nal. For instance, we sometimes just use the word if. So for instance, I might say to you the private investigator is eating lunch at New Havana, if Walter is eating lunch there. There. I just connected two propositions using the word if, but what I'm expressing is a truth functional conditional. It's the same thing as I would have been expressing if I had said if Walter is eating lunch at New Havana, then the private investigator is eating lunch at New Havana. Right? Those are just two different ways of saying the same thing. But in one case, I use the words if and then and in the other case, I just use the word if. So sometimes we express the truth functional conditional just by use of the word if. We could also use the phrase only if and that could also express the truth functional conditional. For instance, I might say Walter is eating lunch at New Havana, only if the private investigator is eating lunch at New Havana. That expresses just the same thing as saying if the private investigator is eating lunch at New Havana, then Walter is too. Right? They express just the same thing. They're true or false in just the same circumstances. It's just a different phrasing. Different phrasing, same meaning. So the truth functional conditional, while it's often expressed in English using the words if and then, is sometimes expressed using other words, and it's not always expressed by means of if then, sometimes if then, expresses propositional connectives that are not truth functional. In the next lecture, we're going to talk about by conditionals. Today, we're going to talk about a new propositional connective, a truth functional connective that I'll call the by conditional. Now what's a by conditional? In order to explain what a by conditional, let me start off by telling you the following story. And this is a true story I should add. When I was in eighth grade, I remember my first class in the morning was a math class. And there was a boy who sat next to me in math class. The teacher called him Bob. My third class later in the morning was a science class. And there was a boy who sat next to me in the science class. And the teacher called him George. And I was struck by the fact that Bob and George, who sat next to me in math and science class respectively, had the same last name. And they also bore an uncanny resemblance to each other. Well, one day, I asked Bob if he had a brother named George. And it turned out that he didn't have a brother named George. In fact, Bob was George. It was one and the same boy. And they just went by two different names to the two different teachers. But suppose that all you knew from my description of the situation was that there was a boy who sat next to me in math class named Bob and a boy who sat next to me in science class named George. And you asked me one day, was Bob born in the United States? Now, I might say to you, well, Bob was born in the United States if and only if George was. Now, there I'm using the phrase if and only if. Now, remember how the phrase if works? If I say if to connect two propositions p if q, that's equivalent to saying if q then p. But now remember how only if works. If I say p only if q, that's equivalent to saying if p then q. So when I use the phrase if and only if, I'm conjoining the conditional if p then q and the conditional if q then p. And so I get what I'll call a biconditional. A biconditional is a propositional connective that connects two propositions into a larger proposition. And the larger proposition is true just in case the two propositions that are part of it have the same truth value. In other words, the larger proposition p if and only if q is going to be true just in case p and q are both true or p and q are both false. So I could say George was born in the United States if and only if Bob was. Under what circumstances is that proposition going to be true? Well, it'll be true if George and Bob were both born in the United States, or well I guess it's misleading to say both because really there's only one boy that we're talking about. If George was born in the United States and Bob was born in the United States, it'll be true when both of those propositions are true. But it'll also be true when George was not born in the United States and Bob was not born in the United States. In other words, it'll be true when both of those two propositions are false. What makes true the proposition George was born in the United States if and only if Bob was is simply that the two propositions that are part of it, George was born in the United States and Bob was born in the United States, have the same truth value. Whatever that truth value is, whether it's the truth value true or the truth value false. As long as those two propositions have the same truth value, it's going to be true that George was born in the United States if and only if Bob was. And so the biconditional connecting those two propositions, George was born in the U.S. and Bob was born in the U.S., that biconditional is going to be true. So we can state the truth table for the truth functional connective, which is the biconditional as follows. The biconditional connects any two propositions, let's call them P and Q, doesn't matter what they are. When P is true and Q is true, then the biconditional P if and only if Q is going to be true. When P is true and Q is false, then the biconditional P if and only if Q is going to be false. When P is false and Q is true, then the biconditional P if and only if Q is going to be false. And finally, if P is false and Q is false, then the biconditional P if and only if q is going to be true. So that's the truth table for the biconditional. Now in the last couple of lectures, I described both the conditional and the biconditional as truth-functional