1 00:00:04,640 --> 00:00:11,450 In the last lecture, I said that, we were going to spend this unit learning the 2 00:00:11,450 --> 00:00:17,706 rules for evaluating deductive arguments. Now this week, we're going to learn the 3 00:00:17,706 --> 00:00:23,207 rules for evaluating deductive arguments that involve, what I'm going to call 4 00:00:23,207 --> 00:00:28,850 propositional connectives. Now, what are propositional connectives? 5 00:00:28,850 --> 00:00:33,587 Well, in order to explain what propositional connectives are. 6 00:00:33,587 --> 00:00:38,877 I first have to tell you what propositions are and then I can talk 7 00:00:38,877 --> 00:00:42,035 about connectives. So first, propositions. 8 00:00:42,035 --> 00:00:46,851 What's a proposition? A proposition, is the kind of thing that 9 00:00:46,851 --> 00:00:53,009 can be true, or false and that can serve as the premise, or the conclusion of an 10 00:00:53,009 --> 00:00:56,440 argument. Here let me give you an example. 11 00:00:58,620 --> 00:01:06,320 See this book this book is not a proposition. 12 00:01:06,320 --> 00:01:12,503 It can't be true or false, and it can't serve as the premise or the conclusion of 13 00:01:12,503 --> 00:01:17,319 an argument. See this hand, this hand is not a 14 00:01:17,319 --> 00:01:25,545 proposition it can be true or false, and it can serve as the premise or the 15 00:01:25,545 --> 00:01:30,700 conclusion of a argument. But now suppose I say, 16 00:01:31,720 --> 00:01:39,337 the book is in my hand. Now what I just said, that the book is in 17 00:01:39,337 --> 00:01:46,483 my hand is a proposition, it can be true in fact, it is true, or it can be false 18 00:01:46,483 --> 00:01:52,234 and right now it is false. It can also serve as the premise of an 19 00:01:52,234 --> 00:01:56,127 argument. I could say, the book is in my hand, 20 00:01:56,127 --> 00:02:00,108 therefore, my hand is not free to shake yours. 21 00:02:00,108 --> 00:02:04,709 And it could serve as the conclusion of an argument. 22 00:02:04,709 --> 00:02:10,460 I could say, you just gave me the book and I haven't let go of it, 23 00:02:10,460 --> 00:02:16,365 therefore, the book is in my hand. I just told you what propositions are. 24 00:02:16,365 --> 00:02:22,494 But what's a propositional connective? A propositional connective is something 25 00:02:22,494 --> 00:02:27,288 that takes propositions and makes new propositions out of it. 26 00:02:27,288 --> 00:02:33,731 Let me give you an example to illustrate. Consider a proposition, the book is under 27 00:02:33,731 --> 00:02:38,348 my hand. Now consider the proposition, my foot is 28 00:02:38,348 --> 00:02:43,181 under the book. We can combine those two propositions 29 00:02:43,181 --> 00:02:49,292 using the propositional connective end to make the new proposition. 30 00:02:49,292 --> 00:02:54,780 The book is under my hand, and by foot is under the book. 31 00:02:54,780 --> 00:03:00,380 Now what I've just said that the book is under my hand and my foot is under the 32 00:03:00,380 --> 00:03:05,140 book, that's a proposition. It's the kind of thing that could be true 33 00:03:05,140 --> 00:03:08,360 or false. For instance, right now it's true and 34 00:03:08,360 --> 00:03:12,630 right now it's false. It's also the kind of thing that can be 35 00:03:12,630 --> 00:03:15,570 premise or the conclusion of the argument. 36 00:03:15,570 --> 00:03:20,750 For instance, I could say, what you're seeing right now is really happening, 37 00:03:20,750 --> 00:03:25,370 therefore, the book is under my hand and my foot is under the book. 38 00:03:25,370 --> 00:03:33,079 Or I could say, the book is under my hand, and my foot is under the book. 39 00:03:33,079 --> 00:03:38,880 Therefore, my foot is under my hand. So you see that the book is under my 40 00:03:38,880 --> 00:03:43,094 hand, and my foot is under the book. That's the kind of thing that can be true 41 00:03:43,094 --> 00:03:46,465 or false and it's the kind of thing that can be the premise, 42 00:03:46,465 --> 00:03:50,396 or the conclusion of an argument. So it's a proposition, 43 00:03:50,396 --> 00:03:56,342 but it's a proposition that we made by combining two other propositions. 