1 00:00:04,440 --> 00:00:08,155 In the last lecture. We talked about the truth functional 2 00:00:08,155 --> 00:00:12,261 connective, conjunction. We gave the truth table for conjunction. 3 00:00:12,261 --> 00:00:17,672 And we showed how we could use the truth table for conjunction to figure out which 4 00:00:17,672 --> 00:00:22,300 inferences that use conjunction are valid and which inferences are not. 5 00:00:22,300 --> 00:00:27,506 Today, we're going to talk about the truth functional connective, disjunction. 6 00:00:27,506 --> 00:00:32,577 We're going to give the truth table for dis-junction, and we're going to show how 7 00:00:32,577 --> 00:00:38,122 we can use that truth table to figure out which inferences that use dis-junction 8 00:00:38,122 --> 00:00:41,030 are valid and which are not. Now in English, 9 00:00:41,030 --> 00:00:47,083 we usually express disjunction by using the word or but the word or can be used 10 00:00:47,083 --> 00:00:52,985 in a couple different ways in English. For instance, suppose that Manchester is 11 00:00:52,985 --> 00:00:57,298 playing Barcelona tonight and you ask me, who's going to win? 12 00:00:57,298 --> 00:01:02,594 And I say, well, I have no idea who's going to win but I can tell you this, 13 00:01:02,594 --> 00:01:08,645 it's going to be Manchester or Barcelona. Now, what I'm suggesting when I say, it's 14 00:01:08,645 --> 00:01:13,726 going to be Manchester or Barcelona, is that it's not going to be both. 15 00:01:13,726 --> 00:01:16,980 Manchester might win, Barcelona might win. 16 00:01:16,980 --> 00:01:21,743 But there's no possible way that both of them are going to win. 17 00:01:21,743 --> 00:01:27,380 Sometimes, in English, when you want to say that it's going to be one thing or 18 00:01:27,380 --> 00:01:33,334 the other, but not both, you say, either or either Manchester is going to win, or 19 00:01:33,334 --> 00:01:39,223 Barcelona is going to win. But sometimes when we use the word or, we 20 00:01:39,223 --> 00:01:42,955 mean it could be one, or the other, or both. 21 00:01:42,955 --> 00:01:50,064 So for instance, suppose you ask me what we should have for dinner tonight and I 22 00:01:50,064 --> 00:01:56,836 say well we could have chicken or fish. Well there's no suggestion that we 23 00:01:56,836 --> 00:02:03,639 couldn't have both maybe we could have a little bit of chicken and a little of 24 00:02:03,639 --> 00:02:07,891 fish. So it has to be chicken or fish or both. 25 00:02:07,891 --> 00:02:13,843 When I say chicken or fish, I'm not suggesting it can't be both. 26 00:02:13,843 --> 00:02:21,306 Sometimes in English we use the phrase and, or to express that it could be one 27 00:02:21,306 --> 00:02:26,761 or the other or both. I'll say, we could have the chicken and, 28 00:02:26,761 --> 00:02:31,350 or the fish. The truth functional connective 29 00:02:31,350 --> 00:02:35,740 disjunction is expressed by the second meaning of or. 30 00:02:35,740 --> 00:02:42,532 It's expressed by the English phrase and, or where you mean it could be one or the 31 00:02:42,532 --> 00:02:46,094 other or both. That's what we're going to call 32 00:02:46,094 --> 00:02:51,396 disjunction in this class. Now let's look at the truth table for 33 00:02:51,396 --> 00:02:56,194 disjunction. So lets look at the truth table for 34 00:02:56,194 --> 00:03:01,477 dis-junction. Suppose you're using disjunction to 35 00:03:01,477 --> 00:03:09,614 combine the propositions We eat chicken and we eat fish into the disjunctive 36 00:03:09,614 --> 00:03:17,750 proposition We eat chicken or fish. Well when is that disjunctive proposition 37 00:03:17,750 --> 00:03:22,857 going to be true? If it's true that we eat chicken, and 38 00:03:22,857 --> 00:03:30,112 it's true that we eat fish, then it's going to be true that we eat chicken or 39 00:03:30,112 --> 00:03:36,975 fish because remember, when we use or here, we don't mean either or, but not 40 00:03:36,975 --> 00:03:38,936 both. We mean and or. 41 00:03:38,936 --> 00:03:43,740 Could be one, could be the other, or could be both. 42 00:03:43,740 --> 00:03:48,984 So if it's true that we eat chicken and it's true that we eat fish, it's going to 43 00:03:48,984 --> 00:03:53,757 be true that we eat chicken or fish. Now supposed it's true that we eat 44 00:03:53,757 --> 00:03:57,327 chicken, but its false that we eat fish. Well. 45 00:03:57,327 --> 00:04:02,516 