connectives. And I've given some reason to think that they are truth-functional connectives. But you might worry that there are some examples that suggest that both the conditional and the biconditional are not truth-functional connectives. For instance, consider the sentence, if 2 plus 2 equals 4, then Pierre is the capital of South Dakota. Now there I'm using if then to express the conditional, right? I'm saying if it's true that 2 plus 2 equals 4, then it's true that Pierre is the capital of South Dakota. Now according to the truth table for the conditional, that conditional has got to be true because the first proposition, what we'll call the antecedent, the proposition that occurs right after the if, that antecedent is true. It's true that 2 plus 2 equals 4. And also the consequent, the proposition that occurs right after the then, the consequent is also true. It's true that Pierre is the capital of South Dakota. So according to the truth table for the conditional, it's going to be true that if 2 plus 2 equals 4, then Pierre is the capital of South Dakota. And now you might worry, what am I saying? This is a very strange consequence of the truth table for the conditional. Is it really true that if 2 plus 2 equals 4, then Pierre is the capital of South Dakota? And that's a very baffling thing to say. Now I want to say, I completely agree with this objection. It is a baffling thing to say that if 2 plus 2 equals 4, then Pierre is the capital of South Dakota. But just because it's a baffling thing to say, doesn't mean it's not true. Look, it's baffling that Pierre is the capital of South Dakota. But baffling as that may be, it's still true that Pierre is the capital of South Dakota. Some things are baffling even though they're true. And this is another example of that general kind of thing. It's baffling, nonetheless true. So to review, here's what we've done this week. We've learned about some truth functional connectives, like conjunction, disjunction, and the conditional, and the biconditional. And we've also learned about the truth functional operator negation. We've looked at the truth tables for these connectives and operators. The truth tables that show how the truth or falsity of the proposition that they can be used to create depends strictly on the truth or falsity of the propositions that they're applied to. And then we've looked at how those truth tables can be used to determine whether and why arguments that employ those truth functional connectives or operators are valid. Now, in order to master the skills that we've learned this week, there's no substitute for practice. And so what I'd like to recommend is that all of you practice the skills that you've learned this week using the resources that are listed. Okay, have fun. And I'll see you next week when we talk about categorical logic and quantifiers. Welcome to week five of our course, the week on categorical logic. Remember last week was on a part of deductive logic that we called propositional logic. Last week we tried to understand why certain inferences, certain arguments that use propositional connectives were valid because of the truth tables for the propositional connectives that they use. This week, we're going to be looking at arguments that are valid not because of the propositional connectives that they use. In fact, a lot of the arguments that we'll be looking at this week have no propositional connectives at all. We'll be looking at arguments that are valid for a different reason. We'll try to understand what that different reason is. But first, to review in more detail what we did last week, let's consider an example of a deductive argument that propositional logic can help us to understand. So let's look at this. Consider the following argument. Premise one. Jill is riding her bicycle if and only if John is walking to the park. You can imagine, let's say that Jill has a bicycle, but the only time she ever rides her bicycle is to meet John at the park. And the only way John ever gets to the park is by walking there. Let's say he lives just a block down from the park, so it doesn't make sense for him to get there any other way. So he only goes to the park by walking there and Jill only rides her bicycle when she's meeting him at the park. So according to premise one, Jill is riding her bicycle if and only if John is walking to the park. Premise two. John is walking to the park if and only if premise one is true. And the conclusion of the argument is therefore Jill is riding her bicycle. Now, let me ask you, is that argument valid? Is there any possible way for the premises to be true while the conclusion is false? Well, that's not obvious, is it? It takes a while to see that in fact, that argument is valid. And the way to see it is by using the truth table for the bi-conditional expressed in English by if and only if. We can use the truth table for the bi-conditional to see that that argument really is valid. Here, let me show you what I have in mind. So look here. There's the proposition Jill is riding her bicycle. Now, that proposition could be either true or false. Then there's the proposition John is walking to the park. And of course that proposition could be either true or false. So there are four possible combinations. Either Jill is riding her bicycle is true and John is walking to the park is true. Jill is riding her bicycle is true and John is walking to the park is false. Jill is riding her bicycle is false and John is walking to the park is true. Or both of those propositions are false. Those are the four possible scenarios. Now, according to premise one, Jill is riding her bicycle if and only if John is walking to the park. So, if premise one is true, that shows us that we're either in the first of those four scenarios