Thursday 9/27
Interlude: Puzzles and Paradoxes II
This week I will argue by example that there is a distinction between a puzzle and a paradox and explain by example why this turns out to be a philosophically important distinction.
All readings will be in the form of in-class handouts. That said, expect lengthy synopses after class.
Synopsis
Recall Puzzle 03:
Having been told by the King (who is, presumably, a knight) that the Sorcerer's Apprentice is presently entertaining two guests and that Abercrombie must deduce which of the three is the Sorcerer's Apprentice, we pick up the crucial part story:
A short walk brought the anthropologist to the house. When he entered, there were indeed three people present.
"Which of you is the Sorcerer's Apprentice?" asked Abercrombie.
"I am," replied one.
"I am the Sorcerer's Apprentice!" cried a second.
But the third remained silent.
"Can you tell me anything?" Abercrombie asked.
"It's funny," answered the third one with a sly smile. "At most, only one of the three of us ever tells the truth!"
Can it be deduced which of the three is the Sorcerer's Apprentice? If so, how? If not, why not?
Puzzle 03's Solution
What do we know?
- Person #1 asserts that he is the Sorcerer's Apprentice.
- Person #2 asserts that he is the Sorcerer's Apprentice.
- Person #3 asserts that at most, only one of the three of them is a knight.
Yes, it can be deduced which of the three is the Sorcerer's Apprentice, and here is how.
Since #1 and #2 cannot both be the Sorcerer's Apprentice, either one or both is a knave--they cannot both be knights.
#3 is either a knight or a knave; we don't know which.
Suppose #3 is a knave. Then his assertion that at most only one of the three of them is a knight is false. Consider: Under what conditions is it false that at most one of the three of them is a knight?
The assertion that at most one of the three of them is a knight is TRUE when either
a) NONE of them is a knight, or
b) ONE of them is a knight.
The assertion that at most one of the three of them is a knight is FALSE when either
c) TWO of them are knights, or
d) All THREE of them are knights.
Is (d) possible? Well, no, since #1 and #2's assertions contradict one another (they can't both be the Sorcerer's Apprentice!), so at least one of them must be a knave.
Hence (c) must be the case if #3 is a knave. But this is impossible! Can you see why? If at least one of the first two is a knave, and at least two of the three are, by (c), knights, then it follows that #3 MUST be the other knight.
This contradicts our hypothesis that #3 is knave. So #3 could not be a knave. He must be a knight. Hence (b) is case, and #3 is the lone knight.
Yet remember, we're supposed to be figuring out which of the three is the Sorcerer's Apprentice, not which of them are knights or knaves.
Do we have enough information now to determine which of the three is the Sorcerer's Apprentice? Surely we do. We've determined that both #1 and #2 are knaves, so they are lying when they assert that they are the Sorcerer's Apprentice. Hence #3, the lone knight, is also the Sorcerer's Apprentice.
Puzzle 04
The Apprentice was delighted with Abercrombie's reasoning and informed him that he could meet the Sorcerer.
"He is now upstairs in the tower conferring with the island Astrologer," said the Apprentice. "You may go up and interview them if you like, but please knock before entering."
The anthropologist went upstairs, knocked on the door, and was bidden to enter. When he did, he saw two very curious individuals, one wearing a green conical hat and the other a blue one. He could not tell from their appearance which was the Astrologer and which was the Sorcerer. After introducing himself he asked, "Is the Sorcerer a knight?"
The one in the blue hat answered the question (he answered either yes or no), and the anthropologist was able to deduce which was the Sorcerer.
Which one was the Sorcerer?
Puzzle 04's Solution
We concluded from this that the man in the green hat was the Sorcerer. How?
It's always a good idea to draw pictures, just as it's always a good idea to list what you do know and list what you don't know:
What We Know
- The Sorcerer is in the room with the Astrologer.
- Knights always tell the truth while Knaves always lie.
