1 00:00:00,989 --> 00:00:06,815 [SOUND] All right. If enough of you are still watching this, 2 00:00:06,815 --> 00:00:14,559 then you know what that means. That means that in a matter of weeks, 3 00:00:14,559 --> 00:00:18,772 Walter is going to lose all of his hair. Now, 4 00:00:18,772 --> 00:00:23,287 Walter thinks he's going to have his head shaved. 5 00:00:23,287 --> 00:00:29,847 But, little does he know, that I am going to use this plucker, to pluck out 6 00:00:29,847 --> 00:00:34,377 one, by one, by one hair until he's bald. That's right. 7 00:00:34,377 --> 00:00:43,012 I'm going to pluck out his hair one at a time until he's bald. Bald, I tell you. 8 00:00:43,012 --> 00:00:46,544 Bald. [LAUGH] Yes, bald. 9 00:00:46,544 --> 00:00:52,565 [SOUND] Hello? Yes. Oh. Hey. Hey, yes. 10 00:00:52,565 --> 00:01:04,093 Were you watching, were you watching the Week 9 video just now? Yeah. Yeah, that's 11 00:01:04,093 --> 00:01:08,579 right. That's right. I was just announcing my plans for 12 00:01:08,579 --> 00:01:11,766 Walter. I was going to pluck out one hair at a 13 00:01:11,766 --> 00:01:17,676 time until he was bald. what do you mean that's impossible? How 14 00:01:17,676 --> 00:01:24,261 could that be impossible? Well, that's right, Walter is not bald 15 00:01:24,261 --> 00:01:32,179 now Well yeah, of course. You can't take something that isn't bald 16 00:01:32,179 --> 00:01:37,153 and make it bald by just pulling out one hair. 17 00:01:37,153 --> 00:01:41,644 That doesn't make any sense. So Wait. 18 00:01:41,644 --> 00:01:51,235 Does that mean that it's impossible for Walter to become bald by losing one hair 19 00:01:51,235 --> 00:01:56,456 at a time? Hang on. Let me write that down. 20 00:01:56,456 --> 00:02:05,672 Let me write that down. Don't go away. Okay? Let me think about 21 00:02:05,672 --> 00:02:15,092 this. Okay, so Walter is not bald. Well, that's 22 00:02:15,092 --> 00:02:22,762 true. Walter isn't bald right now. 23 00:02:22,762 --> 00:02:31,112 My whole point, is that I'm going to make him bald, by plucking out one hair at a 24 00:02:31,112 --> 00:02:35,637 time. Okay. But she says that you can't take 25 00:02:35,637 --> 00:02:42,950 something that's not bald and make it bald by just removing one hair. 26 00:02:58,631 --> 00:03:10,720 Okay, that's right, too. So, I can't make Walter bald by just 27 00:03:10,720 --> 00:03:21,875 plucking one hair out of his head. If I just pluck one hair out of his head, 28 00:03:21,875 --> 00:03:29,294 he's still going to be not bald. But then I'll pluck another hair out of 29 00:03:29,294 --> 00:03:33,812 his head. But, wait a second. If I pluck another 30 00:03:33,812 --> 00:03:41,033 hair out of his head, and he starts off being not bald, well then, even after I 31 00:03:41,033 --> 00:03:47,842 pluck another hair out of his head, he's still going to not be bald, because you 32 00:03:47,842 --> 00:03:54,262 can't take something that's not bald and make it bald by plucking just one hair. 33 00:03:54,262 --> 00:04:00,360 So even after I pluck two hairs out of Walters head, he'll still not be bald. 34 00:04:00,360 --> 00:04:06,897 Well, what if I pluck a third hair out of his head? Well, you can't take something 35 00:04:06,897 --> 00:04:12,673 that's not bald and make it bald by plucking one hair out of its head. 36 00:04:12,673 --> 00:04:19,242 So even after I pluck three hairs out of Walter's head, he'll still not be bald. 37 00:04:19,242 --> 00:04:26,064 Okay, but what if I do another hair? Well If he starts off being not bald, and 38 00:04:26,064 --> 00:04:33,106 I can't make him bald by just plucking one hair out of his head, then even after 39 00:04:33,106 --> 00:04:38,425 I pluck four hairs out of his head, he'll still not be bald. 40 00:04:38,425 --> 00:04:42,852 Now, wait a second. I could keep going like this. 41 00:04:42,852 --> 00:04:50,477 If it's true that you can't take a thing that's not bald and make it bald by 42 00:04:50,477 --> 00:04:57,957 plucking one hair out of it's head, then plucking Walter's hairs out one by 43 00:04:57,957 --> 00:05:04,082 one could never turn him from being not bald into being bald. 44 00:05:04,082 --> 00:05:10,972 Because there'd be no point where plucking one hair out of his head would 45 00:05:10,972 --> 00:05:15,255 turn him from being not bald into being bald. 46 00:05:15,255 --> 00:05:18,832 Okay. So this plan is