1 00:00:04,220 --> 00:00:08,649 In the last lecture I explained what propositional connectives are, 2 00:00:08,649 --> 00:00:13,805 I described a particular category of propositional connectives that we called 3 00:00:13,805 --> 00:00:18,301 truth functional connectives, and I gave you an example of one truth 4 00:00:18,301 --> 00:00:23,062 functional connective. And, another word, and in English isn't always 5 00:00:23,062 --> 00:00:27,738 used to mean a truth functional connective, but sometimes it is. 6 00:00:27,738 --> 00:00:33,601 And one thing I'd like to point out right now, is that there are other words in 7 00:00:33,601 --> 00:00:39,464 English that can be used to indicate the very same truth functional connective 8 00:00:39,464 --> 00:00:45,031 that the word and is used to indicate. For instance, think about the English 9 00:00:45,031 --> 00:00:48,000 words also, moreover, furthermore, and but. 10 00:00:48,000 --> 00:00:55,191 Now, you might think the word and and the word but" mean two very different things. 11 00:00:55,191 --> 00:01:00,980 If I say Walter is poor and happy, that seems to mean something very 12 00:01:00,980 --> 00:01:07,498 different from Walter is poor but happy. In particular, when I say Walter is poor 13 00:01:07,498 --> 00:01:13,691 but happy, I'm suggesting that there is contrast between his poverty and his 14 00:01:13,691 --> 00:01:17,765 happiness. But when I say Walter is poor and happy, 15 00:01:17,765 --> 00:01:23,957 I'm not suggesting any such contrast. Still, whatever contrast there might be 16 00:01:23,957 --> 00:01:30,336 between his poverty and his happiness doesn't effect the truth table for the 17 00:01:30,336 --> 00:01:36,807 truth functional connective but. Let's consider when it would be true to 18 00:01:36,807 --> 00:01:42,755 say Walter is poor but happy. To show you what I mean, about the words" 19 00:01:42,755 --> 00:01:48,841 but and and," let's go back to the truth table for the truth functional connective 20 00:01:48,841 --> 00:01:51,184 and. So remember, if you have two 21 00:01:51,184 --> 00:01:56,894 propositions, p1 and p2, and you use the truth functional connective and to put 22 00:01:56,894 --> 00:02:01,873 them together to make another proposition, the proposition p1 and p2. 23 00:02:01,873 --> 00:02:07,583 And now you want to know when is that new proposition, the proposition p1 and p2, 24 00:02:07,583 --> 00:02:13,520 when is that going to be true? Well, the answer is it's going to be true 25 00:02:13,520 --> 00:02:23,472 only when p1 is true and p2 is true. In any other scenario, the proposition p1 26 00:02:23,472 --> 00:02:30,710 and p2 is going to be false. Let's take an example so I can 27 00:02:30,710 --> 00:02:36,267 illustrate. Let's suppose, for p1 we use the 28 00:02:36,267 --> 00:02:46,220 proposition Walter is poor, and for p2 we use the proposition Walter is happy, 29 00:02:48,240 --> 00:02:53,065 then we use the truth functional connective, and, to put those put two 30 00:02:53,065 --> 00:02:58,101 propositions together into a new proposition and the new proposition is 31 00:02:58,101 --> 00:03:04,780 going to be Walter [SOUND] is poor and happy. 32 00:03:09,011 --> 00:03:12,566 Okay. Now, when it going to be true that Walter 33 00:03:12,566 --> 00:03:16,894 is poor and happy? Well, if it's true that Walter is poor 34 00:03:16,894 --> 00:03:23,154 and it's also true that Walter is happy then its going to be true that Walter is 35 00:03:23,154 --> 00:03:27,250 poor and happy. But, if it's false that Walter is poor, 36 00:03:27,250 --> 00:03:32,041 then it is not going to be true that Walter is poor and happy. 