- Abercrombie is able to figure out whether Blue Hat or Green Hat is the Sorcerer from Blue Hat's answer to his question, "Is the Sorcerer a knight?"
What We Don't Know
- Whether Blue Hat answered "yes" or "no".
- Whether Blue Hat or Green Hat is a knight or a knave.
- Whether the Sorcerer or the Astrologer is a knight or a knave.
- Whether Blue Hat or Green Hat is the Sorcerer.
Now as much as possible it helps to conduct our analysis in systematic fashion. That is, let us fix what we don't know by making assumptions to see where they lead us.
Suppose Blue Hat answered "yes".
Then either Blue Hat is a Knight or Blue Hat is a Knave.
Suppose Blue Hat is a Knight.
Then he is truthfully asserting that the Sorcerer is a knight, but since Green Hat could also be a knight, there is no way to tell which one is the Sorcerer if Blue Hat is a Knight and answers "yes". Yet Abercrombie was able to figure out whether Blue Hat or Green Hat is the Sorcerer!
So suppose Blue Hat is a Knave instead.
Then he is lying when he says the Sorcerer is a knight, hence the Sorcerer is not a knight--he is a knave. Blue Hat is in this case a knave, but Green Hat could be a knave too. Yet once again we remind ourselves that Abercrombie was able to figure out whether Blue Hat or Green Hat is the Sorcerer!
Hence if Blue Hat answers "yes", then it doesn't matter whether he's a knight or a knave. Either way, Abercrombie would not have been able to figure out whether Blue Hat or Green Hat is the Sorcerer.
Hence Blue Hat could not have answered "yes"; he must have answered "no"! We've come a long way since our initial bewilderment.
We know, in particular, that Blue Hat must have answered "no". What follows from a "no" answer? Once again, we have to explore the alternatives of whether Blue Hat is a Knight or a Knave:
Suppose Blue Hat is a Knight.
Then he is truthfully denying that the Sorcerer is a Knight, in which case the Sorcerer is a Knave. But if the Sorcerer is a Knave and Blue Hat is a Knight, then Green Hat must be the Sorcerer.
Suppose Blue Hat is a Knave.
Then he is falsely denying that the Sorcerer is a Knight, in which case the Sorcerer is a Knight. But if the Sorcerer is a Knight and Blue Hat is a Knave, then Green Hat must be the Sorcerer.
Hence since Blue Hat answered "no", it doesn't matter whether Blue Hat is a Knight or a Knave: Either way, Green Hat must be the Sorcerer!
So Green Hat is the Sorcerer, and we are able to deduce that Green Hat is the sorcerer only from the little bits of information we were given and without knowing whether Blue Hat or Green Hat is a knight or a knave.
Pretty cool, eh?
Now, obviously you would not be expected to provide quite so comprehensive an explanation as I provide above, but my goal is to be sure everyone can follow my approach in solving the puzzle.
So much for the puzzles and their solutions. I note once again that your explanations need not be as detailed to receive full marks, but they do need to make the crucial steps clear. Grasping which are the crucial steps, and learning just how to make them clear in your explanations, is something you can only learn by practice, over and over again.
I hope you're getting an important point here. Philosophy is not a body of knowledge, to be memorized in lecture and regurgitated on exams. Rather, philosophy is something you do. As Wittgenstein put it, "philosophy is an activity!"
If that's right, and I think it is, philosophy is as close as you can come to a kind of intellectual athletics. Don't think of yourselves as students. Think of yourselves as athletes. Don't think of me as your professor. Think of me as your coach. It's cheesy, I know. Yet you will find yourself frustrated until you make this crucial shift in your frame of reference.
Next today we went on to discuss a series of paradoxes.
Paradox 01
Newcomb's Paradox*
You are confronted with a choice. There are two boxes before you, A and B. You may either open both boxes, or else just open box B. You may keep what is inside any box you open, but you may not keep what is inside any box you do not open. The background is this.
