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 --> 00:00:18,152 rules for evaluating deductive arguments. Now this week, we're gonna learn the rules 3 00:00:18,152 --> 00:00:24,247 for evaluating deductive arguments that involve, what I'm gonna call propositional 4 00:00:24,247 --> 00:00:30,824 connectives. Now, what are propositional connectives? Well, in order to explain 5 00:00:30,824 --> 00:00:36,982 what propositional connectives are. I first have to tell you what propositions 6 00:00:36,982 --> 00:00:42,745 are and then I can talk about connectives. So first, propositions. What's a 7 00:00:42,745 --> 00:00:49,140 proposition? A proposition, is the kind of thing that can be true, or false and that 8 00:00:49,140 --> 00:00:54,740 can serve as the premise, or the conclusion of an argument. Here let me 9 00:00:54,740 --> 00:01:07,236 give you an example. See this book this book is not a proposition. It can't be 10 00:01:07,236 --> 00:01:13,420 true or false, and it can't serve as the premise or the conclusion of an argument. 11 00:01:13,700 --> 00:01:21,706 See this hand, this hand is not a proposition it can be true or false, and 12 00:01:21,706 --> 00:01:30,041 it can serve as the premise or the conclusion of a argument. But now suppose 13 00:01:30,041 --> 00:01:40,555 I say, the book is in my hand. Now what I just said, that the book is in my hand is 14 00:01:40,555 --> 00:01:47,721 a proposition, it can be true in fact, it is true, or it can be false and right now 15 00:01:47,721 --> 00:01:54,446 it is false. It can also serve as the premise of an argument. I could say, the 16 00:01:54,446 --> 00:02:01,258 book is in my hand, therefore, my hand is not free to shake yours. And it could 17 00:02:01,258 --> 00:02:08,602 serve as the conclusion of an argument. I could say, you just gave me the book and I 18 00:02:08,602 --> 00:02:15,030 haven't let go of it, therefore, the book is in my hand. I just told you what 19 00:02:15,030 --> 00:02:20,609 propositions are. But what's a propositional connective? A propositional 20 00:02:20,609 --> 00:02:26,502 connective is something that takes propositions and makes new propositions 21 00:02:26,502 --> 00:02:32,631 out of it. Let me give you an example to illustrate. Consider a proposition, the 22 00:02:32,631 --> 00:02:39,989 book is under my hand. Now consider the proposition, my foot is under the book. We 23 00:02:39,989 --> 00:02:47,468 can combine those two propositions using the propositional connective end to make 24 00:02:47,468 --> 00:02:55,060 the new proposition. The book is under my hand, and by foot is under the book. Now 25 00:02:55,060 --> 00:03:00,730 what I've just said that the book is under my hand and my foot is under the book, 26 00:03:00,730 --> 00:03:06,050 that's a proposition. It's the kind of thing that could be true or false. For 27 00:03:06,050 --> 00:03:11,370 instance, right now it's true and right now it's false. It's also the kind of 28 00:03:11,370 --> 00:03:16,480 thing that can be premise or the conclusion of the argument. For instance, 29 00:03:16,480 --> 00:03:22,080 I could say, what you're seeing right now is really happening, therefore, the book 30 00:03:22,080 --> 00:03:29,279 is under my hand and my foot is under the book. Or I could say, the book is under my 31 00:03:29,279 --> 00:03:37,700 hand, and my foot is under the book. Therefore, my foot is under my hand. So 32 00:03:37,420 --> 00:03:41,465 you see that the book is under my hand, and my foot is under the book. That's the 33 00:03:41,465 --> 00:03:46,016 kinda thing that can be true or false and it's the kind of thing that can be the 34 00:03:46,016 --> 00:03:51,305 premise, or the conclusion of an argument. So it's a proposition, but it's a 35 00:03:51,305 --> 00:03:58,158 proposition that we made by combining two other propositions. See how propositional 36 00:03:58,158 --> 00:04:04,744 connectives work? They're beautiful, aren't they? I just gave you an example of 37 00:04:04,744 --> 00:04:10,862 a propositional connective. I called it the proposition connective and. But in 38 00:04:10,862 --> 00:04:16,981 English, the word and can be used in different ways. It's not always used as a 39 00:04:16,981 --> 00:04:23,418 proposition connective. Let me give you different