1 00:00:00,800 --> 00:00:06,740 OK. Here we are at week five. Now, this is going to be the most difficult week you'll have done. 2 00:00:06,740 --> 00:00:11,870 We're going to be looking at evaluating arguments again, this time deductive arguments. 3 00:00:11,870 --> 00:00:18,260 And in particular, we've got to look at the notion of validity. And this is a difficult notion to wrap your mind around. 4 00:00:18,260 --> 00:00:24,330 So expect to have a hard slog, but if you keep at it, you'll be fine. 5 00:00:24,330 --> 00:00:34,680 Right. And just a recap, as usual, or last week, if you remember, last week, we evaluated inductive arguments. 6 00:00:34,680 --> 00:00:44,420 Remind me what an inductive argument is. A percentage chance of being right now, I'm not sure that's quite the way I put it. 7 00:00:44,420 --> 00:00:53,750 That's more or less what's more or less likely, given the premises, the conclusion is more or less likely. 8 00:00:53,750 --> 00:00:59,170 Exactly. As opposed to a deduction where if the premises of true, the conclusion must be true. 9 00:00:59,170 --> 00:01:03,850 Okay. So we looked at inductive generalisations and causal generalisations, 10 00:01:03,850 --> 00:01:10,360 arguments from analogy and arguments or authority, all different types of inductive arguments. 11 00:01:10,360 --> 00:01:16,540 And if you remember, we looked at these things. Now, I realised at the end of last week that I hadn't added that one on. 12 00:01:16,540 --> 00:01:21,310 So if you've got you might want to go back to the handouts that you had from last 13 00:01:21,310 --> 00:01:27,580 week and just note that this week's handouts will have that one on as well. 14 00:01:27,580 --> 00:01:33,010 And it's it's fairly obvious what that is. And if it isn't, it will be by the end of today. 15 00:01:33,010 --> 00:01:38,530 OK. So those were the tests for an inductive generalisation. 16 00:01:38,530 --> 00:01:46,870 These were the tests for a causal generalisation. These for the argument from analogy and these for the arguments for authority. 17 00:01:46,870 --> 00:01:52,390 I won't go over them again because as we've started a bit late, I would rather get straight on to today. 18 00:01:52,390 --> 00:01:59,590 But if we have time left at the end of today for questions, if you want to go over any of the ones that we did last week and you can't remember them, 19 00:01:59,590 --> 00:02:05,300 or you're just like a reminder, ask me then and we'll we'll certainly come back to those. 20 00:02:05,300 --> 00:02:10,780 Okay. But this week, we're going to be looking at the distinction between validity and truth. 21 00:02:10,780 --> 00:02:21,730 And that's why videl its validity is important. And, well, we're going to finish both this week and next week with evaluating deductive arguments. 22 00:02:21,730 --> 00:02:25,450 We've actually already mentioned the distinction between validity and truth. 23 00:02:25,450 --> 00:02:33,760 Would anyone like to have a go at telling me what it is? It was about three weeks ago, so quite reasonable, if you don't remember, 24 00:02:33,760 --> 00:02:41,100 and quite reasonable if you get it wrong, because we only mentioned at glancingly. Good. 25 00:02:41,100 --> 00:02:45,090 Yeah. Okay. An argument can be valid, but not true. 26 00:02:45,090 --> 00:02:49,020 Why is that the case? You were right. 27 00:02:49,020 --> 00:02:52,620 Okay. You do. Let's not push it. 28 00:02:52,620 --> 00:03:00,450 No, you were doing very well. You're absolutely right. Statements or sentences or propositions can be true or false. 29 00:03:00,450 --> 00:03:05,520 But arguments can't be true or false. Arguments can only be valid or invalid. 30 00:03:05,520 --> 00:03:09,540 Do you remember that? We talked about that in the first week. 31 00:03:09,540 --> 00:03:15,360 And I said then that one of the things that will give you a way every time to a philosopher or a logician 32 00:03:15,360 --> 00:03:22,260 is if you talk about arguments being true or false or if you talk about sentences being valid or invalid, 33 00:03:22,260 --> 00:03:29,610 because it's like my talking about the table being loud, it makes it clear that I don't understand the meaning of loud or I don't 34 00:03:29,610 --> 00:03:34,650 understand the meaning of table or I'm speaking metaphorically might also help. 35 00:03:34,650 --> 00:03:38,460 So and sentences statements can be true or false. 36 00:03:38,460 --> 00:03:40,410 So premises can be true or false. 37 00:03:40,410 --> 00:03:49,800 Conclusions can be true or false, but arguments can't be an argument can be valid or invalid, but they can't be either true or false. 38 00:03:49,800 --> 00:03:55,280 Okay. So we'll be looking at why that is. In a little more depth today. 39 00:03:55,280 --> 00:04:00,750 And we'll be looking at why validity is important, because one of the interesting things about valid arguments, 40 00:04:00,750 --> 00:04:04,650 as you'll see, is that they can have false conclusions. 41 00:04:04,650 --> 00:04:11,750 And you might ask yourself, why should an argument that has a false conclusion be a useful argument? 42 00:04:11,750 --> 00:04:17,790 Never mind a good argument, but we'll see why in this session. 43 00:04:17,790 --> 00:04:25,110 Okay. Right. First, we're going to deal with the idea of sounds nice and an argument, 44 00:04:25,110 --> 00:04:33,240 a good deductive argument is sound if and only if it's both valid and it has two premises. 45 00:04:33,240 --> 00:04:39,480 So soundness is to do with both validity and the truth of the premises. 46 00:04:39,480 --> 00:04:45,860 So. Okay, is this argument sound? 47 00:04:45,860 --> 00:04:51,140 Put up your hands rather than yell out. Put up your hands if you think this argument is sound. 48 00:04:51,140 --> 00:04:56,850 Do you see what I mean? This argument is valid and has false premises. 49 00:04:56,850 --> 00:05:02,450 It could this argument be sound. Put up your hand rather than shout out. 50 00:05:02,450 --> 00:05:06,900 Right. Anyway, you've all done it now, and you're quite right. No, that can't be sounds, can it? 51 00:05:06,900 --> 00:05:11,570 Okay, what about this one? Put up your hand because some people think more quickly than others. 52 00:05:11,570 --> 00:05:16,440 So give everyone the chance to think. So you've decided to put up your hand. 53 00:05:16,440 --> 00:05:21,680 OK. That's most of you. What is it? No, it's got to be unsound, hasn't it? 54 00:05:21,680 --> 00:05:26,960 It couldn't possibly be both. What about this one? Okay. 55 00:05:26,960 --> 00:05:30,910 What are you saying to this one? No. Okay. Again, this one can't be valid. 56 00:05:30,910 --> 00:05:35,920 So what about this one? Yes. You can yell out this time. 57 00:05:35,920 --> 00:05:45,600 Okay. Yes, that's right. That's got to be some good. Okay, so there's the same thing filled in. 58 00:05:45,600 --> 00:05:54,260 Okay. Now. So the two things that are both necessary for a sound argument or a good argument 59 00:05:54,260 --> 00:06:00,410 insofar as sounds and ESCOs truth of the premises and the validity of the argument. 60 00:06:00,410 --> 00:06:03,050 Now the truth of the premises, and I've mentioned this before, 61 00:06:03,050 --> 00:06:09,620 is not an interesting matter for logicians or for you, given that your interest is in critical reasoning. 62 00:06:09,620 --> 00:06:17,240 And the reason for that is that when we determine the truth or falsity of a premise, it's not necessarily a logical matter. 63 00:06:17,240 --> 00:06:27,920 It's not necessarily a philosophical matter at all. So if I say, is the chair blue, how are you going to determine the truth or falsehood of that? 64 00:06:27,920 --> 00:06:32,120 It's dead easy. There's no trick question here. You're going to look at the chair. 65 00:06:32,120 --> 00:06:36,680 That's right. Okay. If I say two plus two equals four. 66 00:06:36,680 --> 00:06:40,490 Is that true or not? Okay. Yes, it is. 67 00:06:40,490 --> 00:06:44,770 Okay. And how do you know? Yes. 68 00:06:44,770 --> 00:06:51,010 I mean, you will you will know your basic arithmetic, okay? And you might say, okay, all swans are white. 69 00:06:51,010 --> 00:06:54,610 How do you determine the truth of that answer? Inductively and so on. 70 00:06:54,610 --> 00:06:59,230 So there are lots of different ways in which a premise would be determined to be true or false. 71 00:06:59,230 --> 00:07:06,220 And my job in teaching critical reasoning is not to teach you how to look at a premise and decide whether it's true or false. 72 00:07:06,220 --> 00:07:11,310 My job is to teach you how to look at an argument and determine whether it's valid. 73 00:07:11,310 --> 00:07:20,530 Okay. Are you with me? So. So when two characteristics of a of a sound argument, one is true premises, one is validity. 74 00:07:20,530 --> 00:07:28,360 I leave it to you to ask about true premises. All I can teach you is what validity looks like, what validity is. 75 00:07:28,360 --> 00:07:42,180 Okay. So. Yeah. Okay, validity is of interest to logicians because validity preserves truth if an argument is valid. 