44 00:03:56,342 --> 00:04:02,360 See how propositional connectives work? They're beautiful, aren't they? 45 00:04:02,360 --> 00:04:06,889 I just gave you an example of a propositional connective. 46 00:04:06,889 --> 00:04:10,306 I called it the proposition connective and. 47 00:04:10,306 --> 00:04:14,915 But in English, the word and can be used in different ways. 48 00:04:14,915 --> 00:04:18,809 It's not always used as a proposition connective. 49 00:04:18,809 --> 00:04:23,418 Let me give you different examples of how and can be used. 50 00:04:23,418 --> 00:04:27,710 Think about the sentence, Jack and Jill finally talked. 51 00:04:27,710 --> 00:04:33,883 Okay now, there are three different ways to understand what that sentence is 52 00:04:33,883 --> 00:04:37,863 saying. Jack and Jill could be the name of a fast 53 00:04:37,863 --> 00:04:44,117 food company that serves a special stew that's very popular with its patrons. 54 00:04:44,117 --> 00:04:50,209 Now, maybe lawyers have been wondering what the ingredients are in Jack and 55 00:04:50,209 --> 00:04:56,870 Jill's special stew because many of Jack and Jill's patrons have been coming down 56 00:04:56,870 --> 00:05:01,640 with an unusual disease. So, lawyers have been asking Jack and 57 00:05:01,640 --> 00:05:07,663 Jill to disclose what's in their stew. Jack and Jill has been refusing to do so. 58 00:05:07,663 --> 00:05:13,380 But finally, the spokesman for Jack and Jill discloses what's in their stew. 59 00:05:13,380 --> 00:05:19,250 I might tell you about that situation by saying Jack and Jill finally talked. 60 00:05:19,250 --> 00:05:24,816 There, I'm telling you there's a particular company, a company called Jack 61 00:05:24,816 --> 00:05:30,000 and Jill, and that company finally talked through their spokesperson. 62 00:05:31,160 --> 00:05:36,321 Here's a second way to understand the sentence, Jack and Jill finally talked. 63 00:05:36,321 --> 00:05:41,890 Suppose that Jack and Jill are a couple, and recently they've been going through a 64 00:05:41,890 --> 00:05:45,421 tough time. They've been angry and resentful towards 65 00:05:45,421 --> 00:05:49,021 each other. And they haven't talked about what's been 66 00:05:49,021 --> 00:05:52,618 bothering them. Now, I might tell you, Jack and Jill 67 00:05:52,618 --> 00:05:56,450 finally talked. And what I mean by that is, that they 68 00:05:56,450 --> 00:06:00,650 finally talked to each other about what's bothering them. 69 00:06:00,650 --> 00:06:07,375 Now that's different, from the first example in which I said, Jack and Jill 70 00:06:07,375 --> 00:06:12,100 finally talked. There, I was saying that a particular 71 00:06:12,100 --> 00:06:18,644 thing the company, Jack and Jill. Finally talked about the ingredients in 72 00:06:18,644 --> 00:06:23,102 its special stew. But in this second example, I'm saying 73 00:06:23,102 --> 00:06:27,485 that two things, Jack and Jill finally talked to each other. 74 00:06:27,485 --> 00:06:33,205 Let me give you a third way to understand the sentence Jack and Jill finally 75 00:06:33,205 --> 00:06:36,473 talked. Imagine that Jack, Jill, and Roger are 76 00:06:36,473 --> 00:06:41,301 having a silence contest. They're having a contest to see who can 77 00:06:41,301 --> 00:06:45,015 go for the longest period of time without talking. 