examples of how and can be used. 40 00:04:23,418 --> 00:04:29,740 Think about the sentence, Jack and Jill finally talked. Okay now, there are three 41 00:04:29,740 --> 00:04:36,320 different ways to understand what that sentence is saying. Jack and Jill could be 42 00:04:36,320 --> 00:04:42,737 the name of a fast food company that serves a special stew that's very popular 43 00:04:42,737 --> 00:04:49,478 with its patrons. Now, maybe lawyers have been wondering what the ingredients are in 44 00:04:49,478 --> 00:04:55,895 Jack and Jill's special stew because many of Jack and Jill's patrons have been 45 00:04:55,895 --> 00:05:02,250 coming down with an unusual disease. So, lawyers have been asking Jack and Jill to 46 00:05:02,250 --> 00:05:07,968 disclose what's in their stew. Jack and Jill has been refusing to do so. But 47 00:05:07,968 --> 00:05:13,990 finally, the spokesman for Jack and Jill discloses what's in their stew. I might 48 00:05:13,990 --> 00:05:20,013 tell you about that situation by saying Jack and Jill finally talked. There, I'm 49 00:05:20,013 --> 00:05:26,188 telling you there's a particular company, a company called Jack and Jill, and that 50 00:05:26,188 --> 00:05:32,722 company finally talked through their spokesperson. Here's a second way to 51 00:05:32,722 --> 00:05:38,155 understand the sentence, Jack and Jill finally talked. Suppose that Jack and Jill 52 00:05:38,155 --> 00:05:43,520 ar e a couple, and recently they've been going through a tough time. They've been 53 00:05:43,520 --> 00:05:49,021 angry and resentful towards each other. And they haven't talked about what's been 54 00:05:49,021 --> 00:05:54,534 bothering them. Now, I might tell you, Jack and Jill finally talked. And what I 55 00:05:54,534 --> 00:06:00,650 mean by that is, that they finally talked to each other about what's bothering them. 56 00:06:00,650 --> 00:06:07,375 Now that's different, from the first example in which I said, Jack and Jill 57 00:06:07,375 --> 00:06:14,554 finally talked. There, I was saying that a particular thing the company, Jack and 58 00:06:14,554 --> 00:06:21,691 Jill. Finally talked about the ingredients in its special stew. But in this second 59 00:06:21,691 --> 00:06:27,782 example, I'm saying that two things, Jack and Jill finally talked to each other. Let 60 00:06:27,782 --> 00:06:33,724 me give you a third way to understand the sentence Jack and Jill finally talked. 61 00:06:33,724 --> 00:06:39,593 Imagine that Jack, Jill, and Roger are having a silence contest. They're having a 62 00:06:39,593 --> 00:06:45,610 contest to see who can go for the longest period of time without talking. And I'm 63 00:06:45,610 --> 00:06:53,129 watching them to see who wins. Well, you call be periodically and ask me as anyone 64 00:06:53,129 --> 00:06:59,620 of them talked? And for the first few hours I might say to you nope, none of 65 00:06:59,620 --> 00:07:07,394 them has talked yet. And then at one point you call me and you ask me, has anyone of 66 00:07:07,394 --> 00:07:14,553 them talked? And I say well Jack and Jill finally talked. Now here I'm not saying 67 00:07:14,553 --> 00:07:23,373 that Jack or Jill talked to each other. I'm saying that Jack finally talked, and 68 00:07:23,373 --> 00:07:32,706 Jill finally talked. So, when I say Jack and Jill finally talked. I'm expressing a 69 00:07:32,706 --> 00:07:40,403 proposition that's made up of two other propositions. The proposition that Jack 70 00:07:40,403 --> 00:07:48,167 finally talked, and the proposition that Jill finally talked. When I say Jack and 71 00:07:48,167 --> 00:07:55,454 Jill finally talked, I'm using the word and, as a propositional connective. It 72 00:07:55,454 --> 00:08:02,087 takes two propositions. First, that Jack finally talked, adn second that Jill 73 00:08:02,087 --> 00:08:09,153 finally talked. And it combines those two propositions into a third proposition. The 74 00:08:09,153 --> 00:08:14,803 proposition that, Jack and Jill. Finally talked. Now that's a third way to 75 00:08:14,803 --> 00:08:20,204 understand the sentence Jack and Jill finally talked and it's different from the 76 00:08:20,204 --> 00:08:25,471 first two. But that's the only way of understanding the word and, so that it's a 77 00:08:25,471 --> 00:08:30,271 propositional