76 00:07:42,180 --> 00:07:47,100 Then if its premises are true, we can be certain that its conclusion is true. 77 00:07:47,100 --> 00:07:53,940 So it's the F here that's important. If the premises are true, then the conclusion must be true. 78 00:07:53,940 --> 00:08:00,120 And so the validity of an argument preserves the truth of the premises. 79 00:08:00,120 --> 00:08:03,900 If there's truth there, it will preserve it. 80 00:08:03,900 --> 00:08:08,850 If there isn't truth there. It won't say anything about the truth of the conclusion. 81 00:08:08,850 --> 00:08:12,300 So we'll see more about that in a minute. Okay. 82 00:08:12,300 --> 00:08:17,280 So validity is truth preserving. That's why it's useful. 83 00:08:17,280 --> 00:08:21,420 So validity is of interest to anyone who's concerned with truth. 84 00:08:21,420 --> 00:08:25,560 It's not just true premises that concern you if you're interested in truth. 85 00:08:25,560 --> 00:08:35,160 It's also the validity of arguments. And that's because we often don't know the truth of our premises and we often test the truth 86 00:08:35,160 --> 00:08:41,430 of our premises by constructing valid arguments and testing the truth of the conclusion. 87 00:08:41,430 --> 00:08:50,490 So let me go through that again. Okay. If an argument is valid, then if the premises are true, the conclusion must be true. 88 00:08:50,490 --> 00:08:57,750 Okay, so we might have a premise that we don't know whether the whether the premise is true or not. 89 00:08:57,750 --> 00:09:07,920 But if we can use that premise as the premise of a valid argument and then test the conclusion of that argument, see whether that's true or false. 90 00:09:07,920 --> 00:09:15,910 And then if we can show that the conclusion of a valid argument is false, what do we know? 91 00:09:15,910 --> 00:09:24,400 If we can show that the conclusion of a valid. Put your hand up, if we can show that the conclusion of a valid argument is false. 92 00:09:24,400 --> 00:09:33,280 What do we know? Let me remind you again of the argument is valid if and only if if the premises are true. 93 00:09:33,280 --> 00:09:40,600 The conclusion must be true. So if we see that the conclusion of a valid argument is false. 94 00:09:40,600 --> 00:09:46,680 What do we know? Put your hands up when you've got the answer. Little bit longer. 95 00:09:46,680 --> 00:09:54,020 It's quite difficult to play with these concepts in your mind if you're not used to so. 96 00:09:54,020 --> 00:09:58,640 Okay. Right. What do we know? Who wants to give me the answer? 97 00:09:58,640 --> 00:10:03,840 And George can have a go. Yeah. Right. 98 00:10:03,840 --> 00:10:07,560 At least one of the premises must be false. That's right. 99 00:10:07,560 --> 00:10:14,010 They might all be false or it might be just one of them is false. But we know that at least one of them must be false. 100 00:10:14,010 --> 00:10:22,470 So if an argument is valid and the conclusion is false, then we know that at least one of the premises must be false. 101 00:10:22,470 --> 00:10:26,820 And here's a couple of examples. OK. Smoking causes cancer. 102 00:10:26,820 --> 00:10:30,720 That's a hypothesis. We've base that. Okay. Where do we get that from? 103 00:10:30,720 --> 00:10:36,600 What sort of arguments might that be? The conclusion of an inductive argument. 104 00:10:36,600 --> 00:10:42,520 Good. So we see that this smokers got cancer, this smokers got cancer, the smokers got cancer, et cetera. 105 00:10:42,520 --> 00:10:46,890 And we formed the hypothesis smoking causes cancer. 106 00:10:46,890 --> 00:10:56,700 Okay. That generates a prediction, doesn't it? If we put that hypothesis as starkly as that smoking causes cancer, it generates a prediction. 107 00:10:56,700 --> 00:11:01,920 If smoking causes cancer, then every smoker will get cancer, doesn't it? 108 00:11:01,920 --> 00:11:03,300 Oh, yes, it does. No. 109 00:11:03,300 --> 00:11:12,280 No, remember I said if we put the conclusion as starkly as that, sorry, the premises starkly is that that's the prediction it generates. 110 00:11:12,280 --> 00:11:19,000 Okay. The test or the conclusion of this argument is each smoker gets cancer. 111 00:11:19,000 --> 00:11:23,940 OK. Now, if we look at smokers, we see that some of them get cancer and some of them don't. 112 00:11:23,940 --> 00:11:31,520 Don't we. OK. So what do we know? This is a valid argument, isn't it? 113 00:11:31,520 --> 00:11:36,020 Yeah. OK. But we've seen that the conclusion is false. 114 00:11:36,020 --> 00:11:40,580 So what do we know that one of the premises is now? 115 00:11:40,580 --> 00:11:49,010 It might be this. It's not the case that smoking causes cancer or it might be the case that smoking could cause cancer without its being the case. 116 00:11:49,010 --> 00:11:56,460 Every smoker would cause cancer. So what do we do if we're researchers at this point? 117 00:11:56,460 --> 00:12:01,010 What might be the case? We've got a correlation between smoking and cancer. 118 00:12:01,010 --> 00:12:07,460 Haven't we? Otherwise, we wouldn't have that first premise. But it was not going to be an exception. 119 00:12:07,460 --> 00:12:19,160 This correlation, is it? We've seen that. So we could say we could decide to jettison the idea that smoke that causation is sufficient. 120 00:12:19,160 --> 00:12:28,430 So we might want to say, well, okay, it's possible for A to cause B without its B in the case that A is sufficient for B, C, what I mean. 121 00:12:28,430 --> 00:12:39,380 So we could jettison premise too. So possibly A can cause B without its being the case that you always get B if you get A. 122 00:12:39,380 --> 00:12:45,230 Actually we probably wouldn't want to jettison that would we. Why not. 123 00:12:45,230 --> 00:12:48,720 OK. I think I'm going about this, looking at your faces. I'm getting a bit wrong. 124 00:12:48,720 --> 00:12:55,650 Okay. Let let's say that we decide to jettison that. Does that mean we're going to say that's false? 125 00:12:55,650 --> 00:13:05,280 What are we going to say? Instead, we could say that all we could say that smoking plus something else we could say we. 126 00:13:05,280 --> 00:13:11,340 We know the hypothesis is going in the right direction, but it's it's smoking. 127 00:13:11,340 --> 00:13:17,940 Plus, I mean, maybe there's some genetic characteristic of people who smoke and get cancer that's 128 00:13:17,940 --> 00:13:23,220 different from those who smoke and don't get cancer or something along those lines. 129 00:13:23,220 --> 00:13:28,200 Do you see what I mean? You've got so we've got something for which we've got we've got reason to believe it's true, 130 00:13:28,200 --> 00:13:33,630 but we're not sure or actually I mean, given what we know, we're pretty sure it isn't entirely true. 131 00:13:33,630 --> 00:13:38,180 But if we put these together in a valid argument, we get that which we know to be false. 132 00:13:38,180 --> 00:13:42,030 So we know that either one of those has got to be false. 133 00:13:42,030 --> 00:13:50,370 We could play around with something that's the business of philosophers to play around with, namely our concept of causation and say, 134 00:13:50,370 --> 00:13:56,700 well, okay, there can be causes that don't always have their effects, that don't necessitate their effects. 135 00:13:56,700 --> 00:14:04,210 Or we could stick to being empirical and say there's something in addition to smoking that that. 136 00:14:04,210 --> 00:14:14,040 Causes cancer. You with me? If smoking always caused cancer, you couldn't have a smoker that didn't get cancer. 137 00:14:14,040 --> 00:14:18,360 So that would be true, wouldn't it? So because that's false. 138 00:14:18,360 --> 00:14:22,500 We know that one or the other of these is true. 139 00:14:22,500 --> 00:14:26,670 And if you're a philosopher, you might want to say, well, maybe causes don't necessitate. 140 00:14:26,670 --> 00:14:30,960 Their effects could premise to be true. Sorry. False. 141 00:14:30,960 --> 00:14:36,840 If you're a scientist, you're much more likely to say actually causes do necessitate their effects. 142 00:14:36,840 --> 00:14:40,740 Therefore, it's not smoking on its own that causes cancer. 143 00:14:40,740 --> 00:14:46,710 There must be something in addition to smoking that goes to causing cancer. 144 00:14:46,710 --> 00:14:54,380 By definition. Necessitates a factory, you. 145 00:14:54,380 --> 00:15:01,700 Well, you say by definition. But, of course, part of the job of a philosopher is to say, well, is is it true that that's the correct definition? 146 00:15:01,700 --> 00:15:07,400 Cause. Could could there be. No. 147 00:15:07,400 --> 00:15:12,710 If it doesn't cause it is not a cause it. But if it doesn't necessitate it, might it cause it. 148 00:15:12,710 --> 00:15:20,610 I mean if we look at quantum mechanics, for example, here, here are some ideas that there may be causation that doesn't necessitate. 149 00:15:20,610 --> 00:15:24,830 And one of the big questions of philosophy and if you ever do an introduction 150 00:15:24,830 --> 00:15:28,550 to philosophy as opposed to an introduction to critical reasoning course, 151 00:15:28,550 --> 00:15:32,750 you might look at questions of could there be backwards causation, for example? 152 00:15:32,750 --> 00:15:38,540 Could the effects come before a cause? Could there be non necessitating causes? 