78 00:06:45,015 --> 00:06:52,162 And I'm watching them to see who wins. Well, you call be periodically and ask me 79 00:06:52,162 --> 00:06:57,940 as anyone of them talked? And for the first few hours I might say 80 00:06:57,940 --> 00:07:05,538 to you nope, none of them has talked yet. And then at one point you call me and you 81 00:07:05,538 --> 00:07:11,725 ask me, has anyone of them talked? And I say well Jack and Jill finally 82 00:07:11,725 --> 00:07:16,144 talked. Now here I'm not saying that Jack or Jill 83 00:07:16,144 --> 00:07:22,886 talked to each other. I'm saying that Jack finally talked, 84 00:07:22,886 --> 00:07:30,197 and Jill finally talked. So, when I say Jack and Jill finally 85 00:07:30,197 --> 00:07:35,044 talked. I'm expressing a proposition that's made 86 00:07:35,044 --> 00:07:41,865 up of two other propositions. The proposition that Jack finally talked, 87 00:07:41,865 --> 00:07:46,250 and the proposition that Jill finally talked. 88 00:07:46,250 --> 00:07:52,770 When I say Jack and Jill finally talked, I'm using the word and, as a 89 00:07:52,770 --> 00:07:57,660 propositional connective. It takes two propositions. 90 00:07:57,660 --> 00:08:03,364 First, that Jack finally talked, adn second that Jill finally talked. 91 00:08:03,364 --> 00:08:08,813 And it combines those two propositions into a third proposition. 92 00:08:08,813 --> 00:08:13,070 The proposition that, Jack and Jill. Finally talked. 93 00:08:13,070 --> 00:08:18,603 Now that's a third way to understand the sentence Jack and Jill finally talked and 94 00:08:18,603 --> 00:08:23,604 it's different from the first two. But that's the only way of understanding 95 00:08:23,604 --> 00:08:27,671 the word and, so that it's a propositional connective in that 96 00:08:27,671 --> 00:08:30,605 sentence. So I've just explained to you what 97 00:08:30,605 --> 00:08:35,339 propositional connectives are. I gave you an example of a propositional 98 00:08:35,339 --> 00:08:40,339 connective and, and I showed how the English word and, can sometimes be used 99 00:08:40,339 --> 00:08:43,540 as a propositional connective but sometimes not. 100 00:08:43,540 --> 00:08:49,546 Now the English language like every other natural language, contains lots and lots 101 00:08:49,546 --> 00:08:54,454 of different phrases that can be used as propositional connectives. 102 00:08:54,454 --> 00:08:57,970 For instance, consider the phrase, I believe that. 103 00:08:57,970 --> 00:09:04,595 You tell me Jack and Jill finally talked. Well, I might say I believe that Jack and 104 00:09:04,595 --> 00:09:09,281 Jill finally talked. There, I'm taking one proposition Jack 105 00:09:09,281 --> 00:09:15,583 and Jill finally talked, and attaching the phrase to it, I believe that to make 106 00:09:15,583 --> 00:09:21,320 another proposition namely, I believe that Jack and Jill finally talked. 107 00:09:21,320 --> 00:09:27,349 Or consider the phrase, I hate it when. You might tell me, it's raining, and I 108 00:09:27,349 --> 00:09:33,458 might say, I hate it when it's raining. See there, I take the proposition it's 109 00:09:33,458 --> 00:09:38,844 raining, and I attach a phrase to it, I hate it when, to make another 110 00:09:38,844 --> 00:09:44,694 proposition, I hate it when it's raining. So these are other examples of how 111 00:09:44,694 --> 00:09:49,738 phrases in English, I believe that, or I hate it when, can be used as 112 00:09:49,738 --> 00:09:54,856 propositional connectives. They can take other propositions and make 113 00:09:54,856 --> 00:09:59,845 new propositions out of them. But and is a very different kind of 114 00:09:59,845 --> 00:10:04,264 propositional connective from I believe that or I hate it when. 115 00:10:04,264 --> 00:10:09,385 And it's different in two ways. One way it's different is that it makes a 116 00:10:09,385 --> 00:10:13,945 new proposition out of two other propositions, not just one other 117 00:10:13,945 --> 00:10:17,452 proposition. So when you make a new proposition by 118 00:10:17,452 --> 00:10:22,853 attaching I hate it when to another proposition, what you're doing is turning 119 00:10:22,853 --> 00:10:28,609 one proposition into another proposition. But when you make a new proposition by 120 00:10:28,609 --> 00:10:32,447 combining two other propositions using the word and, 121 00:10:32,447 --> 00:10:37,025 you're making a new proposition out of two other propositions. 