1 00:00:04,600 --> 00:00:10,760 Today's lecture is about conditionals, to introduce the topic of conditionals. 2 00:00:10,760 --> 00:00:16,559 Let me start by telling you a story. Imagine that my next door neighbor Travis 3 00:00:16,559 --> 00:00:22,328 has been breaking into my house while I'm gone, and stealing my stuff. 4 00:00:22,328 --> 00:00:27,750 I get suspicious about this, so I hire a private investigator to follow Travis 5 00:00:27,750 --> 00:00:31,851 everywhere he goes. He follows Travis to his house at night. 6 00:00:31,851 --> 00:00:35,327 He follows Travis out of his house in the morning. 7 00:00:35,327 --> 00:00:39,660 And he follows Travis everywhere he goes during the day. 8 00:00:39,660 --> 00:00:46,521 Now, you might ask me is the private investigator right now having lunch at 9 00:00:46,521 --> 00:00:53,212 that Cuban Restaurant, New Havana?. I'll say, well if Travis is having lunch 10 00:00:53,212 --> 00:00:57,609 there, then the private investigator is having lunch there. 11 00:00:57,609 --> 00:01:03,295 Notice that we just used the words if and then to connect two propositions. 12 00:01:03,295 --> 00:01:08,980 We connected the proposition, Travis is eating lunch at New Havana, with the 13 00:01:08,980 --> 00:01:14,060 proposition, the private investigator is eating lunch at New Havana. 14 00:01:14,060 --> 00:01:17,485 By saying, if Travis is eating lunch at New Havana, 15 00:01:17,485 --> 00:01:21,595 then the private investigator is eating lunch at New Havana. 16 00:01:21,595 --> 00:01:27,007 Thus, we built a new proposition, out of those first two propositions connecting 17 00:01:27,007 --> 00:01:32,323 them with the words if and then. So, the words if and then together work 18 00:01:32,323 --> 00:01:37,850 as a propositional connective. Let's consider how well you understand 19 00:01:37,850 --> 00:01:43,056 the propositional connective if then. Suppose you have four cards, 20 00:01:43,056 --> 00:01:48,983 this card has the number three on it, this card has the number eight on it, 21 00:01:48,983 --> 00:01:54,590 this card has the letter A on it, and this card has the letter X on it. 22 00:01:54,590 --> 00:01:59,636 Now consider this claim, if a card has a vowel on one side, then 23 00:01:59,636 --> 00:02:07,769 it has an even number on the other side. Now I ask you, which card or cards, would 24 00:02:07,769 --> 00:02:14,480 you need to flip over in order to verify that this claim is true? 25 00:02:26,240 --> 00:02:34,304 Well, the right answer is this card with the letter A on it and this card with the 26 00:02:34,304 --> 00:02:39,910 number three on it. Now, statistics tell me that 70% of you 27 00:02:39,910 --> 00:02:48,680 answered that you would need to flip over the card with the number eight on it. 28 00:02:48,680 --> 00:02:53,219 And for that reason, you need to learn something, about 29 00:02:53,219 --> 00:02:56,752 conditionals. But do they work as a truth functional 30 00:02:56,752 --> 00:03:00,239 connective? Well, remember they do workers of 31 00:03:00,239 --> 00:03:05,138 truth-functional connective. Only if the truth value of the 32 00:03:05,138 --> 00:03:11,863 proposition that you use them to build depends solely on the truth values of the 33 00:03:11,863 --> 00:03:15,350 propositions that you use them to connect. 34 00:03:15,350 --> 00:03:20,702 So, does the truth value of if Travis is eating lunch at new Havana, then the 35 00:03:20,702 --> 00:03:26,195 private investigator is eating lunch at New Havana, depends solely on the truth 36 00:03:26,195 --> 00:03:29,575 value of Travis's is eating lunch at new Havana? 37 00:03:29,575 --> 00:03:34,787 And on the truth value of the private investigator is eating lunch at new 38 00:03:34,787 --> 00:03:36,686 Havana? Well, let's see. 39 00:03:36,686 --> 00:03:42,941 We can try to construct a truth table and see if the truth value of the whole 40 00:03:42,941 --> 00:03:47,583 proposition depends solely on the truth values of its two ingredient 41 00:03:47,583 --> 00:03:50,879 propositions. So, to see whether the propositional 42 00:03:50,879 --> 00:03:56,396 connective if then is a truth functional connective or not, let's see if we can do 43 00:03:56,396 --> 00:04:00,500 a truth table for it. So, we consider four possible situations. 