153 00:15:38,540 --> 00:15:45,680 What actually is the causal relation? These are all philosophical questions that are true. 154 00:15:45,680 --> 00:15:53,180 No, no, they don't have a different concept, of course. What they do is they unpack the concept, of course, that we think we have. 155 00:15:53,180 --> 00:15:59,240 And I quite agree that the common concept, the cause is that causes necessitating. 156 00:15:59,240 --> 00:16:06,340 Absolutely. Well, then what's the call what do you mean by random here? 157 00:16:06,340 --> 00:16:15,720 Because, well, then your say you're suggesting that a cause isn't necessitating, which is what? 158 00:16:15,720 --> 00:16:20,740 I've got your name. Sorry, Brian is is questioning. Call. 159 00:16:20,740 --> 00:16:29,560 Well, so you disagree on the meaning of the word, cause you can have that conversation later in the common room. 160 00:16:29,560 --> 00:16:33,040 But I'll tell you now that if you want to find out what the answer to that is, 161 00:16:33,040 --> 00:16:39,070 you should come to a philosophy weekend on the nature of causation, because there it's a big issue. 162 00:16:39,070 --> 00:16:52,840 That one. Very big one. You had a question. Well, that that would lead you. 163 00:16:52,840 --> 00:16:55,640 That's why you know that each smoker doesn't. 164 00:16:55,640 --> 00:17:04,540 But you if you're evaluating the inductive argument that leads to that, you might want to deal with percentages and so on. 165 00:17:04,540 --> 00:17:09,790 Exactly. But what I'm pointing out here is that you form a hypothesis, 166 00:17:09,790 --> 00:17:16,540 you generate a prediction which is going to be based on on your understanding of causation and so on. 167 00:17:16,540 --> 00:17:24,190 And then you test that prediction. And look, you've what you've done is you've constructed a deductive argument. 168 00:17:24,190 --> 00:17:33,070 And as long as it's valid and you test the conclusion, if the conclusion comes out false, then you know that at least one of your premises is false. 169 00:17:33,070 --> 00:17:37,270 Either your hypothesis is false or there's something fishy about causation. 170 00:17:37,270 --> 00:17:43,000 And you're almost certainly not going to go for that one in this. You're a philosopher. Okay, here's another one. 171 00:17:43,000 --> 00:17:47,900 All women are passive. Mrs. Thatcher is a woman. Therefore, Mrs. Thatcher is passive. 172 00:17:47,900 --> 00:17:52,030 OK. You've got a hypothesis here. OK, you've generated that on. 173 00:17:52,030 --> 00:17:57,130 On what grounds? None at all. 174 00:17:57,130 --> 00:18:04,130 None at all, he says. I think the grounds of prejudice or something like that, say inductive grounds as well here. 175 00:18:04,130 --> 00:18:10,150 This is observation, isn't it? And do we have a valid argument here? 176 00:18:10,150 --> 00:18:14,090 Yeah. If these premises are true, this conclusion must be true. 177 00:18:14,090 --> 00:18:18,130 Okay. The the conclusion isn't true. I think you'd probably agree with me. 178 00:18:18,130 --> 00:18:21,530 Therefore, one or other of these must be false. 179 00:18:21,530 --> 00:18:26,400 And as we know, an awful lot of people talked about, Mrs. Thatcher has been the best man in the cabinet. 180 00:18:26,400 --> 00:18:35,330 Door to door, in effect, they were relying on this argument and suggesting that instead of that one being false, it's this one that's false. 181 00:18:35,330 --> 00:18:42,740 And here's another one. Tony Blair is sorry. All Labour Party members are left to your all socialists, the left wing. 182 00:18:42,740 --> 00:18:47,990 Tony Blair is a socialist. Therefore, Tony Blair's left wing. 183 00:18:47,990 --> 00:18:51,710 And if you think Tony Blair is far too right-wing to be a socialist, 184 00:18:51,710 --> 00:19:00,050 you're going to say that Tony Blair is into socialist rather than not all socialists da off that came off the top of my head. 185 00:19:00,050 --> 00:19:07,040 I may have got bits of that wrong, but I hope you can see the general thrust of it as we're going to see in a minute. 186 00:19:07,040 --> 00:19:11,750 You can have a valid arguments, the conclusion of which is false. So you might ask. 187 00:19:11,750 --> 00:19:14,180 Well then what's what's the point of a valid argument? 188 00:19:14,180 --> 00:19:20,510 Why should we be interested in validity if it can generate arguments, the conclusion to which is false. 189 00:19:20,510 --> 00:19:31,040 And I'm saying, well, actually, the discovery of a valid argument that its conclusion is false can be hugely important. 190 00:19:31,040 --> 00:19:41,840 And the reason it can be important is that we often test our premises by constructing a valid arguments and testing the conclusion. 191 00:19:41,840 --> 00:19:52,820 Okay, so I just. Well, I just made it clear just to make it clear that we were testing a hypothesis by constructing a valid argument 192 00:19:52,820 --> 00:20:00,060 showing the conclusion was false and then saying that we can question either or both of the premises. 193 00:20:00,060 --> 00:20:06,120 OK, so you can just substitute premise, premise, conclusion, no problem. 194 00:20:06,120 --> 00:20:10,770 Exactly the same. So it's just that we use premises in different ways, don't we? 195 00:20:10,770 --> 00:20:18,180 Sometimes we use premises as hypotheses. Sometimes we use them as predictions and so on. 196 00:20:18,180 --> 00:20:25,170 Okay. So I hope I've convinced you that the relation of validity is important to you. 197 00:20:25,170 --> 00:20:33,630 If you're concerned about truth and what makes it important is that validity preserves truth if there's truth in the premises. 198 00:20:33,630 --> 00:20:37,680 There will be truth in the conclusion if there isn't truth in the premises. 199 00:20:37,680 --> 00:20:47,250 There won't necessarily be truth in the conclusion. And that might tell you that the premises are not true, which might be very useful information. 200 00:20:47,250 --> 00:20:50,490 In fact, as is often very useful information, because, of course, 201 00:20:50,490 --> 00:20:58,980 the whole of science involves testing hypotheses and and hoping, some would say, to falsify them. 202 00:20:58,980 --> 00:21:05,190 Okay. So we're. So why is this relationship have validity so important? 203 00:21:05,190 --> 00:21:11,280 What exactly is it? Well, there are different theories of the nature of validity. 204 00:21:11,280 --> 00:21:15,960 And there are some problems, as we'll see later on, with the notion of validity. 205 00:21:15,960 --> 00:21:21,300 But what I'm going to give you now is the best theory that we can come up with. 206 00:21:21,300 --> 00:21:22,740 An argument is valid. 207 00:21:22,740 --> 00:21:33,450 If and only if a notice this logical phrase here, if and only if there is no possible situation in which all its premises are true. 208 00:21:33,450 --> 00:21:42,630 Remember, it must be all of them. And its conclusion false. Okay, that's that's a claim about a valid argument. 209 00:21:42,630 --> 00:21:45,990 Okay, let's move on. Okay. Two things to beware of. 210 00:21:45,990 --> 00:21:55,240 Firstly, it's the possibility of the combination of true premises and false conclusion that's ruled out by an argument's being valid. 211 00:21:55,240 --> 00:22:04,050 Okay. So it's not that. It must have true premises or it must have a true conclusion. 212 00:22:04,050 --> 00:22:08,250 What it must have in order to be valid is true premises. 213 00:22:08,250 --> 00:22:16,640 So what it must not have in order to be valid is the combination of true premises and false conclusion. 214 00:22:16,640 --> 00:22:22,480 That's completely ruled out by an argument's being valid. 215 00:22:22,480 --> 00:22:26,160 And we're going to do some exercise in a minute. 216 00:22:26,160 --> 00:22:31,800 Also, the second thing to note and second thing to be aware of, and this is going to trip you up, boy, 217 00:22:31,800 --> 00:22:39,840 are you going to leave today having felt you've really had a mental gymnastics because you're all going to get this wrong? 218 00:22:39,840 --> 00:22:43,590 Probably. Maybe you won't. I mean, maybe you're cleverer than most people are. 219 00:22:43,590 --> 00:22:50,370 But every class I've ever taught this to starts off by getting it wrong in almost every aspect. 220 00:22:50,370 --> 00:22:52,740 So don't be worried if you do. 221 00:22:52,740 --> 00:23:03,780 It's the possibility of the combination of true premises and false conclusions that that's ruled out, not just the actuality of that combination. 222 00:23:03,780 --> 00:23:08,070 Okay, so it's not just the premises, as a matter of fact, are true. 223 00:23:08,070 --> 00:23:12,330 And the conclusion, as a matter of fact, is false. That's ruled out. 224 00:23:12,330 --> 00:23:16,770 It's the very possibility of the premises being true. 225 00:23:16,770 --> 00:23:20,340 And the conclusion false that's ruled out. Okay. 226 00:23:20,340 --> 00:23:28,070 Big difference between actuality and possibility. Marianne is actually wearing a skirt today, but she might have been wearing jeans. 227 00:23:28,070 --> 00:23:35,250 Okay. So there is a possible world in which Marion's wearing jeans, even though in the actual world she's wearing a skirt. 