37 00:03:32,041 --> 00:03:37,837 And if it's false that Walter is happy, then it's not going to be true that 38 00:03:37,837 --> 00:03:42,808 Walter is poor and happy. So the proposition Walter is poor and 39 00:03:42,808 --> 00:03:48,261 happy is going to be true, only when Walter is poor is true and Walter is 40 00:03:48,261 --> 00:03:52,078 happy is true. In any other possible scenario, the 41 00:03:52,078 --> 00:03:57,220 proposition Walter is poor and happy will end up being false. 42 00:03:57,220 --> 00:04:06,680 So, lets compare that to the proposition that we get by combining Walter is poor 43 00:04:06,680 --> 00:04:14,840 and Walter is happy with the connective, but, Walter is poor but happy. 44 00:04:16,340 --> 00:04:20,335 Now, when is it going to be true to say Walter is poor but happy? 45 00:04:20,335 --> 00:04:25,379 Well, it's not going to be true to say Walter is poor but happy in any situation 46 00:04:25,379 --> 00:04:28,195 where it's false that Walter is poor. Right? 47 00:04:28,195 --> 00:04:33,173 If it's false that Walter is poor, then it's also going to be false that Walter 48 00:04:33,173 --> 00:04:36,645 is poor but happy. It's also not going to be true to say 49 00:04:36,645 --> 00:04:41,885 Walter is poor but happy in any situation where it's false that Walter is happy. 50 00:04:41,885 --> 00:04:46,797 If it's false that Walter is happy, then it's going to be false that Walter is 51 00:04:46,797 --> 00:04:50,949 poor but happy. So when is it going to be true that 52 00:04:50,949 --> 00:04:55,237 Walter is poor but happy? The only possible situation where it 53 00:04:55,237 --> 00:05:00,839 could be true is the situation where it's true that Walter is poor and it's also 54 00:05:00,839 --> 00:05:04,989 true that Walter is happy. Now, you might think, wait a second. 55 00:05:04,989 --> 00:05:10,660 When I say Walter is poor but happy, I'm saying more than just that Walter is poor 56 00:05:10,660 --> 00:05:15,087 and that Walter is happy. I'm also suggesting a contrast between 57 00:05:15,087 --> 00:05:19,928 his poverty and his happines. And maybe that suggestion is misleading, 58 00:05:19,928 --> 00:05:24,151 maybe poor people are often happy. But notice, 59 00:05:24,151 --> 00:05:28,925 what you say can be misleading even if it's true. 60 00:05:28,925 --> 00:05:36,420 For example, suppose someone comes up to me with a car that's sputtering. 61 00:05:36,420 --> 00:05:41,028 They might say, do you know where there's a gas station around here? 62 00:05:41,028 --> 00:05:46,118 I need to fill up this car with gas. And I might say, there's a gas station 63 00:05:46,118 --> 00:05:50,289 just around the corner. Now, what I say might be true, 64 00:05:50,289 --> 00:05:55,264 there might really be a gas station just around the corner even if I know that 65 00:05:55,264 --> 00:05:59,566 that gas station has been closed for three years and has no gas. 66 00:05:59,566 --> 00:06:04,944 So, what I say is misleading because I've lead them to believe falsely, that they 67 00:06:04,944 --> 00:06:09,045 can get gas if they can just get their car around the corner. 68 00:06:09,045 --> 00:06:14,087 But even though what I've said is misleading, it's still true because there 69 00:06:14,087 --> 00:06:17,650 is a gas station around the corner, only a closed one. 70 00:06:17,650 --> 00:06:24,416 So what you say can be true but misleading and I suggest that when you 71 00:06:24,416 --> 00:06:29,936 say Walter is poor but happy. That can be true even if it's misleading 72 00:06:29,936 --> 00:06:34,912 to suggest that poverty and happiness are somehow at odds with each other. 73 00:06:34,912 --> 00:06:40,090 I've just said that the word but in English can be used to indicate the same 74 00:06:40,090 --> 00:06:45,200 truth functional connective that the word and is sometimes used to indicate. 