PHIL 126 Lecture 7 Enumerative Induction

0:00 All right everybody welcome. Today we're going to be talking about enumerative induction. So we're now in the portion of the course where we will be talking about inductive arguments. which are arguments that intend to provide logically probable but not necessarily conclusive support for their conclusions. meaning the best we can get is very likely conclusions with enumerative with inductive arguments.

0:30 Now, with deductive arguments, they're all the same. They all adhere to the same standard of validity, meaning it must be that if the premises are true, it is impossible for the conclusion to also be false. If that's the case, then the argument is valid. With inductive arguments, things are a bit different. In particular with inductive arguments, whether the argument counts as strong or not, which remember is the analog for valid, the strength, whether the argument counts as strong or not depends on what kind of argument is being presented.

1:12 So it's not the case that all inductive arguments are going to be judged by the same standard. they'll actually be judged by quite different standards depending on the type of argument that is being presented. This is most easily seen with our first kind of inductive argument which is innumera induction. So we'll see what the standards are for counting an enumerative induction argument strong meaning if the premises are true the conclusion is likely to be true. Okay. So, enumerative induction arguments reason from premises about individual members of a group to conclusions about the group as a whole, from particular to general or from part to whole. In these cases, we begin with some observations about some group members, members of the group, and end with a generalization about all of them.

2:13 Okay, this is a pretty common way we reason. Here are some instances of enumerative induction that sort of happen all the time and we don't even recognize it. Most peace activists I know are kindhearted. So probably all of them are kindhearted. Every Gizmo computer I've bought in the last two years has had a faulty monitor. Therefore, all Gizmo computers probably have faulty monitors. 40% of the pickles that you've pulled out of the barrel are exceptionally good. So 40% of all the pickles in the barrel are exceptionally good.

2:50 Let's focus in on that last one because it identifies I think most clearly the structure of enumerative induction arguments. Here's how they work. We say something like here's the general pattern. X percentage of the observed members of group A have some property P. And then we infer from that that X percentage of all members of group A, the observed and the unobserved probably have that property P as well. So if we take a look at that pickle example and formalize it, put it in standard form, it would look something like this. A1 says 40% of the observed pickles from the barrel are exceptionally good, meaning the ones that we've pulled out and tasted.

3:38 Therefore, 40% of all the pickles in the barrel are probably exceptionally good. Enumerative induction comes with some useful terminology, not just for this class, but for your life more generally. The group as a whole, the whole collection of individuals that are in question is called the target population or target group. The observed members of the group are called the sample members or the sample. In the property we're interested in is called the relevant property or the property in question.

4:13 In the example we just went over, the target group is the pickles in the barrel. The sample is the observed pickles that we've taken out and tried and the property is the quality of being exceptionally good. Okay, with this terminology we can study enumerative induction arguments a little more closely. Now remember with validity it's an onoff switch. Something is either valid or it ain't nothing in between. But with strength, arguments can vary in their strength. Some something can be strong but not very strong. An argument can be weak but not very weak.

4:59 So there will be differences in the degree of support that an inductive argument gives for its conclusion. So what this means is that the argument especially with enumerative induction can vary in strength with the premises as well as what is being claimed in the conclusion as we'll see. So let's look at some examples in a little bit more detail. Here's the B argument. It goes like this. All of the corporate executives that Jacques has worked for have been crooks.

5:33 So all corporate executives are probably crooks. I want you to ask yourself, is this a strong just intuitively a strong or weak argument? Is this good reason? Is B2 justified on the basis of B1? Well, the target group here is corporate executives. The sample is the corporate executives that Jacques has worked for. and the relevant property is being a crook.

6:08 Now, we don't know how many corporate executives Jacques has worked for, but we have to assume from what we know about normal lives that the numbers probably pretty small, no more than a dozen certainly. And probably also that the kinds of corporate executives that a person works for are in one port of a corporation, right? Sales, engineering, finance, right? You don't people usually don't hop from one to another to another.

6:43 So the corporate execs that Jacques has worked for have been maybe 10, maybe 20 and mostly probably of a similar stripe. Now we don't know how many corporate execs there are in America or whatever range this this argument is going for, but we can safely guess that there are thousands, hundreds of thousands of corporate executives. should be obvious to us that this enumerative induction argument falls short on at least one score.