not going to work. 47 00:05:18,832 --> 00:05:26,058 I know. I'm going to do five hairs at a time, instead of one hair at a time. 48 00:05:26,058 --> 00:05:34,110 Okay, let me get the big, let me get the big tweezers here and sharpen those up. 49 00:05:34,110 --> 00:05:40,302 [SOUND] Wait a second. If I do five hairs at a time, that means 50 00:05:40,302 --> 00:05:47,707 that I could get Walter to lose hair five times faster than if I do one hair at a 51 00:05:47,707 --> 00:05:51,967 time. But no matter how fast I get him to lose 52 00:05:51,967 --> 00:05:59,308 hair, I could never get him to lose more hair, five hairs at a time, than I could 53 00:05:59,308 --> 00:06:03,891 eventually get him to lose, by plucking out one hair at a time. I mean, however 54 00:06:03,891 --> 00:06:08,524 many hairs I can get him to lose pulling out five of his hairs at a time, I could 55 00:06:08,524 --> 00:06:13,563 still get him to lose the same number of hairs pulling out one hair at a time. So 56 00:06:13,563 --> 00:06:21,333 if it's true that a thing that's not bald can't become bald by losing one hair off 57 00:06:21,333 --> 00:06:29,442 it's head, then that means a thing that's not bald can't become bald by losing five 58 00:06:29,442 --> 00:06:35,655 hairs off it's head. Which means there's no way that Walter 59 00:06:35,655 --> 00:06:41,667 could become bald. You have just seen an example of what's 60 00:06:41,667 --> 00:06:49,420 called the paradox of vagueness. Let me explain to you what just happened. 61 00:06:49,420 --> 00:06:55,014 Consider this argument, that I was going through in the previous 62 00:06:55,014 --> 00:06:58,817 segment. Walter is not bald, there's premise one. 63 00:06:58,817 --> 00:07:02,620 That's obviously true. Right? Walter is not bald. 64 00:07:02,620 --> 00:07:08,321 Premise two, you can't turn something that's not bald into something that's 65 00:07:08,321 --> 00:07:11,567 bald by just removing a single hair. All right? 66 00:07:11,567 --> 00:07:17,597 Removing one hair is not going to make the difference, between something that's 67 00:07:17,597 --> 00:07:23,825 not bald and something that's bald. But the problem is that if both of those 68 00:07:23,825 --> 00:07:29,282 premises are true, then it follows, by mathematical deduction, 69 00:07:29,282 --> 00:07:35,194 that, no matter how many hairs Walter loses, he will not be bald. 70 00:07:35,194 --> 00:07:42,666 What do I mean by mathematical deduction? Well, an intuitive way of explaining it 71 00:07:42,666 --> 00:07:47,076 is this. Think about something that's not bald, 72 00:07:47,076 --> 00:07:50,222 like Walter. If premise two is true, 73 00:07:50,222 --> 00:07:55,511 then you can't turn that not bald thing into a bald thing by removing one hair. 74 00:07:55,511 --> 00:08:00,243 It's still going to not be bald. And so you can't turn that non-bald thing 75 00:08:00,243 --> 00:08:03,229 into a bald thing by removing one more hair. 76 00:08:03,229 --> 00:08:07,784 It's still going to not be bald. And you can't turn that non-bald thing 77 00:08:07,784 --> 00:08:11,124 into a bald thing by removing one, one more hair. 78 00:08:11,124 --> 00:08:16,703 It's still going to not be bald. And so on, for all the hairs it has. 79 00:08:16,703 --> 00:08:24,927 So, if these two premises are true, then it's gotta be true that no matter how 80 00:08:24,927 --> 00:08:29,753 many hairs Walter loses, he will not be bald. 81 00:08:29,753 --> 00:08:35,531 This argument is apparently valid. But here's the problem. 82 00:08:35,531 --> 00:08:41,780 If the argument is valid, then there's no possible way that premise one and premise 83 00:08:41,780 --> 00:08:45,793 two could both be true while the conclusion is false. 84 00:08:45,793 --> 00:08:51,953 You remember what a valid argument is. A valid argument is an argument where the 85 00:08:51,953 --> 00:08:56,502 premises can't all be true while the conclusion is false. 86 00:08:56,502 --> 00:09:01,438 But premise one is apparently true. You look at Walter, he's not bald. 87 00:09:01,438 --> 00:09:06,380 Premise two is apparently true. There's no occasion on which you could 88 00:09:06,380 --> 00:09:12,344 take something that's not bald, and turn it into something that's bald by removing 89 00:09:12,344 --> 00:09:15,883 a single hair. So premise two is apparently true. 90 00:09:15,883 --> 00:09:19,228 But the conclusion seems to be obviously false. 