1 00:00:00,000 --> 00:00:08,549 So let's see how uncertainty or statements which are not 100% true, can really wreak 2 00:00:08,549 --> 00:00:16,876 havoc with our whole notion of logic. Let's suppose we have predicates or 3 00:00:16,876 --> 00:00:23,036 relationships A, B, and C; and rules such as, you know for all values of X. 4 00:00:23,036 --> 00:00:31,816 If A of X is true, then B of X is true. Similarly, if we have for all values of X; 5 00:00:31,816 --> 00:00:40,739 B of X is true and C of X is true, then normal logic allows us to entail that for 6 00:00:40,739 --> 00:00:47,317 all X, A of X is true, so B of X is true, B of X is true, then C of X is true. 7 00:00:47,317 --> 00:00:54,250 Since this holds for every value of X it holds for all values of X So we have the 8 00:00:54,626 --> 00:01:02,252 inferred rule that A of X implies C of X. And this logical entailment of one rule 9 00:01:02,252 --> 00:01:09,595 from a pair of rules is fundamental. We can simply not reason if we don't have 10 00:01:09,595 --> 00:01:14,020 such freedom to entail new rules from old ones. 11 00:01:15,280 --> 00:01:23,628 But this fundamental principle that we rely on has a problem if the statements 12 00:01:23,628 --> 00:01:30,074 are no longer certain. For example, say that for most X, E of X 13 00:01:30,074 --> 00:01:36,097 implies B of X. An example might be that most firemen are 14 00:01:36,097 --> 00:01:43,573 men. Another statement might be that, for most 15 00:01:43,573 --> 00:01:50,416 X, B of X implies C of X. An example might be, most men have safe 16 00:01:50,416 --> 00:02:02,840 jobs. The, inferred rule for most X, A of X 17 00:02:02,840 --> 00:02:10,380 implies C of X. Does not follow, it is not true that most 18 00:02:10,380 --> 00:02:18,625 firemen have safe jobs. Where is if we'd said that all firemen are 19 00:02:18,625 --> 00:02:26,241 men and all men have safe jobs, we could say that all firemen have safe jobs. 20 00:02:26,241 --> 00:02:31,525 But obviously that's not true. The job of a fireman is not safe and the 21 00:02:31,525 --> 00:02:43,047 reason we have this confusion is that while for most X almost all of A, which is 22 00:02:43,047 --> 00:02:46,800 a set of firemen. We have, 23 00:02:46,800 --> 00:02:53,611 They are men, right, so very few of them are women, because this, this piece over 24 00:02:53,611 --> 00:02:58,436 here. And, for most of these B s, which are the, 25 00:02:58,436 --> 00:03:03,485 those that have, which are the men, most of them have safe jobs. 26 00:03:03,485 --> 00:03:09,673 Which, which essentially include all of these people, including possibly some 27 00:03:09,673 --> 00:03:16,985 firemen, who basically don't go on fire calls but, do only administration The 28 00:03:16,985 --> 00:03:24,393 trouble is that because of this uncertain relationship, you don't have a situation 29 00:03:24,393 --> 00:03:30,520 that most A have safe jobs, only a small number of A have safe jobs. 30 00:03:30,940 --> 00:03:38,425 So this statement, for most A, A of X implies C of X, is simply not true. 31 00:03:38,425 --> 00:03:46,431 So, this basic entailment of A implies C from A implies B and B implies C, a very 32 00:03:46,431 --> 00:03:53,009 fundamental piece of logical inference, simply doesn't hold when we have uncertain 33 00:03:53,009 --> 00:03:57,180 statements. And this creates much more problems than 34 00:03:57,180 --> 00:04:02,795 the fundamental limits of logic. Another problem one can get into with 35 00:04:02,795 --> 00:04:09,534 normal logic if one is not careful is the notion of one event causing another event, 36 00:04:09,534 --> 00:04:16,252 that is causality. That is statements are not necessarily true all the time, but 37 00:04:16,252 --> 00:04:23,071 their truth values change over time because one event causes another event to 38 00:04:23,071 --> 00:04:29,066 become true. For example, if we have a statement like, 39 00:04:29,066 --> 00:04:33,351 if the sprinkler was on, then the grass is wet. 40 00:04:33,351 --> 00:04:39,500 Which we, we might write this as