Wednesday 1/22
Introduction
Readings
(none for today)
Synopsis
We began today by reviewing the syllabus and discussing the mechanics of the course. The requirements are simple: five examinations of increasing weight, early exams counting least, later most. Hopefully this will give you the space you need to 'get the hang' of the material and begin to demonstrate improving skills.
I pause to note that last semester's Intro (Fall 2019) was the first time I'd attempted this plan for exams. It took some students a long time to grasp that doing poorly on an early exam (as happens) has little impact on their overall course grade given how little the early exams count for their overall course grade. Obviously I need to figure out a better way to explain it, but I struggle because the math just seems obvious to me. In any case, students made many useful suggestions as we progressed during the semester, two of which I adopted and for which I am grateful:
- Permit a single 4x6 (inch!) handwritten notecard; and,
- Publish a collection of essay questions in advance from which the examination's essay questions will be selected.
We'll follow this approach again this semester, although I really do need to figure out an effective way to explain missing many points on an exam worth relatively little is not predestination so much as a not-so-subtle hint.
Now, as I was thinking a few years ago about how to run this course, I thought back to all the times I'd taught it previously and realized I was never very happy with how it went. Then I thought all the way back to when I took Intro to Philosophy and realized with horror that I couldn't remember it. At all. Not a bit of it.
Something is amiss, and I think the problem has to do with how we think about philosophy. To wit, thinking of historical or topical surveys as appropriate for an introductory philosophy course is a mistake--elementary and common, perhaps, but a mistake nonetheless.
It's technically a category mistake, like asking to see the U.S. after driving through all fifty states, as if the U.S. were the same kind of thing as (or in the same category as--hence "category mistake") just another state.
You see, despite the fact that the University is organized according to various disciplines like Biology, Theater, Chemistry, English, Mathematics, Philosophy, Education, Business, and so on, putting Philosophy on the grand list of disciplines is like thinking of the U.S. as just another state like Texas or Delaware that in fact constitutes it.
Philosophy, I would argue, is not a distinct discipline in the same way Biology is a distinct discipline from English with its own subject matter and its own history. Fundamentally, the subject matter of philosophy is the subject matter of every discipline; the history of philosophy is the history of every discipline.
Yet if philosophy has neither its own set of topics nor its own history, what, then, is philosophy?
I think in this at least Wittgenstein was right: Philosophy, he said, is an activity. It is something you do.
Think about riding a bicycle. We're not going to study bicycle mechanics. We're not going to learn about famous cyclists. We're not going to study the history of cycling, or memorize who won what race when.
We're going to get on bicycles and ride.
Some of you will pick it up quickly. You'll be zooming around the room in no time.
Others will fall, a lot.
Eventually, though, we should all be up and riding.
Just don't try to explain to anyone else what you're doing in Intro to Philosophy. Make something up. It doesn't matter what.
It may get bumpy at times, but I hope that, unlike me, you'll remember your introduction to philosophy. If Wittgenstein was right, you will.
The upshot of all this is that this semester, we won't study philosophy or philosophers, we will do philosophy.
To start us on that path, today I distributed a series of puzzles which prove to be simple to state but frustratingly challenging to solve and to explain as well.
Why would I do such a thing? (Judging by the stunned looks I saw, many of you were asking this. Some even came up afterward, wondering if this class was for them. Rest assured, it is.)
Well, go back to the home page for a minute and look at the title banner. What does the little man in the painting have in his hands? Where is he sitting? If you look long enough, you come to realize that he is busy examining an impossible object, while he himself is perched on an impossible ledge. How can this be?
This feeling you may have looking carefully at the painting has many names: Bewilderment, puzzlement, perplexity, astonishment, wonderment. The Ancient Greeks had a special term for it: Aporia. Think of aporia as that sense one has of encountering something which is not merely surprising, but astonishing and, seemingly, inexplicable. Unsurprisingly, aporia--bewilderment, or wonderment--is often associated with a sense of fear and discomfort. We like thinking we understand; being shown we don't understand what we thought we did is often seen as offensive or disruptive. Hence the looks I saw in class.
Let us recall that "philosophy," Aristotle announced, "begins in wonder." Without fear or discomfort, philosophy embraces aporia, finding more of value in the puzzle than in any seemingly satisfactory solution. Philosophy is special in this regard. Where other disciplines seek solutions, philosophy revels in perplexity. Philosophy invites us to wonder and, unflinching, encourages our bewilderment. Philosophy, you see, is much more about what we don't know than what we do!
So my goal in giving you these puzzles was not to 'show you up' or make you feel stupid. Anyone, very much myself included, can be given a puzzle that will vex and stymie them. No, the point rather was to hopefully give everyone a glimpse of that sense of aporia, of not knowing how to go on or how to see your way to a solution. It is that sense which is characteristic of philosophy and which we must bravely learn how to embrace.
Thus far the feeling of aporia: what about the feeling of understanding?
Let me be clear that the feeling you understand can deceive. You don't really understand a solution unless and until you can explain it to someone who doesn't understand. 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 working through the solutions to the puzzles we considered in class today: 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.
Okay, let's solve some 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.
The First Puzzle
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?
The First Puzzle'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: if in responding to the question "would he tell me to go left?" the guard says "yes!", go right. If "no!", go left.
How does this solve the puzzle? 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 faithfully 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.
The Second Puzzle
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?
The Second Puzzle'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.
The Third Puzzle
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?
The Third Puzzle'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 a knave. So #3 could not be a knave. He must be a knight. Hence (b) is the 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.
The Fourth Puzzle
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?
The Fourth Puzzle'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 a solution to each puzzle.
So much for the puzzles and solving them. 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. Again, as Wittgenstein put it, "philosophy is an activity!" (Our question, then, will in part be, what sort of activity is philosophy?)
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 these 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.)