44 00:04:00,500 --> 00:04:05,478 In one situation, it's true that Travis is at New Havana, and true that the 45 00:04:05,478 --> 00:04:09,755 private eye is at New Havana. In the second possible situation, it's 46 00:04:09,755 --> 00:04:14,346 true that Travis is at New Havana, but false that the private eye is at New 47 00:04:14,346 --> 00:04:17,222 Havana. And the third, it's false that Travis is 48 00:04:17,222 --> 00:04:20,528 at New Havana, but true that the private eye is at New 49 00:04:20,528 --> 00:04:23,221 Havana. And in the fourth possible situation, 50 00:04:23,221 --> 00:04:28,239 it's false that Travis is at New Havana, and also false that the private eye is at 51 00:04:28,239 --> 00:04:31,422 New Havana. Now, what can we say about the situations 52 00:04:31,422 --> 00:04:36,196 in which it's true that if Travis is at New Havana, then the private eye is at 53 00:04:36,196 --> 00:04:40,813 New Havana. Well, one thing that's clear is that if 54 00:04:40,813 --> 00:04:49,002 it's true that Travis is at New Havana, but it's false that the private eye is at 55 00:04:49,002 --> 00:04:54,917 New Havana. Then this proposition right here is going 56 00:04:54,917 --> 00:05:00,234 to be false. So in the situation represented on the 57 00:05:00,234 --> 00:05:08,783 second line, the proposition if Travis is at New Havana, then the private eye is at 58 00:05:08,783 --> 00:05:16,060 New Havana is going to be false. But what about the other situations? 59 00:05:16,060 --> 00:05:21,580 Suppose, Travis is at New Havana and the private eye is at New Havana. 60 00:05:21,580 --> 00:05:27,260 Well, if they're both at New Havana, then it seems pretty clear that the 61 00:05:27,260 --> 00:05:33,500 statement if Travis is at New Havana, then the private eye is at New Havana is 62 00:05:33,500 --> 00:05:39,125 going to turn out to be true. But you might worry. 63 00:05:39,125 --> 00:05:43,509 Wait a second, just because the first proposition is 64 00:05:43,509 --> 00:05:50,252 true and the second proposition is true, does that mean that combining them with 65 00:05:50,252 --> 00:05:55,900 the propositional connective if then results in a true proposition? 66 00:05:55,900 --> 00:06:01,885 You might think not necessarily, because consider the following example. 67 00:06:01,885 --> 00:06:07,132 Consider this example, if two plus two equals four, then Pierre 68 00:06:07,132 --> 00:06:12,899 is the capital of South Dakota. Now what I've done there is used the 69 00:06:12,899 --> 00:06:19,260 propositional connective if then to connect two propositions, both of which 70 00:06:19,260 --> 00:06:23,416 are true. It's true that two plus two equals four, 71 00:06:23,416 --> 00:06:28,760 and it's also true that Pierre is that capital of South Dakota. 72 00:06:28,760 --> 00:06:33,336 But just because I've used the proportional connective if then to 73 00:06:33,336 --> 00:06:38,398 connect two propositions that are true, does that mean that the resulting 74 00:06:38,398 --> 00:06:41,865 proposition, if two plus two equals four then Pierre 75 00:06:41,865 --> 00:06:47,160 is the capital of South Dakota. That the resulting proposition is true. 76 00:06:47,160 --> 00:06:51,727 Well, it's not clear. If I tell you, if two plus two equals 77 00:06:51,727 --> 00:06:55,573 four, then Pierre is the capital of South Dakota. 78 00:06:55,573 --> 00:06:59,820 Do you want to say that's true? Or that's false? 79 00:06:59,820 --> 00:07:04,627 Well, it's not obviously true, but it's also not obviously false. 80 00:07:04,627 --> 00:07:09,511 It's just kind of baffling. But just because this proposition is 81 00:07:09,511 --> 00:07:15,845 baffling doesn't mean it's not true, look it's baffling that Pierre is that capital 82 00:07:15,845 --> 00:07:21,644 of South Dakota but that doesn't mean it's not true, as it happens it's true 83 00:07:21,644 --> 00:07:26,910 that Pierre is the capital of South Dakota, baffling as that might be. 84 00:07:26,910 --> 00:07:30,801 So, just because this proposition of two2=4, plus two euqals four, then Pierre 85 00:07:30,801 --> 00:07:35,591 is the capital of South Dakota. Just because that proposition is 86 00:07:35,591 --> 00:07:38,660 baffling, doesn't mean that it's not true. 87 00:07:38,660 --> 00:07:41,951 So is it true? Let's find out. 88 00:07:41,951 --> 00:07:47,400 So suppose you have two propositions, P and Q. 89 00:07:47,400 --> 00:07:52,196 P could be the proposition, Travis is at New Havana and Q could be the 90 00:07:52,196 --> 00:07:55,940 proposition, the private eye is in New Havana. 