1 00:00:04,340 --> 00:00:10,674 Today's lecture is about negation. The negation that we're going to be 2 00:00:10,674 --> 00:00:15,469 talking about today is a truth functional connective. 3 00:00:15,469 --> 00:00:22,075 It differs from disjunction and conjunction in that instead of combining 4 00:00:22,075 --> 00:00:29,314 two propositions into a new proposition, it just converts one proposition into a 5 00:00:29,314 --> 00:00:34,472 new proposition. Negation is expressed in English usually 6 00:00:34,472 --> 00:00:40,494 by the word not. No, no, no, not that kind of knot. 7 00:00:40,494 --> 00:00:46,752 Not, not knot. Now in English, the word not can be used 8 00:00:46,752 --> 00:00:52,111 in a variety of ways. For instance, you might say, hey Rom 9 00:00:52,111 --> 00:00:53,165 catch. Oh, no. 10 00:00:53,165 --> 00:00:59,403 Not the, Not the jacket, not, oh, no, not the pencil, not the sticky pad. 11 00:00:59,403 --> 00:01:06,344 But as we're going to be using the word not, it functions as a truth functional 12 00:01:06,344 --> 00:01:10,649 connective. It takes one proposition, for instance, 13 00:01:10,649 --> 00:01:20,333 the proposition I am drinking coffee. And creates another proposition for 14 00:01:20,333 --> 00:01:25,220 instance the proposition I am not drinking coffee.. 15 00:01:25,220 --> 00:01:32,295 [SOUND] Let's consider the truth table for this true functional connective. 16 00:01:32,295 --> 00:01:39,850 If it's true that I'm drinking coffee, then it's going to be false that I am not 17 00:01:39,850 --> 00:01:45,761 drinking coffee. And if it's false that I'm drinking 18 00:01:45,761 --> 00:01:52,396 coffee, then it's going to be true that I'm not drinking coffee. 19 00:01:52,396 --> 00:02:00,469 In general, for any proposition P, whenever P is true, then the negation of 20 00:02:00,469 --> 00:02:07,776 P, call it not P is going to be false. And whenever P is false, then the 21 00:02:07,776 --> 00:02:16,265 negation of P, not P is going to be true. We can abbreviate the negation of P by 22 00:02:16,265 --> 00:02:23,557 using this symbol. That stands for negation and that's the 23 00:02:23,557 --> 00:02:28,366 negation of P. Sometimes, it's not obvious how to negate 24 00:02:28,366 --> 00:02:32,912 a proposition. For instance, consider the proposition, 25 00:02:32,912 --> 00:02:39,557 Walter has stopped beating his dogs. What's the negation of that proposition? 26 00:02:39,557 --> 00:02:43,580 Is it Walter has not stopped beating his dogs? 27 00:02:43,580 --> 00:02:48,574 Well, let's see. If it's true that Walter has stopped 28 00:02:48,574 --> 00:02:53,760 beating his dogs, then it's going to be false that Walter 29 00:02:53,760 --> 00:02:58,196 has not stopped beating his dogs. So far, so good. 30 00:02:58,196 --> 00:03:03,937 But now, what if it's false that Walter has stopped beating his dogs? 31 00:03:03,937 --> 00:03:10,607 Does that mean that it's going to be true that Walter has not stopped beating his 32 00:03:10,607 --> 00:03:12,801 dogs? Well, not necessarily. 33 00:03:12,801 --> 00:03:19,296 Suppose the reason is false that Walter has stopped beating his dogs, is that 34 00:03:19,296 --> 00:03:25,622 Walter doesn't have any dogs or suppose it's false that Walter has stopped 35 00:03:25,622 --> 00:03:30,346 beating his dogs because Walter never did beat his dogs. 36 00:03:30,346 --> 00:03:36,926 In either of those two cases, it would not be true that Walter has not stopped 37 00:03:36,926 --> 00:03:41,529 beating his dogs. So Walter has not stopped beating his 38 00:03:41,529 --> 00:03:46,868 dogs is not the negation of Walter has stopped beating his dogs. 39 00:03:46,868 --> 00:03:53,291 See, if it were the negation of Walter has stopped beating his dogs then we'd 40 00:03:53,291 --> 00:03:58,129 have the truth table for negation right here, but, we don't. 41 00:03:58,129 --> 00:04:04,719 Since it could be false that Walter has stopped beating his dogs, but still, not 42 00:04:04,719 --> 00:04:08,890 true that Walter has not stopped beating his dogs. 43 00:04:08,890 --> 00:04:16,593 So this is not the negation of this. In English, the only phrase I know of 44 00:04:16,593 --> 00:04:21,504 that reliably expresses negations is the phrase it is not the case that.... 45 00:04:21,504 --> 00:04:26,347 So if you want to negate some proposition, for instance the proposition 46 00:04:26,347 --> 00:04:32,531 that Walter has stopped beating his dogs. You just apend it is not the case that, 47 00:04:32,531 --> 00:04:35,921 to the beginning of it nd you can reliably negate it. 48 00:04:35,921 --> 00:04:41,123 So it is not the case that Walter has stopped beating his dogs is the negation 49 00:04:41,123 --> 00:04:45,996 of Walter has stopped beating his dogs, because whether Walter has not stopped 50 00:04:45,996 --> 00:04:51,066 beating his dogs, or whether Walter never did beat his dogs, or whether Walter 51 00:04:51,066 --> 00:04:55,873 doesn't even have any dogs, it's still going to be true that it is not the case 52 00:04:55,873 --> 00:05:01,273 that Walter has stopped beating his dogs. So remember, the truth table for negation 53 00:05:01,273 --> 00:05:05,751 show us what it is the negation does. Negation takes a proposition, 54 00:05:05,751 --> 00:05:09,356 call it P, and creates a new proposition, the 55 00:05:09,356 --> 00:05:14,424 negation of P. That's false when P is true and that's 56 00:05:14,424 --> 00:05:23,592 true when P is false, but what is the negation of the negation 57 00:05:23,592 --> 00:05:28,918 of P? Well, applying negation to the negation 58 00:05:28,918 --> 00:05:36,281 of P, we created a new proposition. That's true whenever the negation of P is 59 00:05:36,281 --> 00:05:42,420 false and that's false whenever the negation of P is true. 60 00:05:42,420 --> 00:05:48,578 And notice what this means. This means that the negation of the 61 00:05:48,578 --> 00:05:56,692 negation of P is going to be true when P is true and it's going to be false when P 62 00:05:56,692 --> 00:06:01,384 is false. In other words, double negation cancels 63 00:06:01,384 --> 00:06:05,333 itself out. I just showed you how negation can 64 00:06:05,333 --> 00:06:10,987 combine with itself to cancel itself out. Let me now talk about how negation can 65 00:06:10,987 --> 00:06:14,900 combine with other truth functional connectives. 66 00:06:14,900 --> 00:06:18,532 Remember, this is the truth table for conjunction. 67 00:06:18,532 --> 00:06:24,759 If you want to conjoin two propositions P and Q into the new proposition P & Q, we 68 00:06:24,759 --> 00:06:29,726 end up with this truth table. When P is true and Q is true, then the 69 00:06:29,726 --> 00:06:34,915 conjunction of P & Q is true. When P is true and Q is false, then the 70 00:06:34,915 --> 00:06:40,178 conjunction of P & Q is false. When P is false and Q is true, then the 71 00:06:40,178 --> 00:06:45,590 conjunction of P & Q is false. And when both P and Q are false, then the 72 00:06:45,590 --> 00:06:50,602 conjunction P & Q is false. So this is the truth table for 73 00:06:50,602 --> 00:06:57,388 conjunction. But now, suppose we apply negation to the 74 00:06:57,388 --> 00:07:05,438 conjunction we end up with this, the negation of the conjunction of P & Q. 75 00:07:05,438 --> 00:07:11,578 That's going to be false when the conjunction of P & Q is true and it's 76 00:07:11,578 --> 00:07:16,680 going to be true whenever the conjunction of P & Q is false. 