7:21 That's the sample size. The sample is way too small. We simply cannot draw reliable conclusions about all corporate executives based on just a handful of them. So the argument is pretty clearly weak. Additionally, the argument is weak because the sample is likely not representative of the target group. With thousands of corporate executives working for thousands of corporations, we have to assume that corporate executives in their temperament, in their morality, in their demographics, and every other factor are a pretty diverse lot.

7:58 It's therefore pretty unlikely that Jacques himself has had bosses that are representative of all these different corporate executives in their crookedness, which is the relevant property. And if the sample isn't representative of the whole, then we can't use it to draw accurate conclusions about the whole target population. So argument, this argument is weak for that second reason. Also, let's move on to another example and then we'll make some general claims.

8:30 Consider this one. All of the blue herand that we've examined at many different sites in the nature preserve, which is about 200 birds that we've looked at, have had birth defects. So, most of the blue herand in the nature preserve, probably have birth defects. So again in this argument the target group is the blue herand in the nature preserve the sample is the 200 blue herand examined and the relevant property is having birth defects based on your intuition assuming I don't know a thousand blue herand in the group in the in the entire nature preserve 2,000 maybe is this a strong argument just intuitive Would you consider this to be a strong argument?

9:22 I think probably. Assuming that the premise is true, we would be really surprised to find out that only a tiny minority of the target group had birth defects. If it was less than half, I think we'd be pretty surprised. The sample was drawn from many parts of the preserve. That gives us some reason to think that it's representative of the target group, right? It's not just like there's one corner of the preserve that's polluted. It's it's everywhere. And due to the general uniformity of characteristics among birds in the wild, I think we would assume that a sample of 200 birds would be large enough to strongly support the conclusion for at least a thousand birds, maybe 2,000, maybe even more. So, I think this example, this argument is pretty strong.

10:14 Now notice that the premise says all of the herand that we've examined have birth defects and it concludes merely that most of the ones overall have birth defects. Imagine the conclusion asserted all of the herand in the reserve had birth defects. That would probably not be a strong argument, right? That's way too much to ask for in a conclusion on the basis of the premise that's given. There could easily be some blue herand in the preserve that don't have birth defects even if most of them do.

10:52 So through this exercise we've seen that enumerative induction arguments can go wrong primarily in one of two ways. It can either have too small of a sample or its sample can be can fail to be representative. So let's talk about sample size first. There's a name for when in an enumerative induction argument you have too small of a sample size.

11:25 It's called a hasty generalization. So, we're guilty of hasty generalization whenever we draw a conclusion about a target group based on an inadequate sample size. Oops. What's going on? So, let me give an example. Actually, let's just look at so sorry.

12:01 We can see hasty generalization in a really silly sort of way. Let's say you try to conduct a survey of college students to determine their attitude towards federal deficits or some whatever issue. So you stand around in the yuyu and you ask the first five people that pass you. Four out of the five people say that deficits don't really matter. So, you conclude that 80% of the student body believes that deficits don't matter. You send it to the Mustang Daily and they laugh you out of the room, right?

12:29 The survey is a joke. The sample's way too small to draw reliable conclusions about 20,000 plus students. This is hasty generalization. These sort of things pop up everywhere in like professional sort of claims being made, right? like polls, opinion polling, political polls, consumer opinion surveys, scientific studies, these sorts of things. But hasty generalization, I think, is really, really popular among normal thinking. So, here are some everyday experiences that we casually make or hear people commit hasty generalizations. Someone might say, "Hey, you should buy a Dell. They're great. I bought one last year and it's been nothing but good to me." Hasty generalization. You have one instance.

13:22 You cannot conclude that Dells are good on the basis of that. The only male professor I've had this year was a chauvinist pig. Hope you're not including me. All the male professors at the school therefore must be chauvinist pigs. Psychology majors are incredibly ignorant about psychology. Believe me, I know what I'm talking about. My friend is a psych major. He's an idiot. The French are snobby and rude. Remember those two high and mighty guys with really bad manners? They're French.