conn ective in that sentence. So I've just explained to you 78 00:08:30,271 --> 00:08:35,339 what propositional connectives are. I gave you an example of a propositional 79 00:08:35,339 --> 00:08:40,539 connective and, and I showed how the English word and, can sometimes be used as 80 00:08:40,539 --> 00:08:46,176 a propositional connective but sometimes not. Now the English language like every 81 00:08:46,176 --> 00:08:52,329 other natural language, contains lots and lots of different phrases that can be used 82 00:08:52,329 --> 00:08:57,603 as propositional connectives. For instance, consider the phrase, I believe 83 00:08:57,603 --> 00:09:03,868 that. You tell me Jack and Jill finally talked. Well, I might say I believe that 84 00:09:03,868 --> 00:09:10,008 Jack and Jill finally talked. There, I'm taking one proposition Jack and Jill 85 00:09:10,008 --> 00:09:16,229 finally talked, and attaching the phrase to it, I believe that to make another 86 00:09:16,229 --> 00:09:22,606 proposition namely, I believe that Jack and Jill finally talked. Or consider the 87 00:09:22,606 --> 00:09:28,956 phrase, I hate it when. You might tell me, it's raining, and I might say, I hate it 88 00:09:28,956 --> 00:09:35,307 when it's raining. See there, I take the proposition it's raining, and I attach a 89 00:09:35,307 --> 00:09:41,416 phrase to it, I hate it when, to make another proposition, I hate it when it's 90 00:09:41,416 --> 00:09:47,479 raining. So these are other examples of how phrases in English, I believe that, or 91 00:09:47,479 --> 00:09:52,749 I hate it when, can be used as propositional connectives. They can take 92 00:09:52,749 --> 00:09:58,231 other propositions and make new propositions out of them. But and is a 93 00:09:58,231 --> 00:10:03,703 very different kind of propositional connective from I believe that or I hate 94 00:10:03,703 --> 00:10:09,385 it when. And it's different in two ways. One way it's different is that it makes a 95 00:10:09,385 --> 00:10:13,945 new proposition out of two other propositions, not just one other 96 00:10:13,945 --> 00:10:19,416 proposition. So when you make a new proposition by attaching I hate it when to 97 00:10:19,416 --> 00:10:24,888 another proposition, what you're doing is turning one proposition into another 98 00:10:24,888 --> 00:10:30,085 proposition. But when you make a new proposition by combining two other 99 00:10:30,085 --> 00:10:36,065 propositions using the word and, you're making a new proposition out of two other 100 00:10:36,065 --> 00:10:41,528 propositions. Jack finally talked, and Jill finally talked. You make the new 101 00:10:41,528 --> 00:10:47,655 proposition, Jack and Jill finally talked. That's one way in which the propositional 102 00:10:47,655 --> 00:10:53,340 connective and is different from the propositional connective, I hate it when. 103 00:10:53,880 --> 00:10:59,295 But there's a second wa y, in which the propositional connective, and, differs 104 00:10:59,295 --> 00:11:04,999 from the propositional connective, I hate it when. And that is, that when you use 105 00:11:04,999 --> 00:11:10,630 the propositional connective and, to combine two other propositions into a new 106 00:11:10,630 --> 00:11:16,623 proposition. Whether that proposition is true or false doesn't depend upon anything 107 00:11:16,623 --> 00:11:22,038 other than whether the two original propositions that you used to build it 108 00:11:22,038 --> 00:11:28,708 were true or false, that's all it depends upon. Let me illustrate this point using 109 00:11:28,708 --> 00:11:35,362 something we call a truth table. A truth table is a way of representing various 110 00:11:35,362 --> 00:11:42,353 possible situations, and how the truth of a proposition depends upon which of those 111 00:11:42,353 --> 00:11:49,091 various possible situations is real. For instance, consider the two possibilities 112 00:11:49,091 --> 00:11:55,830 there are with Jack finally talked. The proposition Jack finally taught, could be 113 00:11:55,830 --> 00:12:01,913 true. Or it could be false. The proposition Jill finally talked could be 114 00:12:01,913 --> 00:12:08,499 true or it could be false. So, there are four possibilities we have to consider. 