77 00:07:16,680 --> 00:07:22,054 Remember, that's what negation does. It flips the truth values of whatever 78 00:07:22,054 --> 00:07:26,912 proposition it's negating. Okay now, notice that in the truth table 79 00:07:26,912 --> 00:07:33,096 that I just put on the board, I had the negation of P & Q where the conjunction P 80 00:07:33,096 --> 00:07:37,734 & Q is in parentheses. Now why did I use those parentheses to 81 00:07:37,734 --> 00:07:42,520 isolate the conjunction of P & Q and then apply negation to it? 82 00:07:42,520 --> 00:07:47,421 Why didn't I just write it this way, the negation of P & Q? 83 00:07:47,421 --> 00:07:52,538 Well, the reason I didn't do this is because these two expressions mean 84 00:07:52,538 --> 00:07:56,719 different things. This truth table shows you how those two 85 00:07:56,719 --> 00:08:02,290 expressions mean different things. Alright. Here is P and Q and here is the 86 00:08:02,290 --> 00:08:08,289 truth table for their conjunction, P & Q. Okay? Now, the negation of the 87 00:08:08,289 --> 00:08:15,848 conjunction of P & Q is going to be false whenever P is true and Q is true and it's 88 00:08:15,848 --> 00:08:19,540 going to be true in every other situation. 89 00:08:21,180 --> 00:08:28,035 In contrast, the negation of P & Q is going to be false when P and Q are both 90 00:08:28,035 --> 00:08:34,259 true, it's going to be false when P is true and Q is false, it's going to be 91 00:08:34,259 --> 00:08:41,114 true when P is false and Q is true, and it's going to be false when P and Q are 92 00:08:41,114 --> 00:08:45,804 both false. The only situation where the negation of 93 00:08:45,804 --> 00:08:52,208 P & Q is going to be true is the situation in which P is false and Q is 94 00:08:52,208 --> 00:08:58,252 true. Since the truth table for not P & Q is 95 00:08:58,252 --> 00:09:04,090 different from the truth table for not P & Q. 96 00:09:04,090 --> 00:09:09,952 It follows that not P & Q must mean something different from not P & Q. 97 00:09:09,952 --> 00:09:15,112 Let me give you an example. Suppose the proposition P is Walter is 98 00:09:15,112 --> 00:09:19,646 busy working and the proposition Q is Rom is busy working. 99 00:09:19,646 --> 00:09:25,743 Well then, the conjunction of those two is going to be Walter and Ram are both 100 00:09:25,743 --> 00:09:29,333 busy working. Now, the negation of that conjunction is 101 00:09:29,333 --> 00:09:33,901 going to say it's not the case that Walter and Rahm are both busy working. 102 00:09:33,901 --> 00:09:38,726 Now, that statement could be true if Walter's not busy working, and it could 103 00:09:38,726 --> 00:09:43,680 be true if Rom is not busy working, and it could be true if both of them are 104 00:09:43,680 --> 00:09:47,219 not busy working. Now, that's a very different statement 105 00:09:47,219 --> 00:09:52,495 from the statement it's not the case that Walter is busy working and Rom is busy 106 00:09:52,495 --> 00:09:55,567 working. And that statement could not be true if 107 00:09:55,567 --> 00:09:59,788 Ram is not busy working, because part of what you're saying there 108 00:09:59,788 --> 00:10:04,839 is Ram is busy working and Walter isn't. That's a very different statement from 109 00:10:04,839 --> 00:10:09,060 saying it's not the case that Walter and Ram are both busy working. 110 00:10:09,060 --> 00:10:15,746 So, there's an example to illustrate the difference between negating P and then 111 00:10:15,746 --> 00:10:21,567 conjoining that with Q on the one hand or negating the conjunction of P & Q on the 112 00:10:21,567 --> 00:10:25,579 other hand. Now, just as you can apply negation to a 113 00:10:25,579 --> 00:10:31,558 conjunction of propositions, you can also apply negation to a disjunction of 114 00:10:31,558 --> 00:10:35,334 propositions. Now remember how disjunction works, 115 00:10:35,334 --> 00:10:39,110 if you have a proposition P and a proposition Q, 116 00:10:39,110 --> 00:10:46,062 the disjunction of those two propositions P or Q is going to be true whenever P and 117 00:10:46,062 --> 00:10:50,508 Q are both true. Its going to be true when P is true and Q 118 00:10:50,508 --> 00:10:54,308 is false. It's going to be true when P is false and 119 00:10:54,308 --> 00:10:59,240 Q is true and its going to be false when P and Q are both false. 120 00:10:59,240 --> 00:11:03,101 So that's the truth table for disjunction. 