A very powerful being, who has been invariably accurate in his predictions about your behavior in the past, has already acted in the following way:
He has put $1,000.00 in box A.
If he has predicted that you will open just box B, he has in addition put $1,000,000.00 in box B.
If he has predicted that you will open both boxes, he has put nothing in box B.
Should you open both boxes A and B? Or just B? Justify your answer.
*adapted from Sainsbury, R.M., Paradoxes, 2nd ed (Cambridge University Press, 1995).
As we discussed, there are equally good reasons to be a one-boxer as a two-boxer. The one-boxer will say, by choosing to open just box B, I'm doing what the predictor would have predicted, so I'll get the million dollars. The two-boxer says, sure, but the the predictor has already come and gone. Either there is a million and a thousand dollars in the boxes as you stand there wondering whether to open both, or there's not. So you might as well open both.
In this case, we seem to have two equally rational yet incompatible courses of action at hand. Unlike a puzzle, where we can grasp the solution once we discover and demonstrate it, in this case there seems no final solution at hand.
Paradox 02
Setting a Surprise Exam*
On Friday before the last week of class, the Classical Formal Logic class are told that they are to have a quiz on what they have learned so far this term, particularly Aristotle's 256 logical forms. This is because they are a rather slow and lazy class, teacher adds, offensively. The class are not pleased and start muttering. 'When is it, anyway?' they ask sullenly.
Teacher smirks. 'That is up to me. I may have it at any point between now and next Friday. However, let me assure you of this, when I do have the quiz it will be a surprise!'
After school, Bob and Patricia are discussing the bad news. Bob is very worried as he has a poor memory. 'I could pass, I'm sure,' says, 'if I knew on which day the test would be, then I could learn everything the night before.'
'Don't worry, Bob,' says Patricia, 'I think teacher is having a bit of a joke at our expense - you see, I don't think there can be a quiz!'
And she explains that the test cannot be held next Friday, the last day of the term, because by then the class will know that it must be going to be held, and will therefore quickly memorize the material for the test the night before. That's great" says Bob sarcastically. 'So it's any day between now and Thursday then?'
Patricia explains patiently. 'It can't be Thursday either, because if it can't be on Friday, and it is already Wednesday night, we'd all know it would have to be coming up on Thursday, the next day!'
Bob gets it now. 'Nor could it be Wednesday, nor Tuesday - nor indeed Monday! Hey! What a joke - teacher trying to get us worried! - and now he can't hold the pop quiz without having to back down on the surprise bit. Silly old fool!'
They don't tell the others, who spend ages trying to memorize the 256 aristotelian logical forms and other nonsense, much to Bob and Patricia's secret amusement. Then on Wednesday, teacher comes in and announces the quiz.
'You can't do this!' says Bob.
'Why not?' replies teacher, surprised, but not very.
'Because it's got to be a surprise - and you can only hold the test when we're not expecting it!'
'Yes, but Bob, you're surprised, and I am holding the test' says teacher in a teachery sort of way.
Is there a flaw in Bob and Patricia's reasoning? If not, how was teacher able to surprise them with a pop quiz? If so, what is the mistake in their reasoning?
*adapted from Cohen, M. 1999. 101 Philosophy Problems. London: Routledge.
If there is a flaw in their reasoning, there must be a step in it to which we can point and say, "ah ha! That's where you've gone wrong." Yet as we think about it, where would the step be? We all agreed the exam can't be Friday, but since it can't be Friday we know that once Thursday rolls around it won't be a surprise on Thursday. Since it can't be Thursday any more than it could have been Friday, it also can't be Wednesday. Nor Tuesday. Nor Monday. In thinking about it, we see no particular step in the argument where the kids went wrong. Yet here they are, surprised all the same.