122 00:10:37,025 --> 00:10:40,273 Jack finally talked, and Jill finally talked. 123 00:10:40,273 --> 00:10:44,555 You make the new proposition, Jack and Jill finally talked. 124 00:10:44,555 --> 00:10:50,387 That's one way in which the propositional connective and is different from the 125 00:10:50,387 --> 00:10:56,623 propositional connective, I hate it when. But there's a second way, in which the 126 00:10:56,623 --> 00:11:02,255 propositional connective, and, differs from the propositional connective, I hate 127 00:11:02,255 --> 00:11:05,287 it when. And that is, that when you use the 128 00:11:05,287 --> 00:11:10,630 propositional connective and, to combine two other propositions into a new 129 00:11:10,630 --> 00:11:15,973 proposition. Whether that proposition is true or false doesn't depend upon 130 00:11:15,973 --> 00:11:21,388 anything other than whether the two original propositions that you used to 131 00:11:21,388 --> 00:11:25,760 build it were true or false, that's all it depends upon. 132 00:11:25,760 --> 00:11:31,403 Let me illustrate this point using something we call a truth table. 133 00:11:31,403 --> 00:11:37,046 A truth table is a way of representing various possible situations, 134 00:11:37,046 --> 00:11:43,027 and how the truth of a proposition depends upon which of those various 135 00:11:43,027 --> 00:11:47,912 possible situations is real. For instance, consider the two 136 00:11:47,912 --> 00:11:52,039 possibilities there are with Jack finally talked. 137 00:11:52,039 --> 00:11:56,251 The proposition Jack finally taught, could be true. 138 00:11:56,251 --> 00:12:01,674 Or it could be false. The proposition Jill finally talked could 139 00:12:01,674 --> 00:12:07,482 be true or it could be false. So, there are four possibilities we have 140 00:12:07,482 --> 00:12:11,806 to consider. Either Jack finally talked is true and 141 00:12:11,806 --> 00:12:17,572 Jill finally talked is true. Or Jack finally talked is true and Jill 142 00:12:17,572 --> 00:12:23,083 finally talked is false. Or Jack finally talked is false and Jill 143 00:12:23,083 --> 00:12:28,425 finally talked is true. Or finally, Jack finally talked is false 144 00:12:28,425 --> 00:12:34,530 and Jill finally talked is false. Those are the four possible situations. 145 00:12:34,530 --> 00:12:37,592 Now, if the first of those four situations is 146 00:12:37,592 --> 00:12:42,637 the real situation, so it's true that Jack finally talked and it's true that 147 00:12:42,637 --> 00:12:47,484 Jill finally talked, then is it going to be true or false that Jack and Jill 148 00:12:47,484 --> 00:12:50,927 finally talked? It's going to be true. 149 00:12:50,927 --> 00:12:58,221 All right, because Jack finally talked, and Jill finally talked, so Jack and Jill 150 00:12:58,221 --> 00:13:02,988 finally talked. Now suppose it's true that Jack finally 151 00:13:02,988 --> 00:13:06,405 talked, but it's false that Jill finally talked. 152 00:13:06,405 --> 00:13:11,349 Then is it going to be true or false that Jack and Jill finally talked? 153 00:13:11,349 --> 00:13:16,513 It'll be false, and suppose it's false that Jack finally 154 00:13:16,513 --> 00:13:20,097 talked, but it's true that Jill finally talked. 155 00:13:20,097 --> 00:13:25,394 Then, is it going to be true or false that Jack and Jill finally talked, 156 00:13:25,394 --> 00:13:30,439 again it's going to be false. And finally, suppose it's false that Jack 157 00:13:30,439 --> 00:13:34,716 finally talked and it's also false that Jill finally talked. 158 00:13:34,716 --> 00:13:38,994 Then is it going to be true or false that Jack and Jill finally talked? 159 00:13:38,994 --> 00:13:44,026 Well obviously in that situation, it's going to be false that Jack and Jill 160 00:13:44,026 --> 00:13:48,596 finally talked. So what this truth table demonstrates is 161 00:13:48,596 --> 00:13:54,170 that the truth of the proposition Jack and Jill finally talked. 