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.
Series 1
9
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Series 1
Agree
Strongly Agree
b. I am familiar with mathematical paradoxes.
Responses
10
9
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
c. I am familiar with the mathematics of
infinity.
Responses
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
d. I am familiar with theories of knowledge
and knowing.
Responses
10
9
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
e. I am familiar with metaphysical theories of
possibility and necessity.
Responses
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
f. I am familiar with the philosophy of
language.
Responses
9
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
g. I am familiar with mathematical theories of
probability.
Responses
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
h. I am familiar with conditional (“if… then…”)
statements.
Responses
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
i. I am familiar with methods for reasoning
about causes and effects.
Responses
9
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
j. I am familiar with the properties of logical
systems (like soundness and completeness).
Responses
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
Part II: Basic Mathematics
True or False
True or False. Circle T if the statement is true and circle F if it is false.
1.
2.
3.
38%
T
T
T
62%
F
1 = 0.999999…
F
0.5 is a natural number
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.
1.
2.
3.
38%
T
T
T
62%
F
1 = 0.999999…
F
0.5 is a natural number
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.
1.
2.
3.
23%
T
T
T
76%
F
1 = 0.999999…
F
0.5 is a natural number
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.
5.
6.
43 = __________ 100% Correct
4½ = __________
The binary notation expression ‘01010’ is equivalent to what
decimal notation expression? __________
Short Answer
Fill in the blanks with the correct answer(s).
4.
5.
6.
43 = __________
4½ = __________ 88% Correct
The binary notation expression ‘01010’ is equivalent to what
decimal notation expression? __________
Short Answer
Fill in the blanks with the correct answer(s).
4.
5.
6.
43 = __________
4½ = __________
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. England
b. France
c. Germany
d. Holland
e. Italy
Multiple Choice
1. Immanuel Kant was a philosopher from which country?
a. England
b. France
c. Germany
d. Holland
e. Italy
No Answer
5%
5%
79%
0%
0%
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)
b. 17th (1600-1699)
c. 18th (1700-1799)
d. 19th (1800-1899)
e. 20th (1900-1999)
No answer
11%
21%
47%
5%
0%
16%
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
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
No answer
84%
5%
0%
5%
0%
5%
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
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
No answer
0%
0%
21%
5%
47%
26%
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
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
No answer
5%
0%
5%
68%
0%
21%
Multiple Choice
6. Epistemology is the study of:
a. Being
b. Knowledge
c. Truth
d. Beauty
e. The Good
Multiple Choice
6. Epistemology is the study of:
a. Being
b. Knowledge
c. Truth
d. Beauty
e. The Good
None/ b&c
5%
79%
0%
0%
0%
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
F
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.
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. 2/3
b. 3/2
c. 1/3
d. 1/2
e. 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
b. 3/2
c. 1/3
d. 1/2
e. 3/5
None
35%
5%
0%
0%
50%
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. 5%
e. All of the above.
79%
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%
b. 98%
c. 99%
d. 100%
e. There is not enough information to answer this question.
0%
0%
32%
5%
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)
b. Pr(x is an animal/ x is a dog)
42%
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.
3.
4.
5.
6.
YES
YES
YES
YES
YES
NO
NO
NO
NO
NO
Conclusion: If P, then Q
Conclusion: If Q, then P
Conclusion: If not-Q, then P
Conclusion: If not-Q, then not-P
Conclusion: If P, then P
Yes/ No
Premise: P if and only if Q
2.
3.
4.
5.
6.
YES
YES
YES
YES
YES
76% NO
NO
NO
NO
NO
13% Conclusion: If P, then Q
Conclusion: If Q, then P
Conclusion: If not-Q, then P
Conclusion: If not-Q, then not-P
Conclusion: If P, then P
Yes/ No
Premise: P if and only if Q
2.
3.
4.
5.
6.
YES
YES
YES
YES
YES
NO
NO
NO
NO
NO
Conclusion: If P, then Q
Conclusion: If Q, then P
Conclusion: If not-Q, then P
Conclusion: If not-Q, then not-P
Conclusion: If P, then P
Yes/ No
Premise: P if and only if Q
2.
3.
4.
5.
6.