Then, it's still going to be true that we eat chicken or fish. 46 00:04:02,516 --> 00:04:08,332 Suppose it's false that we eat chicken, but true that we eat fish. 47 00:04:08,332 --> 00:04:13,522 Then, it's still going to be true that we eat chicken or fish. 48 00:04:13,522 --> 00:04:20,232 But suppose it's false that we eat chicken and it's also false that we eat 49 00:04:20,232 --> 00:04:24,675 fish. Then, is it going to be true that we eat 50 00:04:24,675 --> 00:04:26,305 chicken or fish? No! 51 00:04:26,305 --> 00:04:32,226 Because we won't be eating either. So then it'll be false that we eat 52 00:04:32,226 --> 00:04:36,828 chicken or fish. This is the truth table for disjunction. 53 00:04:36,828 --> 00:04:43,205 And, like the truth table that we saw for conjunction, it's going to work no matter 54 00:04:43,205 --> 00:04:47,321 what propositions we put into here, or here, or here. 55 00:04:47,321 --> 00:04:52,245 So, no matter what proposition you have right here, call it P1. 56 00:04:52,245 --> 00:04:57,249 And, no matter what proposition you have right here, call it P2. 57 00:04:57,249 --> 00:05:01,850 When you use the truth functional connective disjunction. 58 00:05:01,850 --> 00:05:09,594 To create a new proposition out of those two proposition's, so you got a new 59 00:05:09,594 --> 00:05:16,013 proposition P one or P two. That new disjunctive proposition is going 60 00:05:16,013 --> 00:05:20,640 to be true. Whenever P1 is true, and it's also going 61 00:05:20,640 --> 00:05:26,290 to be true whenever P2 is true. So unlike conjunction. 62 00:05:26,290 --> 00:05:32,243 Where you need both of the two ingredient propositions to be true in order for the 63 00:05:32,243 --> 00:05:37,622 conjunctive proposition to be true. In disjunction, you only need for one of 64 00:05:37,622 --> 00:05:43,289 the of the two ingredient propositions to be true in order for the disjunctive 65 00:05:43,289 --> 00:05:47,951 proposition to be true. The disjunctive proposition is false only 66 00:05:47,951 --> 00:05:51,268 when. Both of the two ingredient propositions 67 00:05:51,268 --> 00:05:54,819 are false. That's the only time a disjunction is 68 00:05:54,819 --> 00:05:57,999 false. So now, let me give you an example, of 69 00:05:57,999 --> 00:06:01,550 how you can use the truth table for disjunction. 70 00:06:01,550 --> 00:06:05,397 Just show that a particular kind of argument is valid. 71 00:06:05,397 --> 00:06:09,887 We're going to discuss, a kind of argument that is sometimes known. 72 00:06:09,887 --> 00:06:14,348 As process of elimination. Here's how it goes. 73 00:06:14,348 --> 00:06:18,562 Suppose, that you have to solve. A murder mystery. 74 00:06:18,562 --> 00:06:22,864 Mister Jones, has been stabbed in his living room. 75 00:06:22,864 --> 00:06:28,594 With a knife in the back. Now, you figured out that there were only 76 00:06:28,594 --> 00:06:33,572 two people in the house at the time of his stabbing, the butler and the 77 00:06:33,572 --> 00:06:36,587 accountant. You also know that the knife is 78 00:06:36,587 --> 00:06:40,373 positioned in Mr. Johnson's back in such a way that he 79 00:06:40,373 --> 00:06:45,141 couldn't possibly have stabbed himself. So it had to be someone else. 80 00:06:45,141 --> 00:06:50,680 And whoever else it was it had to be someone who's in the house at the time of 81 00:06:50,680 --> 00:06:54,466 the stabbing. So it could only have been, the butler or 82 00:06:54,466 --> 00:07:00,676 the accountant, or maybe both. So you know that the butler did it, or 83 00:07:00,676 --> 00:07:06,293 the accountant did it. Now you find out that the accountant is a 84 00:07:06,293 --> 00:07:10,551 quadriplegic, so the accountant couldn't have stabbed Mr. 85 00:07:10,551 --> 00:07:15,830 Jones in the back. So now you know that the account didn't 86 00:07:15,830 --> 00:07:20,410 do it. And so, from the two premises, the butler 87 00:07:20,410 --> 00:07:26,028 did it, or the accountant did it. And the accountant didn't do it. 88 00:07:26,028 --> 00:07:31,964 You can conclude, the butler did it. Now, why is that argument valid? 89 00:07:31,964 --> 00:07:35,134 Here's why. Think about the truth table for 90 00:07:35,134 --> 00:07:39,483 disjunction again. So remember the first premise, the butler 91 00:07:39,483 --> 00:07:43,094 did it or the accountant did it is a disjunction. 