or the last of those four scenarios. We're either in a scenario where it's true that Jill is riding her bicycle and it's true that John is walking to the park or else we're in a scenario where it's false that Jill is riding her bicycle and it's false that John is walking to the park. Okay, now consider the proposition. John is walking to the park if and only if that last statement, the statement to the left that Jill is riding her bicycle if and only if John is walking to the park, John is walking to the park if and only if that statement is true. Now, what's the truth table for that going to be? Well, the truth table for that is going to be as follows. It's going to be true whenever it's true that John is walking to the park and it's also true that Jill is riding her bicycle if and only if John is walking to the park. It's also going to be true whenever it's false that John is walking to the park and it's false that Jill is riding her bicycle if and only if John is walking to the park. So, that means that that last biconditional that John is walking to the park if and only if premise one is true. That's going to be true in the first two of our scenarios. It's going to be true when it's true that Jill is riding her bicycle and true that John is walking to the park. It's also going to be true when it's true that Jill is riding her bicycle and false that John is walking to the park. Okay, so now under what conditions will premise one and premise two of our argument be both true? Remember, premise one says Jill is riding her bicycle if and only if John is walking to the park. Premise two says John is walking to the park if and only if premise one is true. So under what conditions will those two premises be true? Well, those two premises will be true only in the first of our four scenarios because in the first of our four scenarios it'll be true that Jill is riding her bicycle if and only if John is walking to the park and in the first of our four scenarios it'll also be true that John is walking to the park if and only if that last statement premise one is also true. So that's the only scenario under which both of those premises are true. But now notice in that scenario when both of those premises are true Jill is riding her bicycle and so the argument that we just looked at is going to be valid because in any situation in which the two premises are true the conclusion is going to have to be true and we learn that by looking at the truth table for the bi-conditional. Now see that's an example of how we can use the truth table for a propositional connective like the bi-conditional to discover that a particularly tricky argument is valid. Right? We have a tricky deductive argument right here. It's not obvious whether or not it's valid but we can use the truth table for the bi-conditional to discover that the argument is valid. But remember I said we don't just use truth tables to discover when arguments are valid or that they're valid. We can also use truth tables to explain why they're valid even in cases where they're obviously valid. Right? So last week we looked at lots of examples of arguments that were obviously valid and in some cases obviously invalid and we used the truth table not to figure out that they were valid or invalid. It was already obvious that they were valid or invalid as the case may be. We used the truth table to understand why they were valid or invalid. That's what the truth table was for in those cases. Now, this week in our study of categorical logic, we want to find a method that can function like the method of truth tables to help us discover whether particular inferences are valid and why particular inferences are valid. But it's not going to be the same as the method of truth tables because truth tables only work for inferences or arguments that use truth functional connectives. But not every valid argument uses a truth functional connective. Consider a couple of the examples we looked at last week. Consider this argument from a week ago. No fish have wings. All birds have wings. All animals with gills are fish. Therefore no birds have gills. Is that argument valid? It's not immediately obvious whether or not it's valid. But it turns out that it is valid and we have a method for proving that it's valid and this method is what we'll be talking about this week. It's the central method of categorical logic. Notice also that there are other inferences which are obviously valid but the validity of which truth tables don't help us to understand. Consider for instance this example. Mary has a child who is pregnant. Only daughters can become pregnant. Therefore, Mary has at least one daughter. Now that argument is pretty obviously valid. But why? What is it about the argument that makes it valid? I said at the beginning of week four that there's something about the form of the argument. Something about the use of the terms only and at least that makes that argument valid. And any argument no matter what it's about that uses the terms only and at least in the way that this argument does is also going to be valid. But what is it about the use of those terms that makes the argument valid? Well this week in our study of categorical logic we're going to discover a way of understanding what's going on with that use of those terms only and at least those terms that we call quantifiers. We're going to discover a method of understanding quantifiers that can help us to understand why this particular argument and others that are of the same form are valid. Okay, so that's what we'll be doing this week in categorical logic. We'll be understanding how quantifiers work. But what are quantifiers? And after all, if the central words that we're going to be concerned about this week, the central concept this