[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.