Paradox 03
Protagoras' Problem*
Euathlos has learned from Protagoras how to be a lawyer, under a very generous arrangement whereby he doesn't need to pay anything for his tuition until and unless he wins his first court case. Rather to Protagoras' annoyance, however, after giving up hours of his time training Euathlos, the pupil decides to become a musician and never takes any court cases. Protagoras demands that Euathlos pay him for his trouble and, when the musician refuses, decides to sue him in court. Protagoras reasons that if Euathlos loses the case, he, Protagoras, will have won, in which case he will get his money back, and furthermore, that even if he loses, Euathlos will then have won a case, despite his protestations about being a musician now, and will therefore still have to pay up.
Euathlos reasons a little differently however. If I lose, he thinks, then I will have lost my first court case, in which event, the original agreement releases me from having to pay any tuition fees. And, even if he wins, Protagoras will still have lost the right to enforce the contract, so he will not need to pay anything.
They can't both be right. So who's making the mistake?
*adapted from Cohen, M. 1999. 101 Philosophy Problems. London: Routledge.
With Paradox 02, we found a chain of reasoning whose logic we couldn't pierce, but which nevertheless led us (the kids) astray. With Paradox 03 we have apparently equally good reasons for completely different outcomes. Indeed, presented with these arguments, it is unclear how any judge could justify ruling in favor of Protagoras over Euathlos, or Euathlos over Protagoras. Protagoras' reasons for why he should be awarded the tuition money are just as good as Euathlos' reasons why he owes no tuition money. Yet it cannot both be the case that Euathlos owes tuition and Euathlos does not owe tuition!
Paradox 04
The Ship Theseus*
This is not what Ray North had bargained for. As an international master criminal he prided himself on being able to get the job done. His latest client had demanded that he steal the famous yacht Theseus, the vessel from which British newspaper magnate Lucas Grub had thrown himself to his death and which more recently had been the scene of the murder of LA rapper Daddy Iced Tea.
But here he was in the dry dock where the boat had just finished being repaired, confronted by two seemingly identical yachts. North turned to the security man, who was being held at gunpoint by one of his cronies.
“If you want to live, you'd better tell me which of these is the real Theseus,” demanded Ray.
“That kinda depends,” came the nervous reply. “You see, when we started to repair the ship, we needed to replace lots of parts. Only, we kept all the old parts. But as the work progressed, we ended up replacing virtually everything. When we had finished, some of the guys thought it would be good to use all the old parts to reconstruct another version of the ship. So that's what we've got. On the left, the Theseus repaired with new parts and on the right, the Theseus restored from old parts.”
“But which one is the genuine Theseus?” demanded Ray.
“I've told you all I know!” screamed the guard, as the crony tightened his grip. Ray scratched his head and started to think about how he could get away with both...
Which is the real ship Theseus? Which ship should Ray steal, if he can only steal one?
*adapted from Julian Baggini, “The Pig That Wants to Be Eaten: 100 Experiments for the Armchair Philosopher”, (New York: Plume Books) pp. 31-32
If the dock workers had not rebuilt the Theseus as they tossed aside old parts being replaced on the ship Theseus, we would not be in the difficult position of determine which of the two ships is the real ship Theseus--the ship, that is, from which Lucas Grub threw himself to his death and upon which Daddy Iced Tea was murdered. As it stands, we can make as much a case for Ray to steal the Theseus built from new parts as for him to steal the Theseus built from the old parts of the ship. Yet there can't be two ships Theseus, can there? These-i? From both of which Grub killed himself and upon both of which Tea was murdered?
Next time we'll discuss the difference between puzzles and paradoxes and the importance they have for philosophical inquiry. We'll also examine the first of the major philosophical puzzles (or would it be paradoxes?) we shall take up this semester: Why is there something and not nothing?
Please note that the Knights and Knaves puzzles are drawn from the logician (and magician!) Raymond Smullyan's "Satan, Cantor, and Infinity" (Smullyan, R. 1992. Satan, Cantor, and Infinity: And Other Mind-Boggling Puzzles. New York: Alfred A. Knopf, Inc.)