13:56 And one that I think happens a lot that we don't really realize, the food at Papy's restaurant is awful. I had a sandwich there once and it did not taste good. Well, you've had one sandwich one time. Is that really enough to justify the belief. Okay, so these are hasty generalizations. In general, generally speaking, the larger the sample, the more likely it is to reliably reflect the nature of the group. That's what we think. In many cases, our common sense sort of tells us when a sample is or is not large enough to draw reliable conclusions about a particular group.

14:36 Here is a test sort of to tell whether how much of a sample size we need. The more homogeneous similar the target group is in traits that are relevant to the property in question, the smaller the sample size can be in order to draw a good conclusion. So here's an example. Let's say I want to know two things about whatever the current generation of iPhone is. First thing I want to know is what the dimensions the screen size of the newest iPhone is.

15:17 Well, if I wanted to know that, how many samples do I need? How many newest edition iPhones do I need to have in front of me in order to know what the size of it is? The answer is one. You just need one. Just got to look at one and it'll tell, right? Because these things are all they're made in a factory to be precisely the same in terms of their specifications. What if I wanted to know how many photos the current generation of iPhones actually have on them?

15:54 Well, then I need a lot of samples in order to draw any sort of reliable conclusion about this claim. Right? People have different numbers of photos on their phone and so to guess anything I need a ton of examples. So I'll read the the the bumper sticker again. The more homogeneous a target group is in traits relevant to the property in question, the smaller the sample size can be. The less homogeneous, the larger the sample size should be. So because iPhones are very homogeneous with respect to their physical characteristics, we don't need very many to draw conclusions about them. Whereas they are not homogeneous with respect to what is on them like what software, what files are on them.

16:49 And so we need a ton more in order to draw reliable conclusions. Okay, sample size isn't the only way that these sorts of arguments can go wrong. They can also go wrong because the sample, even if it's a large enough sample size, doesn't resemble the group in particular ways. What we need is a representative sample, right? A representative sample resembles the target group in all the ways that matter. if it doesn't properly represent the target group is what's called a biased sample.

17:29 Anu an enumerative sorry that's wild and an enumerative induction argument is only strong if the sample is representative of the whole. Many many arguments with unrepresentative samples are silly but some of them are more subtle. So let's look at a couple examples. College students are glad that Congress is controlled by Republicans. Why? Surveys of the members of Turning Points USA clubs on dozens of college campuses prove this. Clearly absurd by a sample, right? Most nurses in this hospital are burnt out, stressed out, and overworked. Just ask the ones who work in the emergency department.

18:10 They'll tell you they're absolutely miserable. Well, yeah, the ones in the are going to be more stressed out than ones that work in other places. No one is happy. Almost everyone is complaining about something. Just look at the letters to the editor in any big city newspaper. Complaints, complaints, complaints. Right? To truly be representative, the sample must be like the target group by having all the same relevant characteristics and having these characteristics in roughly the same proportions that the target group does.

18:50 The relevant characteristics are features that could influence the property in question. Okay, so our sample size needs to be both large enough and representative of the target population in order for the argument to be strong. Little interesting thing about so that's the the main bits on enumerative induction. There's a couple things that follow from this discussion that I think are important to draw out and you'll become a better human being and reasoner if you understand these things.

19:27 So opinion polls are a form of enumerative induction. So as inductive arguments, opinion polls should one be strong, meaning they have a unbiased and repres they have they have a representative and large enough sample size and they have to have true premises. The claims they make about things must be correct. to be strong the poll must have a representative in large blah blah blah.

19:58 Okay, strong polling. So what counts as good polling in opinion polls? Well, national polling samples, right? If you pull people largely, you you get a a big portion of the population answering. Representativeness is a little bit nuanced here because if you for example put it in an ad for a newspaper that you're conducting a poll well then only the only people who are going to respond to the poll are people who read the newspaper who might have a specific set of opinions as compared to the general population. So oftent times what we do in opinion polling is we do random sampling. Get a list of phone numbers. You call them and you ask people questions.

