1. Background Knowledge Results!

2. Part I For each of the following topics, rate each according to your level of prior knowledge of (familiarity with) that topic. (Circle just one number.) 1. Strongly disagree. 2. Disagree. 3. Neither agree nor disagree. 4. Agree. 5. Strongly agree.

3. a. I am familiar with set theory.

4. b. I am familiar with mathematical paradoxes.

5. c. I am familiar with the mathematics of infinity.

6. d. I am familiar with theories of knowledge and knowing.

7. e. I am familiar with metaphysical theories of possibility and necessity.

8. f. I am familiar with the philosophy of language.

9. g. I am familiar with mathematical theories of probability.

10. h. I am familiar with conditional (“if… then…”) statements.

11. i. I am familiar with methods for reasoning about causes and effects.

12. j. I am familiar with the properties of logical systems (like soundness and completeness).

13. Part II: Basic Mathematics

14. True or False True or False. Circle T if the statement is true and circle F if it is false. 38%62% 1. TF1 = 0.999999… 2. TF0.5 is a natural number 3.TFThere are two natural numbers p and q where: √2 = p ÷ q

15. True or False True or False. Circle T if the statement is true and circle F if it is false. 38%62% 1. TF1 = 0.999999… 2. TF0.5 is a natural number 3.TFThere are two natural numbers p and q where: √2 = p ÷ q

16. True or False True or False. Circle T if the statement is true and circle F if it is false. 23%76% 1. TF1 = 0.999999… 2. TF0.5 is a natural number 3.TFThere are two natural numbers p and q where: √2 = p ÷ q

17. Short Answer Fill in the blanks with the correct answer(s). 4.4 3 = __________100% Correct 5.4 ½ = __________ 6.The binary notation expression ‘01010’ is equivalent to what decimal notation expression? __________

18. Short Answer Fill in the blanks with the correct answer(s). 4.4 3 = __________ 5.4 ½ = __________88% Correct 6.The binary notation expression ‘01010’ is equivalent to what decimal notation expression? __________

19. Short Answer Fill in the blanks with the correct answer(s). 4.4 3 = __________ 5.4 ½ = __________ 6.The binary notation expression ‘01010’ is equivalent to what decimal notation expression? __________ 42% Correct

20. Part III: Philosophy Familiarity

21. Multiple Choice 1. Immanuel Kant was a philosopher from which country? a. England b. France c. Germany d. Holland e. Italy

22. Multiple Choice 1. Immanuel Kant was a philosopher from which country? a. England5% b. France5% c. Germany79% d. Holland0% e. Italy0% No Answer11%

23. Multiple Choice 2. Immanuel Kant wrote his most important works in which century? a. 16 th (1500-1599) b. 17 th (1600-1699) c. 18 th (1700-1799) d. 19 th (1800-1899) e. 20 th (1900-1999)

24. Multiple Choice 2. Immanuel Kant wrote his most important works in which century? a. 16 th (1500-1599)11% b. 17 th (1600-1699)21% c. 18 th (1700-1799)47% d. 19 th (1800-1899)5% e. 20 th (1900-1999)0% No answer16%

25. Multiple Choice 3. Which of the following was written by Immanuel Kant? a. The Critique of Pure Reason b. The Ethics c. On What Matters d. Naming and Necessity e. The Wealth of Nations

26. Multiple Choice 3. Which of the following was written by Immanuel Kant? a. The Critique of Pure Reason84% b. The Ethics5% c. On What Matters0% d. Naming and Necessity5% e. The Wealth of Nations0% No answer5%

27. Multiple Choice 4. W.V.O. Quine was a philosopher from which country? a. Australia b. Canada c. England d. South Africa e. The United States of America

28. Multiple Choice 4. W.V.O. Quine was a philosopher from which country? a. Australia0% b. Canada0% c. England21% d. South Africa5% e. The United States of America47% No answer26%

29. Multiple Choice 5. Saul Kripke wrote which of the following books? a. The Critique of Pure Reason b. The Ethics c. On What Matters d. Naming and Necessity e. The Wealth of Nations

30. Multiple Choice 5. Saul Kripke wrote which of the following books? a. The Critique of Pure Reason5% b. The Ethics0% c. On What Matters5% d. Naming and Necessity68% e. The Wealth of Nations0% No answer21%

