1. Paradoxes of Knowledge

2. The Surprise Exam

3. The Announcement
“I will give one and only one exam this month. The exam will be a surprise: no student will know the day before the exam that the exam will be given on that day.”

4. The Reasoning
Obviously, the exam has to happen on a day that there’s class.

5. The Reasoning
Obviously, the exam has to happen on a day that there’s class.

6. The Reasoning
Suppose the exam occurs on the 24th.

7. The Reasoning
Then the students on the 23rd would know that it was happening on the 24th, since it had not so far occurred earlier.
Thus it would not be a surprise.

8. The Reasoning
So we can rule out the 24th.

9. The Reasoning
Now suppose that the exam occurs on the 17th.

10. The Reasoning
Now on the 16th, the students would know that the exam must occur on the 17th, because it has not happened so far, and we’ve ruled out the 24th.

11. The Reasoning Once again, the exam would not be a surprise.
Therefore we can rule out the 17th.

12. The Reasoning
By similar reasoning we can rule out the 10th and the 3rd, leaving us with no day on which the exam can possibly occur.

13. The Paradox
But surprise exams are possible! They happen all the time. People have memed them.

14. Saul Kripke American philosopher (1940- present)
Proved important logic results as a teenager
Wrote one of the great classics of Anglophone (“analytic”) philosophy, Naming and Necessity
Has a solution to The Liar

15. Kripke on the Paradox
“Here, more so than with typical philosophical problems, we are in the kind of ‘intellectual cramp’ that Wittgenstein describes—”

16. Kripke on the Paradox
“One in which all the facts seem to be before us, there does not seem to be any new information to be gained, and yet we don’t quite know what is going wrong with our picture of the problem.”

17. The Guarantees

18. The Guarantees
The surprise exam paradox does not show that a surprise exam is impossible. If it shows anything, it shows that a surprise exam is possible WHEN:
There is guaranteed to be an exam.
It is guaranteed to be a surprise.

19. No Guaranteed Exam
Suppose the teacher says the following: “Every day I will write an exam. Then, at the beginning of the class, I will roll a die. If it lands 6, I will give the exam; otherwise, I will not give an exam.”

20. No Guaranteed Exam
Here, the exam, if it happens, will always be a surprise– to both teacher and students. But there is no guaranteed exam: the die might never turn up 6.

21. No Guaranteed Exam
Therefore, the students cannot start by reasoning that the exam will (won’t) happen on the last day: no-one knows whether it will (won’t).

22. No Guaranteed Surprise
Suppose the teacher says the following: “Every day I will write an exam. Then, at the beginning of the class, I will roll a die. If it lands 6, I will give the exam; otherwise, I will not give an exam, unless we reach the last day of class, at which point we will have an exam regardless.”

23. No Guaranteed Surprise
Here again, there is no paradox. The students could try to assume that the exam will happen on the last day and not be a surprise. But they cannot derive a contradiction from this, because this is one of the allowable situations: no surprise is guaranteed.

24. The Number of Days

25. The One-Day Announcement
“I will give one and only one exam and it will happen next class period. The exam will be a surprise: no student will know the day before the exam that the exam will be given on that day.”

26. The One-Day Announcement
This is clearly a ridiculous statement. If there is only one day that the exam can occur on, it will not be a surprise that it occurs on that day.

27. The Two-Day Announcement
“I will give one and only one exam and it will happen on one of the next two class periods. The exam will be a surprise: no student will know the day before the exam that the exam will be given on that day.”

28. The Two-Day Announcement
This is still a very bizarre announcement. Clearly, if it happens on the 2nd day, it won’t be a surprise after the 1st day has ended. So if the professor wants to make it a surprise, he must put it on the 1st day. But then it is not a surprise– we just figured out when it is.

29. The Twelve-Day Announcement
“I will give one and only one exam and it will happen on one of the next twelve class periods. The exam will be a surprise: no student will know the day before the exam that the exam will be given on that day.”

30. Somehow this is totally fine
Somehow this is totally fine. The argument that is used in the surprise exam paradox gets worse and worse the more days there are. This requires explanation.

31. A Logical Analysis

32. Today’s Lesson
I think there’s nothing very deep about the surprise exam paradox. Solving it won’t give us some amazing philosophical insight. But I do think that one thing it can help us see is how careful, logical analysis can help us think through difficult philosophical problems.

33. The Announcement
“I will give one and only one exam this month. The exam will be a surprise: no student will know the day before the exam that the exam will be given on that day.”