228 00:23:35,250 --> 00:23:42,330 Okay. So faced with an argument whose validity we're trying to determine, we've got to ask. 229 00:23:42,330 --> 00:23:49,230 Cannot just all the premises true. And the conclusion false together in actuality. 230 00:23:49,230 --> 00:23:56,040 But could the premises be true? And the conclusion false together in some situation. 231 00:23:56,040 --> 00:24:02,970 Okay. Do you see the difference? You're all looking very worried. 232 00:24:02,970 --> 00:24:08,600 Okay, let's let's try a few. Let's test a few, okay? 233 00:24:08,600 --> 00:24:12,230 I'm going to leave that with you. You have a lot. 234 00:24:12,230 --> 00:24:16,250 Put your hand up when you think, well, actually, let's do the first one. 235 00:24:16,250 --> 00:24:24,650 Do you think the arguments could an argument could be valid if all its premises are false? 236 00:24:24,650 --> 00:24:28,790 Put your hands up. If you think you've got the answer, don't yell out. 237 00:24:28,790 --> 00:24:33,070 And don't worry, if you're a bit slower. Everyone thinks different speeds. 238 00:24:33,070 --> 00:24:39,710 OK. If the premises of an argument are false. Could the argument be valid? 239 00:24:39,710 --> 00:24:46,720 Could the argument be valid if the premises are false? 240 00:24:46,720 --> 00:24:51,760 OK. Those who said yes are right. OK. 241 00:24:51,760 --> 00:24:57,580 You could have a valid argument in which the premises are false. 242 00:24:57,580 --> 00:25:02,800 The only thing that's ruled out by an argument is being valid is the possibility 243 00:25:02,800 --> 00:25:11,070 of the combination of the premises being true and the conclusion false. 244 00:25:11,070 --> 00:25:18,950 OK, so the only thing that's ruled out by an argument's being valid is the possibility, 245 00:25:18,950 --> 00:25:26,090 not the actuality of the combination of the premises being true and the conclusion false. 246 00:25:26,090 --> 00:25:31,040 So you could have a valid argument. The premises, which are all false. 247 00:25:31,040 --> 00:25:37,550 If it's Friday, Marianne's wearing jeans. It is Friday. Therefore, Marianne is wearing jeans. 248 00:25:37,550 --> 00:25:43,450 Isn't that an argument that has the two false premises but is valid? 249 00:25:43,450 --> 00:25:50,660 Okay. Right. What about the fact that the premises of the argument are true and the conclusion is true? 250 00:25:50,660 --> 00:25:56,810 OK. Remember the question. Do you think an argument of this sort could be valid? 251 00:25:56,810 --> 00:26:02,190 The premises are true. And the conclusion is true. OK. 252 00:26:02,190 --> 00:26:06,360 Could it be valid or not? Well done. 253 00:26:06,360 --> 00:26:10,770 Well done. Getting there. OK. That's pretty spectacular. 254 00:26:10,770 --> 00:26:15,630 OK, good. What about the last one? The premises of the argument are true and the conclusion false. 255 00:26:15,630 --> 00:26:19,750 Put your hands up. Don't yell out the answer. OK. 256 00:26:19,750 --> 00:26:24,910 Some of you say yes, some of you say no. The ones who say no in this case are right. 257 00:26:24,910 --> 00:26:33,490 Okay. The only thing that's ruled out by the definition of validity is an argument that's valid where there's a possibility. 258 00:26:33,490 --> 00:26:39,640 And in this case, there's an actuality of the combination of premises that are true and conclusion false. 259 00:26:39,640 --> 00:26:43,270 That's the only one that's ruled out. And that is this one. 260 00:26:43,270 --> 00:26:47,860 So an argument of this kind couldn't be valid. 261 00:26:47,860 --> 00:26:54,690 The other two could. Okay. That's why there's plenty of time to practise with its finesses. 262 00:26:54,690 --> 00:27:00,940 OK. So so let's think again if the premises could be true together with. 263 00:27:00,940 --> 00:27:04,750 So do you remember I said two things you need to beware of. What are they? 264 00:27:04,750 --> 00:27:08,260 Two things I said you've got to beware of. What are they? 265 00:27:08,260 --> 00:27:13,990 Look back on your notes if you can't remember. Because I really meant it that you must be aware of these things. 266 00:27:13,990 --> 00:27:17,140 Two things you must be aware of. 267 00:27:17,140 --> 00:27:24,490 It's the combination of true premises and false conclusion that's ruled out not either one or the other, but both together. 268 00:27:24,490 --> 00:27:30,040 So if the premises could be true, together with the conclusions being false. 269 00:27:30,040 --> 00:27:37,330 Okay, that's that's one of the things you're looking out for. What's the other one? It's the possibility of the combination rather than the actuality. 270 00:27:37,330 --> 00:27:41,680 So if the conclusion could be false. 271 00:27:41,680 --> 00:27:46,450 Together with the premises being true, if they could be the case, not just that. 272 00:27:46,450 --> 00:27:53,250 If it is the case, if it could be the case, then the argument. 273 00:27:53,250 --> 00:27:59,970 Is valid, is that right? Sorry I lost track myself there. 274 00:27:59,970 --> 00:28:09,600 Yes, because there's only deductive arguments that are truth, preserving that where the conclusion is guaranteed by the truth of the premises. 275 00:28:09,600 --> 00:28:15,690 Some people do talk about inductive validity and deductive validity. 276 00:28:15,690 --> 00:28:24,300 But if they do, they're using valid in two different senses. And I think it's actually much better to stick to the idea of validity with deductive 277 00:28:24,300 --> 00:28:31,400 arguments and talk about inductive strengths rather than inductive validity. 278 00:28:31,400 --> 00:28:36,390 Because validity is an either all thing, whereas strength is a matter of degree. 279 00:28:36,390 --> 00:28:43,590 An inductive arguments are more or less strong, whereas deductive arguments are either valid or invalid. 280 00:28:43,590 --> 00:28:47,980 Okay. What is it what does an argument have to have to be sounds? 281 00:28:47,980 --> 00:28:57,610 Can anyone tell me two things it needs in order to be sound to them to premises and be valid? 282 00:28:57,610 --> 00:29:02,050 That's right. So that's Sounness. And that's fairly easy. We can put that on one side. 283 00:29:02,050 --> 00:29:14,160 We're now looking at validity. One of those characteristics. Last night. 284 00:29:14,160 --> 00:29:18,420 Because, well, as I'm about to show you, it doesn't mean it necessarily is valid. 285 00:29:18,420 --> 00:29:24,600 It could be valid. Valid. Is an either or thing. 286 00:29:24,600 --> 00:29:33,480 For every argument. But if you if you have a situation where the premises have one truth, 287 00:29:33,480 --> 00:29:38,910 value and the conclusion, another truth value, that could encompass different arguments. 288 00:29:38,910 --> 00:29:43,290 I'll show you exactly what I mean. OK. So just keep this in mind. 289 00:29:43,290 --> 00:29:47,070 If the premises could be true together with the conclusions being false. 290 00:29:47,070 --> 00:29:51,420 Then the argument is invalid. Otherwise, it could be valid. 291 00:29:51,420 --> 00:29:58,950 Okay. Now, this is a bit of a trick question. But and I'll explain why. 292 00:29:58,950 --> 00:30:03,120 Actually, it isn't a trick question afterwards. OK, here's an argument. 293 00:30:03,120 --> 00:30:09,240 Now, remember, I told you right at the very beginning that anything can be an argument depending on the context. 294 00:30:09,240 --> 00:30:14,430 Okay, so you may not find this a very convincing argument, but believe me, it's an argument. 295 00:30:14,430 --> 00:30:19,860 These two plus two equals five. Therefore, grass is green. 296 00:30:19,860 --> 00:30:26,530 Okay. That's no arguments. Could that be valid? 297 00:30:26,530 --> 00:30:32,170 OK. Now, don't try and be clever here. Don't try and be clever here. 298 00:30:32,170 --> 00:30:38,020 Work it out actually from the definition which you can't read there, but you can read in your notes. 299 00:30:38,020 --> 00:30:46,600 So have a look at that. Is there a situation in which the premises could be true and the conclusion false? 300 00:30:46,600 --> 00:30:51,370 Is there a situation, any situation in which that is true? 301 00:30:51,370 --> 00:31:01,140 And that is false. Is it could it be that that is true and that false in any possible world? 302 00:31:01,140 --> 00:31:06,800 No. OK. So is that argument valid? 303 00:31:06,800 --> 00:31:12,260 Who said yes? Somebody said yes. And if they would admit to it, I would then give them a clap. 304 00:31:12,260 --> 00:31:23,960 Well done. It is. That's a valid argument. And what that shows you is that actually for an argument to be good, it means it needs more than validity. 305 00:31:23,960 --> 00:31:29,750 OK. That is a perfectly valid argument because there is no possible situation in which that premise is true. 306 00:31:29,750 --> 00:31:35,720 And that conclusion false. And the reason for that is there is no possible situation in which that premise is true. 307 00:31:35,720 --> 00:31:44,490 Is the. That premise is a contradiction. And you may have heard the saying anything follows from a contradiction. 308 00:31:44,490 --> 00:31:51,360 And the reason anything follows from a contradiction is that a contradiction must be false. 