115 00:12:08,499 --> 00:12:14,859 Either Jack finally talked is true and Jill finally talked is true. Or Jack 116 00:12:14,859 --> 00:12:21,812 finally talked is true and Jill finally talked is false. Or Jack finally talked is 117 00:12:21,812 --> 00:12:28,765 false and Jill finally talked is true. Or finally, Jack finally talked is false and 118 00:12:28,765 --> 00:12:35,334 Jill finally talked is false. Those are the four possible situations. Now, if the 119 00:12:35,334 --> 00:12:40,911 first of those four situations is the real situation, so it's true that Jack finally 120 00:12:40,911 --> 00:12:45,824 talked and it's true that Jill finally talked, then is it going to be true or 121 00:12:45,824 --> 00:12:52,775 false that Jack and Jill finally talked? It's going to be true. All right, cuz Jack 122 00:12:52,775 --> 00:13:00,370 finally talked, and Jill finally talked, so Jack and Jill finally talked. Now 123 00:13:00,370 --> 00:13:05,896 suppose it's true that Jack finally talked, but it's false that Jill finally 124 00:13:05,896 --> 00:13:11,349 talked. Then is it going to be true or false that Jack and Jill finally talked? 125 00:13:11,349 --> 00:13:18,149 It'll be false, and suppose it's false that Jack finally talked, but it's true 126 00:13:18,149 --> 00:13:23,836 that Jill finally talked. Then, is it going to be true or false that Jack and 127 00:13:23,836 --> 00:13:29,810 Jill finally talked, again it's going to be false. And finally, suppose it's false 128 00:13:29,810 --> 00:13:35,219 that Jack finally talked and it's also false that Jill final ly talked. Then is 129 00:13:35,219 --> 00:13:40,126 it going to be true or false that Jack and Jill finally talked? Well obviously in 130 00:13:40,126 --> 00:13:45,677 that situation, it's going to be false that Jack and Jill finally talked. So what 131 00:13:45,677 --> 00:13:52,843 this truth table demonstrates is that the truth of the proposition Jack and Jill 132 00:13:52,843 --> 00:13:58,519 finally talked. Just depends, it depends on nothing other than the truth of the 133 00:13:58,519 --> 00:14:03,835 proposition Jack finally talked, and the truth of the proposition Jill finally 134 00:14:03,835 --> 00:14:08,667 talked. In other words, the truth of the proposition that we've use the 135 00:14:08,667 --> 00:14:14,204 propositional connective and to build. Depends on nothing other than the truth of 136 00:14:14,204 --> 00:14:19,496 the two ingredient propositions that we connected by means of the propositional 137 00:14:19,496 --> 00:14:24,920 connective and. Because the propositional connective and, works that way because it 138 00:14:24,920 --> 00:14:30,145 builds new propositions whose truth depends on nothing other than the truth of 139 00:14:30,145 --> 00:14:35,569 the ingredient propositions that go into building them. That kind of propositional 140 00:14:35,569 --> 00:14:41,110 connective is one that we're going to call a truth functional connective. Now and, 141 00:14:41,110 --> 00:14:47,261 the propositional connective and, is a truth-functional connective. But not all 142 00:14:47,261 --> 00:14:52,230 propositional connectives are truth-functional connectives. For 143 00:14:52,230 --> 00:14:58,065 instance, suppose we try to construct a truth table for I hate it when, the 144 00:14:58,065 --> 00:15:04,728 propositional connective I hate it when. Well, so consider the proposition, it's 145 00:15:04,728 --> 00:15:11,365 raining. Now that proposition could be true or it could be false. Sometimes its 146 00:15:11,365 --> 00:15:17,748 true, sometimes its false. So lets consider these two possible situations. So 147 00:15:17,748 --> 00:15:24,555 suppose the proposition Its raining is true. In that situation is it going to be 148 00:15:24,555 --> 00:15:31,802 true or false that I hated when its raining. Could be either one, it could be 149 00:15:31,802 --> 00:15:38,770 raining even though I enjoyed the rain or it could be raining even though I hate the 150 00:15:38,770 --> 00:15:45,040 rain. Or it could be raining even though I'm indifferent to the rain. So the truth, 151 00:15:45,040 --> 00:15:53,602 of I hate it when it's raining isn't determined by its raining. So, if it's 152 00:15:53,602 --> 00:16:00,012 true that it's raining. It's unclear whether, I hate it when it's raining. 