75 00:06:45,200 --> 00:06:50,580 And there are other words in English that can be used to indicate that same truth 76 00:06:50,580 --> 00:06:54,076 functional connective, also, furthermore, moreover, and 77 00:06:54,076 --> 00:06:56,430 sometimes we even use the word too, too. 78 00:06:56,430 --> 00:07:02,210 But now I want to introduce a term that's going to describe that truth functional 79 00:07:02,210 --> 00:07:07,701 connective no matter what word in ordinary language we use to indicate that 80 00:07:07,701 --> 00:07:10,230 connective. The term is conjunction. 81 00:07:10,230 --> 00:07:16,896 And the term conjunction, as I'm using it here and as philosophers use it, is not 82 00:07:16,896 --> 00:07:23,480 the same term that grammarians use when they talk about conjunctive terms like 83 00:07:23,480 --> 00:07:28,313 but, or, and, therefore. Here's something that could help you 84 00:07:28,313 --> 00:07:33,480 understand what conjunctions in the grammarian sense are like. 85 00:07:33,480 --> 00:07:39,522 All of those terms are conjunctions in the grammarian sense, but they're not 86 00:07:39,522 --> 00:07:45,879 conjunctions in the philosopher's sense. A conjunction in the philosopher's sense 87 00:07:45,879 --> 00:07:51,922 is just the truth functional connective that has this particular truth table. 88 00:07:51,922 --> 00:07:58,220 You can use the conjunction to create a new proposition out of joining two other 89 00:07:58,220 --> 00:08:03,945 propositions and that new proposition that you create using conjunction is 90 00:08:03,945 --> 00:08:08,586 going to be true, only when the other two propositions are true. 91 00:08:08,586 --> 00:08:13,151 In any other case, the new proposition is going to be false, 92 00:08:13,151 --> 00:08:18,180 that's what a conjunction is. And we can use the symbol ampersand, 93 00:08:18,180 --> 00:08:21,330 like that, in order to signify conjunction. 94 00:08:21,330 --> 00:08:27,255 Now that we know the truth table for the conjunction, let's consider how we can 95 00:08:27,255 --> 00:08:33,030 use that truth table to figure out when an argument that uses conjunction is 96 00:08:33,030 --> 00:08:37,226 valid. Consider the argument Walter is poor but 97 00:08:37,226 --> 00:08:40,150 happy, therefore, Walter is happy. 98 00:08:40,150 --> 00:08:45,369 Is that argument valid or invalid? Well, pretty obviously, that argument is 99 00:08:45,369 --> 00:08:50,946 valid. There's no possible way for the premise to be true while the conclusion 100 00:08:50,946 --> 00:08:54,164 is false. But, can you see why the argument is 101 00:08:54,164 --> 00:08:57,310 valid using the truth table for conjunction? 102 00:08:57,310 --> 00:09:03,333 You should be able to in a situation in which the premise is true, Walter is poor 103 00:09:03,333 --> 00:09:06,679 but happy, there are going to have to be two other 104 00:09:06,679 --> 00:09:11,810 propositions that are true, namely Walter is poor and Walter is happy. 105 00:09:11,810 --> 00:09:17,461 So, if its true that Walter is true but happy, then its going to have to be true 106 00:09:17,461 --> 00:09:23,484 that Walter is happy and that's why the argument is valid. That's why there is no 107 00:09:23,484 --> 00:09:28,690 possible way for the premise to be true while the conclusion is false. 108 00:09:28,690 --> 00:09:34,581 Let's consider some other arguments that involve conjunction. 109 00:09:34,581 --> 00:09:39,699 Consider the argument Walter is poor, walter is happy, 110 00:09:39,699 --> 00:09:45,210 therefore, Walter is poor and happy. Is that argument valid? 