91 00:07:55,940 --> 00:08:04,197 Well, let's suppose, that the negation of P, and the negation of Q is true. 92 00:08:04,197 --> 00:08:08,856 Suppose that, for a moment. What follows? 93 00:08:08,856 --> 00:08:14,899 What, what follows is that P and the negation of Q, that proposition right 94 00:08:14,899 --> 00:08:19,370 there is false. Remember, because every proposition has 95 00:08:19,370 --> 00:08:24,089 the truth value opposite the truth value of its negation. 96 00:08:24,089 --> 00:08:30,794 So, if the negation of that proposition is true then that proposition itself must 97 00:08:30,794 --> 00:08:35,308 be false. Now if that proposition is false, then 98 00:08:35,308 --> 00:08:41,727 what does that tell us? Well it tells us that P and not Q cannot 99 00:08:41,727 --> 00:08:47,488 both be true. So if P is true, than not Q has got to be 100 00:08:47,488 --> 00:08:51,970 false. Right, if P and not Q were both true, 101 00:08:51,970 --> 00:08:57,305 then this conjunction would be true and not false. 102 00:08:57,305 --> 00:09:03,320 So if P is true, then not Q has got to be false. 103 00:09:03,320 --> 00:09:11,710 But if not Q is false, then Q is true. because Q is got to have the opposite 104 00:09:11,710 --> 00:09:17,029 truth value of not Q. So what can we conclude? 105 00:09:17,029 --> 00:09:26,838 We can conclude that if the negation of P and not Q is true, then if P is true then 106 00:09:26,838 --> 00:09:32,842 Q is true. If this proposition right here is true, 107 00:09:32,842 --> 00:09:41,812 then this proposition right here is true. But notice, we could conclude something 108 00:09:41,812 --> 00:09:45,968 else. Suppose that if P is true, then Q is 109 00:09:45,968 --> 00:09:50,720 true. What follows from that? 110 00:09:50,720 --> 00:09:57,283 Well, one thing that follows from it IS that, you can't have it be the case. 111 00:09:57,283 --> 00:10:00,920 That P, ends up being true but Q is false. 112 00:10:02,040 --> 00:10:09,286 Let's write that down. It can't be the case, 113 00:10:09,286 --> 00:10:20,400 it's not the case that P is true and Q is false, in other words not Q is true. 114 00:10:22,200 --> 00:10:27,020 It can't be the case that P is true and that not Q is true. 115 00:10:29,080 --> 00:10:35,274 But that's just what we started with. So here's what we've shown. 116 00:10:35,274 --> 00:10:40,654 We've shown that if the negation, of P and not Q is true. 117 00:10:40,654 --> 00:10:45,362 Then it follows, that if P is true then Q is true. 118 00:10:45,362 --> 00:10:49,590 And we've also shown, that if the proposition. 119 00:10:49,590 --> 00:10:54,970 If P is true then Q is true, if that proposition is true? 120 00:10:54,970 --> 00:10:58,237 Then it follows, that the negation. 121 00:10:58,237 --> 00:11:11,599 of P and not Q is going to be true. So, this proposition implies this one and 122 00:11:11,599 --> 00:11:19,375 this one implies this one. Which means, that whenever this one is 123 00:11:19,375 --> 00:11:27,085 true this one is got to be true and whenever this one is true this one is got 124 00:11:27,085 --> 00:11:32,026 to be true. So in other words, they are true in all 125 00:11:32,026 --> 00:11:38,253 the same situations. So if there is a truth table for this its 126 00:11:38,253 --> 00:11:43,393 going to be the very same as the truth table for this. 127 00:11:43,393 --> 00:11:49,873 But is there a truth table for this? Lets figure out whether there is a truth 128 00:11:49,873 --> 00:11:53,316 table for this. So remember if you're going to do the 129 00:11:53,316 --> 00:11:58,962 truth table for this we have to start by considering the four possible situations 130 00:11:58,962 --> 00:12:04,333 we have with the propositions P and Q. It could be true that P and true that Q, 131 00:12:04,333 --> 00:12:09,635 true that P and false that Q, false that P and true the Q or false that P and 132 00:12:09,635 --> 00:12:12,733 false that Q. Okay, so those are our four, four 133 00:12:12,733 --> 00:12:16,727 possible situations. Well consider the truth table for the 134 00:12:16,727 --> 00:12:21,112 negation of Q, for not Q. Well that's going to be false whenever Q 135 00:12:21,112 --> 00:12:24,779 is true, and it's going to be true whenever Q is false. 