34. The Reasoning

35. Premise #1
Let “E3” mean “the exam happens on 3 February,” “E10” mean “the exam happens on 10 February, and so on. Here is what the students know: Premise #1: E3 or E10 or E17 or E24

36. Premise #2
The professor says he is giving one and only one exam, so the students also know: Premise #2: Not: E3 and E10 Not: E10 and E17 Not: E3 and E17 Not: E10 and E24 Not: E3 and E24 Not: E17 and E24

37. Premise #2: Not: Ex and Ey for any x, y in { 3, 10, 17, 24}

38. Premise #3
We also need to formulate the claim that the exam is a surprise. Let Kx(Ey) mean that the students know on x February that the exam happens on y February. Then: Premise #3: Not: K2(E3) Not: K16(E17) Not: K9(E10) Not: K23(E24)

39. Premise #3: Not: Kx-1(Ex) for any x in { 3, 10, 17, 24 }

40. Premise #4
Furthermore, we need to formalize this general principle: if the exam is not given on day x, then the students know every day after x that the exam is not given on x. Premise #4 IF not Ex, THEN Ky(not Ex) for any x in { 3, 10, 17, 24 } and any 29 > y > x

41. Reductio ad Absurdum
In general, if we can show that the logical consequences of some statement P are false, then we can conclude that P itself is false. (This is the logical principle known as Modus Tollens.) If rains, then the sidewalks will be wet. The sidewalks are not wet. Therefore it did not rain.

42. Reductio ad Absurdum
Reductio ad absurdum is an argument form that makes use of this fact. If we want to show that P is false, we assume that P is true, show that P has some obviously false consequence, and then conclude that P was false all along. Suppose that there is a smallest rational number N. For every number M, there is a number M/2 such that M/2 < M. So N/2 < N. Therefore there is no smallest rational number.

43. An Attempted Proof (by Reductio)

44. An Attempted Proof
Suppose: E24 Premise #2: Not: Ex and Ey for any x, y in { 3, 10, 17, 24} Therefore, not: E3 and not: E10 and not: E17 Premise #4: IF not Ex, THEN Ky(not Ex) for any x in { 3, 10, 17, 24 } and any 29 > y > x Therefore, K23(not E3) and K23(not E10) and K23(not E17)

45. An Attempted Proof
K23(not E3) and K23(not E10) and K23(not E17) Premise #1: E3 or E10 or E17 or E24 Therefore, …E24?... K23(E24)?

46. Quine’s Solution to the Paradox

47. W.V.O. Quine American philosopher (1908- 2000)
Did important work in logic and the philosophy of language
Wrote the classic paper, “Two Dogmas of Empiricism”
Also wrote the book, The Ways of Paradox and Other Essays

48. Quine’s Solution to the Paradox
Premise #1 says that the exam happens on one of the 4 class days. But the premise we need is that the students know that the exam happens on one of the 4 days. Quine argues that they don’t know this.

49. The One-Day Announcement
“I will give one and only one exam and it will happen next class period. The exam will be a surprise: no student will know the day before the exam that the exam will be given on that day.”

50. Quine’s Solution to the Paradox
In the one-day announcement case, you certainly cannot say you know what the professor says is true. He is saying that there is an exam next class, but you don’t know it. This is a… performative contradiction.

51. Brief Aside: Moore’s Paradox

52. G.E. Moore One of the founders of Anglophone (“analytic”) philosophy
Argued for pluralism and realism, when those things weren’t cool
Argued for common sense in philosophy

53. Moore’s Paradox
In general, the following is possible: for person X and proposition P: P but X does not believe P.

54. Moore’s Paradox
Example: It is unacceptable to grab women who don’t want to be grabbed, but Donald Trump does not believe that it is unacceptable to grab women who don’t want to be grabbed.

55. Moore’s Paradox
However, although it is in general possible, it is in general not OK to assert about yourself (fine for others to say it, or you to say it about others): P, but I don’t believe P

56. Moore’s Paradox
Example: I say, “Philosophy is interesting, but I don’t believe philosophy is interesting.”

57. Moore’s Paradox Extended
Also consider: I say, “Philosophy is interesting, but I don’t know philosophy is interesting.”

58. Analysis
To assert that P is to represent yourself as believing/ knowing that P. So while “P, but I don’t believe/ know that P”– while not a contradiction– is what we might call a “performative contradiction.” Your first statement, P, represents you as believing/ knowing P, then your second statement immediately says the opposite of the thing you’ve represented.