111 00:09:45,210 --> 00:09:49,156 Clearly it is. And again, you can use the truth table 112 00:09:49,156 --> 00:09:55,076 for conjunctions to see why it's valid. In a situation where the first premise 113 00:09:55,076 --> 00:10:01,072 Walter is poor is true, and in which the second premise Walter is happy is true. 114 00:10:01,072 --> 00:10:06,992 In that situation, the conclusion Walter is poor and happy, is going to have to be 115 00:10:06,992 --> 00:10:10,028 true. So there's no possible way for the 116 00:10:10,028 --> 00:10:16,404 premises of that argument to both be true while the conclusion is false and so that 117 00:10:16,404 --> 00:10:21,054 argument is also valid. Now notice, just as we can combine two 118 00:10:21,054 --> 00:10:24,646 propositions with each other using conjunction, 119 00:10:24,646 --> 00:10:30,454 we can then also combine the resulting proposition with another proposition 120 00:10:30,454 --> 00:10:36,407 using conjunction. So, consider the proposition Walter is 121 00:10:36,407 --> 00:10:43,448 poor but happy and popular. That proposition uses two conjunctions to 122 00:10:43,448 --> 00:10:49,413 combine three other propositions into a single conjunctive proposition. 123 00:10:49,413 --> 00:10:55,126 To understand how that works, let's look at the truth table for that. 124 00:10:55,126 --> 00:11:01,259 So when is it going to be true that Walter is poor but happy and popular? 125 00:11:01,259 --> 00:11:04,200 When is that going to be true? Well, 126 00:11:04,200 --> 00:11:11,391 if it's false that Walter is poor, then it's definitely not going to be true that 127 00:11:11,391 --> 00:11:18,316 Walter is poor but happy and popular. So in all of these situations right down 128 00:11:18,316 --> 00:11:23,835 here, walter is poor but happy and popular, is 129 00:11:23,835 --> 00:11:28,598 going to to be false. [SOUND] If it's false that Walter is 130 00:11:28,598 --> 00:11:36,116 happy, then it's definitely not going to be true that Walter is poor but happy and 131 00:11:36,116 --> 00:11:39,417 popular, because he's not going to be happy. 132 00:11:39,417 --> 00:11:46,476 So, in these situations right here where it's false that Walter is happy, it's 133 00:11:46,476 --> 00:11:52,161 also going to be false [SOUND] that Walter is poor but happy and popular. 134 00:11:52,161 --> 00:11:58,198 And if it's false that Walter is popular, then of course, it's also going to be 135 00:11:58,198 --> 00:12:01,415 false that Walter is poor but happy and popular. 136 00:12:01,415 --> 00:12:06,575 So, in this situation right here, it'll be false that Walter is poor but happy 137 00:12:06,575 --> 00:12:10,217 and popular. So, is it ever going to be true that 138 00:12:10,217 --> 00:12:16,665 Walter is poor but happy and popular? Yes. It'll be true just when it's true 139 00:12:16,665 --> 00:12:22,683 that Walter is poor, it's true that Walter is happy, and it's true that 140 00:12:22,683 --> 00:12:27,325 Walter is popular. That's the only situation when it's 141 00:12:27,325 --> 00:12:32,140 going to be true that Walter is poor but happy and popular. 142 00:12:32,140 --> 00:12:38,017 In general, this is the kind of truth table that we get when we combine three 143 00:12:38,017 --> 00:12:43,208 propositions using conjunction. So now, considered how we can use the 144 00:12:43,208 --> 00:12:48,781 truth table for conjunctions of three propositions to figure out whether 145 00:12:48,781 --> 00:12:52,140 certain deductive arguments are valid or not. 146 00:12:52,140 --> 00:12:55,069 So consider the following deductive argument. 147 00:12:55,069 --> 00:12:59,953 From the premises Paris is the capital of France, Jakarta is the capital of 148 00:12:59,953 --> 00:13:04,120 Indonesia, and Washington DC is the capital of the United States. 149 00:13:04,120 --> 00:13:09,731 Let's conclude Paris is the capital of France, and Jakarta is the capital of 150 00:13:09,731 --> 00:13:14,530 Indonesia, and Washington D.C. is the capitol of the United States. 