162 00:13:54,170 --> 00:13:59,692 Just depends, it depends on nothing other than the truth of the proposition Jack 163 00:13:59,692 --> 00:14:04,318 finally talked, and the truth of the proposition Jill finally talked. 164 00:14:04,318 --> 00:14:08,667 In other words, the truth of the proposition that we've use the 165 00:14:08,667 --> 00:14:14,005 propositional connective and to build. Depends on nothing other than the truth 166 00:14:14,005 --> 00:14:18,570 of the two ingredient propositions that we connected by means of the 167 00:14:18,570 --> 00:14:23,200 propositional connective and. Because the propositional connective and, 168 00:14:23,200 --> 00:14:28,028 works that way because it builds new propositions whose truth depends on 169 00:14:28,028 --> 00:14:32,857 nothing other than the truth of the ingredient propositions that go into 170 00:14:32,857 --> 00:14:36,495 building them. That kind of propositional connective is 171 00:14:36,495 --> 00:14:40,480 one that we're going to call a truth functional connective. 172 00:14:40,480 --> 00:14:46,315 Now and, the propositional connective and, is a truth-functional connective. 173 00:14:46,315 --> 00:14:51,914 But not all propositional connectives are truth-functional connectives. 174 00:14:51,914 --> 00:14:58,065 For instance, suppose we try to construct a truth table for I hate it when, the 175 00:14:58,065 --> 00:15:04,728 propositional connective I hate it when. Well, so consider the proposition, it's 176 00:15:04,728 --> 00:15:08,898 raining. Now that proposition could be true or it 177 00:15:08,898 --> 00:15:13,493 could be false. Sometimes its true, sometimes its false. 178 00:15:13,493 --> 00:15:17,492 So lets consider these two possible situations. 179 00:15:17,492 --> 00:15:21,492 So suppose the proposition Its raining is true. 180 00:15:21,492 --> 00:15:29,180 In that situation is it going to be true or false that I hated when its raining. 181 00:15:29,180 --> 00:15:35,491 Could be either one, it could be raining even though I enjoyed the rain or it 182 00:15:35,491 --> 00:15:39,500 could be raining even though I hate the rain. 183 00:15:39,500 --> 00:15:44,092 Or it could be raining even though I'm indifferent to the rain. 184 00:15:44,092 --> 00:15:52,080 So the truth, of I hate it when it's raining isn't determined by its raining. 185 00:15:52,080 --> 00:15:59,371 So, if it's true that it's raining. It's unclear whether, I hate it when it's 186 00:15:59,371 --> 00:16:01,133 raining. Could be true, 187 00:16:01,133 --> 00:16:05,640 could be false. Suppose it's false, that it's raining. 188 00:16:05,640 --> 00:16:10,674 Then, is it going to be true or false that I hate it when it's raining. 189 00:16:10,674 --> 00:16:15,709 Again, could be either one. The truth of I hate it when it's raining 190 00:16:15,709 --> 00:16:19,466 isn't determined by the falsehood of it's raining. 191 00:16:19,466 --> 00:16:24,726 So, even if it's not raining, that doesn't mean anything one way or the 192 00:16:24,726 --> 00:16:28,183 other for whether I hate it when it's raining. 193 00:16:28,183 --> 00:16:31,640 So once again, if it's false that it's raining, 194 00:16:32,880 --> 00:16:37,896 I hate it when it's raining, could be true or could be false. 195 00:16:37,896 --> 00:16:41,789 So the proposition, I hated when it's raining. 196 00:16:41,789 --> 00:16:48,881 Whether that proposition is true or false doesn't just depend on the truth or the 197 00:16:48,881 --> 00:16:55,282 falsehood of the proposition it's raining, that you built this proposition 198 00:16:55,282 --> 00:17:00,460 out of using the propositional connective, I hate it when. 199 00:17:00,460 --> 00:17:06,073 Because of that, the propositional connective I hate it when is not a 200 00:17:06,073 --> 00:17:11,442 truth-functional connective. It's different from the propositional 201 00:17:11,442 --> 00:17:15,835 connective and which is a truth-functional connective. 