136 00:12:24,779 --> 00:12:30,388 So that's the, the truth table for not Q. Now how about the conjunction of P and 137 00:12:30,388 --> 00:12:34,072 not Q. That's only going to be true when P is 138 00:12:34,072 --> 00:12:38,323 true and not Q is true. In other words, in this situation. 139 00:12:38,323 --> 00:12:42,422 This is the only situation where P and Q will be true. 140 00:12:42,422 --> 00:12:48,418 In every other situation it'll be false. It'll be false wherever P is false and 141 00:12:48,418 --> 00:12:51,440 then will be false wherever not Q is false. 142 00:12:51,440 --> 00:12:56,160 Now how about the negation of P and not Q. 143 00:12:56,160 --> 00:13:02,632 Well the negation of P and not Q, is going to true, whenever P and not Q is 144 00:13:02,632 --> 00:13:07,919 false and its going to be false whenever P and not Q is true. 145 00:13:07,919 --> 00:13:13,120 So, its true here, false here, true here and true here. 146 00:13:13,120 --> 00:13:17,800 Now does this truth table look familiar to you? 147 00:13:17,800 --> 00:13:24,332 Yeah, you remember, it's the same as the truth table for not P or Q. 148 00:13:24,332 --> 00:13:30,240 The truth table that we constructed in last lecture. 149 00:13:30,240 --> 00:13:37,579 So remember, the negation of P and not Q is true. 150 00:13:37,579 --> 00:13:44,920 In just those cases, when if P then Q is true. 151 00:13:44,920 --> 00:13:51,563 These two propositions are going to be true or false in all the same situations. 152 00:13:51,563 --> 00:13:57,431 So if there is a truth table for the negation of P and not Q, then there is 153 00:13:57,431 --> 00:14:04,420 going to be a truth table for if P then Q and its going to be the same truth table. 154 00:14:04,420 --> 00:14:10,434 So lets get back to our Travis example and see how to fill it out. 155 00:14:10,434 --> 00:14:17,450 So applying that discussion to this example, we can now say that whenever its 156 00:14:17,450 --> 00:14:24,370 false that Travis is at New Havana. That no matter where the private eye is, 157 00:14:24,370 --> 00:14:31,926 its going to be, true that if Travis is at New Havana, then the private eye is at 158 00:14:31,926 --> 00:14:36,590 New Havana. So, this is the truth table for if then. 159 00:14:36,590 --> 00:14:40,537 And so if then is a truth-functional connective. 160 00:14:40,537 --> 00:14:45,966 I've just argued that in English the phrase if then can work as a 161 00:14:45,966 --> 00:14:52,546 truth-functional connective. But there are other phrases in English that work in 162 00:14:52,546 --> 00:14:58,879 exactly the same way, to express exactly the same truth-functional connective. 163 00:14:58,879 --> 00:15:01,840 One such phrase is only if. If I say, 164 00:15:01,840 --> 00:15:07,324 if Travis is in New Havana, then the private eye is in New Havana. 165 00:15:07,324 --> 00:15:13,998 What I'm saying means the same thing as Travis is in New Havana only if the 166 00:15:13,998 --> 00:15:18,948 private eye is in New Havana. Those two propositions are going to be 167 00:15:18,948 --> 00:15:24,936 true, under just the same circumstances. Another word that we can use in English 168 00:15:24,936 --> 00:15:30,468 to express the very same truth functional connective is just the word if. 169 00:15:30,468 --> 00:15:35,850 We could say the private eye is in New Havana, if Travis is in New Havana. 