Introduction to Philosophy

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Tuesday, 09-12-17: Logic I

Synopsis:

Today I introduced the Propositional Calculus (PC). We also took the second of the quizzes that we've been working on this semester, which was a follow-up to the Knights and Knaves puzzles we've been considering. Let me lay out the solutions to the first two of these puzzles we've considered thus far.

Now, 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 hope that distinction came across loud and clear when I explained the THOG puzzle (Quiz 1) on the synopsis of our previous discussion.

(Let me hasten to add, post Thog-quiz, that finding yourself in the Zed-10 group is not a punishment. It is, rather, an opportunity for you to i) earn extra credit and ii) get up to speed on the subjects at hand.)

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.

One last point before explaining the solutions to those puzzles: 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 two puzzles!

The First 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 First Puzzle's Solution

What do we know?

  1. Arthur asserts that Bernard and Charles are both knights.
  2. 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 Second 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 Second Puzzle's Solution

What do we know?

  1. Person #1 asserts that he is the Sorcerer's Apprentice.
  2. Person #2 asserts that he is the Sorcerer's Apprentice.
  3. 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.

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.

In keeping with this, today I introduced a new playing field--a new pitch, if you will--upon which you will continue developing your intellectual prowess. The Propositional Calculus (PC) is a very simple logic, but it provides us with lots of room to develop your abilities with an absolutely important concept: Entailment.

Today we defined the Propositional Calculus by specifying its syntax and semantics and we gave a general definition of entailment. Next time we'll give a specific definition for entailment and use it to develop a method for determining when entailments obtain: The Method of Truth Tables. We'll then set to completing exercises in class.