20:53 Now this helps unbias the poll, but it doesn't do it perfectly because who answers their phone to spend 15 minutes talking to a poll researcher? Well, only a certain type of person. And so there's sort of a self- selection bias going on with who even answers these polls that skews them. So the polling is really really tough to get an unbiased and big enough sample. Additionally, you got to capture good data and there are very easy ways to get bad data.

21:30 These have been studied in sort of I don't want to say excruciating detail by political scientists but something like excretionary exc ex excl excruciating detail. You can get bad data in polls by having poor question phrasing, poor question ordering, restricted choices. So question phrasing, there's sort of obvious examples of this. Things like, "Are you in favor of a woman's right to kill her unborn child?" Sort of skews people towards one way when you ask it like that. Whereas if you ask the question in a different way, are you in favor of a woman's right to let's see, I'm trying to come up with one that that would clearly get a different result. Are you in favor of a woman's right to I don't even want to say choose to.

22:41 Sorry. Let me let me just pause my brain for a second. Okay, I came up with it's been 10 seconds. I came up with two possible questions. One of them is, "Are you in favor of a woman's right to remove unwanted things from her body? Clearly get a different response, right?" that one will do. Okay. Question ordering. It's been shown that the order in which you see questions can affect how you answer them. And so, you have to be really careful with that. And finally, restricted choices. is if you only give people A, B, and C and they actually think something different, they're going to be forced to say something they don't really believe. And so you'll get bad data. So you'll say 93% of those people we surveyed believe X, therefore a large portion of the population believes X. But that doesn't that's not necessarily a cogent argument because if you have bad data you can't even truthfully say 93% of people pled said blank. Right? Okay.

23:51 Finally, even if you have a big unbiased sample size, and you have good questions, good data, your poll is likely to at least be a little bit off since each random sample will be different. These differences are identified in what's called the margin of error. So if a poll says candidate X will get 52% of the vote with a plus three plus or minus three margin of error means that the true value is very likely to be between 49% and 55%. The confidence level so there's two numbers. There's the margin of error which is it's going to be within this range. The confidence level is the probability that the sample will accurately represent the target group within the margin of error. Right? So let's say that it's between 49 and 55%.

24:46 Well, what's your confidence level that it's going to actually be within that? Is it like 60% confident that it's going to be within that, meaning just over half the time, or are you 95% confident that it's going to be within that range? Changes things. I want to go over a couple ways in which enumerative induction arguments go wrong. So I'm actually going to scroll down and do the second one first. So there's a bias that we have when we're doing sort of collecting data.

25:23 one of them is called the survivorship bias. The idea is that you only observe survivors. So the sample can systematically exclude failures and you'll get biased data. So here's a famous instance of this. I think this was in World War II. the Allied forces would send out whatever this plane is and it would come home and be riddled with bullet holes looking like this. So, every red dot represents a bullet hole. And so, the thought is, if you see a plane coming home like this, where are you going to put extra metal plating to, to bolster that part of a plane when it gets shot? An initial thought might be, well, you should do the bits with the red on them.

26:16 They're clearly getting shot a lot right there. But that's actually incorrect. You should you don't need to put extra armor on the spots where the red the red bullet holes are. Why not? Well, this plane came home. Meaning, this plane can fly when it's got bullet holes in these areas. No planes are coming home that have bullet holes in the other areas, meaning that's where they get shot down. If the if one of the planes gets shot in the areas without the red dot, then that's an indication that that plane is going down.

26:59 And so that's where you need the extra armor is in the place where you don't see the red the red dots. That's survivorship bias. You need to make sure that you're correctly interpreting the information that you're getting. okay. The other one I want to talk about is what's called the base rate. I want you to turn on your math brain for the next several minutes. This is going to be fun. and it's relevant to your life.

27:28 Okay. So, I'm going to walk you through a scenario. Suppose the following. There's some disease. COVID 19. No, not CO. Don't do CO. It'll bring us back in the wrong sort of way. Suppose there's this disease that affects one in a thousand people in the general population. It's got a 0.1% prevalence within the population. Suppose there's this test for the disease that is 99% sensitive. Meaning if you have the disease, it returns a positive rate. The the test comes out positive 99% of the time. Very accurate.