92 00:07:43,094 --> 00:07:48,254 It's going to be true whenever one of it's disjuncts is true, one of it's 93 00:07:48,254 --> 00:07:53,192 ingredient propositions is true. So it's going to be true whenever the 94 00:07:53,192 --> 00:07:57,910 butler did it, and it's going to be true whenever the butler did it. 95 00:07:57,910 --> 00:08:02,800 The second premise tells you that the accountant didn't do it. 96 00:08:02,800 --> 00:08:08,849 So the only way for the first premise to be true, given that the accountant didn't 97 00:08:08,849 --> 00:08:14,645 do it, is for the butler to have done it. And so you know, since the accountant 98 00:08:14,645 --> 00:08:18,908 couldn't have done it. That the only way for the dis-junction, 99 00:08:18,908 --> 00:08:24,499 the butler did it or the accountant did it to be true, is for the butler to have 100 00:08:24,499 --> 00:08:30,020 done it and that's why you can conclude the butler did it and your argument is 101 00:08:30,020 --> 00:08:33,101 valid. That's one example of a process of 102 00:08:33,101 --> 00:08:37,218 elimination argument. Of course there are lots of others, but 103 00:08:37,218 --> 00:08:42,708 with all of those others you can see why they are valid by looking at the truth 104 00:08:42,708 --> 00:08:46,688 table for dis-junction. Remember how you can use the truth 105 00:08:46,688 --> 00:08:51,972 functional connective conjunction to build a new proposition out of not just 106 00:08:51,972 --> 00:08:56,226 two other propositions but sometimes three other propositions. 107 00:08:56,226 --> 00:09:00,550 You can conjoin one proposition with a second and with a third. 108 00:09:00,550 --> 00:09:05,080 Well, you can do the same thing with disjunction. 109 00:09:05,080 --> 00:09:11,877 You can disjoin one proposition with a second and a third, to create the 110 00:09:11,877 --> 00:09:17,502 proposition. Either this, or that or the other or any 111 00:09:17,502 --> 00:09:24,802 combination of the three. What does the truth table for that look 112 00:09:24,802 --> 00:09:27,240 like? Here it is. 113 00:09:27,240 --> 00:09:35,110 The disjunctive proposition, P1 or P2 or P3, is going to be true. 114 00:09:35,110 --> 00:09:40,684 Whenever P1 is true, it's also going to be true whenever P2 is true. 115 00:09:40,684 --> 00:09:44,865 And it's also going to be true whenever P3 is true. 116 00:09:44,865 --> 00:09:51,747 In fact, the only time that P1 or P2 or P3, the only time that, that disjunctive 117 00:09:51,747 --> 00:09:58,367 proposition is going to be false is when all these ingredient propositions are 118 00:09:58,367 --> 00:10:03,094 false. So here's what the truth table for P1, or 119 00:10:03,094 --> 00:10:08,257 P2, or P3 looks. Now let's use the truth table for our 120 00:10:08,257 --> 00:10:12,913 triple disjunction to show how a particular process of elimination 121 00:10:12,913 --> 00:10:17,089 argument can be valid. Let's go back to our murder mystery in 122 00:10:17,089 --> 00:10:21,128 order to do that. Now suppose that you find out contrary to 123 00:10:21,128 --> 00:10:26,263 what you had previously believed, that Butler and the accountant were not the 124 00:10:26,263 --> 00:10:29,206 only people in the house, at the time of Mr. 125 00:10:29,206 --> 00:10:33,793 Jonathan's death. In addition, the maid was in the house 126 00:10:33,793 --> 00:10:36,395 and the cook was in the house. Alright. 127 00:10:36,395 --> 00:10:40,967 Well, now, you know, that the butler or the maid or the cook did it. 128 00:10:40,967 --> 00:10:46,604 We don't yet know which of them did it, but we know that the butler or the maid 129 00:10:46,604 --> 00:10:50,404 or the cook did it. Now suppose that you find out that the 130 00:10:50,404 --> 00:10:55,356 maid and the cook, at the time of the stabbing we're off in the opposite corner 131 00:10:55,356 --> 00:10:58,051 of the house doing something else together. 132 00:10:58,051 --> 00:11:00,872 Well now you know, that the maid didn't do it. 133 00:11:00,872 --> 00:11:06,474 And you know that the cook didn't do it. So what can you conclude from those three 134 00:11:06,474 --> 00:11:09,937 premises? Premise one, the butler or the maid or 135 00:11:09,937 --> 00:11:13,622 the cook did it. Premise two, the maid didn't do it. 136 00:11:13,622 --> 00:11:19,516 And premise three: the cook didn't do it. Well, lets use the truth table to figure 137 00:11:19,516 --> 00:11:24,241 this out. Premise one of the truth table tells you 138 00:11:24,241 --> 00:11:28,210 that the butler or the maid or the cook did it. 139 00:11:28,210 --> 00:11:35,134 So the situation in which it falls that the butler or the maid or the cook did it 140 00:11:35,134 --> 00:11:38,765 that situation is ruled out by premise one. 141 00:11:38,765 --> 00:11:44,760 So premise one tells you at that situation is not the actual situation. 