[SOUND] All right. If enough of you are still watching this,
then you know what that means. That means that in a matter of weeks,
Walter is going to lose all of his hair. Now,
Walter thinks he's going to have his head shaved.
But, little does he know, that I am going to use this plucker, to pluck out
one, by one, by one hair until he's bald. That's right.
I'm going to pluck out his hair one at a time until he's bald. Bald, I tell you.
Bald. [LAUGH] Yes, bald.
[SOUND] Hello? Yes. Oh. Hey. Hey, yes.
Were you watching, were you watching the Week 9 video just now? Yeah. Yeah, that's
right. That's right. I was just announcing my plans for
Walter. I was going to pluck out one hair at a
time until he was bald. what do you mean that's impossible? How
could that be impossible? Well, that's right, Walter is not bald
now Well yeah, of course. You can't take something that isn't bald
and make it bald by just pulling out one hair.
That doesn't make any sense. So Wait.
Does that mean that it's impossible for Walter to become bald by losing one hair
at a time? Hang on. Let me write that down.
Let me write that down. Don't go away. Okay? Let me think about
this. Okay, so Walter is not bald. Well, that's
true. Walter isn't bald right now.
My whole point, is that I'm going to make him bald, by plucking out one hair at a
time. Okay. But she says that you can't take
something that's not bald and make it bald by just removing one hair.
Okay, that's right, too. So, I can't make Walter bald by just
plucking one hair out of his head. If I just pluck one hair out of his head,
he's still going to be not bald. But then I'll pluck another hair out of
his head. But, wait a second. If I pluck another
hair out of his head, and he starts off being not bald, well then, even after I
pluck another hair out of his head, he's still going to not be bald, because you
can't take something that's not bald and make it bald by plucking just one hair.
So even after I pluck two hairs out of Walters head, he'll still not be bald.
Well, what if I pluck a third hair out of his head? Well, you can't take something
that's not bald and make it bald by plucking one hair out of its head.
So even after I pluck three hairs out of Walter's head, he'll still not be bald.
Okay, but what if I do another hair? Well If he starts off being not bald, and
I can't make him bald by just plucking one hair out of his head, then even after
I pluck four hairs out of his head, he'll still not be bald.
Now, wait a second. I could keep going like this.
If it's true that you can't take a thing that's not bald and make it bald by
plucking one hair out of it's head, then plucking Walter's hairs out one by
one could never turn him from being not bald into being bald.
Because there'd be no point where plucking one hair out of his head would
turn him from being not bald into being bald.
Okay. So this plan is not going to work.
I know. I'm going to do five hairs at a time, instead of one hair at a time.
Okay, let me get the big, let me get the big tweezers here and sharpen those up.
[SOUND] Wait a second. If I do five hairs at a time, that means
that I could get Walter to lose hair five times faster than if I do one hair at a
time. But no matter how fast I get him to lose
hair, I could never get him to lose more hair, five hairs at a time, than I could
eventually get him to lose, by plucking out one hair at a time. I mean, however
many hairs I can get him to lose pulling out five of his hairs at a time, I could
still get him to lose the same number of hairs pulling out one hair at a time. So
if it's true that a thing that's not bald can't become bald by losing one hair off
it's head, then that means a thing that's not bald can't become bald by losing five
hairs off it's head. Which means there's no way that Walter
could become bald. You have just seen an example of what's
called the paradox of vagueness. Let me explain to you what just happened.
Consider this argument, that I was going through in the previous
segment. Walter is not bald, there's premise one.
That's obviously true. Right? Walter is not bald.
Premise two, you can't turn something that's not bald into something that's
bald by just removing a single hair. All right?
Removing one hair is not going to make the difference, between something that's
not bald and something that's bald. But the problem is that if both of those
premises are true, then it follows, by mathematical deduction,
that, no matter how many hairs Walter loses, he will not be bald.
What do I mean by mathematical deduction? Well, an intuitive way of explaining it
is this. Think about something that's not bald,
like Walter. If premise two is true,
then you can't turn that not bald thing into a bald thing by removing one hair.
It's still going to not be bald. And so you can't turn that non-bald thing
into a bald thing by removing one more hair.
It's still going to not be bald. And you can't turn that non-bald thing
into a bald thing by removing one, one more hair.
It's still going to not be bald. And so on, for all the hairs it has.
So, if these two premises are true, then it's gotta be true that no matter how
many hairs Walter loses, he will not be bald.
This argument is apparently valid. But here's the problem.
If the argument is valid, then there's no possible way that premise one and premise
two could both be true while the conclusion is false.
You remember what a valid argument is. A valid argument is an argument where the
premises can't all be true while the conclusion is false.
But premise one is apparently true. You look at Walter, he's not bald.
Premise two is apparently true. There's no occasion on which you could
take something that's not bald, and turn it into something that's bald by removing
a single hair. So premise two is apparently true.