week is the concept of a quantifier, why is this called categorical logic anyway? Why not quantifier logic? Categorical logic is the logic of categories. Quantifiers are words like all, some, none, only, at least, and so forth. What do these things have to do with each other? We'll talk about that in the next lecture. In the last lecture, I raised the question why it is that we call the subject that we're going to study this week categorical logic, the logic of categories, when really what we're going to be doing is learning about quantifiers, quantifiers like all, some, none, only, at least, and so forth. What do categories and quantifiers have to do with each other? That's the topic of today's lecture. So first, let me talk a bit about categories and what role they can play in arguments. Now frequently, we give arguments that make use of categories of kinds of thing, consider the following argument. Brazilians speak Portuguese, Portuguese speakers understand Spanish, and therefore Brazilians understand Spanish. Now there's an argument with two premises in a conclusion, and it's an argument that talks about various categories of thing. One category of thing it talks about is Brazilians. Another category is Portuguese speakers, people who speak Portuguese, and a third category is people who understand Spanish. The argument brings those three categories into relation with each other, but how does it do that exactly? What precisely is it that the argument is telling us? Well, here's one interpretation of the argument. You could understand the argument of saying that some Brazilians speak Portuguese, certainly that's true. Maybe not all Brazilians speak Portuguese, or maybe there are some Brazilian citizens who've only recently acquired citizenship and have never learned to speak Portuguese. So some Brazilians speak Portuguese, that much, at least, is true. And some Portuguese speakers understand Spanish, of course that's true. But suppose the argument concludes from those two premises that some Brazilians understand Spanish. Would that be a valid argument? Well, although I'm confident that the conclusion of that argument is true, the argument itself wouldn't be valid, because there is a way for both premises of the argument to be true while the conclusion is false. Just imagine this. While it's true that some Brazilians speak Portuguese and some Portuguese speakers understand Spanish, you could imagine that all of the Portuguese speakers who do understand Spanish are in Portugal, not in Brazil, or in any case are in some other part of the world than Brazil, and that none of the Brazilians who speak Portuguese are among the Portuguese speakers who understand Spanish. In that circumstance, the premises could both be true while the conclusion that some Brazilians understand Spanish would be false. So the argument, even if its conclusion is true, the argument is not valid. But maybe that's not how to understand the argument that we just considered a moment ago, the argument from Brazilians speak Portuguese and Portuguese speakers understand Spanish to Brazilians understand Spanish. Maybe there's a different way to understand the argument. Maybe the right way to understand the argument is saying this. Most Brazilians speak Portuguese. Certainly that's true. And maybe the argument says is supposed to say also that most Portuguese speakers understand Spanish. I don't know if that's true, but I gather from people that it's probably true. And maybe the argument intends to draw the conclusion from those two premises that most Brazilians understand Spanish. Now would that argument be valid? Well, no, it wouldn't. Because even if the conclusion of the argument is true, there's still a possible scenario in which the premises are true while the conclusion is false. I could describe such a scenario to you. Suppose that while most Brazilians do speak Portuguese and most Portuguese speakers do understand Spanish, most Portuguese speakers, let's suppose, live outside Brazil. And the Portuguese speakers who understand Spanish are just those Portuguese speakers who live outside Brazil. So while then it would be true that most Brazilians speak Portuguese and most Portuguese speakers understand Spanish, it might not be true that most Brazilians understand Spanish because maybe that minority of Portuguese speakers who live in Brazil don't understand Spanish. The only Portuguese speakers who understand Spanish we could suppose are that majority of Portuguese speakers who live outside Brazil. Now, of course, that isn't the actual scenario, but it's a possible scenario. And since it's a possible scenario, there's a possible scenario in which the premises of the argument are true and the conclusion false. And so the argument is not valid. But maybe that's not even the correct way to understand our original argument about Brazilians. Maybe the correct way to understand our original argument is like this. Maybe the argument was intended to say all Brazilians speak Portuguese and all Portuguese speakers understand Spanish, therefore, all Brazilians understand Spanish. Now, one thing that this argument has to its credit is that it is valid. If the premises of this argument are true, if it's true that all Brazilians speak Portuguese, and it's true that all Portuguese speakers understand Spanish, then it's got to be true that all Brazilians understand Spanish. So this argument, in contrast to the last two arguments that we looked at, this argument is valid. Unfortunately, it's not sound because it's almost certainly not the case that the two premises are both true. Be that as it may, though, the argument is valid. Now, notice what we did. We started off with an initial argument that used three categories in the argument, the category of Brazilian, the