202 00:17:15,835 --> 00:17:22,099 A moment ago, we built a truth table for a proposition that was built using a 203 00:17:22,099 --> 00:17:28,363 truth-functional connective, specifically the truth-functional connective and. 204 00:17:28,363 --> 00:17:33,163 But I'd like us to notice something about that truth table. 205 00:17:33,163 --> 00:17:39,832 Notice that if we replace the particular propositions that we are putting together 206 00:17:39,832 --> 00:17:45,604 using the truth-functional connective and, to make a different resultant 207 00:17:45,604 --> 00:17:49,212 proposition. Even if we change the ingredient 208 00:17:49,212 --> 00:17:52,740 propositions the truth table looks the same. 209 00:17:52,740 --> 00:17:57,710 Let me show you what I mean. Suppose instead of having Jack and Jill 210 00:17:57,710 --> 00:18:01,512 finally talked. We have Jack finally walked and Jill 211 00:18:01,512 --> 00:18:05,386 finally talked. So now, we're connecting two different 212 00:18:05,386 --> 00:18:09,407 propositions using the truth-functional connective and. 213 00:18:09,407 --> 00:18:13,501 There's Jack finally walked, there's Jill finally talked. 214 00:18:13,501 --> 00:18:19,130 And then we connect them up into Jack finally walked and Jill finally talked. 215 00:18:19,130 --> 00:18:21,832 Okay, now the truth of that resultant 216 00:18:21,832 --> 00:18:25,777 proposition, Jack finally walked and Jill finally talked. 217 00:18:25,777 --> 00:18:30,599 How does that depend on the truth or false sort of the ingredient 218 00:18:30,599 --> 00:18:34,324 propositions?Jack Jack finally walked and Jill finally talked. 219 00:18:34,324 --> 00:18:40,083 Well it's the same patent we saw earlier. If it's true that Jack finally walked, 220 00:18:40,083 --> 00:18:42,774 and it's also true that Jill finally talked. 221 00:18:42,774 --> 00:18:49,029 and it's also true that Jill finally talked. then it's going to be true that 222 00:18:49,029 --> 00:18:52,230 Jack finally walked and Jill finally talked. 223 00:18:52,230 --> 00:18:57,385 If it's true that Jack finally walked, but it's false that Jill finally talked, 224 00:18:57,385 --> 00:19:02,276 then it's going to be false that Jack finally walked and Jill finally talked. 225 00:19:02,276 --> 00:19:07,431 If it's false that Jack finally walked but it's true that Jill finally talked, 226 00:19:07,431 --> 00:19:12,323 then it's going to be false that Jack finally walked and Jill finally talked. 227 00:19:12,323 --> 00:19:17,346 And if it's false that Jack finally walked and it's false that Jill finally 228 00:19:17,346 --> 00:19:22,237 talked, then it's going to be false that Jack finally walked and Jill finally 229 00:19:22,237 --> 00:19:25,698 talked. So even if we change one of the 230 00:19:25,698 --> 00:19:31,690 ingredient propositions as long as we're combining propositions using the truth 231 00:19:31,690 --> 00:19:36,440 functional connective and, the overall truth table looks the same. 232 00:19:38,060 --> 00:19:43,736 We could change them some more to illustrate this point. I suppose if we 233 00:19:43,736 --> 00:19:47,520 changed Jill finally talked to the zebra escaped. 234 00:19:59,693 --> 00:20:09,102 Change it here, so notice what we have here. 235 00:20:09,102 --> 00:20:15,205 We take two propositions, the proposition Jack finally walked, 236 00:20:15,205 --> 00:20:23,140 and the proposition the zebra escaped. And we put them together with the truth 237 00:20:23,140 --> 00:20:29,244 functional connective, and to create a resultant proposition 238 00:20:29,244 --> 00:20:33,516 Jack finally walked and the zebra escaped. 