Tuesday 9/25
Interlude: Puzzles and Paradoxes I
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
We began today drawing our all-too-brief discussion of truth-phobic language to a close. To be sure, there are many more fallacies than those we described, and identifying stretches of fallacious reasoning for the fallacies they contain can be difficult. The very point of a fallacy is often to guide or manipulate reasoning without doing so obviously or transparently. Truth-phobic language works best in the dark. And unlike truth-tropic language, we don't have tests for it.
Next today we began working 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, such as what you've been developing studying logic, 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 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.
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?
We did not, alas, get to the solution for Puzzle 03. We'll pick up there next time after you've had a couple of days to think on it.
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.)