309 00:31:51,360 --> 00:31:54,600 So there's no possible situation in which the premise is true. 310 00:31:54,600 --> 00:32:00,660 Therefore, there couldn't possibly be a situation in which the premise is true and the conclusion false. 311 00:32:00,660 --> 00:32:06,330 Could there? So that satisfies the definition of validity. 312 00:32:06,330 --> 00:32:14,460 And that's what I want you to go on. I did. Because what shall I tell you why you're all sitting there looking so worried? 313 00:32:14,460 --> 00:32:18,680 I feel for you. I really feel for you. But I guess I went through this. I honestly, I did. 314 00:32:18,680 --> 00:32:22,530 I tell you, when I was an undergraduate, we did logic in my first term. 315 00:32:22,530 --> 00:32:30,510 And I sat there looking bit like some of you looking through the whole term thinking, oh, what have I got myself in for? 316 00:32:30,510 --> 00:32:35,790 And then during the vacation, I sat there with my logic book and I worked through it step by step by step. 317 00:32:35,790 --> 00:32:39,780 And I went over it again and again and again and again. 318 00:32:39,780 --> 00:32:48,030 And the things that I had been missing all along was that I should have been taking more account of definitions than I had been. 319 00:32:48,030 --> 00:32:54,150 I was doing what you're almost certainly doing now. True is a good thing, isn't it? 320 00:32:54,150 --> 00:33:02,470 True sort of gives you a nice warm feeling, all sorts of warm and cuddly and validity gives you that same sort of nice, warm feeling, doesn't it? 321 00:33:02,470 --> 00:33:07,380 You know, so therefore, anything that's true is valid and anything that's valid is true. 322 00:33:07,380 --> 00:33:11,590 And all this business about false premises, you know, has nothing to do with. 323 00:33:11,590 --> 00:33:15,690 Is that do you think this might be behind what. Yeah. Yes. 324 00:33:15,690 --> 00:33:27,420 Okay. Does that help? So try and drop the fact that both truth and validity make you feel all warm inside and just go for the definition. 325 00:33:27,420 --> 00:33:33,200 What does this argument satisfy the definition of validity or does it not? 326 00:33:33,200 --> 00:33:37,830 Could this argument satisfies definition validity or could it not? 327 00:33:37,830 --> 00:33:41,850 That's what you answer on, not on your nice, warm feelings of truth and validity. 328 00:33:41,850 --> 00:33:50,940 Going together because sadly, they don't. Except in the one case of truth being preserved by true premises. 329 00:33:50,940 --> 00:33:56,950 Okay. Does everyone see why this is a valid argument? Would you like me to go over it again? 330 00:33:56,950 --> 00:34:03,580 Yeah. OK, sorry. OK. 331 00:34:03,580 --> 00:34:08,800 OK. The definition of validity is there is no possible sitt to an argument is valid. 332 00:34:08,800 --> 00:34:13,090 If and only if there is no possible situation in which the premises are true. 333 00:34:13,090 --> 00:34:19,420 And the conclusion false. And what matters is that it's a possible situation, not an actual situation. 334 00:34:19,420 --> 00:34:24,710 And that is the combination of true premises and false conclusion that that's ruled out. 335 00:34:24,710 --> 00:34:31,410 OK, that's the definition of validity. Now, here we have an argument. 336 00:34:31,410 --> 00:34:36,180 OK. Could this be valid? Well, is there a situation? 337 00:34:36,180 --> 00:34:40,570 Could it be the case? Could it be the case? That's what I'm asking. 338 00:34:40,570 --> 00:34:47,040 That this premise is true together with this premise being false. 339 00:34:47,040 --> 00:34:56,360 Well, you look at this and you see that there actually isn't any situation ever anywhere in which this could be true. 340 00:34:56,360 --> 00:35:06,070 Is that true? So how could there ever be a situation in which that's true and that's false? 341 00:35:06,070 --> 00:35:11,500 Couldn't be. Could the. With me. Yeah, I can see it's getting there, isn't it? 342 00:35:11,500 --> 00:35:18,250 I can see understanding dawning on about 50 percent of faces. 343 00:35:18,250 --> 00:35:25,290 The other 50 percent are looking pretty sick. Okay. 344 00:35:25,290 --> 00:35:29,410 Yes. OK. 345 00:35:29,410 --> 00:35:36,560 If I say that something's P and Q. And then I say that, OK. 346 00:35:36,560 --> 00:35:47,450 Is that true? You don't know do. But if I then say not pee and say, is this true, what do you know? 347 00:35:47,450 --> 00:35:51,440 No, you don't. No, not you. But do you know the truth value of this? 348 00:35:51,440 --> 00:35:57,230 If PE's falls. What's the truth? Value of P and Q? 349 00:35:57,230 --> 00:36:05,120 False. It must be, mustn't it? Because a necessary condition for P and Q being true is that the P and Q True. 350 00:36:05,120 --> 00:36:09,280 Sorry, but it isn't that you knew that immediately in the same way. 351 00:36:09,280 --> 00:36:15,830 If I say there is no possible situation where this is true and this is false. 352 00:36:15,830 --> 00:36:22,550 Well if there's no possible situation in which this is true. Then how could there be a situation which this is true. 353 00:36:22,550 --> 00:36:32,060 And this is false. Another couple of nights went on, then I'll do another one. 354 00:36:32,060 --> 00:36:39,220 Let's let's see another one. Could this argument be valid? 355 00:36:39,220 --> 00:36:51,110 Put your hands up when you've got the answer. Don't yell it out. Don't cheque it out. 356 00:36:51,110 --> 00:37:06,280 Just. Nobody's got it yet. 357 00:37:06,280 --> 00:37:18,850 Oh, yeah. One hands up to. Well done. 358 00:37:18,850 --> 00:37:25,450 So you're asking yourself, could there be a possible situation in which that is true and that's false? 359 00:37:25,450 --> 00:37:31,590 Is there any possible situation in which that's true and that's false together? 360 00:37:31,590 --> 00:37:37,570 So any situation which that's true and that's false together. 361 00:37:37,570 --> 00:37:53,720 Put your hand up when you've got the answer. Yes. 362 00:37:53,720 --> 00:37:58,370 No, you're you're doing exactly what I'm warning against. They're yours. 363 00:37:58,370 --> 00:38:05,240 Did you hear what lady said? She said she thought a valid argument was where the premise was were true and the conclusion true. 364 00:38:05,240 --> 00:38:11,360 Now, that's a classic example of thinking that because validity is good and truth is good. 365 00:38:11,360 --> 00:38:15,300 Therefore, a valid argument must have true premises and true conclusion. 366 00:38:15,300 --> 00:38:22,730 Do you see how that works? But actually, a valid argument doesn't have true premises and true conclusions. 367 00:38:22,730 --> 00:38:30,710 What is a valid argument? Give me give you give me the definition of a valid argument. 368 00:38:30,710 --> 00:38:34,620 It's. You've got it written down in front of you. Oh, no, you haven't got it written there anymore. 369 00:38:34,620 --> 00:38:40,880 It's on slide 19. 370 00:38:40,880 --> 00:38:52,930 I'm glad you asked me that question, because you will be far from the only person in the room who's thinking, well, exactly what you're thinking. 371 00:38:52,930 --> 00:38:59,980 And the complete unknown. The two things that are important is that we're looking at the combination of true premises and false conclusion. 372 00:38:59,980 --> 00:39:05,860 And we're looking at the possibility of not the actuality of this is why the definition is important. 373 00:39:05,860 --> 00:39:13,690 And as soon as you understand that, as soon as you stop thinking of a valid arguments as having true premises and true conclusion, 374 00:39:13,690 --> 00:39:18,270 you will see what's going on here. I promise you. 375 00:39:18,270 --> 00:39:25,260 It's a bit rash, isn't it? Addenda don't give me the answer yet. 376 00:39:25,260 --> 00:39:31,520 OK. Put your hands up if you've got the answer to that. Is that all? Or could that argument be valid? 377 00:39:31,520 --> 00:39:41,880 Hand up. If you got it right up so I can see you. OK. 378 00:39:41,880 --> 00:39:47,490 Is it valid? Good. 379 00:39:47,490 --> 00:39:51,960 Yes, it is valid. Those who said yes have got it right. OK. 380 00:39:51,960 --> 00:39:59,280 Now, why is it valid? OK. Would anyone like to give me a crisp and clear account of why it's valuable or should I do it? 381 00:39:59,280 --> 00:40:07,210 Would you like to have a go? That's right. 382 00:40:07,210 --> 00:40:14,440 This can't possibly be false. Can it? So there is no possible situation in which the conclusion is false. 383 00:40:14,440 --> 00:40:21,700 So how could there be a possible situation in which the premise is true? And the conclusion false? 384 00:40:21,700 --> 00:40:28,060 See what I mean? Could this argument be valid? Answer yes, actually, as a matter of fact, this argument must be valid. 385 00:40:28,060 --> 00:40:33,820 Can anyone tell me why it must be valid rather than just could be valid? 386 00:40:33,820 --> 00:40:40,210 That can never be false. That's necessarily true, isn't it, given that that's necessarily true? 387 00:40:40,210 --> 00:40:45,580 How could there be a possible possible situation in which that's true and that's false. 388 00:40:45,580 --> 00:40:48,940 Given that that cannot be false. Cannot be false. 