239 00:20:33,516 --> 00:20:37,891 Now, when is that proposition going to be true? 240 00:20:37,891 --> 00:20:43,648 Well, again it depends just on, when these propositions are true. 241 00:20:43,648 --> 00:20:48,952 So if it's true, that Jack finally walked, and it's also true that the zebra 242 00:20:48,952 --> 00:20:54,184 escaped, then it's going to be true that Jack finally walked and the zebra 243 00:20:54,184 --> 00:20:57,062 escaped. If it's true that Jack finally walked, 244 00:20:57,062 --> 00:21:01,505 but it's false that the zebra escaped, then it's going to be false that Jack 245 00:21:01,505 --> 00:21:06,130 finally walked and the zebra escaped. If it's false that Jack finally walked 246 00:21:06,130 --> 00:21:10,511 and it's true that the zebra escaped, then it's going to be false that Jack 247 00:21:10,511 --> 00:21:14,954 finally walked and the zebra escaped. And finally, if it's false that Jack 248 00:21:14,954 --> 00:21:19,396 finally walked and it's false that the zebra escaped, then of course it's going 249 00:21:19,396 --> 00:21:23,170 to be false that Jack finally walked and the zebra escaped. 250 00:21:23,170 --> 00:21:28,803 So once again, same truth table even if we change the ingredient propositions 251 00:21:28,803 --> 00:21:34,290 that we're putting together with the functional connective and to make the 252 00:21:34,290 --> 00:21:39,994 resultive proposition. Now, since the truth table stays the same 253 00:21:39,994 --> 00:21:44,552 even when we change these propositions up on top. 254 00:21:44,552 --> 00:21:51,434 We could represent that fact, by replacing these propositions altogether 255 00:21:51,434 --> 00:21:56,413 with variables that can range over any proposition. 256 00:21:56,413 --> 00:22:03,735 So, for instance, instead of saying Jack finally walked, we could just have a 257 00:22:03,735 --> 00:22:08,323 variable here call it P1 Our first proposition. 258 00:22:08,323 --> 00:22:16,133 Instead of saying the zebra escaped we can have a variable there, call it P2 our 259 00:22:16,133 --> 00:22:17,988 second proposition. And finally, when we put those two 260 00:22:17,988 --> 00:22:28,397 propositions together using the truth functional connective and. 261 00:22:28,397 --> 00:22:31,552 We'll have P1 one and. P2. 262 00:22:31,552 --> 00:22:38,750 So that's going to be our resultant proposition, P1 and P2. 263 00:22:38,750 --> 00:22:46,085 And whatever exactly that is, is going to depend of course on what P1 is and what 264 00:22:46,085 --> 00:22:50,460 P2 is. But whether this third proposition is 265 00:22:50,460 --> 00:22:57,780 true or false again is only going to depend on the truth or falsehood of P1 266 00:22:57,780 --> 00:23:04,900 and of P2, when P1 whatever exactly that is, is true and P2 is true then. 267 00:23:04,900 --> 00:23:09,512 The proposition p1 and p2 is going to be true. 268 00:23:09,512 --> 00:23:14,426 Whatever proposition that is, is going to be true. 269 00:23:14,426 --> 00:23:22,047 And in every other possible situation, that proposition is going to be false. 270 00:23:22,047 --> 00:23:30,370 So no matter what proposition we have for P1 and P2 their conjunction P1 and P2 is 271 00:23:30,370 --> 00:23:37,490 going to be true, just in those situations when P1 and P2 are both true. 272 00:23:37,490 --> 00:23:43,404 That's the truth table for the truth functional connective and, which we'll 273 00:23:43,404 --> 00:23:47,880 also call conjunction. In the next lecture, we're going to see 274 00:23:47,880 --> 00:23:53,285 how we can use the truth table for the truth functional connective and. 275 00:23:53,285 --> 00:23:59,070 To figure out the rules for evaluating deductive arguments that rely on the 276 00:23:59,070 --> 00:24:04,246 truth functional connected and. And in the following three lectures, 277 00:24:04,246 --> 00:24:10,412 we'll see how we can use the truth tables for other truth functional connectives, 278 00:24:10,412 --> 00:24:16,502 to figure out the rules for evaluating deductive arguments that use those other 279 00:24:16,502 --> 00:24:18,938 connectives. See you in next lecture.