153 00:16:00,012 --> 00:16:06,466 Could be true, could be false. Suppose it's false, that it's raining. Then, is it 154 00:16:06,466 --> 00:16:12,327 going to be true or false that I hate it when it's raining. Again, could be either 155 00:16:12,327 --> 00:16:18,264 one. The truth of I hate it when it's raining isn't determined by the falsehood 156 00:16:18,264 --> 00:16:23,900 of it's raining. So, even if it's not raining, that doesn't mean anything one 157 00:16:23,900 --> 00:16:29,836 way or the other for whether I hate it when it's raining. So once again, if it's 158 00:16:29,836 --> 00:16:37,377 false that it's raining, I hate it when it's raining, could be true or could be 159 00:16:37,377 --> 00:16:44,211 false. So the proposition, I hated when it's raining. Whether that proposition is 160 00:16:44,211 --> 00:16:51,390 true or false doesn't just depend on the truth or the falsehood of the proposition 161 00:16:51,390 --> 00:16:57,964 it's raining, that you built this proposition out of using the propositional 162 00:16:57,964 --> 00:17:04,690 connective, I hate it when. Because of that, the propositional connective I hate 163 00:17:04,690 --> 00:17:10,303 it when is not a truth-functional connective. It's different from the 164 00:17:10,303 --> 00:17:16,893 propositional connective and which is a truth-functional connective. A moment ago, 165 00:17:16,893 --> 00:17:23,482 we built a truth table for a proposition that was built using a truth-functional 166 00:17:23,482 --> 00:17:29,014 connective, specifically the truth-functional connective and. But I'd 167 00:17:29,014 --> 00:17:35,583 like us to notice something about that truth table. Notice that if we replace the 168 00:17:35,583 --> 00:17:40,633 particular propositions that we are putting together using the 169 00:17:40,633 --> 00:17:47,208 truth-functional connective and, to make a different resultant proposition. Even if 170 00:17:47,208 --> 00:17:53,617 we change the ingredient propositions the truth table looks the same. Let me show 171 00:17:53,617 --> 00:17:59,757 you what I mean. Suppose instead of having Jack and Jill finally talked. We have Jack 172 00:17:59,757 --> 00:18:05,386 finally walked and Jill finally talked. So now, we're connecting two different 173 00:18:05,386 --> 00:18:10,942 propositions using the truth-functional connective and. There's Jack finally 174 00:18:10,942 --> 00:18:16,863 walked, there's Jill finally talked. And then we connect them up into Jack finally 175 00:18:16,863 --> 00:18:22,709 walked and Jill finally talked. Okay, now the truth of that resultant proposition, 176 00:18:22,709 --> 00:18:28,261 Jack finally walked and Jill finally talked. How does that depend on the truth 177 00:18:28,261 --> 00:18:32,936 or false sort of the ingredient propositions?Jack Jack finally walked and 178 00:18:32,936 --> 00:18:38,629 Jill finally talked. Well it's the same patent we saw earlier. If it's true that 179 00:18:38,629 --> 00:18:44,520 Jack finally walked, and it's also true that Jill finally talked. and it's also 180 00:18:44,520 --> 00:18:49,393 true that Jill finally tal ked. Then it's going to be true that Jack 181 00:18:49,393 --> 00:18:55,006 finally walked and Jill finally talked. If it's true that Jack finally walked, but 182 00:18:55,006 --> 00:19:00,227 it's false that Jill finally talked, then it's going to be false that Jack finally 183 00:19:00,227 --> 00:19:05,449 walked and Jill finally talked. If it's false that Jack finally walked but it's 184 00:19:05,449 --> 00:19:10,736 true that Jill finally talked, then it's going to be false that Jack finally walked 185 00:19:10,736 --> 00:19:16,156 and Jill finally talked. And if it's false that Jack finally walked and it's false 186 00:19:16,156 --> 00:19:21,378 that Jill finally talked, then it's going to be false that Jack finally walked and 187 00:19:21,378 --> 00:19:28,182 Jill finally talked. So even if we change one of the ingredient propositions as long 188 00:19:28,182 --> 00:19:33,882 as we're combining propositions using the truth functional connective and, the 189 00:19:33,882 --> 00:19:41,607 overall truth table looks the same. We could change them some more to illustrate 190 00:19:41,607 --> 00:19:47,520 this point. I suppose if we changed Jill finally talked to the zebra escaped. 