142 00:11:46,600 --> 00:11:51,300 Premise two tells you that the maid did not do it. 143 00:11:51,300 --> 00:11:56,768 So any situation in which its true that the maid did it is also not the actual 144 00:11:56,768 --> 00:12:00,091 situation. So this situation is one in which its 145 00:12:00,091 --> 00:12:05,767 true that the maid did it so that's not the actual situation according to premise 146 00:12:05,767 --> 00:12:08,813 two. This situation is one in which its true 147 00:12:08,813 --> 00:12:12,689 that the maid did it. So that's not the actual situation 148 00:12:12,689 --> 00:12:16,992 according to premise two. This situation is one in which its true 149 00:12:16,992 --> 00:12:20,593 that the maid did it. So that's not the actual situation 150 00:12:20,593 --> 00:12:25,481 according to premise two, and this situation is one in which it is true that 151 00:12:25,481 --> 00:12:28,632 the maid did it. So that's not the actual situation 152 00:12:28,632 --> 00:12:33,752 according to premise two. Premise three tells you that the cook 153 00:12:33,752 --> 00:12:37,706 didn't do it. So, that rules out any situation in which 154 00:12:37,706 --> 00:12:42,582 it's true that the cook did it. Well, here's a situation in which it's 155 00:12:42,582 --> 00:12:46,821 true that the cook did it. So, that situation is ruled out by 156 00:12:46,821 --> 00:12:50,425 premise three. And, here's a situation in which it's 157 00:12:50,425 --> 00:12:54,664 true that the cook did it. So, that situation is ruled out by 158 00:12:54,664 --> 00:12:58,480 premise three. So, premise one rules out this situation. 159 00:12:58,480 --> 00:13:04,160 Premise two, rules out this, this, this and this situation. 160 00:13:04,160 --> 00:13:10,450 And premise three, rules out this, this, this and this situation. 161 00:13:10,450 --> 00:13:17,915 Well, whats left? The only situation left that could be the 162 00:13:17,915 --> 00:13:26,505 actual situation is this one. See cause in this situation, it's true 163 00:13:26,505 --> 00:13:32,156 that the butler or the maid or the cook did it just as premise one tells us. 164 00:13:32,156 --> 00:13:37,808 Its false that the maid did just as premise two tells us, and its false that 165 00:13:37,808 --> 00:13:41,920 the cook did just as premise three tells us. 166 00:13:41,920 --> 00:13:46,885 But, that's the situation in which it's true that the butler did it. 167 00:13:46,885 --> 00:13:52,370 So, the conclusion that we can draw, based on the situations that are ruled 168 00:13:52,370 --> 00:13:57,928 out by premises one, two, and three, is that the actual situation is this one, 169 00:13:57,928 --> 00:14:02,523 and in that actual situation, it's true that the butler did it. 170 00:14:02,523 --> 00:14:10,312 So, the butler did it That's why the process of elimination reasoning that we 171 00:14:10,312 --> 00:14:15,463 just considered is valid. If premise one says, the butler or the 172 00:14:15,463 --> 00:14:20,012 maid or the cook did it. Premise two says the maid didn't do it, 173 00:14:20,012 --> 00:14:23,622 and premise three says that the cook didn't do it. 174 00:14:23,622 --> 00:14:29,399 Then by process of elimination we can draw the valid conclusion that the butler 175 00:14:29,399 --> 00:14:34,057 did it and this is why. Let me give you another example of how 176 00:14:34,057 --> 00:14:39,591 you can use the truth table for disjunction in order to show whether or 177 00:14:39,591 --> 00:14:43,358 not the process of elimination argument is valid. 178 00:14:43,358 --> 00:14:48,123 Suppose we know that Walter is a professional football player. 179 00:14:48,123 --> 00:14:52,658 Well, that means that he plays either American football, U.S. 180 00:14:52,658 --> 00:14:58,192 Football, or European football, which Americans call soccer, or Australian 181 00:14:58,192 --> 00:15:02,420 rules football. But now suppose we find out that Walter 182 00:15:02,420 --> 00:15:08,618 does not play American football. And you conclude from that, that he must 183 00:15:08,618 --> 00:15:12,722 play European football. So you argue as follows. 