121 00:11:03,101 --> 00:11:07,790 Now, suppose you apply negation to this disjunction, 122 00:11:07,790 --> 00:11:15,117 well then, you get this truth table. The negation of P or Q is going to be 123 00:11:15,117 --> 00:11:22,654 false when P and Q are both true. It's going to be false when P is true and 124 00:11:22,654 --> 00:11:27,574 Q is false. It's going to be false when P is false 125 00:11:27,574 --> 00:11:31,827 and Q is true. And the only situation in which it's 126 00:11:31,827 --> 00:11:36,800 going to be true is the situation in which P and Q are both false. 127 00:11:36,800 --> 00:11:42,753 Now, you may have noticed, that on that last truth table, I wrote the negation of 128 00:11:42,753 --> 00:11:45,993 P or Q where the P or Q was in parentheses. 129 00:11:45,993 --> 00:11:51,493 And you might wonder, well, why didn't I just write the negation of P or Q? 130 00:11:51,493 --> 00:11:56,843 Well again, it's because these two expression mean two different things. 131 00:11:56,843 --> 00:12:02,570 Here's a truth table that shows how those two expressions differ in meaning. 132 00:12:02,570 --> 00:12:07,031 So, the negation of the disjunction P or Q, 133 00:12:07,031 --> 00:12:11,979 that's going to be false in the situation when P and Q are both true. 134 00:12:11,979 --> 00:12:17,405 It's going to be false in the situation where P is true and Q is false. 135 00:12:17,405 --> 00:12:22,831 It's going to be false in the situation where P is false and Q is true. 136 00:12:22,831 --> 00:12:28,417 And the only situation where that proposition is going to be true is the 137 00:12:28,417 --> 00:12:35,573 situation where P and Q are both false. In contrast, the proposition the negation 138 00:12:35,573 --> 00:12:40,898 of P or Q is going to be true when P and Q are both true. 139 00:12:40,898 --> 00:12:45,828 It's going to be false when P is true and Q is false. 140 00:12:45,828 --> 00:12:53,421 It's going to be true when P is false and Q is true and it's going to be true when 141 00:12:53,421 --> 00:12:59,097 P and Q are both false. So, this proposition and this proposition 142 00:12:59,097 --> 00:13:05,082 are true in different situations and so they must mean different things. 143 00:13:05,082 --> 00:13:08,573 Here's an example to illustrate the point. 144 00:13:08,573 --> 00:13:14,474 Again, suppose that the proposition P is Walter is busy working and the 145 00:13:14,474 --> 00:13:20,693 proposition Q is Rom is busy working. Well, then, disjunction P or Q is 146 00:13:20,693 --> 00:13:24,006 going to be Walter or Ram is busy working. 147 00:13:24,006 --> 00:13:30,803 And the negation of that disjunction is going to be it's not the case that Walter 148 00:13:30,803 --> 00:13:38,280 or Rom is busy working or in other words, neither Walter nor Rom is busy working. 149 00:13:38,280 --> 00:13:44,393 That's what this says. This says neither Walter nor Rom is busy 150 00:13:44,393 --> 00:13:49,148 working, but, the disjunction of not P and Q says, 151 00:13:49,148 --> 00:13:56,757 Walter is not busy working or Rom is busy working and that means something very 152 00:13:56,757 --> 00:14:00,383 different from neither Walter nor Rom is busy working. 153 00:14:00,383 --> 00:14:05,723 So that's an example to illustrate how this proposition and this proposition are 154 00:14:05,723 --> 00:14:13,145 going to mean two very different things. A moment ago, I showed you the truth 155 00:14:13,145 --> 00:14:18,660 table for the disjunction of the negation of P and Q. 156 00:14:18,660 --> 00:14:24,324 Alright? That disjunction is going to be true whenever P and Q are both true and 157 00:14:24,324 --> 00:14:27,541 it's also going to be true whenever P is false. 158 00:14:27,541 --> 00:14:32,716 The only situation in which that disjunction is going to be false is the 159 00:14:32,716 --> 00:14:35,793 situation in which P is true and Q is false. 160 00:14:35,793 --> 00:14:41,178 Now, I want you to remember this truth table, because it's going to be relevant 161 00:14:41,178 --> 00:14:43,696 in our next lecture on conditionals.