background knowledge results!. part i for each of the following topics, rate each according to your...

Background Knowledge

Results!

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.

a. I am familiar with set theory.

Strongly Disagree Disagree Neither Agree Strongly Agree0

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b. I am familiar with mathematical paradoxes.


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c. I am familiar with the mathematics of infinity.


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


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


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f. I am familiar with the philosophy of language.


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g. I am familiar with mathematical theories of probability.


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


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


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


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Part II: Basic Mathematics

True or False

True or False. Circle T if the statement is true and circle F if it is false.

38% 62%1. T F 1 = 0.999999…2. T F 0.5 is a natural number3. T F There are two natural numbers p and q where: √2 =

p ÷ q

True or False

True or False. Circle T if the statement is true and circle F if it is false.

23% 76%1. T F 1 = 0.999999…2. T F 0.5 is a natural number3. T F There are two natural numbers p and q where: √2 =

p ÷ q

Short Answer

Fill in the blanks with the correct answer(s). 4. 43 = __________ 100% Correct5. 4½ = __________6. The binary notation expression ‘01010’ is equivalent to what decimal notation expression? __________

Short Answer

Fill in the blanks with the correct answer(s). 4. 43 = __________5. 4½ = __________ 88% Correct6. The binary notation expression ‘01010’ is equivalent to what decimal notation expression? __________

Short Answer

Fill in the blanks with the correct answer(s). 4. 43 = __________5. 4½ = __________6. The binary notation expression ‘01010’ is equivalent to what decimal notation expression? __________

42% Correct

Part III: Philosophy Familiarity

Multiple Choice

1. Immanuel Kant was a philosopher from which country? a. Englandb. Francec. Germanyd. Hollande. Italy

Multiple Choice

1. Immanuel Kant was a philosopher from which country? a. England 5%b. France 5%c. Germany 79%d. Holland 0%e. Italy 0%No Answer 11%

Multiple Choice

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

Multiple Choice

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

Multiple Choice

3. Which of the following was written by Immanuel Kant? a. The Critique of Pure Reasonb. The Ethicsc. On What Mattersd. Naming and Necessitye. The Wealth of Nations

Multiple Choice

3. Which of the following was written by Immanuel Kant? a. The Critique of Pure Reason 84%b. The Ethics 5%c. On What Matters 0%d. Naming and Necessity 5%e. The Wealth of Nations 0%No answer 5%

Multiple Choice

4. W.V.O. Quine was a philosopher from which country? a. Australiab. Canadac. Englandd. South Africae. The United States of America

Multiple Choice

4. W.V.O. Quine was a philosopher from which country? a. Australia 0%b. Canada 0%c. England 21%d. South Africa 5%e. The United States of America 47%No answer 26%

Multiple Choice

5. Saul Kripke wrote which of the following books? a. The Critique of Pure Reasonb. The Ethicsc. On What Mattersd. Naming and Necessitye. The Wealth of Nations

Multiple Choice

5. Saul Kripke wrote which of the following books? a. The Critique of Pure Reason 5%b. The Ethics 0%c. On What Matters 5%d. Naming and Necessity 68%e. The Wealth of Nations 0%No answer 21%

Multiple Choice

6. Epistemology is the study of: a. Beingb. Knowledgec. Truthd. Beautye. The Good

Multiple Choice

6. Epistemology is the study of: a. Being 5%b. Knowledge 79%c. Truth 0%d. Beauty 0%e. The Good 0%None/ b&c 16%

Part IV: Probability

True or False

For each of the following statements, circle T if it is true, and circle F if it is false.

T F

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.

True or False

For each of the following statements, circle T if it is true, and circle F if it is false.

T F32% 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.

True or False

For each of the following statements, circle T if it is true, and circle F if it is false. T F

2. If there are two possibilities, A and not-A, then each has a 50% chance of happening.

True or False

For each of the following statements, circle T if it is true, and circle F if it is false. T F

11% 89%2. If there are two possibilities, A and not-A, then each has a 50% chance of happening.