category of Portuguese speaker, and the category of person who understands Spanish. Then we saw that by the use of certain modifiers, the modifiers sum, most, or all, we could make that original argument about Brazilians understanding Spanish. We could make that original argument more precise. And once we made it more precise, we could test whether or not it was valid. Okay. Now what I want to introduce is a way of testing whether or not an argument is valid when the argument is one that uses categories and quantifiers to modify those categories. The method involves the use of what we'll call Venn diagrams. A Venn diagram is a diagram that represents a number of different categories. In this simple Venn diagram, we have a representation of the category of Brazilians and we have a representation of the category of Portuguese speakers. Premise one of our original arguments stated a relation between Brazilians and Portuguese speakers. But what relation did it state? Well, that was left vague by the original statement of the argument. But then, when we looked at three different ways of making the argument more precise, we saw three different relations that could be stated. It could have said that some Brazilians are Portuguese speakers. The way of representing that would be by drawing an X right here to show that there is something, there is someone who is both a Brazilian and a Portuguese speaker. The second modification we made of the statement said most Brazilians are Portuguese speakers. Now, this week we're not going to discuss a way of representing the quantifier most in Venn diagrams. I just call it to your attention as a quantifier that can be used and frequently is used to modify categories and argument. But the third way that we interpreted our original argument about Brazilians understanding Spanish. The third way was as an argument that began by stating that all Brazilians speak Portuguese. Now, how would we represent the fact that all Brazilians speak Portuguese? Well, what are you saying when you say all Brazilians speak Portuguese? Well, you're saying that there is no Brazilian who falls outside the category of Portuguese speakers. Right? This circle right here represents the category of Portuguese speakers. And when you say all Brazilians speak Portuguese, what you're saying is that whatever Brazilians there are, they have to be inside this circle. They can't be outside the circle. They've got to be inside the circle of Portuguese speakers. So, the way to represent that is to shade out the portion of the Brazilian circle that's outside the Portuguese speaker circle. There, you're indicating that there isn't anything in the Brazilian circle. There isn't anything in the category of Brazilians that falls outside the category of Portuguese speakers. In other words, all Brazilians speak Portuguese. That's how you could represent that. And that's what we're going to do. To represent the statement, all Brazilians speak Portuguese, we simply shade out the part of the Brazilian circle that's outside the Portuguese speaker circle. And to represent some Brazilians speak Portuguese, we put an X in the Brazilian circle that's also inside the Portuguese speaker circle. And we used a Venn diagram to represent the information contained in the first premise of the argument about Brazilians. But how could we use a Venn diagram to represent the information contained in both premises and the conclusion? Well, here's how. So, here are three circles corresponding to the categories of Brazilians, Portuguese speakers, and those who understand Spanish. Now, suppose we interpret the first premise of our argument, Brazilians speak Portuguese, as saying that some Brazilians speak Portuguese. Well, as we saw already, the way to represent that is with an X right here, that's both in the circle of Brazilians and in the circle of Portuguese speakers. Okay, that represents that there is something, that X, that is both a Brazilian and a Portuguese speaker. Now, how would we represent the information that some Portuguese speakers understand Spanish? Again, we could do that with an X that is both in the circle of Portuguese speakers and in the circle of those who understand Spanish. Alright, so that represents that there is something that is both a Portuguese speaker and an understander of Spanish. But now notice, we've represented the information that some Brazilians speak Portuguese and some Portuguese speakers understand Spanish. Now, does that information imply that some Brazilians understand Spanish? If the information that some Brazilians are Portuguese speakers and some Portuguese speakers understand Spanish, if that information is true, does it follow from that, that some Brazilians understand Spanish? Well, one look at this diagram should tell us that no, it doesn't follow, because look here, right? You have a thing that is a Brazilian and a Portuguese speaker, representing the fact that some Brazilians speak Portuguese, then you have a thing that is a Portuguese speaker and understand Spanish, representing the fact that some Portuguese speakers understand Spanish. But do you have anything that is both a Brazilian and something that understands Spanish? No, not necessarily. So this diagram right here shows you that if the premises of our argument are some Brazilians are Portuguese speakers and some Portuguese speakers understand Spanish, from those premises it doesn't follow that some Brazilians understand Spanish. If you want to conclude that some Brazilians understand Spanish, your argument is not going to be valid. There's going to be a possible scenario in which the premises of your argument are true and the conclusion is false. And so we can say about this argument right here that this argument is not valid and we