B). He sees only one mode of escape from the difficulty created by the impossibility alleged above, on the one hand, and, on the other, the undoubted fact that the major premiss virtually includes the conclusion? 5. Show, by an example, whether of your own or taken from the text, that a catena of syllogisms is really a catena of inductions. 6. (a). What is the generic difference between the sciences that can be made deductive and those that must yet remain inductive? (8). How did Acoustics become deductive? (7). What logical rule, less general than (a), does this particular case of transformation illustrate? (8). One branch of Mathematics illustrates the same rule slightly modified? 7. (a). In the controversy as to the grounds of our belief in the axioms of geometry, the burden of proof falls on the a priori, not the empirical, school? (8). What are Whewell's two great arguments in behalf of the intuitive nature of these beliefs? (7). Reply to the first of these, and notice the coincidence between Mr. Bain's explanation of the apparent intuition, and a remark quoted subsequently from Sir John Herschell. 8. Although Euripides never saw the sacred streams flow upward, the fact that he imagined them doing so is quite reconcileable with the system of empiricism? 9. Whewell was obliged, by the facts he himself cited, to alter the received definition of Necessary truths? (a). Give the alteration, and (B). The facts. (y). Write a brief note on the paradox that increased knowledge often imposes limits on our powers of conception. 10. (a). What two peculiarities have made arithmetic and algebra the stronghold of Nominalism? (B). What facts, however, (1st) are fatal to the Nominalist theory of numbers; and (2nd) prove that we are conscious of numbers in their character of things, and not of mere signs? f MR. ABBOTT. 1. Discuss the question of the definition of Logic. 2. What is meant by connotative and non-connotative names? Explain the distinction fully. 3. How does the connotation of a term come to be first agreed upon, and how is it subsequently learned ? 4. Under what class of Nameable Things does Mill bring Quantity, and why? 5. What is the theory of predication stated in Murray's Logic? State the objections to it suggested by Mill's discussion of the subject, or otherwise. 6. What is Mill's own theory of predication? Apply it to the proposition, "Avarice is a mean vice." 7. What view does Mr. Bain take of Definition, and how does it differ from that of Mill? 8. What is the distinction drawn by Mill between genus and species ? MR. MAHAFFY, 1. What are the genus, species, difference, property, and accident of an Examination? 2. What mode does Murray exclude by the special rules, and why? 3. State in a syllogism, 4. Are Fallacies in their nature distinct from each other? Give reasons for your answer. 5. Are they necessarily distinct from sound arguments? 6. To what terms is logical definition inapplicable? 7. Prove the two rules of sorites. 8. How is a conclusion to be drawn from probable premises? 9. In ostensive reduction, when is conversion alone sufficient? 10. When is an induction from a single case valid? Give examples. 1. Test the following arguments :— (a). A is the son of B; B is the son of C; (8). A is descended from B; B is descended from C; 2. Write a note on the treatment of singulars in Logic. 3. If we admitted such terms as most X into Logic, as a separate class, what changes would be required in syllogistic rules? 4. Murray says, a dilemma is invalid, if it can be retorted." Criticise his argument in proof of this, and discuss the question generally. 5. If a premiss and conclusion are interchanged, what will be the logical results? 6. If the predicate were quantified, what changes must be admitted in the rules of Opposition? 7. Write out rules for the other figures corresponding with the rule de omni for the first. 8. If the middle term be twice universal, determine mode and figure. 9. Show that subcontrariety cannot be employed in reductio ad impossibile. 1. Prove the 47th Proposition, B. I., of Euclid, by showing by a figure that the square of the hypothenuse consists of four times the area of the triangle, together with the square of the difference of the sides. 2. Given base, sum of sides, and difference of base angles of a triangle; construct it. 3. Calculate the lengths of the lines joining the vertices to the middle points of opposite sides in the triangle whose sides are 7, 8, 9. 4. Reduce the expression (1 − x2) (1 − y2) (1 − z2) + (x − yz) (y − zx) (z — xy) to its simplest form. I- xyz 5. Construct a circle equal to the ring enclosed between two concentric circles. 6. Calculate the value of √ II +√/72 - √ II - √/72 √8+/28 +1/8/28 to three places of decimals, by first taking its square. MR. BURNSIDE. 7. The sum of the squares of the sides of any quadrilateral is equal to the sum of the squares of its two diagonals, together with four times the square of the line joining the middle points of the diagonals ? 8. In a triangle, being given the ratio of the sides, and the segments of the base made by the perpendicular from the vertical angle; construct it. 9. Bisect the area of a triangle by a perpendicular to one of the sides. 10. Solve the equation 13. Construct a rectangle which shall be equal to a given square, and the difference of whose sides shall be equal to a given line. 14. Given base and vertical angle of a triangle, find the locus of intersection of perpendiculars from the extremities of base on the opposite sides. 15. Prove that the excess of the sum of the radii of the three exscribed circles of a triangle over the radius of the inscribed circle is equal to twice the diameter of the circumscribed circle. 16. Calculate to four decimal places the value of √27-1875-50 X ✓ 28 √20 √63+ √45 - 17. The difference of squares of the seventh and twenty-first terms of an arithmetical series is 1512, and the first term is unity; find the series. 18. Given y+z-3x=2a, 2+x-3y=2b, x+y-3x= 2c, it is required to express the product (a + 2x) (b +2y) (c + 2x) in terms of a, b, c. |