1. Kinds of Truths

2. In-Class Assignment #2

3. The null set is a proper subset of every set?
FALSE “A is a subset of B” = “You can’t find a member of A that isn’t also in B” So the null set is a subset of every set. “A is a proper subset of B” = “A is a subset of B and not vice versa.” But null set is a subset of itself and vice versa.

4. Names for Sets

5. Extensive Notation
Bracket symbols

6. Extensive Notation Names of the set’s members
John , Paul , George , Ringo

7. Extensive Notation
Order doesn’t matter
Paul , Ringo , John , George

8. Extensive Notation Any name will do
Paul , Ringo , Dr. Winston O’Boogie , George

9. Intensive Notation
Variable
(your choice: x, y, z, etc.) x

10. Intensive Notation
Up and down line x

11. Intensive Notation Condition that uniquely picks out the set’s members
x x is a member of The Beatles

12. Intensive Notation Condition that uniquely picks out the set’s members
x x had the most #1 British albums

13. Intensive Notation Condition that uniquely picks out the set’s members
x x sang a song on Rubber Soul

14. Intensive Notation Condition that uniquely picks out the set’s members
x x = John or x = Paul or x = George
or x= Ringo

15. Converting Notations
Write names for the following set in extensive notation: { x | 3x = 4 } Wrong answers: 4/3 ‘4/3’ x ⊆ { 4/3 } ‘{ 4/3 }’ { x | 4/3} { 3x = 4 } 4

16. Converting Notations
Write names for the following set in extensive notation: { x | 3x = 4 } Right answer: { 4/3 } { 1 1/3 } { 8/6 } { 2 – 2/3 }

17. Converting Notations
Write names for the following set in extensive notation: { x | x is the name of the instructor of PHIL 2000 } Wrong answers: Michael ‘Michael’ { Michael } ‘{ Michael }’

18. Converting Notations
Write names for the following set in extensive notation: { x | x is the name of the instructor of PHIL 2000 } Right answer: { ‘Michael’ } { ‘Michael Johnson’ } { ‘Dr. Johnson’ }

19. Converting Notations
Write names for the following set in intensive notation: { } Wrong answers: The null set the empty set Ø { Ø } { x | x is empty } { x | x is { } }

20. Converting Notations
Write names for the following set in intensive notation: { } Right answers: { x | x is a member of { } } { x | x = 0 & x = 1 } { x | x > 5 & x < 5 } { x | x rhymes with ‘orange’ & x ≠ ‘orange’ }

21. Converting Notations
Write names for the following set in intensive notation: { Canada, Kennedy Town, Run Run Shaw Tower } Wrong answers: Places Places on Earth The set whose members are Canada, K-Town, and Run Run Shaw Twr. { x | x = Canada, x | x = K-Town, x | x = Run Run Shaw Tower }

22. Converting Notations
Write names for the following set in intensive notation: { Canada, Kennedy Town, Run Run Shaw Tower } Right answers: { x | x is a member of { Canada, Kennedy Town, Run Run Shaw Tower } } { x | x = Canada OR x = K-Town OR x = Run Run Shaw Tower } { x | x is a place where the police are looking for Michael }

23. Power Sets
What is the power set of { A, B, C } ? STEP 1: Do the calculation. How many members does the set have? Three. So the power set has 23 members = 8. STEP 2: Make a chart with 1 column for each member and 1 row for each subset.

24. A B C -

25. Power Set
STEP 3: Put set brackets around each row. These are the subsets of the original set.

26. A B C -

27. Power Set
STEP 4: Put set brackets around all the subsets. This is the power set.

28. A B C - , , , , , , ,

29. What We’re Skipping

30. Chapter 3 Real numbers and the power set of the natural numbers
The power set theorem
The continuum hypothesis

31. Two Distinctions

32. Analytic/ Synthetic Distinction
Examples of analytic truths: geometric terms, kinship terms, animal terms (boar, sow, piglet, drift, pork)
True in virtue of meaning?
Relation to definitions?
Synthetic truths: “truth depends on actual facts”

33. A priori/ A posteriori Epistemological distinction
A priori “can be known prior to the experience of facts”

34. A Priori Knowledge

35. A priori/ A posteriori Epistemological distinction
A priori “can be known prior to the experience of facts”
Examples: analytic truths
Some experience necessary: learning the concepts
Other examples: Cartesian truths
“I exist” true in virtue of meaning?
A priori known vs. knowable: computing sums w/ calculator
A posteriori “can only be known as a result of relevant experiences”

36. Synthetic A Priori?
Truth depends on actual facts, not just word meanings/ can be known without investigating actual facts
Examples? “All triangles have interior angles that sum to π radians”?
“The real numbers can’t be paired one-to-one with the integers”?
“The future will resemble the past”?
Universal Grammar?

37. The Knowledge of Babies
“In a few domains, babies seem to have intuitions that guide their expectations about how important entities in the world (e.g., objects, people) act and interact. For example, babies appear to be born knowing that objects cannot magically appear or disappear, that they cannot pass through each other, and that they cannot move unless contacted by another object. These expectations hold for objects, but not for non-object entities like substances (e.g., liquid, sand).” --Kristy vanMarle

38. Explanation 1: “Innate Ideas from God”
God gives us some knowledge at birth.
Problem: methodological naturalism.
The problem of epistemological evil– why doesn’t God give us more or better knowledge?

39. Explanation 2: “Preconditions of Experience”
Experience is a product of the “things in themselves” AND the way our mind structures them.
Our mind imposes a Euclidean space-time structure on experience.
Problem?: space-time isn’t part of external reality.
Problem: empirically false.

40. Explanation 3: Evolution
Just as we are born with various innate physical traits that are the product of evolution, so too are we born with innate mental traits that are the product of evolution.

41. Explanation 3: Evolution
Problem: evolution satisfices, doesn’t optimize.
In fact, many innate principles are only approximately true– consider optical illusions.