31. Multiple Choice 6. Epistemology is the study of: a. Being b. Knowledge c. Truth d. Beauty e. The Good

32. Multiple Choice 6. Epistemology is the study of: a. Being5% b. Knowledge79% c. Truth0% d. Beauty0% e. The Good0% None/ b&c16%

33. Part IV: Probability

34. True or False For each of the following statements, circle T if it is true, and circle F if it is false. TF 1. If there is a 10% chance of rain on Monday and a 10% chance of rain on Tuesday, then there is a 20% chance that it will rain on either Monday or Tuesday.

35. True or False For each of the following statements, circle T if it is true, and circle F if it is false. TF 32%68% 1. If there is a 10% chance of rain on Monday and a 10% chance of rain on Tuesday, then there is a 20% chance that it will rain on either Monday or Tuesday.

36. True or False For each of the following statements, circle T if it is true, and circle F if it is false. TF 2. If there are two possibilities, A and not-A, then each has a 50% chance of happening.

37. True or False For each of the following statements, circle T if it is true, and circle F if it is false. TF 11%89% 2. If there are two possibilities, A and not-A, then each has a 50% chance of happening.

38. Multiple Choice 3. Suppose the odds of Medic Swordsman (a horse) winning the race are 3-2. What is the probability that Medic Swordsman will win? a. 2/3 b. 3/2 c. 1/3 d. 1/2 e. 3/5

39. Multiple Choice 3. Suppose the odds of Medic Swordsman (a horse) winning the race are 3-2. What is the probability that Medic Swordsman will win? a. 2/335% b. 3/25% c. 1/30% d. 1/20% e. 3/550% None10%

40. Multiple Choice 4. Suppose that wealthy people score higher on intelligence tests. Which of the following would be a possible explanation of this? (Circle one.) a. Having more wealth increases intelligence. b. Having less wealth decreases intelligence. c. Having less intelligence decreases wealth. d. Having high social status increases wealth and increases intelligence. e. All of the above.

41. Multiple Choice 4. Suppose that wealthy people score higher on intelligence tests. Which of the following would be a possible explanation of this? (Circle one.) a. Having more wealth increases intelligence.5% b. Having less wealth decreases intelligence.0% c. Having less intelligence decreases wealth.0% d. Having high social status increases wealth and increases intelligence. e. All of the above.79% 5%

42. Multiple Choice 5. Suppose that I have an AIDS test. If someone has AIDS and they take the test, then they will test positive 99% of the time. Suppose you take the test and test positive. What is the probability that you have AIDS? a. 1% b. 98% c. 99% d. 100% e. There is not enough information to answer this question.

43. Multiple Choice 5. Suppose that I have an AIDS test. If someone has AIDS and they take the test, then they will test positive 99% of the time. Suppose you take the test and test positive. What is the probability that you have AIDS? a. 1%0% b. 98%0% c. 99%32% d. 100%5% e. There is not enough information to answer this question.58%

44. Multiple Choice 6. Which of the following is equal to 100%? a. Pr(x is a dog/ x is an animal) b. Pr(x is an animal/ x is a dog)

45. Multiple Choice 6. Which of the following is equal to 100%? a. Pr(x is a dog/ x is an animal)42% b. Pr(x is an animal/ x is a dog)58%

46. Multiple Choice 7. Which of the following is most likely to happen? a. There will not be a final exam in this class. b. There will not be a final exam in this class, because the instructor has to leave the country. c. HKU closes and there will not be a final exam in this class. d. There is not enough information to answer this question.

47. Multiple Choice 7. Which of the following is most likely to happen? a. There will not be a final exam in this class.47% b. There will not be a final exam in this class, because the instructor has to leave the country.0% c. HKU closes and there will not be a final exam in this class.5% d. There is not enough information to answer this question.47%

48. Multiple Choice 8. Which of the following is more reasonable to believe? a. If Michael Johnson (the professor in this class) didn’t write this exam, then someone else did. b. If Michael Johnson hadn’t written this exam, then someone else would have.