But the conclusion seems to be obviously false.
It's ridiculous to say, no matter how many hairs Walter loses, he will not be
bald. What if he lost all his hairs? So this
conclusion seems absurd, and yet the argument seems valid.
So this is why it's paradoxical. It's paradoxical because, any solution to
this problem is going to be something that's contrary to appearances.
Either premise one is false, and Walter really is bald, contrary to appearances.
Or premise two is false and there really is some occasion on which you could take
a non-bald thing and make it bald by removing a single hair.
Or, the conclusion really is true, no matter how many hairs Walter loses, he
will not be bald. That seems ridiculous.
Or, alternatively, the argument isn't valid.
And yet, as we saw, the argument seems plainly valid.
So something is contrary to appearances here.
And that's why this is a paradox, because no matter how we solve this,
no matter whether we say premise one contrary to appearances, is false, or
premise two, contrary to appearances, is false,
or the conclusion, contrary to appearances, is true,
or the argument, contrary to appearances, is invalid, no matter which of those
things we say, what we're saying is something that's contrary to appearances.
And so, that's why we call this a paradox.
And why is it a paradox of vagueness? Here's why.
The reason this paradox arises, is because the expression bald, doesn't
have, a precise definition. Suppose I give a precise definition of
bald. Suppose that I said, a person is bald, if
they have exactly, 102, or fewer hairs on their head.
Okay so I give a precise definition of bald.
Well now that I've defined the word bald precisely, by specifying a precise number
of hairs, or less, that a creature has to have on it's head in order to be bald,
well, now that I've defined the word bald precisely, now, notice, premise two is
obviously no longer true. You can take a non-bald thing and make it
bald by just removing a single hair. An example of that is when you take a
creature with 103 hairs on his head, and remove a single hair from those 103,
well, now you've made it bald. Because now it has only 102 hairs, and
according to our precise definition of bald, 102 hairs or fewer, that's bald.
So, the reason the paradox arises is because bald is a vague expression.
It doesn't have a precise definition. That's why this paradox arises.
And to prove that to you, we could consider other examples of this
same kind of paradox that use other vague expressions besides bald.
For instance, suppose I take these letter magnets right here, and I start stacking
them up, and adding one, by one, by one to the
stack, in order to make a heap. Okay?
Now, I'm trying to make a heap, or maybe I don't have a heap just yet.
But I think, well if I keep adding them one, by one, by one, eventually I'll have
a heap. Well, since heap is a vague term also,
since there's no precise definition of heap,
the problem is that you get the same kind of paradox with heap that you do with
bald. Here, let me show you what I mean One
magnet is not a heap. Right? You'll all agree with that.
A single magnet by itself if not a heap of magnets.
But, it seems, it seems that you can not take, you can not take a non-heap and
turn it into a heap by adding a single magnet.
All right? A single magnet is not going to make the
difference between a non-heap and a heap. There's no definite point where you turn
a non-heap into a heap by just adding one magnet.
But again, if that's true, and the single magnet is not a heap, well, then two
magnets will not be a heap. Right? Because you can't take a non-heap
and turn it into a heap by adding a single magnet.
And if two magnets is not a heap, and it's true that you can't turn a non-heap
into a heap by adding a single magnets, a single magnet,
then that means three magnets will not be a heap.
And if three magnets is not a heap, and you can't turn a non-heap into a heap by
adding a single magnet, then that means four magnets is not a heap.
And we could continue this reasoning indefinitely.
Which means that no amount of magnets makes a heap.
So again, this argument appears to be valid by the reasoning I just went
through. It seems that there's no way that both of
the premises could be true while the conclusion's false.
Premise one seems obviously true. One magnet is not a heap.
Premise two seems obviously true, as well.
You can't turn a non heap into a heap by just adding a single magnet.
But the conclusion seems obviously false. If you have a billion magnets all stacked
up, that's gotta make a heap. So, what's the problem here? If the
argument is valid, then you can't have the premises all
being true and the conclusion being false.
And yet, the premises both seem true and the conclusion seems false.
So what's the solution? Again,
a paradox. Because no matter what we say about this, it's going to be something
contrary to appearances, and once again, as in the previous case, the paradox
arises because the expression heap is a vague expression. It has no precise
definition. See, if we give a precise definition of
heap, if we say that a heap is 102 things or more, then premise two is going to end
up being false. You can turn a non-heap into a heap by
adding just a single thing, namely, when you have 101 things and you just add one
more thing to it. Then you've turned a non-heap into a
heap. Because we've defined heap as precisely
102 things or more. Since the word, heap, in ordinary
English, doesn't have a precise definition like that, premise two seems
to be true. And since premise one seems to be true
and the conclusion seems to be false, and yet the argument seems to be valid, we
have a paradox. That's the paradox of vagueness.