170 00:15:35,850 --> 00:15:40,440 And that means the very same thing. And it expresses the very same 171 00:15:40,440 --> 00:15:45,518 proposition, as if Travis is in New Havana, then the private eye is in New 172 00:15:45,518 --> 00:15:48,162 Havana. We're going to use a horseshoe to 173 00:15:48,162 --> 00:15:51,640 symbolize the truth functional connective, if then. 174 00:15:51,640 --> 00:15:58,822 And we can use the truth table for if then to show that a certain kind of 175 00:15:58,822 --> 00:16:06,101 inference, a kind of inference that we're going to call modus ponens is always 176 00:16:06,101 --> 00:16:10,080 valid. Now a modus ponens inference is an 177 00:16:10,080 --> 00:16:17,165 inference that has two premises. The first premise has the form if P then 178 00:16:17,165 --> 00:16:20,973 Q. The second premise has the form P, 179 00:16:20,973 --> 00:16:28,626 in other words the second premise just says that the first proposition in if P 180 00:16:28,626 --> 00:16:34,920 then Q is true. And the conclusion is here. 181 00:16:34,920 --> 00:16:42,245 So the first premise tells us that we're in a situation in which P horseshoe Q is 182 00:16:42,245 --> 00:16:45,908 true. So the first premise rules out this 183 00:16:45,908 --> 00:16:51,179 situation right here. It tells us that this situation right 184 00:16:51,179 --> 00:16:57,790 here isn't the actual situation. The second premise tells us that we're in 185 00:16:57,790 --> 00:17:03,710 a situation in which P is true. So, the second premise rules out this 186 00:17:03,710 --> 00:17:10,181 situation where P is false, and it rules out this situation where P is false. 187 00:17:10,181 --> 00:17:16,737 So, the two premises together rule out all three of these possible situations. 188 00:17:16,737 --> 00:17:23,634 In other words, according to the premise P horseshoe Q and the premise P together, 189 00:17:23,634 --> 00:17:31,100 the only possible situation is this one. But notice, if this is the only possible 190 00:17:31,100 --> 00:17:36,930 situation, then Q has gotta be true, because Q is true in this situation. 191 00:17:36,930 --> 00:17:44,195 So whenever we have an inference, the first premise of which is P horseshoe Q, 192 00:17:44,195 --> 00:17:49,450 the second premise of which is P. We know we can draw a conclusion to the 193 00:17:49,450 --> 00:17:53,230 effect that Q and that inference is going to be valid. 194 00:17:53,230 --> 00:17:58,410 There's no possible way for the two premises of that inference to be true 195 00:17:58,410 --> 00:18:03,240 while the conclusion is false. And that's why modus ponens is a valid 196 00:18:03,240 --> 00:18:07,370 inference. There's another kind of inference called 197 00:18:07,370 --> 00:18:13,041 the modus tollens. A modus tollens inference is one in which 198 00:18:13,041 --> 00:18:19,868 the first premise says if P then Q. The second premise says not Q, 199 00:18:19,868 --> 00:18:26,458 and the conclusion is not P. Now we can use the truth table for if 200 00:18:26,458 --> 00:18:32,314 then to show that a modus tollens inference is always valid. 201 00:18:32,314 --> 00:18:36,403 Here's how, in order to see that modus tollens is a 202 00:18:36,403 --> 00:18:42,092 valid form of inference, we need a slightly more extensive truth table than 203 00:18:42,092 --> 00:18:46,870 we used to show that modus ponens is a valid form of inference. 204 00:18:46,870 --> 00:18:50,738 So again, remember we have four possible situations. 205 00:18:50,738 --> 00:18:56,881 P and Q are both true, P is true and Q is false, P is false and Q is true or P and 206 00:18:56,881 --> 00:19:01,053 Q are both false. Now, when P and Q are both true, then P 207 00:19:01,053 --> 00:19:05,173 horseshoe Q is true. When P is true and Q is false, then P 208 00:19:05,173 --> 00:19:10,780 horseshoe Q is false and whenever P is false then P horseshoe Q is true. 209 00:19:10,780 --> 00:19:16,026 Now the negation of Q is going to be false whenever Q is true and true 210 00:19:16,026 --> 00:19:20,193 whenever Q is false. And the negation of P is going to be 211 00:19:20,193 --> 00:19:24,800 false whenever P is true and true whenever P is false. 212 00:19:24,800 --> 00:19:31,417 Now, using this truth table, we can prove that modus tollens is a valid form of 213 00:19:31,417 --> 00:19:35,489 inference. Remember, the first premise of a modus 214 00:19:35,489 --> 00:19:39,476 tollens inference is, if P then Q, P horseshoe Q. 215 00:19:39,476 --> 00:19:46,093 So, that first premise rules out the possible situation where P horseshoe Q is 216 00:19:46,093 --> 00:19:49,995 false. So, we can rule out that situation right 217 00:19:49,995 --> 00:19:56,674 here. The second premise of a modus tollens 218 00:19:56,674 --> 00:20:04,094 inference, says, not Q and that rules out the situation in which not Q is false. 