184 00:15:12,722 --> 00:15:18,082 Premise 1- Walter plays either American football or European football or 185 00:15:18,082 --> 00:15:23,663 Australian Rules football, premise 2- he does not play American football and 186 00:15:23,663 --> 00:15:27,334 therefore you conclude he plays European football. 187 00:15:27,334 --> 00:15:33,282 Well, that argument is invalid and we can use the truth table for disjunction to 188 00:15:33,282 --> 00:15:36,660 show why it's invalid. Look at this truth table. 189 00:15:36,660 --> 00:15:43,403 Premise one, recall, is that Walter plays American or European or Australian rules 190 00:15:43,403 --> 00:15:47,649 football. So premise one rules out the situation in 191 00:15:47,649 --> 00:15:53,976 which it's false that Walter plays American or European or Australian rules 192 00:15:53,976 --> 00:15:57,056 football. And that's all it rules out. 193 00:15:57,056 --> 00:16:03,300 It just rules out the situation in which it's false that Walter plays any of 194 00:16:03,300 --> 00:16:08,304 those. Premise two, Walter doesn't play American 195 00:16:08,304 --> 00:16:12,138 Football. That rules out the situation in which 196 00:16:12,138 --> 00:16:15,891 it's true that Walter plays American football. 197 00:16:15,891 --> 00:16:24,200 So it rules out this situation. And rules out this situation. 198 00:16:24,200 --> 00:16:34,440 And it rules out this situation. And it rules out this situation. 199 00:16:35,740 --> 00:16:41,390 So, premise one rules out the situation represented at the bottom. 200 00:16:41,390 --> 00:16:48,171 Premise two rules out the situations represented by these four columns at the 201 00:16:48,171 --> 00:16:51,735 top. So, can we conclude that Walter plays 202 00:16:51,735 --> 00:16:56,300 European football? No. 203 00:16:56,300 --> 00:17:02,401 He might play European football but he might also play Australian Rules 204 00:17:02,401 --> 00:17:06,214 football. He's looked all the premise one and 205 00:17:06,214 --> 00:17:10,874 premise two together rule out is these five situations. 206 00:17:10,874 --> 00:17:15,196 But there is no three situations that are possible. 207 00:17:15,196 --> 00:17:22,060 In one of them Walter plays both European football and Australian rules football. 208 00:17:22,060 --> 00:17:27,805 In another one of them, Walter plays European football, but not Australian 209 00:17:27,805 --> 00:17:33,550 rules football, and in the third situation, this left open by premises one 210 00:17:33,550 --> 00:17:39,688 and two, it's false that Walter plays European football but true that he plays 211 00:17:39,688 --> 00:17:45,040 Australian rules football. So based on the information that premises 212 00:17:45,040 --> 00:17:50,942 one and two give us, we cannot conclude that Walter plays European football. 213 00:17:50,942 --> 00:17:54,720 He might play Australian rules football instead. 214 00:17:54,720 --> 00:17:59,989 So the argument that you made is invalid. In the next lecture, we're going to 215 00:17:59,989 --> 00:18:05,547 consider a truth functional connective that's different from conjunction and 216 00:18:05,547 --> 00:18:10,757 disjunction in the following way. While conjunction and disjunction are 217 00:18:10,757 --> 00:18:16,412 connectives that can be used to build propositions out of two or more other 218 00:18:16,412 --> 00:18:20,356 propositions. Negation, the connective that we'll talk 219 00:18:20,356 --> 00:18:26,309 about next time, is the connective that is used to build new propositions out of 220 00:18:26,309 --> 00:18:31,815 just one single other proposition. Negation, in other words, is a connective 221 00:18:31,815 --> 00:18:36,950 that you apply to one proposition to build a second proposition. 222 00:18:36,950 --> 00:18:40,150 And that's what we'll talk about next time. 223 00:18:40,150 --> 00:18:42,950 Now, there's some exercise's for you to do. 224 00:18:42,950 --> 00:18:48,551 These exercises test your understanding of the truth table for disjunction and of 225 00:18:48,551 --> 00:18:54,085 how the truth table for dis-junction can be used to determine whether a particular 226 00:18:54,085 --> 00:18:58,020 argument that uses disjunction is a valid argument or not.