28:12 Let's also say that the test is 95% specific. Meaning if you do not have the disease, it returns negative 95% of the time. Meaning that the false positive rate is 5%. Okay? So the false negative rate is 1%, the false positive rate is 5%. you test positive, what is the probability that you actually have the disease?

28:49 Now, you might think, well, look, test is 95 99% accurate or 95% accurate, whichever one something like 99 to 99% accurate. So, that means the probability that I actually have the disease is probably 95 to 99% circle. Don't know which one of those it it is, but it's it's very probable that I have this disease if I test positive, right? Well, we can do the math on this. Let's do the math.

29:20 So, let's work with a sample of 10,000 people. The base rate of one in a thousand means that what we can expect for a random sample of 10,000 people is that there are 10 people who have the disease. The number of people who don't have the disease we can expect from any given random sample of 10,000 is 9,990. So these are our expected amount of people with and without the disease for any given 10,000 person chunk of the population. Now let's apply the test performance to each group and see what results among the 10 diseased people.

30:06 The number of true positives is 99.9 or is 99%. So, we should expect that there are 9.9 true positive test results, right? It's 99% accurate. Meaning, if you have the disease, 99% of the time you will test positive. The number of false negatives, well, there's 1% of the time that if you have the disease, you'll test negative.

30:36 Meaning 10 * 0.01 01 is a 0.1 expected amount of times that somebody with the disease will not test positive. Now consider the nine 9,990 people who do not have the disease. Well, the false positive rate we know is 5%. Right? It's 95% accurate. The false positive rate is 90 is 5%. So we multiply 9,990 by 0.05 and we get the result 499.5.

31:12 That's the amount of false positives we should expect. The amount of true negatives we should expect, well 95% accurate. 9,990 * 0.95 gives us 9,490.5. Okay. All right. Let's go through how what we're supposed to interpret from all this. So, a positive test, if you get a positive test, that means that can be one of two things. It can either be a true positive, meaning you are diseased and a positive test, or it can be a false positive, meaning you are not diseased, but it's a positive test.

31:59 The total number of positives in our 10,000 person run as given by this math here is 9.9 plus 499.5 which gives us 59.4 people. That's how many people that we should expect to test positive. So the probability that you have the disease given a positive test is the true positives over the false positives. Meaning 9.9 divided by 509.4 which is roughly 1.9%.

32:45 So merely having a positive test leaves the disease very unlikely because the disease is really rare and false positives dominate the true positives. There's way more false positives than there are true positives because the prevalence of the disease is so rare. Okay, so this is the base rate fallacy. It's this idea that if if a test is 99% accurate, then if I test positive, then there's a 99% chance that I have that disease. Now, this sort of reasoning applies to a lot of things we see.

33:30 Here's one instance. I don't know about you, but I surf sometimes. And there have been instances where I'm surfing in like low tide or something and I feel my leash get yanked. Okay. Now, every surfer knows that you automatically think that you're going to die, that it's a shark, that you're gone. What I have is this one bit of evidence, my leash getting pulled. Now, that does happen in shark attacks.

34:08 However, what is the likelihood that there's a shark attack happening right now? That there's a shark right there? Incredibly low. So, the mere fact that I have some evidence that would be explained by there being a shark attack right now does not make it reasonable to believe that's what's happened. All I have is one bit of evidence. And given that the prevalence is so extremely low, it's extremely unlikely that it's happening, right? Okay. So, this sort of base rate fallacy happens all the time.

34:43 Second, don't go too far with the base rate fallacy in the other direction. So, let's say that you are you take a COVID test and let's say the COVID test matches the 99% 95% thing that we got going on here. and let's say it's got the same base rate, only 1% at any given point. And so you think, oh well, there's only according to the math here, there's a 2% chance that I have COVID. Well, let's add usually if you're taking a COVID test, you're not taking it randomly.

35:26 You're taking it because you have symptoms or you know people who have COVID. That is going to ext extremely quickly raise the probability that you have COVID. Maybe not to 50% but definitely most of the time that we are doing tests like this it's because there's a prior reason which should raise our our likelihood that we have the the disease.

35:56 Okay. So that is what I got on the numerative induction. We'll see you next time.