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