219 00:20:04,094 --> 00:20:09,020 There're two subsituations. There's this one, 220 00:20:09,020 --> 00:20:15,380 that's one where not Q is false and there's this one where not Q is false. 221 00:20:16,640 --> 00:20:24,500 So, the two premises of a modus tollens inference rule out these three possible 222 00:20:24,500 --> 00:20:31,068 situations, and they leave the only possible situation as this one. 223 00:20:31,068 --> 00:20:38,431 This situation where P is false, Q is false, P horseshoe Q is true, not Q is 224 00:20:38,431 --> 00:20:43,502 true and not P is true. But since that's the only possible 225 00:20:43,502 --> 00:20:49,413 situation left open by the premises of a modus tollens inference, and since that's 226 00:20:49,413 --> 00:20:55,511 the situation in which not P is true. The conclusion of a modus tollens 227 00:20:55,511 --> 00:21:01,776 inference, namely not P is going to have to be true whenever the premises of that 228 00:21:01,776 --> 00:21:06,194 inference are true. So every modus tollens inference is 229 00:21:06,194 --> 00:21:09,890 valid. The truth functional connective if then, 230 00:21:09,890 --> 00:21:15,808 is what we call a conditional. Each conditional has two propositions 231 00:21:15,808 --> 00:21:22,102 that it connects, a proposition that comes before the horseshoe, right after 232 00:21:22,102 --> 00:21:25,205 the if part. That's called the antecedent. 233 00:21:25,205 --> 00:21:29,686 And then there's a proposition that comes after the horseshoe. 234 00:21:29,686 --> 00:21:33,661 Right after the then part, that's called the consequent. 235 00:21:33,661 --> 00:21:37,780 For every conditional has an antecedent and a consequent. 236 00:21:37,780 --> 00:21:43,056 And the conditional is true whenever the antecedent is false, or when the 237 00:21:43,056 --> 00:21:46,741 antecedent is true and the consequent is also true. 238 00:21:46,741 --> 00:21:52,234 The only time a conditional is false is when the antecedent is true and the 239 00:21:52,234 --> 00:21:54,330 consequence is false. Finally, 240 00:21:54,330 --> 00:22:00,385 I want to wrap up by introducing another kind of truth functional connective that 241 00:22:00,385 --> 00:22:07,330 we call the biconditional. The biconditional is really a conjunction 242 00:22:07,330 --> 00:22:14,238 of two conditionals. It's the conjunction of if P then Q and 243 00:22:14,238 --> 00:22:20,823 if Q then P. What the by conditional says, is, two 244 00:22:20,823 --> 00:22:27,220 propositions P and Q. Have exactly the same truth value. 245 00:22:27,220 --> 00:22:34,877 P is true if and only if Q is true. We were going to use these three 246 00:22:34,877 --> 00:22:38,450 horizontal lines to symbolize the biconditional. 247 00:22:38,450 --> 00:22:41,831 Now, here's the truth table for the biconditional. 248 00:22:41,831 --> 00:22:44,672 Remember we have four possible situations. 249 00:22:44,672 --> 00:22:50,150 P and Q are both true, P is true and Q is false, P is false and Q is true, or P and 250 00:22:50,150 --> 00:22:55,766 Q are both false. Well P biconditional Q is true whenever P 251 00:22:55,766 --> 00:23:01,822 and Q have the same truth value. So, P biconditional Q is going to be true 252 00:23:01,822 --> 00:23:04,975 in this situation, when P and Q are both true. 253 00:23:04,975 --> 00:23:09,879 And it's going to be true in this situation, where P and Q are both false. 254 00:23:09,879 --> 00:23:14,643 But in these two situations, the propositions, P and Q, have different 255 00:23:14,643 --> 00:23:18,357 truth values. So P biconditional Q is going to be false 256 00:23:18,357 --> 00:23:21,370 in both of those two situations. In English, 257 00:23:21,370 --> 00:23:26,504 we express the biconditional using the phrase if and only if. 258 00:23:26,504 --> 00:23:32,963 That concludes our lectures on truth functional connectives and propositional 259 00:23:32,963 --> 00:23:36,176 logic. Next week we'll talk about categorical 260 00:23:36,176 --> 00:23:36,594 logic.