91 00:09:19,228 --> 00:09:24,567 It's ridiculous to say, no matter how many hairs Walter loses, he will not be 92 00:09:24,567 --> 00:09:27,597 bald. What if he lost all his hairs? So this 93 00:09:27,597 --> 00:09:31,802 conclusion seems absurd, and yet the argument seems valid. 94 00:09:31,802 --> 00:09:39,460 So this is why it's paradoxical. It's paradoxical because, any solution to 95 00:09:39,460 --> 00:09:46,347 this problem is going to be something that's contrary to appearances. 96 00:09:46,347 --> 00:09:54,542 Either premise one is false, and Walter really is bald, contrary to appearances. 97 00:09:54,542 --> 00:10:01,918 Or premise two is false and there really is some occasion on which you could take 98 00:10:01,918 --> 00:10:07,500 a non-bald thing and make it bald by removing a single hair. 99 00:10:07,500 --> 00:10:14,965 Or, the conclusion really is true, no matter how many hairs Walter loses, he 100 00:10:14,965 --> 00:10:20,086 will not be bald. That seems ridiculous. 101 00:10:20,086 --> 00:10:25,589 Or, alternatively, the argument isn't valid. 102 00:10:25,589 --> 00:10:32,081 And yet, as we saw, the argument seems plainly valid. 103 00:10:32,081 --> 00:10:36,182 So something is contrary to appearances here. 104 00:10:36,182 --> 00:10:40,822 And that's why this is a paradox, because no matter how we solve this, 105 00:10:40,822 --> 00:10:46,102 no matter whether we say premise one contrary to appearances, is false, or 106 00:10:46,102 --> 00:10:49,367 premise two, contrary to appearances, is false, 107 00:10:49,367 --> 00:10:53,032 or the conclusion, contrary to appearances, is true, 108 00:10:53,032 --> 00:10:59,107 or the argument, contrary to appearances, is invalid, no matter which of those 109 00:10:59,107 --> 00:11:06,321 things we say, what we're saying is something that's contrary to appearances. 110 00:11:06,321 --> 00:11:10,132 And so, that's why we call this a paradox. 111 00:11:10,132 --> 00:11:14,592 And why is it a paradox of vagueness? Here's why. 112 00:11:14,592 --> 00:11:21,453 The reason this paradox arises, is because the expression bald, doesn't 113 00:11:21,453 --> 00:11:27,546 have, a precise definition. Suppose I give a precise definition of 114 00:11:27,546 --> 00:11:31,785 bald. Suppose that I said, a person is bald, if 115 00:11:31,785 --> 00:11:37,325 they have exactly, 102, or fewer hairs on their head. 116 00:11:37,325 --> 00:11:40,527 Okay so I give a precise definition of bald. 117 00:11:40,527 --> 00:11:46,560 Well now that I've defined the word bald precisely, by specifying a precise number 118 00:11:46,560 --> 00:11:52,144 of hairs, or less, that a creature has to have on it's head in order to be bald, 119 00:11:52,144 --> 00:11:57,981 well, now that I've defined the word bald precisely, now, notice, premise two is 120 00:11:57,981 --> 00:12:02,988 obviously no longer true. You can take a non-bald thing and make it 121 00:12:02,988 --> 00:12:08,605 bald by just removing a single hair. An example of that is when you take a 122 00:12:08,605 --> 00:12:14,574 creature with 103 hairs on his head, and remove a single hair from those 103, 123 00:12:14,574 --> 00:12:20,034 well, now you've made it bald. Because now it has only 102 hairs, and 124 00:12:20,034 --> 00:12:26,869 according to our precise definition of bald, 102 hairs or fewer, that's bald. 125 00:12:26,869 --> 00:12:33,060 So, the reason the paradox arises is because bald is a vague expression. 126 00:12:33,060 --> 00:12:39,325 It doesn't have a precise definition. That's why this paradox arises. 127 00:12:39,325 --> 00:12:45,543 And to prove that to you, we could consider other examples of this 128 00:12:45,543 --> 00:12:52,056 same kind of paradox that use other vague expressions besides bald. 129 00:12:52,056 --> 00:13:00,899 For instance, suppose I take these letter magnets right here, and I start stacking 130 00:13:00,899 --> 00:13:04,970 them up, and adding one, by one, by one to the 131 00:13:04,970 --> 00:13:10,561 stack, in order to make a heap. Okay? 132 00:13:10,561 --> 00:13:18,802 Now, I'm trying to make a heap, or maybe I don't have a heap just yet. 133 00:13:18,802 --> 00:13:26,081 But I think, well if I keep adding them one, by one, by one, eventually I'll have 134 00:13:26,081 --> 00:13:30,402 a heap. Well, since heap is a vague term also, 135 00:13:30,402 --> 00:13:35,202 since there's no precise definition of heap, 136 00:13:35,202 --> 00:13:47,849 the problem is that you get the same kind of paradox with heap that you do with 137 00:13:47,849 --> 00:13:55,127 bald. Here, let me show you what I mean One 138 00:13:55,127 --> 00:14:05,597 magnet is not a heap. Right? You'll all agree with that. 139 00:14:05,597 --> 00:14:14,942 A single magnet by itself if not a heap of magnets. 