389 00:40:48,940 --> 00:40:54,550 It's not just that it is false. It's that it can't be false. And actually, the same is true of the other one. 390 00:40:54,550 --> 00:41:01,120 If you look at this, given that that is is a contradiction, it's necessarily false. 391 00:41:01,120 --> 00:41:04,930 Again, there is no possible situation in which that's true. 392 00:41:04,930 --> 00:41:09,310 So how could it be the case that that's true and that's false? It couldn't be. 393 00:41:09,310 --> 00:41:15,190 So if we say could this argument be valid in each of those cases, the answer is no. 394 00:41:15,190 --> 00:41:21,910 It could. Sorry. Yes, it is. Not only is it valid, but it must be valid. 395 00:41:21,910 --> 00:41:29,410 And why is that? Because there is no possible situation in which the premises are true and the conclusion false. 396 00:41:29,410 --> 00:41:36,360 So when you ask me as the answer to those two questions different, there are different questions. 397 00:41:36,360 --> 00:41:46,000 OK. Let me just tell you that the two arguments I've given you here are called the paradoxes of entailment, the paradoxes of entailment. 398 00:41:46,000 --> 00:41:50,500 And a paradox is something that you cannot understand. 399 00:41:50,500 --> 00:41:58,570 It's just something you can't wrap your mind around. And the reason you'd want to look at these and tell me that they're not arguments, okay. 400 00:41:58,570 --> 00:42:07,870 Is because they couldn't convince you of anything. Could they? I mean, as arguments, they would be pretty lousy because they wouldn't be convincing. 401 00:42:07,870 --> 00:42:12,640 But they're still valid because they satisfy the definition of validity. 402 00:42:12,640 --> 00:42:19,060 So in the same way, we might say, if you think, you know, whales, for example, aren't fish, are they? 403 00:42:19,060 --> 00:42:22,900 Well, they jolly well should be, shouldn't they? I mean, it seems to me they jolly well should be. 404 00:42:22,900 --> 00:42:31,150 They they go swimming around and, you know, if they're fish only, but they're not because they don't satisfy the definition of fish. 405 00:42:31,150 --> 00:42:33,430 They satisfy the definition of mammal. 406 00:42:33,430 --> 00:42:42,550 Well, you might say, well, surely we should fiddle with our definitions so that whales come out fish because they're fish, obviously, but they're not. 407 00:42:42,550 --> 00:42:52,390 And do you see what I mean? Sometimes things that really rather irritating are thrown up by definitions that work for every other case. 408 00:42:52,390 --> 00:42:56,080 And therefore you think, oh, sod it, we'll just put up with these anomalies. 409 00:42:56,080 --> 00:43:04,420 Well, these are the paradoxes of entailment because we wish that our definition of validity didn't generate these two arguments as valid. 410 00:43:04,420 --> 00:43:12,850 But they do. And actually, given that think about everything follows from a contradiction that that is that makes sense, doesn't it? 411 00:43:12,850 --> 00:43:20,980 Because if I say Marion's wearing a skirt and it's not the case, Marion's wearing a skirt that leaves open every possibility, doesn't it? 412 00:43:20,980 --> 00:43:29,380 Everything follows from a contradiction because a contradiction doesn't rule out any possibility at all. 413 00:43:29,380 --> 00:43:35,800 Okay, so those are the paradoxes of entailment. Actually, these are a pretty good test of your understanding of validity, 414 00:43:35,800 --> 00:43:41,800 because when you see that those two arguments, the paradoxes of entailment must be valid. 415 00:43:41,800 --> 00:43:57,300 Then you'll you'll have understood validity. So that's quite a good little way of testing your own understanding of validity. 416 00:43:57,300 --> 00:44:01,030 Yes. And the argument is still. Yes. Yes, absolutely. 417 00:44:01,030 --> 00:44:06,220 That's right. It really doesn't matter. And actually, that's true anywhere because. 418 00:44:06,220 --> 00:44:12,160 Let me. This is actually a theorem. 419 00:44:12,160 --> 00:44:21,340 If you have an argument. I've just seen a can of worms opening up in front of me. 420 00:44:21,340 --> 00:44:31,530 I haven't thought to do this, but let's see if I can do it. OK. If you have a premise here and the premise here, I'm sorry. 421 00:44:31,530 --> 00:44:36,480 Let's. Let's do it this way. 422 00:44:36,480 --> 00:44:46,820 And then to draw a truth table, I haven't actually introduced you to truth tables, but it doesn't really matter because they're very easy. 423 00:44:46,820 --> 00:44:51,950 True, true, false, false, he. 424 00:44:51,950 --> 00:44:59,810 Q Okay, I've got each of these structures is a different world, okay. 425 00:44:59,810 --> 00:45:04,980 This is the world in which Q is true and P is true. 426 00:45:04,980 --> 00:45:10,970 Okay. What's the truth value of P and Q in this world. 427 00:45:10,970 --> 00:45:17,120 True. Good. Well done. Okay, this is the world in which Q is true and P is false. 428 00:45:17,120 --> 00:45:22,540 What's the truth. Value of P and Q in this world. Brilliant. 429 00:45:22,540 --> 00:45:27,140 Okay. This is the world in which Q is false and P is true. 430 00:45:27,140 --> 00:45:32,320 Okay. What's the truth. Value of P and Q in this world. 431 00:45:32,320 --> 00:45:36,530 And here's the world where P and Q are both false. What's the truth. 432 00:45:36,530 --> 00:45:41,570 False again. Okay. So we had true. False, false, false. 433 00:45:41,570 --> 00:45:46,430 Didn't we remember that. Okay. Now this is the world where Q is true and is true. 434 00:45:46,430 --> 00:45:50,120 What's the truth. Five Q In this world. True. 435 00:45:50,120 --> 00:45:55,130 What about this world. True. What about this world? 436 00:45:55,130 --> 00:45:58,730 False. False. That's just Q. 437 00:45:58,730 --> 00:46:02,450 So in the world where Q is false, it's got to be false, doesn't it? Okay. 438 00:46:02,450 --> 00:46:08,320 What about P? True. 439 00:46:08,320 --> 00:46:12,760 False, true, false. OK. That's just copying out that bit of the truth. 440 00:46:12,760 --> 00:46:16,630 Now, here's an argument. Q and P. 441 00:46:16,630 --> 00:46:21,670 Q therefore P. Okay, you're with me now. 442 00:46:21,670 --> 00:46:25,990 We can use this truth table to test whether this argument is valid here. 443 00:46:25,990 --> 00:46:29,710 The two premises are true and the conclusion is true. 444 00:46:29,710 --> 00:46:36,820 So that's okay. That that seems to be valid here. One premise is true, the other is false and the conclusions false. 445 00:46:36,820 --> 00:46:40,240 So that seems to be valid as well. Is that reasonable? 446 00:46:40,240 --> 00:46:46,300 That's not that's not a possible situation in which the premises are both true and the conclusion false. 447 00:46:46,300 --> 00:46:54,140 Yeah. Is this a situation in which the premises are both true and the conclusion false? 448 00:46:54,140 --> 00:46:58,380 No, because the premises are both false and the conclusion true. The other way round. 449 00:46:58,380 --> 00:47:02,180 And what about this? The premises are both false and the conclusion false. So that's. 450 00:47:02,180 --> 00:47:06,500 That could also be valid. So as you'd expect, that argument is valid. 451 00:47:06,500 --> 00:47:12,000 But notice, if I put in any other premise here, it's going to stay valid. 452 00:47:12,000 --> 00:47:17,480 Any other premise? What would you like me to put in here in order to make the argument invalid? 453 00:47:17,480 --> 00:47:22,340 Not P. Yeah. OK. If I've put in not P. 454 00:47:22,340 --> 00:47:27,200 OK. This is the world in which P is true. So not P will be. 455 00:47:27,200 --> 00:47:33,180 No. This is the world in which P is true. So not P will be false. 456 00:47:33,180 --> 00:47:38,360 Okay. This is the world where P is false. So not P will be true. 457 00:47:38,360 --> 00:47:44,150 Good. This is a world where P is true. So not P will be. 458 00:47:44,150 --> 00:47:49,700 And it's false here. So it will be true. So let's test for validity again. 459 00:47:49,700 --> 00:47:55,700 Okay. Now this one in this world. The premises are true, true and false. 460 00:47:55,700 --> 00:48:00,620 And the conclusion is true. Okay. Is that all right or is that a counterexample? 461 00:48:00,620 --> 00:48:07,690 Is that still valid? So it's not the case that all the premises are true. 462 00:48:07,690 --> 00:48:11,880 OK. The conclusion is true. That's okay. It's still valid, isn't it? 463 00:48:11,880 --> 00:48:19,800 What about false? True, true as the premises and the conclusion is false. 464 00:48:19,800 --> 00:48:26,620 Still a case that we haven't yet found, one where we've got true, true, true and then false. 465 00:48:26,620 --> 00:48:33,570 The conclusion, which is what we'd be looking for. Here we've got false, false, false, true. 466 00:48:33,570 --> 00:48:41,470 Yes. And here we've got false, false, true, false. 467 00:48:41,470 --> 00:48:47,380 So, you see, I could add in even a negation of the conclusion, and it hasn't changed the validity. 468 00:48:47,380 --> 00:48:55,090 Once you've got a valid argument, you can add anything you like and it'll still be valid. 469 00:48:55,090 --> 00:48:59,380 I don't know why I went into telling you all that is this is this is information you really didn't need to know. 470 00:48:59,380 --> 00:49:04,060 But it's so interesting, isn't it? I think it's interesting. 