S implies W, sprinkler implies wet. 41 00:04:40,540 --> 00:04:47,800 We might have other statement that if the grass is wet, then it had rained last 42 00:04:47,800 --> 00:04:55,336 night which we might write as if the grass is wet then R, that is it rained last 43 00:04:55,336 --> 00:04:59,656 night. Logical inference would probably blindly 44 00:04:59,656 --> 00:05:05,853 combine these two statements as implies R through logical entailment, 45 00:05:05,853 --> 00:05:11,917 Which states that if the sprinkler is on, then it rained last night which is 46 00:05:11,917 --> 00:05:16,384 blatantly false. So, something has gone wrong because our 47 00:05:16,384 --> 00:05:22,607 statements are no longer about things which are always true, but they're about 48 00:05:22,607 --> 00:05:29,711 things which cause other things. The problem is that causality was treated 49 00:05:29,711 --> 00:05:35,740 differently in each statement, which resulted in an absurdity. 50 00:05:36,120 --> 00:05:43,183 Well it turns out that we've seen causality earlier, without actively having 51 00:05:43,183 --> 00:05:47,180 realized it, when we studied classification. 52 00:05:48,860 --> 00:05:53,100 Let's look at classification again. What were we doing then? 53 00:05:55,520 --> 00:06:02,446 A statement like, if the sprinkler is on then the grass is wet, is also a statement 54 00:06:02,446 --> 00:06:08,905 saying that, The fact the grass is wet is an observable 55 00:06:08,905 --> 00:06:13,720 feature of the event that the sprinkler was on. 56 00:06:15,980 --> 00:06:26,193 Similarly, If it had rained the grass being wet is 57 00:06:26,193 --> 00:06:32,235 again. That W is an observable feature, of the 58 00:06:32,235 --> 00:06:40,629 event, of raining. The trouble is the statement that if W is 59 00:06:40,629 --> 00:06:49,270 observed, then R happened in the past is not a statement about the forward cause 60 00:06:49,270 --> 00:06:56,383 and effect of R having caused wetness, or S having caused wetness, but it's 61 00:06:56,383 --> 00:07:01,285 statement what might've happened if one observed W. 62 00:07:01,285 --> 00:07:09,071 Remember in classification, we were doing something similar, we were observing the 63 00:07:09,071 --> 00:07:15,895 features that, when found in the world and trying to infer their causes. 64 00:07:15,895 --> 00:07:21,570 So, in some sense. This reasoning about R having happened, 65 00:07:21,570 --> 00:07:30,362 having observed W, is like concluding which class of event actually is being 66 00:07:30,362 --> 00:07:33,483 observed. Is it sprinkler or rain? 67 00:07:33,483 --> 00:07:40,197 We are observing the features and concluding what kind of event we are 68 00:07:40,197 --> 00:07:45,683 actually observing. A kind of classification or prediction 69 00:07:45,683 --> 00:07:51,140 using a classifier. Not exactly, but something very similar. 70 00:07:52,060 --> 00:08:00,861 This is an example of abductive reasoning, where one tries to infer the most likely 71 00:08:00,861 --> 00:08:05,740 cause given a set of observations or features. 72 00:08:05,740 --> 00:08:12,910 Abductive reasoning is exactly what we do when we compute the class of an 73 00:08:12,910 --> 00:08:18,821 observation using a classifier. It's also a form of reasoning. 74 00:08:18,821 --> 00:08:26,185 It's not deductive that it is going from sprinkler or rain to wetness, which is 75 00:08:26,185 --> 00:08:33,355 actually the likelihood computation. But is the A Posteriori calculation of 76 00:08:33,355 --> 00:08:38,440 having observed W. What is the most likely cause? 77 00:08:38,440 --> 00:08:45,005 Is it sprinkler or rain? If you view this in the language of 78 00:08:45,005 --> 00:08:53,708 classification, the confusion about having incorrectly concluded that sprinkler 79 00:08:53,708 --> 00:08:59,084 implies, rain goes away. So one needs to, distinguish between 80 00:08:59,084 --> 00:09:04,460 deduction and abduction. Fairly deep, but in the language of 81 00:09:04,460 --> 00:09:09,380 classification, it becomes fairly simple. Let's see how.