191 00:19:59,693 --> 00:20:10,322 Change it here, so notice what we have here. We take two propositions, the 192 00:20:11,950 --> 00:20:20,190 proposition Jack finally walked, and the proposition the zebra escaped. And we put 193 00:20:20,190 --> 00:20:28,023 them together with the truth functional connective, and to create a resultant 194 00:20:28,023 --> 00:20:35,246 proposition Jack finally walked and the zebra escaped. Now, when is that 195 00:20:35,246 --> 00:20:43,012 proposition gonna be true? Well, again it depends just on, when these propositions 196 00:20:43,012 --> 00:20:48,244 are true. So if it's true, that Jack finally walked, and it's also true that 197 00:20:48,244 --> 00:20:53,760 the zebra escaped, then it's going to be true that Jack finally walked and the 198 00:20:53,760 --> 00:20:58,523 zebra escaped. If it's true that Jack finally walked, but it's false that the 199 00:20:58,523 --> 00:21:03,269 zebra escaped, then it's going to be false that Jack finally walked and the zebra 200 00:21:03,269 --> 00:21:07,895 escaped. If it's false that Jack finally walked and it's true that the zebra 201 00:21:07,895 --> 00:21:12,763 escaped, then it's going to be false that Jack finally walked and the zebra escaped. 202 00:21:12,763 --> 00:21:17,327 And finally, if it's false that Jack finally walked and it's false that the 203 00:21:17,327 --> 00:21:22,074 zebra escaped, then of course it's going to be false that Jack finally walked and 204 00:21:22,074 --> 00:21:27,047 the zebra escaped. So once again, same truth table even if we change the 205 00:21:27,047 --> 00:21:33,412 ingredient propositions that we're putting together with the functional connective an 206 00:21:33,412 --> 00:21:40,459 d to make the resultive proposition. Now, since the truth table stays the same even 207 00:21:40,459 --> 00:21:47,530 when we change these propositions up on top. We could represent that fact, by 208 00:21:47,530 --> 00:21:55,242 replacing these propositions altogether with variables that can range over any 209 00:21:55,242 --> 00:22:03,052 proposition. So, for instance, instead of saying Jack finally walked, we could just 210 00:22:03,052 --> 00:22:11,057 have a variable here call it P1 Our first proposition. Instead of saying the zebra 211 00:22:11,057 --> 00:22:17,988 escaped we can have a variable there, call it P2 our second proposition. And finally, 212 00:22:17,988 --> 00:22:26,852 when we put those two propositions together using the truth functional 213 00:22:26,852 --> 00:22:36,019 connective and. We'll have P1 one and. P2. So that's going to be our resultant 214 00:22:36,019 --> 00:22:43,264 proposition, P1 and P2. And whatever exactly that is, is going to depend of 215 00:22:43,264 --> 00:22:51,262 course on what P1 is and what P2 is. But whether this third proposition is true or 216 00:22:51,262 --> 00:22:59,284 false again is only going to depend on the truth or falsehood of P1 and of P2, when 217 00:22:59,284 --> 00:23:07,507 P1 whatever exactly that is, is true and P2 is true then. The proposition p1 and p2 218 00:23:07,507 --> 00:23:15,729 is going to be true. Whatever proposition that is, is going to be true. And in every 219 00:23:15,729 --> 00:23:23,852 other possible situation, that proposition is going to be false. So no matter what 220 00:23:23,852 --> 00:23:32,075 proposition we have for P1 and P2 their conjunction P1 and P2 is going to be true, 221 00:23:32,075 --> 00:23:39,619 just in those situations when P1 and P2 are both true. That's the truth table for 222 00:23:39,619 --> 00:23:46,053 the truth functional connective and, which we'll also call conjunction. In the next 223 00:23:46,053 --> 00:23:52,143 lecture, we're going to see how we can use the truth table for the truth functional 224 00:23:52,143 --> 00:23:58,156 connective and. To figure out the rules for evaluating deductive arguments that 225 00:23:58,156 --> 00:24:04,246 rely on the truth functional connected and. And in the following three lectures, 226 00:24:04,246 --> 00:24:10,640 we'll see how we can use the truth tables for other truth functional connectives, to 227 00:24:10,640 --> 00:24:16,502 figure out the rules for evaluating deductive arguments that use those other 228 00:24:16,502 --> 00:24:18,938 connectives. See you in next lecture.