471 00:49:04,060 --> 00:49:09,900 Let's move on. Okay. So let's let's test you again. 472 00:49:09,900 --> 00:49:18,810 OK? Don't shout out your answer. So is the argument valid if we've got true premises and a true conclusion? 473 00:49:18,810 --> 00:49:22,020 Could it be valid? Is what I should've asked. Yeah. Hands up if you think. 474 00:49:22,020 --> 00:49:27,400 Yes. Yeah. Okay. You're quite right. False premises. True conclusion. 475 00:49:27,400 --> 00:49:30,790 Hand up. OK. 476 00:49:30,790 --> 00:49:35,610 The premise is a false. The conclusions true. Could it still be valid? 477 00:49:35,610 --> 00:49:40,650 Yep, good. Well done. What about false conclusion, false premises? 478 00:49:40,650 --> 00:49:45,150 Could it still be valid? Yes. 479 00:49:45,150 --> 00:49:48,990 Yes. And true premise is false conclusion. 480 00:49:48,990 --> 00:49:53,850 Could it be valid? No. Some of you guessing this. 481 00:49:53,850 --> 00:49:59,000 Some of you. I thought I'd got there. I've got myself lost in the meantime. 482 00:49:59,000 --> 00:50:03,540 OK, let's. OK. I'm going to look at this another way of looking at it. 483 00:50:03,540 --> 00:50:12,140 Just because you're already probably exhausted. I'm already exhausted. We're going to look at Venn diagrams. 484 00:50:12,140 --> 00:50:16,800 And we're actually going to run out of time in this session. OK. 485 00:50:16,800 --> 00:50:21,390 Here's an example here. Here we have two arguments. 486 00:50:21,390 --> 00:50:27,420 In both cases, the premises are actually true. And the conclusion is actually true. 487 00:50:27,420 --> 00:50:37,250 OK. But one of them is valid and the other one isn't valid. And I'm going to show that to you by means Venn diagrams, if I can find my pen. 488 00:50:37,250 --> 00:50:43,440 So all cat's miaow. OK, here's the class of me hours. 489 00:50:43,440 --> 00:50:48,630 Right? All Katsumi hours. 490 00:50:48,630 --> 00:50:57,870 So where where should I draw the class of all? Inside, outside or overlapping inside? 491 00:50:57,870 --> 00:51:02,940 If all cats me out and then all cats must be inside the class of things that miaow. 492 00:51:02,940 --> 00:51:15,270 Is that right? OK, Bo. Incidentally, I hope you noticed my topical reference to the dog of the United States of America. 493 00:51:15,270 --> 00:51:19,140 Less has died since I. Okay. Bo does not miaow. 494 00:51:19,140 --> 00:51:24,750 So where do I draw the bow here. 495 00:51:24,750 --> 00:51:29,480 Okay. Bo is not a cat. Well is that true. 496 00:51:29,480 --> 00:51:37,500 Yes. Okay, that's true. So if all cat's miaow on bo does not miaow then Bo is not a cat. 497 00:51:37,500 --> 00:51:42,030 There's no possibility is there, in which those are true. And that's not true. 498 00:51:42,030 --> 00:51:47,880 If those are true, that's got to be true. Okay. So that's a valid argument. 499 00:51:47,880 --> 00:51:53,160 Accepted. Okay. And yet the truth value of that is true. 500 00:51:53,160 --> 00:51:59,280 Yes. True, Sally. That is true. Assume that don't doesn't me out. 501 00:51:59,280 --> 00:52:06,450 And that's true as well. Okay. So you accept my claim about the truth values and you accept my claim about the validity. 502 00:52:06,450 --> 00:52:14,730 Okay. Let's have a look at this one. All Cat's miaow. So here's the me hours. 503 00:52:14,730 --> 00:52:20,080 Where do I draw the thing of cats? OK, cats. 504 00:52:20,080 --> 00:52:27,860 OK. Second premise. Dogs are not cats. They're outside the class of caps, aren't they? 505 00:52:27,860 --> 00:52:35,090 But but they they could be either inside or outside or overlapping. 506 00:52:35,090 --> 00:52:41,810 The class of me hours. Couldn't they? So there is a possible situation in which that's true. 507 00:52:41,810 --> 00:52:49,370 And that's true. But that isn't true. And that's the possible situation in which Cat's Miaow. 508 00:52:49,370 --> 00:53:00,110 Now we know that cats don't actually miaow. So glad you're all with me. 509 00:53:00,110 --> 00:53:03,500 But you see, dogs could me out. Could they. As far as that. 510 00:53:03,500 --> 00:53:09,740 I mean, these do not guarantee that. Do they. In the way that these guarantee that it's that. 511 00:53:09,740 --> 00:53:14,790 Right. OK. And yet would you agree with me that that's true? 512 00:53:14,790 --> 00:53:22,470 Yeah, that's true. Is that true? Oh, come on. 513 00:53:22,470 --> 00:53:27,240 Is it true? Is it actually true? Yes, it is. Dogs do not miaow. 514 00:53:27,240 --> 00:53:32,070 But. But that's not a necessary truth, is it? Dogs could miaow. 515 00:53:32,070 --> 00:53:37,350 So there is a possible situation and this does not rule out that possible situation. 516 00:53:37,350 --> 00:53:42,720 So here we have premises where the actual truth values the premises are all true. 517 00:53:42,720 --> 00:53:47,670 The conclusion is true. But here's one of an argument exactly like that that's valid. 518 00:53:47,670 --> 00:53:52,860 And his argument exactly like that is not valid. What's the difference between them? 519 00:53:52,860 --> 00:53:59,910 Answer. In this case, if the premises are true, the conclusion couldn't be false. 520 00:53:59,910 --> 00:54:06,830 OK. Whereas here, if the premises are true, the conclusion could still be false. 521 00:54:06,830 --> 00:54:10,400 OK. If we didn't know, I'm talking about dogs here. 522 00:54:10,400 --> 00:54:17,460 But what if I was talking about. Name an animal, you wouldn't know anything about Griffin's. 523 00:54:17,460 --> 00:54:27,360 Achieve watched your Harry Potter. So that's a bad example, too. But okay, I'm talking about a mythological animal or an animal Venu just discovered. 524 00:54:27,360 --> 00:54:34,440 I can tell you it's a. You unicorns are not cats, but unicorns don't miaow. 525 00:54:34,440 --> 00:54:42,340 You actually wouldn't know the truth value of that, would you? And you'd have to recognise that the truth of those two leaves that one open. 526 00:54:42,340 --> 00:54:48,540 Okay, that's the that's the importance of validity. Here's the next one. 527 00:54:48,540 --> 00:54:55,890 OK, I've got two arguments again. The premises are actually false, I hope, and the conclusion is actually true. 528 00:54:55,890 --> 00:55:01,620 OK. Do you accept that that's false? Is that false? Is that true? 529 00:55:01,620 --> 00:55:06,950 It is. Is that false? 530 00:55:06,950 --> 00:55:15,650 There are fish without scales. I looked it up and and Jewish people either can or can't eat them. 531 00:55:15,650 --> 00:55:20,540 I can't remember which is. But there's a whole table of which fish have scales in which don't. 532 00:55:20,540 --> 00:55:26,030 Okay. Whales have scales. Is that false? Good whales are not fish. 533 00:55:26,030 --> 00:55:32,250 True or false. True. Okay, good. So you you agree with me that the truth value is the actual truth. 534 00:55:32,250 --> 00:55:43,730 Values are like that. Yes. OK, as I described, now let's look at the validity of these arguments, all fish have lungs. 535 00:55:43,730 --> 00:55:53,540 This is the class of lungs, things. Where do I draw the class of fish inside his fish? 536 00:55:53,540 --> 00:56:02,270 If whales have lungs, where have I got to draw the. 537 00:56:02,270 --> 00:56:07,620 Which all fish have lungs? Whales off? 538 00:56:07,620 --> 00:56:17,050 Oh, yeah. Whales are fish. It's got to be there, isn't it? Okay, so it's got to be then if it's there, what's got to be the case? 539 00:56:17,050 --> 00:56:24,550 It's got to have lungs, haven't it, hasn't it? So if those two are true, that's got to be true, except that. 540 00:56:24,550 --> 00:56:29,250 Now, let's have a look at the. Final one. 541 00:56:29,250 --> 00:56:38,400 Sorry, not the fun one. All fish scales. This is class of scales things class fish in that. 542 00:56:38,400 --> 00:56:44,790 OK. Fish whales have scales. So where do I draw whales? 543 00:56:44,790 --> 00:56:51,980 It's got to be within there. Has it got to be here or here. It could be any either. 544 00:56:51,980 --> 00:56:58,170 Couldn't sit and whales are not fish. OK. That tells us that it it is there. 545 00:56:58,170 --> 00:57:02,780 But it could be either, couldn't it, again, with the disease. 546 00:57:02,780 --> 00:57:08,160 The truth of these two. If they were true. Doesn't guarantee the truth of that. 547 00:57:08,160 --> 00:57:12,610 Because you can have scales without being a fish. OK. 548 00:57:12,610 --> 00:57:18,400 So, again, if you look at the explanation underneath. In both cases, the premises are actually false. 549 00:57:18,400 --> 00:57:27,100 And the conclusion is actually true. But in the first place, if the premises were true, the truth is a print of the conclusion. 550 00:57:27,100 --> 00:57:31,510 I should have said there. Can you see scrub it out on your copy. 551 00:57:31,510 --> 00:57:33,550 That should be conclusion. 552 00:57:33,550 --> 00:57:43,780 The truth is, the conclusion would be guaranteed in the second case, even if the premises were true, the conclusion could still be false. 553 00:57:43,780 --> 00:57:52,700 OK. It isn't false, but it could be. Are you beginning to follow? 