140 00:14:14,942 --> 00:14:51,607 But, it seems, it seems that you can not take, you can not take a non-heap and 141 00:14:51,607 --> 00:14:56,100 turn it into a heap by adding a single magnet. 142 00:14:56,100 --> 00:15:00,935 All right? A single magnet is not going to make the 143 00:15:00,935 --> 00:15:09,707 difference between a non-heap and a heap. There's no definite point where you turn 144 00:15:09,707 --> 00:15:14,870 a non-heap into a heap by just adding one magnet. 145 00:15:16,082 --> 00:15:24,293 But again, if that's true, and the single magnet is not a heap, well, then two 146 00:15:24,293 --> 00:15:28,873 magnets will not be a heap. Right? Because you can't take a non-heap 147 00:15:28,873 --> 00:15:32,213 and turn it into a heap by adding a single magnet. 148 00:15:32,213 --> 00:15:37,147 And if two magnets is not a heap, and it's true that you can't turn a non-heap 149 00:15:37,147 --> 00:15:40,968 into a heap by adding a single magnets, a single magnet, 150 00:15:40,968 --> 00:15:44,022 then that means three magnets will not be a heap. 151 00:15:44,022 --> 00:15:50,548 And if three magnets is not a heap, and you can't turn a non-heap into a heap by 152 00:15:50,548 --> 00:15:56,190 adding a single magnet, then that means four magnets is not a heap. 153 00:15:56,190 --> 00:16:00,702 And we could continue this reasoning indefinitely. 154 00:16:00,702 --> 00:16:11,640 Which means that no amount of magnets makes a heap. 155 00:16:11,640 --> 00:16:25,721 So again, this argument appears to be valid by the reasoning I just went 156 00:16:25,721 --> 00:16:29,399 through. It seems that there's no way that both of 157 00:16:29,399 --> 00:16:33,633 the premises could be true while the conclusion's false. 158 00:16:33,633 --> 00:16:37,693 Premise one seems obviously true. One magnet is not a heap. 159 00:16:37,693 --> 00:16:40,632 Premise two seems obviously true, as well. 160 00:16:40,632 --> 00:16:45,492 You can't turn a non heap into a heap by just adding a single magnet. 161 00:16:45,492 --> 00:16:54,267 But the conclusion seems obviously false. If you have a billion magnets all stacked 162 00:16:54,267 --> 00:17:01,052 up, that's gotta make a heap. So, what's the problem here? If the 163 00:17:01,052 --> 00:17:05,635 argument is valid, then you can't have the premises all 164 00:17:05,635 --> 00:17:09,039 being true and the conclusion being false. 165 00:17:09,039 --> 00:17:14,462 And yet, the premises both seem true and the conclusion seems false. 166 00:17:14,462 --> 00:17:16,952 So what's the solution? Again, 167 00:17:16,952 --> 00:17:22,512 a paradox. Because no matter what we say about this, it's going to be something 168 00:17:22,512 --> 00:17:27,943 contrary to appearances, and once again, as in the previous case, the paradox 169 00:17:27,943 --> 00:17:33,349 arises because the expression heap is a vague expression. It has no precise 170 00:17:33,349 --> 00:17:36,917 definition. See, if we give a precise definition of 171 00:17:36,917 --> 00:17:44,002 heap, if we say that a heap is 102 things or more, then premise two is going to end 172 00:17:44,002 --> 00:17:48,498 up being false. You can turn a non-heap into a heap by 173 00:17:48,498 --> 00:17:55,034 adding just a single thing, namely, when you have 101 things and you just add one 174 00:17:55,034 --> 00:17:59,447 more thing to it. Then you've turned a non-heap into a 175 00:17:59,447 --> 00:18:03,118 heap. Because we've defined heap as precisely 176 00:18:03,118 --> 00:18:07,359 102 things or more. Since the word, heap, in ordinary 177 00:18:07,359 --> 00:18:12,935 English, doesn't have a precise definition like that, premise two seems 178 00:18:12,935 --> 00:18:16,088 to be true. And since premise one seems to be true 179 00:18:16,088 --> 00:18:20,828 and the conclusion seems to be false, and yet the argument seems to be valid, we 180 00:18:20,828 --> 00:18:23,681 have a paradox. That's the paradox of vagueness.