Tuesday 7/03
Practice 01
Readings
None! We worked from handouts today.
Synopsis
Today we worked through a series of puzzles for amusement and to illustrate several important features of the work we'll be doing this session.
First, it is important to distinguish between the context of discovery and the context of justification. The context of discover is largely inexplicable. It is that flash of insight or 'ah ha!' moment we all enjoy when we see a solution. Discovery is many things: Creative, inspired, insightful, and mysterious.
Yet it is one thing to see a solution, quite another to justify that it is the correct solution. Here we emphasize explanation, understanding, demonstration, illustration, and proof.
Second, analysis is crucial in all that we do. That is, we must cultivate the ability to break problems down into manageable parts, taking care that we haven't missed anything along the way.
Analysis is usually thought to be primarily a function of the context of justification. A case can be made, I submit, that having a sharply honed analytic ability helps immensely in the context of discovery. So analysis, on the one hand, and the distinction between discovery and justification, really are two different, but complementary points to consider as you read about these puzzles.
Now again, remember what I said in class: It is one thing to grasp or have the sense that you understand the solution to a puzzle, quite another to explain the solution.
I will go further and say that the feeling you understand can deceive. You don't really understand a solution unless and until you can explain it. At most, you have a vague sense of 'seeing' the solution. When you can explain it, clearly and in detail, then and only then can you be sure you understand the puzzle and its solution. Discovery is not about understanding--I can see a solution without having much idea why it's a solution.
One last point before explaining the solutions to the puzzles we considered in class: Explanations can be made in different ways and be equally correct. Some explanations may be lengthy; some short. Some may use diagrams, and thus be visualized; some may be completely verbalized. What I provide below are lengthy explanations that detail each and every step. Your explanations need not be anywhere near as involved to be absolutely correct. I realize this can be frustrating, especially for a generation raised on bubble-tests.
Okay, let's solve the puzzles! Compare them to your solutions in class to see if you were at least on the right track, and if not, where your explanation went awry.
Puzzle 01
Suppose you found yourself on the Island of Knights and Knaves needing to find the Castle, where you will find your eternal and truest love. Knights, you know, always tell the truth; Knaves never do. You also know that a Knight and a Knave guard the way to the Castle at a fork in the road and you can ask only one question to find out which way to go. Unfortunately, there is no way to tell Knights from Knaves by the way they dress or look. Nevertheless, if you are extremely clever there is a question you can ask which will get you to the Castle straightaway to be forever united in bliss with your love.
What one question should you ask? And why should you ask that question?
Puzzle 01's Solution
Point to one guard and ask the other, "Would he tell me left (or right, it doesn't matter which way you ask) is the way to the castle?" Go the opposite direction of whatever you're told.
Since one of the guards is a knight and the other a knave, you're either asking the knight or you're asking the knave this question. If the knight, he will faithfully report what the knave would say, but the knave would lie, so you do the opposite. If the knave, he will lie about what the knight truthfully says, so again you do the opposite.
The solution trades on the fact that a lie truthfully reported is still a lie, while the truth lied about is, again, a lie. No matter what you're told, it will be a lie. Hence the instruction to do the opposite.
Puzzle 02
With a twinge of apprehension such as he had never felt before, an anthropologist named Abercrombie stepped onto the Island of Knights and Knaves. He knew that this island was populated by most perplexing people: knights, who make only true statements, and knaves, who make only false ones. “How,” Abercrombie wondered, “am I ever to learn anything about this island if I can't tell who is lying and who is telling the truth?”
Abercrombie knew that before he could find out anything he would have to make one friend, someone whom he could always trust to tell him the truth. So when he came upon the first group of natives, three people, presumably named Arthur, Bernard, and Charles, Abercrombie thought to himself, “This is my chance to find a knight for myself.” Abercrombie first asked Arthur, “Are Bernard and Charles both knights?” Arthur replied, “Yes.” Abercrombie then asked: “Is Bernard a knight?” To his great surprise, Arthur answered: “No.”
Is Charles a knight or a knave?
Puzzle 02's Solution
What do we know?
- Arthur asserts that Bernard and Charles are both knights.
- Arthur also asserts that Bernard is not a knight.
Arthur himself must be a knave: Bernard cannot both be a knight and not a knight. Hence Arthur is lying when he makes both of the above assertions.
So it is false that Bernard is not a knight, which entails that he is a knight.
It is also false that Bernard and Charles are both knights. Specifically, Arthur's first assertion is false just in case neither of them are knights or only one of them is a knight.
Yet we've already shown Bernard is a knight since Arthur's second assertion is false. So it must be the case that the Charles is not a knight. He is a knave.
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 be systematic. 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.
Please note that the four puzzles are drawn from 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.)