151 00:13:14,530 --> 00:13:16,656 Valid or not? Well, clearly, 152 00:13:16,656 --> 00:13:21,680 that argument is valid and the truth table shows us why. 153 00:13:21,680 --> 00:13:27,554 The conclusion Paris is the capital of France, and Jakarta is the capital of 154 00:13:27,554 --> 00:13:31,984 Indonesia, and Washington D.C. is the capital of the United States is 155 00:13:31,984 --> 00:13:37,196 true just when it's true that Paris is the capital of France, and it's also true 156 00:13:37,196 --> 00:13:42,407 that Jakarta is the capital of Indonesia, and it's also true that Washington D.C. 157 00:13:42,407 --> 00:13:47,228 is the capital of the United States. So whenever the premises are true, the 158 00:13:47,228 --> 00:13:51,268 conclusion is also true, and that's why that argument is valid. 159 00:13:51,268 --> 00:13:54,590 The truth table explains why the argument is valid. 160 00:13:54,590 --> 00:14:00,867 Now consider a different one. From the premise Mick Jagger is a singer, 161 00:14:00,867 --> 00:14:07,944 a man, and a septuagenarian. We could draw the conclusion Mick Jagger is a 162 00:14:07,944 --> 00:14:11,722 septuagenarian. Now is that argument valid? 163 00:14:11,722 --> 00:14:16,877 Yes, it is and the truth table for conjunction explains why it's valid. 164 00:14:16,877 --> 00:14:22,842 Think about the situation in which it's true that Mick Jagger is a man, a singer, 165 00:14:22,842 --> 00:14:27,334 and a septuagenarian. The only situation in which that's true 166 00:14:27,334 --> 00:14:33,225 is the situation in which it's true that Mick Jagger is a man, it's true that Mick 167 00:14:33,225 --> 00:14:38,380 Jagger is a singer, and it's true that Mick Jagger is a septuagenarian. 168 00:14:38,380 --> 00:14:43,738 But that means that if the premises is true, then the conclusion has got to be 169 00:14:43,738 --> 00:14:46,897 true. The premise is only true in a situation 170 00:14:46,897 --> 00:14:52,461 in which the conclusion is true and so that argument has got to be valid and the 171 00:14:52,461 --> 00:14:58,094 truth table for conjunction explains why. I have said that conjunction can be used 172 00:14:58,094 --> 00:15:02,010 to connect two other propositions into a new proposition. 173 00:15:02,010 --> 00:15:07,230 And conjunction can also be used to connect three other propositions into a 174 00:15:07,230 --> 00:15:10,909 new proposition. But there's no limit to the number of 175 00:15:10,909 --> 00:15:15,865 propositions that can be connected using the truth functional connective 176 00:15:15,865 --> 00:15:21,799 conjunction, or as we could say, there's no limit to the number of propositions 177 00:15:21,799 --> 00:15:26,530 that can be conjoined. You can conjoin four propositions, five 178 00:15:26,530 --> 00:15:32,658 propositions, or however many you like, and notice that there's a pattern to the 179 00:15:32,658 --> 00:15:35,994 truth tables for all of these conjunctions. 180 00:15:35,994 --> 00:15:40,712 In every case, the conjoined proposition is going to be 181 00:15:40,712 --> 00:15:45,810 true only when all of the propositions that are conjoined in it are true. 182 00:15:45,810 --> 00:15:50,838 Now, I'ld like you to take several minutes and look at the following truth 183 00:15:50,838 --> 00:15:54,099 tables, and identify which of these truth tables 184 00:15:54,099 --> 00:15:58,108 are truth tables for conjunction and which of them are not. 185 00:15:58,108 --> 00:16:02,728 Well, that's it for our discussion of conjunction and reasoning with 186 00:16:02,728 --> 00:16:06,329 conjunctions. In the next lecture we'll introduce the 187 00:16:06,329 --> 00:16:10,066 topic of disjunction and reasoning with disjunctions. 188 00:16:10,066 --> 00:16:11,493 See you next lecture.