49. Multiple Choice 8. Which of the following is more reasonable to believe? a. If Michael Johnson (the professor in this class) didn’t write this exam, then someone else did.68% b. If Michael Johnson hadn’t written this exam, then someone else would have.21% No response11%

50. Part V: Logic

51. Logic 1. Consider the following argument: Premise 1: CY Leung is a cat. Premise 2: Cats have 17 legs. Conclusion: Therefore, CY Leung has 17 legs.

52. Logic This argument is (circle all that apply): True2Untrue12 Valid17Invalid1 Sound1Unsound17

53. Yes/ No Premise: P if and only if Q 2.YESNOConclusion: If P, then Q 3. YESNOConclusion: If Q, then P 4.YESNOConclusion: If not-Q, then P 5.YESNOConclusion: If not-Q, then not-P 6.YESNOConclusion: If P, then P

54. Yes/ No Premise: P if and only if Q 2.YES76%NO13%Conclusion: If P, then Q 3. YESNOConclusion: If Q, then P 4.YESNOConclusion: If not-Q, then P 5.YESNOConclusion: If not-Q, then not-P 6.YESNOConclusion: If P, then P

55. Yes/ No Premise: P if and only if Q 2.YESNOConclusion: If P, then Q 3. YESNOConclusion: If Q, then P 4.YESNOConclusion: If not-Q, then P 5.YESNOConclusion: If not-Q, then not-P 6.YESNOConclusion: If P, then P

56. Yes/ No Premise: P if and only if Q 2.YESNOConclusion: If P, then Q 3. YES94%NO6%Conclusion: If Q, then P 4.YESNOConclusion: If not-Q, then P 5.YESNOConclusion: If not-Q, then not-P 6.YESNOConclusion: If P, then P

57. Yes/ No Premise: P if and only if Q 2.YESNOConclusion: If P, then Q 3. YESNOConclusion: If Q, then P 4.YESNOConclusion: If not-Q, then P 5.YESNOConclusion: If not-Q, then not-P 6.YESNOConclusion: If P, then P

58. Yes/ No Premise: P if and only if Q 2.YESNOConclusion: If P, then Q 3. YESNOConclusion: If Q, then P 4.YES0%NO100%Conclusion: If not-Q, then P 5.YESNOConclusion: If not-Q, then not-P 6.YESNOConclusion: If P, then P

59. Yes/ No Premise: P if and only if Q 2.YESNOConclusion: If P, then Q 3. YESNOConclusion: If Q, then P 4.YESNOConclusion: If not-Q, then P 5.YESNOConclusion: If not-Q, then not-P 6.YESNOConclusion: If P, then P

60. Yes/ No Premise: P if and only if Q 2.YESNOConclusion: If P, then Q 3. YESNOConclusion: If Q, then P 4.YESNOConclusion: If not-Q, then P 5.YES89%NO11%Conclusion: If not-Q, then not-P 6.YESNOConclusion: If P, then P

61. Yes/ No Premise: P if and only if Q 2.YESNOConclusion: If P, then Q 3. YESNOConclusion: If Q, then P 4.YESNOConclusion: If not-Q, then P 5.YESNOConclusion: If not-Q, then not-P 6.YESNOConclusion: If P, then P

62. Yes/ No Premise: P if and only if Q 2.YESNOConclusion: If P, then Q 3. YESNOConclusion: If Q, then P 4.YESNOConclusion: If not-Q, then P 5.YESNOConclusion: If not-Q, then not-P 6.YES78%NO22%Conclusion: If P, then P

63. Multiple Choice 7. Which of the following is not equivalent to (P v Q)? (Circle one.) a. ~(~P & ~Q) b. ~(P → ~Q) c. ~P → Q d. (~(P ↔ Q) v (P & Q)) e. They are all equivalent to (P v Q)

64. Multiple Choice 7. Which of the following is not equivalent to (P v Q)? (Circle one.) a. ~(~P & ~Q)16% b. ~(P → ~Q)21% c. ~P → Q11% d. (~(P ↔ Q) v (P & Q))0% e. They are all equivalent to (P v Q)5% No answer/ b&d47%

65. Part VI: Personal Interest

66. a. I am interested in learning about set theory.

67. b. I am interested in learning about mathematical paradoxes.

68. c. I am interested in learning about the mathematics of infinity.