59. Historical Aside
From Wikipedia: “In addition to Moore's own work on the paradox, the puzzle also inspired a great deal of work by Ludwig Wittgenstein, who described the paradox as the most impressive philosophical insight that Moore had ever introduced. It is said that when Wittgenstein first heard this paradox one evening (which Moore had earlier stated in a lecture), he rushed round to Moore's lodgings, got him out of bed and insisted that Moore repeat the entire lecture to him.”

60. Quine’s Solution to the Paradox
Let’s return to Quine’s solution to the paradox. We might say that to assert that P is to represent yourself as believing/ knowing that P and trying to get your audience to believe/ know that p.

61. Quine’s Solution to the Paradox
So if you say, “The exam is next time, but you don’t know that it is next time,” this is another performative contradiction. Your first assertion tries to get your audience to know that the exam is next time, and then you immediately deny that you succeeded.

62. Quine’s Solution to the Paradox
Quine thinks that someone behaving so bizarrely cannot be trusted. So you don’t know what they’re saying is true, and this is needed for the paradox to work.

63. An Attempted Proof
K23(not E3) and K23(not E10) and K23(not E17) Premise #1: E3 or E10 or E17 or E24 Therefore, …E24?... K23(E24)?

64. The Paradox Revisited

65. Quine’s Solution to the Paradox
The problem with Quine’s solution is that the four-day case is different from the one-day case, and even the two-day case.

66. Kripke Replies to Quine
“Often, I think, you do know something simply because a good teacher has told you so. If a teacher were to announce a surprise exam to be given within a month, a student who did badly could not excuse herself by saying that she did not know that there was going to be an exam.”

67. Revised Premise #1
Old Premise #1: E3 or E10 or E17 or E24 New Premise #1: K1(E3 or E10 or E17 or E24)

68. An Attempted Proof
K23(not E3) and K23(not E10) and K23(not E17) New Premise #1: K1(E3 or E10 or E17 or E24) Therefore, …E24?... K1(E24)… K23(E24)? Some things we still need: Logical closure: students know the logical consequences of things they know Knowledge persistence: what students know on 1 February they know every day afterward.

69. Closure

70. Logical Closure
We can formulate the principle of logical closure as follows: If Kx(P) and Kx(if P, then Q), then Kx(Q) If you know that P is true, and you know that Q is a logical consequence of P, then you know that Q is true.

71. Closure is (Generally) False
If closure were true, we’d know every answer to every math problem– provided the problem gave us enough information to solve it.

72. Closure is (Sometimes) True
If students know the exam is on one of four days, and they know it’s not on the first three, then probably they know it’s on the last one.

73. Persistence

74. Persistence
We can formulate the principle of knowledge persistence as follows: If Kx(P), then Ky(P) for any 29> y > x

75. Persistence is (Generally) False
Many things you knew once, you don’t know now. Most obviously, the things you knew but then forgot.

76. Persistence is (Sometimes) True
Students who know there’s an exam in their class, probably aren’t going to forget that.

77. Finally… the Proof

78. An Attempted Proof
Suppose: E24 Premise #2: Not: Ex and Ey for any x, y in { 3, 10, 17, 24} Therefore, not: E3 and not: E10 and not: E17 Premise #4: IF not Ex, THEN Ky(not Ex) for any x in { 3, 10, 17, 24 } and any 29 > y > x Therefore, K23(not E3) and K23(not E10) and K23(not E17)

79. An Attempted Proof
K23(not E3) and K23(not E10) and K23(not E17) New Premise #1: K1(E3 or E10 or E17 or E24) Knowledge Persistence: If Kx(P), then Ky(P) for any 29> y > x K23(E3 or E10 or E17 or E24) Knowledge Closure: If Kx(P) and Kx(if P, then Q), then Kx(Q) K23(E24)

80. An Attempted Proof
K23(E24) Premise #3: Not: Kx-1(Ex) for any x in { 3, 10, 17, 24 } So, Not: K23(E24) Contradiction! Obviously, we need to do this for every other day.

81. Kripke’s Solution

82. Persistence is (Generally) False
Many things you knew once, you don’t know now. Most obviously, the things you knew but then forgot.

83. Persistence is (Generally) False
But there are other ways to lose knowledge!

84. Other Ways of Losing Knowledge
You all think I’m the instructor of this course. But am I?