YES
YES
YES
YES
YES
NO
94% NO
NO
NO
NO
6%
Conclusion: If P, then Q
Conclusion: If Q, then P
Conclusion: If not-Q, then P
Conclusion: If not-Q, then not-P
Conclusion: If P, then P
Yes/ No
Premise: P if and only if Q
2.
3.
4.
5.
6.
YES
YES
YES
YES
YES
NO
NO
NO
NO
NO
Conclusion: If P, then Q
Conclusion: If Q, then P
Conclusion: If not-Q, then P
Conclusion: If not-Q, then not-P
Conclusion: If P, then P
Yes/ No
Premise: P if and only if Q
2.
3.
4.
5.
6.
YES
YES
YES
YES
YES
0%
NO
NO
NO
NO
NO
Conclusion: If P, then Q
Conclusion: If Q, then P
100% Conclusion: If not-Q, then P
Conclusion: If not-Q, then not-P
Conclusion: If P, then P
Yes/ No
Premise: P if and only if Q
2.
3.
4.
5.
6.
YES
YES
YES
YES
YES
NO
NO
NO
NO
NO
Conclusion: If P, then Q
Conclusion: If Q, then P
Conclusion: If not-Q, then P
Conclusion: If not-Q, then not-P
Conclusion: If P, then P
Yes/ No
Premise: P if and only if Q
2.
3.
4.
5.
6.
YES
YES
YES
YES
YES
NO
NO
NO
89% NO
NO
Conclusion: If P, then Q
Conclusion: If Q, then P
Conclusion: If not-Q, then P
11% Conclusion: If not-Q, then not-P
Conclusion: If P, then P
Yes/ No
Premise: P if and only if Q
2.
3.
4.
5.
6.
YES
YES
YES
YES
YES
NO
NO
NO
NO
NO
Conclusion: If P, then Q
Conclusion: If Q, then P
Conclusion: If not-Q, then P
Conclusion: If not-Q, then not-P
Conclusion: If P, then P
Yes/ No
Premise: P if and only if Q
2.
3.
4.
5.
6.
YES
YES
YES
YES
YES
NO
NO
NO
NO
78% NO
Conclusion: If P, then Q
Conclusion: If Q, then P
Conclusion: If not-Q, then P
Conclusion: If not-Q, then not-P
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 → Q
d. (~(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)
b. ~(P → ~Q)
c. ~P → Q
d. (~(P ↔ Q) v (P & Q))
e. They are all equivalent to (P v Q)
No answer/ b&d
16%
21%
11%
0%
5%
47%
Part VI: Personal Interest
a. I am interested in learning about set
theory.
Responses
12
10
8
6
4
2
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
b. I am interested in learning about
mathematical paradoxes.
Responses
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
c. I am interested in learning about the
mathematics of infinity.
Responses
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
d. I am interested in learning about theories
of knowledge and knowing.
Responses
12
10
8
6
4
2
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
e. I am interested in learning about metaphysical
theories of possibility and necessity.
Responses
9
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
f. I am interested in learning about the
philosophy of language.
Responses
9
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
g. I am interested in learning about
mathematical theories of probability
Responses
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
h. I am interested in learning about
conditional (“if… then…”) statements.
Responses
10
9
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
i. I am interested in learning about methods
for reasoning about causes and effects.
Responses
9
8
7
6
5
4
3
2
1
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
j. I am interested in learning about the properties
of logical systems (like soundness and
completeness).
Responses
12
10
8
6
4
2
0
Strongly Disagree
Disagree
Neither
Responses
Agree
Strongly Agree
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
Assumption
2. B did not shave any villager X where X shaved X
Assumption
3. B did not shave B
1,2 Logic
Suppose B Did Not Shave B
1. B did not shave B
Assumption
2. B shaved every villager X where X did not shave X
Assumption
3. 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 B
A implies C
B implies C
Therefore, C
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.
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 B
A implies C
B 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
2. R ∈ {x: ¬x ∈ x}
3. ¬R ∈ R
Yes?
1, Def of R
2, Reduction
4. ¬R ∈ R
5. R ∈ {x: ¬x ∈ x}
6. R ∈ R
No?
4, Abstraction
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