69. d. I am interested in learning about theories of knowledge and knowing.

70. e. I am interested in learning about metaphysical theories of possibility and necessity.

71. f. I am interested in learning about the philosophy of language.

72. g. I am interested in learning about mathematical theories of probability

73. h. I am interested in learning about conditional (“if… then…”) statements.

74. i. I am interested in learning about methods for reasoning about causes and effects.

75. j. I am interested in learning about the properties of logical systems (like soundness and completeness).

76. Topics for Discussion

77. What are sets? Where are they? How do they relate to their members? Do they exist? How are they different from Venn diagrams? How many sets are there? Which axiom tells us this? How many empty sets are there?

78. The Barber Paradox Once upon a time there was a village, and in this village lived a barber named B.

79. The Barber Paradox B shaved all the villagers who did not shave themselves, And B shaved none of the villagers who did shave themselves.

80. The Barber Paradox Question, did B shave B, or not?

81. Suppose B Shaved B 1. B shaved BAssumption 2. B did not shave any villager X where X shaved X Assumption 3. B did not shave B1,2 Logic

82. Suppose B Did Not Shave B 1. B did not shave BAssumption 2. B shaved every villager X where X did not shave X Assumption 3. B shaved B1,2 Logic

83. Contradictions with Assumptions We can derive a contradiction from the assumption that B shaved B. We can derive a contradiction from the assumption that B did not shave B.

84. The Law of Excluded Middle Everything is either true or not true. Either P or not-P, for any P. Either B shaved B or B did not shave B, there is no third option.

85. It’s the Law Either it’s Tuesday or it’s not Tuesday. Either it’s Wednesday or it’s not Wednesday. Either killing babies is good or killing babies is not good. Either this sandwich is good or it is not good.

86. Disjunction Elimination A or B A implies C B implies C Therefore, C

87. Example Either Michael is dead or he has no legs If Michael is dead, he can’t run the race. If Michael has no legs, he can’t run the race. Therefore, Michael can’t run the race.

88. Contradiction, No Assumptions B shaves B or B does not shave B [Law of Excluded Middle] If B shaves B, contradiction. If B does not shave B, contradiction. Therefore, contradiction

89. Contradictions Whenever we are confronted with a contradiction, we need to give up something that led us into the contradiction.

90. Give up Logic? For example, we used Logic in the proof that B shaved B if and only if B did not shave B. So we might consider giving up logic. A or B A implies C B implies C Therefore, C

91. No Barber In this instance, however, it makes more sense to give up our initial acquiescence to the story: We assumed that there was a village with a barber who shaved all and only the villagers who did not shave themselves.

92. The Barber Paradox The paradox shows us that there is no such barber, and that there cannot be.

93. Set Theoretic Rules Reduction: a ∈ {x: COND(x)} Therefore, COND(a) Abstraction: COND(a) Therefore, a ∈ {x: COND(x)}

94. Examples Reduction: Mt. Everest ∈ {x: x is a mountain} Therefore, Mt. Everest is a mountain. Abstraction: Mt. Everest is a mountain. Therefore, Mt. Everest ∈ {x: x is a mountain}

95. Self-Membered Sets It’s possible that some sets are members of themselves. Let S = {x: x is a set}. Since S is a set, S ∈ {x: x is a set} (by abstraction), and thus S ∈ S (by Def of S). Or consider H = {x: Michael hates x}. Maybe I even hate the set of things I hate. So H is in H.

96. Russell’s Paradox Set Most sets are non-self-membered. The set of mountains is not a mountain; the set of planets is not a planet; and so on. Define: R = {x: ¬x ∈ x}

97. Is R in R? 1. R ∈ RYes? 2. R ∈ {x: ¬x ∈ x}1, Def of R 3. ¬R ∈ R2, Reduction 4. ¬R ∈ RNo? 5. R ∈ {x: ¬x ∈ x}4, Abstraction 6. R ∈ R5, Def of R

98. Historical Importance Russell’s paradox was what caused Frege to stop doing mathematics and do philosophy of language instead.

99. Comparison with the Liar Russell thought that his paradox was of a kind with the liar, and that any solution to one should be a solution to the other. Basically, he saw both as arising from a sort of vicious circularity.

100. The von Neumann Heirarchy