554 00:57:52,700 --> 00:58:00,210 And here I'll actually I'll I'll skip over these because. Oh, would you like me to do these with you as I've done the others. 555 00:58:00,210 --> 00:58:04,760 You would. Okay. Your gluttons for punishment aren't you. OK. 556 00:58:04,760 --> 00:58:10,500 Firstly, do you agree with me that the truth values are, as I say. 557 00:58:10,500 --> 00:58:14,910 OK. I've said everything is false. Is that false? All fish have wings. 558 00:58:14,910 --> 00:58:19,890 Yep. Wails of fish. Is that false? Whales have wings. 559 00:58:19,890 --> 00:58:25,210 False. False. Okay. All fish have scales. 560 00:58:25,210 --> 00:58:29,900 False whales have scale's false whales, a fish. 561 00:58:29,900 --> 00:58:33,760 Okay, said all the premises and the conclusions are false here. 562 00:58:33,760 --> 00:58:38,980 But this argument is valid and this argument is invalid. 563 00:58:38,980 --> 00:58:45,870 Okay. Why do I say that? Well, if I find my pen again. 564 00:58:45,870 --> 00:58:55,230 All fish have wings. Here's the class of winged things, class of fish is inside. 565 00:58:55,230 --> 00:59:00,700 Whales have wings. Oh, sorry. 566 00:59:00,700 --> 00:59:04,900 Yes, I thank you have suddenly think, oh, my God, it's not valid. 567 00:59:04,900 --> 00:59:11,590 All fish have wings, whales are fish. Okay. Therefore they've got to have wings, haven't they? 568 00:59:11,590 --> 00:59:18,620 That guarantees it. They've got to be inside the class of winged things. What do we got over here? 569 00:59:18,620 --> 00:59:25,520 We've got all we're all fish have scales, so there's the scales things. 570 00:59:25,520 --> 00:59:29,600 And there's the fish. Whales have scales. 571 00:59:29,600 --> 00:59:34,910 Well, they could be either or. Couldn't they? Whales are not fish. 572 00:59:34,910 --> 00:59:41,270 Well, okay. That tells us that whales are here. But is that guaranteed? 573 00:59:41,270 --> 00:59:47,100 Other than the fact that we're saying that that's true. You know, actually, it could be false. 574 00:59:47,100 --> 00:59:52,890 Couldn't it? Even though it is true, it could be false. 575 00:59:52,890 --> 00:59:57,880 And it's not guaranteed by those. Is it? 576 00:59:57,880 --> 01:00:07,570 Because as far as the fact that all fish are in the class of things that have scales, that doesn't tell us where the whales are out here. 577 01:00:07,570 --> 01:00:13,690 So they have scales but are not fish or whether they're in here and they're fish with scales. 578 01:00:13,690 --> 01:00:19,120 We don't know that. This is the structure where you're talking about burning down. 579 01:00:19,120 --> 01:00:32,300 We see the. No. It happens that all the arguments I'm using are straightforward cases and modus Poland. 580 01:00:32,300 --> 01:00:42,270 Poland's. All piece A, Q, p, therefore Q No, hang on. 581 01:00:42,270 --> 01:00:51,210 No, I'm sorry. Go. I've gone past the stage where I'm thinking your homework this week is to go home and work out what the form of that argument is, 582 01:00:51,210 --> 01:00:59,850 because I'm using the same one for each one. What I'm trying to point out is that what matters is the possibility, not the actuality. 583 01:00:59,850 --> 01:01:09,360 It's the possible truth values, not the actual truth values that matter, possible truth values, not the actual truth values. 584 01:01:09,360 --> 01:01:14,010 Okay, here's another way. We've used Venn diagrams to have a look. 585 01:01:14,010 --> 01:01:23,220 But let's have another look. Another way of determining validity is to create a counter set and then determine consistency. 586 01:01:23,220 --> 01:01:27,930 Let's have a look at what we mean by that. Firstly, to determine the counterexamples set. 587 01:01:27,930 --> 01:01:32,610 We set out the argument logic bookstall. Okay, here are two arguments set out. 588 01:01:32,610 --> 01:01:39,060 Logic bookstall. You familiar with these. You've seen them before. Okay. 589 01:01:39,060 --> 01:01:42,750 Now to create. That's the argument set out logic bookstall. 590 01:01:42,750 --> 01:01:49,590 Now I'm going to negate the conclusion and thereby set out the counterexamples set. 591 01:01:49,590 --> 01:01:55,320 So we negate the conclusion by tacking. It is not the case that in front of the conclusion. 592 01:01:55,320 --> 01:02:01,800 So here we are. This is the counter example set of the two arguments I gave you. 593 01:02:01,800 --> 01:02:10,080 And all I've given you is exactly the same arguments, but with the words, it is not the case that tact in front of them. 594 01:02:10,080 --> 01:02:17,190 The next thing we do is we need to consider whether the set consisting of the premises and the negation of the conclusion. 595 01:02:17,190 --> 01:02:21,920 In other words, the counter example set is consistent. 596 01:02:21,920 --> 01:02:33,030 I could they all be true together. If the counter example set is consistent, then the original argument is invalid. 597 01:02:33,030 --> 01:02:36,990 Can anyone see why that's the case, huh? This is really tough. 598 01:02:36,990 --> 01:02:47,040 Why am I doing this to. If the counter example set isn't consistent, then the original argument isn't valid. 599 01:02:47,040 --> 01:02:51,360 Now, again, let me warn you that consistency is a bit like truth. 600 01:02:51,360 --> 01:02:57,750 You think of it as a nice, good, warm thing. And so you tend to think that if it's valid, it must be consistent. 601 01:02:57,750 --> 01:03:04,090 But what I'm suggesting here is if the counter example set is consistent, in other words, it's a counterexample. 602 01:03:04,090 --> 01:03:09,600 Sets can't all be true together. Then the argument is invalid. 603 01:03:09,600 --> 01:03:14,730 Let's have a look. Why? OK. 604 01:03:14,730 --> 01:03:22,040 This is the counterexample set. OK, consisting of the premises, plus the negation of the conclusion. 605 01:03:22,040 --> 01:03:32,430 Do you accept that you're with me on that? Now, what I'm claiming is if this is consistent, then the original argument must have been valid. 606 01:03:32,430 --> 01:03:39,150 Okay, let's have a look at here with. You'll agree with me that the original argument here is invalid, isn't it? 607 01:03:39,150 --> 01:03:43,570 Do you agree with that? Okay. So if it's snowing, the mail will be late. 608 01:03:43,570 --> 01:03:48,990 The mail will be late. Therefore, it's snowing. That's invalid, isn't it? 609 01:03:48,990 --> 01:03:53,880 Because there is a possible situation where the premises are true and the conclusion false. 610 01:03:53,880 --> 01:04:02,180 Well, what we're doing here. Is we're saying, well, let's assume that the conclusion is false. 611 01:04:02,180 --> 01:04:14,870 And let's see if these can all be true together. Well, if they can, then it can't be the case that the original argument was valid, could it? 612 01:04:14,870 --> 01:04:18,850 The I think this is an argument too far, frankly. 613 01:04:18,850 --> 01:04:26,550 So if you're not with me, don't. But actually what you can do is you can cause this bit is dead simple, actually. 614 01:04:26,550 --> 01:04:33,480 And if you go away yourself and you work out, there's several ways in which you can test your understanding of what I've said today. 615 01:04:33,480 --> 01:04:41,210 The first one is if you understand why the paradoxes of entailment, the two arguments I gave you and told you were the paradoxes of intelligence. 616 01:04:41,210 --> 01:04:46,620 If you can work out why they are valid. You will have understood validity. 617 01:04:46,620 --> 01:04:51,000 Second thing is if you understand why this is the case, 618 01:04:51,000 --> 01:05:02,820 why you can test the validity of an argument by appeal to a counterexample sets and why you and you can understand why this is the case. 619 01:05:02,820 --> 01:05:09,870 In other words, if the counterexample sets is consistent, then the original argument is invalid. 620 01:05:09,870 --> 01:05:21,600 And if the counterexample set isn't consistence. Ow, ow, ow, ow, ow. 621 01:05:21,600 --> 01:05:27,450 So you can still see it. My hands, can't you? That's terrible. Right. OK. 622 01:05:27,450 --> 01:05:34,980 You will all see that I put a knot in where I shouldn't to put a knot, which is the worst thing you can do when you're teaching people logic. 623 01:05:34,980 --> 01:05:40,040 So if I will you please all note that that not shouldn't be there. 624 01:05:40,040 --> 01:05:47,130 And so the second test of whether you've understood today's work is whether you can see how this works. 625 01:05:47,130 --> 01:05:52,170 Okay. So if you haven't understood where you are now and if that you're feeling. 626 01:05:52,170 --> 01:05:58,230 I'm never gonna understand another thing again. I don't ever want to see that told. But woman ever again quite understands. 627 01:05:58,230 --> 01:05:59,970 But go home and have a look at it yourself. 628 01:05:59,970 --> 01:06:08,640 And I promise you, if you work through it step by step and just think of the definitions, you will understand. 629 01:06:08,640 --> 01:06:13,950 And if you don't email me and we'll go through it all again by e-mail. 630 01:06:13,950 --> 01:06:22,498 Okay, I'll stop there. There's five minutes for questions. If anyone has still got some energy.