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. 2/3b. 3/2c. 1/3d. 1/2e. 3/5

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 35%b. 3/2 5%c. 1/3 0%d. 1/2 0%e. 3/5 50%None 10%

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.

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%

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.

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%

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)

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%

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.

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%

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.

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 response 11%

Part V: Logic

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.

Logic

This argument is (circle all that apply): True 2 Untrue 12 Valid 17 Invalid 1 Sound 1 Unsound 17

Yes/ No

Premise: P if and only if Q 2. YES NO Conclusion: If P, then Q3. YES NO Conclusion: If Q, then P4. YES NO Conclusion: If not-Q, then P5. YES NO Conclusion: If not-Q, then not-P6. YES NO Conclusion: If P, then P

Yes/ No

Premise: P if and only if Q 2. YES 76% NO 13% Conclusion: If P, then Q3. YES NO Conclusion: If Q, then P4. YES NO Conclusion: If not-Q, then P5. YES NO Conclusion: If not-Q, then not-P6. YES NO Conclusion: If P, then P

Yes/ No


Yes/ No

Premise: P if and only if Q 2. YES NO Conclusion: If P, then Q3. YES 94% NO 6% Conclusion: If Q, then P4. YES NO Conclusion: If not-Q, then P5. YES NO Conclusion: If not-Q, then not-P6. YES NO Conclusion: If P, then P

Yes/ No


Yes/ No

Premise: P if and only if Q 2. YES NO Conclusion: If P, then Q3. YES NO Conclusion: If Q, then P4. YES 0% NO 100% Conclusion: If not-Q, then P5. YES NO Conclusion: If not-Q, then not-P6. YES NO Conclusion: If P, then P

Yes/ No


Yes/ No

Premise: P if and only if Q 2. YES NO Conclusion: If P, then Q3. YES NO Conclusion: If Q, then P4. YES NO Conclusion: If not-Q, then P5. YES 89% NO 11% Conclusion: If not-Q, then not-P6. YES NO Conclusion: If P, then P

Yes/ No


Yes/ No

Premise: P if and only if Q 2. YES NO Conclusion: If P, then Q3. YES NO Conclusion: If Q, then P4. YES NO Conclusion: If not-Q, then P5. YES NO Conclusion: If not-Q, then not-P6. YES 78% NO 22% Conclusion: If P, then P

Multiple Choice

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

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 → Q 11%d. (~(P ↔ Q) v (P & Q)) 0%e. They are all equivalent to (P v Q) 5%No answer/ b&d 47%

Part VI: Personal Interest

a. I am interested in learning about set theory.


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b. I am interested in learning about mathematical paradoxes.


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


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


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


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


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


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


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


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


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Topics for Discussion

• 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?

The Barber Paradox

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

The Barber Paradox

B shaved all the villagers who did not shave themselves,

And B shaved none of the villagers who did shave themselves.

The Barber Paradox

Question, did B shave B, or not?

Suppose B Shaved B

1. B shaved B Assumption2. B did not shave any villager X where X shaved X

Assumption3. B did not shave B 1,2 Logic

Suppose B Did Not Shave B

1. B did not shave B Assumption2. B shaved every villager X where X did not shave X

Assumption3. B shaved B 1,2 Logic

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.

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.

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.

Disjunction Elimination

A or BA implies CB implies C

Therefore, C

Example

Either Michael is dead or he has no legsIf 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.

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

Contradictions

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

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 BA implies CB implies C

Therefore, C

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.

The Barber Paradox

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

Set Theoretic Rules

Reduction:a {x: COND(x)}∈

Therefore, COND(a)

Abstraction:COND(a)

Therefore, a {x: COND(x)}∈

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}∈

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.

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}∈

Is R in R?

1. R R∈ Yes?2. R {x: ¬x x}∈ ∈ 1, Def of R3. ¬R R∈ 2, Reduction

4. ¬R R∈ No?5. R {x: ¬x x}∈ ∈ 4, Abstraction6. R R∈ 5, Def of R

Historical Importance

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

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.

The von Neumann Heirarchy

background knowledge results!. part i for each of the following topics, rate each according to your...

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