can understand why it's not valid by looking at this Venn diagram right here. That shows why our argument about some Brazilians being Portuguese speakers, why that argument is not valid. Okay, but now let's consider not the argument to the effect that some Brazilians speak Portuguese, let's consider the argument to the effect that all Brazilians speak Portuguese. How would we represent the information in those premises in that conclusion? Well if all Brazilians speak Portuguese what that means is that all of the Brazilians must be inside the category of Portuguese speakers so there can't be any Brazilians outside that category. Okay, so we can shade out the portion of the Brazilians circle that's outside the category of Portuguese speakers. We know there's nothing in there because all the Brazilians there are are inside the category of Portuguese speaker according to premise one of this new argument. Right, according to premise one all Brazilians are Portuguese speakers according to that premise all the Brazilians there are have to be in here. Okay premise two says all Portuguese speakers understand Spanish. Okay, well if all Portuguese speakers understand Spanish then there cannot be any Portuguese speakers who are outside the category of those who understand Spanish. Right, here's the category of those who understand Spanish and all the Portuguese speakers according to premise two have to be inside that circle. So that means that we can shade out the portion of the Portuguese speaker circle that's outside the understand Spanish circle. Right, there's there are no Portuguese speakers that fall outside the category of understanders of Spanish according to premise two of this argument. Right, that all Portuguese speakers understand Spanish. So now we can represent that information that way. Okay, now that we've represented the information contained in those two premises all Brazilians speak Portuguese and all Portuguese speakers understand Spanish. Let's ask does it follow from those two premises that all Brazilians understand Spanish? Well, let's see. Yes, it does. What Brazilians can there be there can't be any out here and there can't be any in here and of course there can't be any in here either. The only Brazilians there can be are these in here. That's the only part of the Brazilian circle that is left unshaded. Once we shaded in the circles that we had to shade in to represent the first two premises of the argument. Right, once we shaded in the part of the Brazilian circle that we had to shade in in order to represent the premise that all Brazilians were Portuguese speakers and then we also shaded in the Portuguese speaker circle outside the understanding of Spanish circle. The only part of the Brazilian circle that's left unshaded is this, which means that whatever Brazilians there are have to be in here. So if there are any Brazilians at all, they've got to understand Spanish because they're in the circle of things that understand Spanish. Therefore, all Brazilians understand Spanish and we've just used this Venn diagram to prove visually that our argument is valid. This argument back here is valid and our three circle Venn diagram shows that it's valid. In the last lecture I said that quantifiers modify categories. That's how quantifiers and categories are related to each other. Categories are kinds of things and quantifiers modify categories. They can modify them in different ways. There are different kinds of quantifiers. In today's lecture we'll talk about some of the different kinds of quantifiers and how they modify categories. We'll also talk about how the representation of these different quantifiers differs both verbally and visually using the Venn diagram. Okay, so what are the different kinds of quantifiers? Well, here are some of the different kinds of quantifiers. There's the quantifier all and we can apply this to categories. Notice by the way in this slide I'm using an uppercase F and an uppercase G to represent any category at all. So in a statement of the form all F's are G's that that's the claim that all things that fall into one category, never mind exactly what that category is. Let's just call it F. Also fall into a second category. Again, never mind what that category is. We can just call it G. There are lots of examples of this kind of claim. When you say all things of one category also fall into a second category. For instance, all dogs are mammals. All squares are rectangles. All Coursera students are human and so on. In all of these cases you're making a claim of the form that all things that fall into one category, the F category also fall into the G category. So all is one kind of quantifier and any statement of the form all F's are G's. We're going to call a statement of type A, a proposition of type A. Then there's the quantifier no, which we can use in a statement of the form no F's are G's. No things of one category also fall into a second category. So for instance, you might say no humans are reptile, no Coursera students are trees, no democracies are at war. So there are examples of statements of the form that we're going to call E and the E statement is a statement to the effect that no things that fall into one category also fall into a second category. Then there's the quantifier sum. Now the quantifier sum can be used to make statements of two very different kinds. You could say that some things that fall into one category also fall into a second category. So if you'd said some humans are female, some Coursera students are American, some Americans speak Spanish. Those are statements to the effect that some things falling into one category, the F's, also fall into a second category, the G's. Some F's are G's. Propositions of that form will call I propositions. Finally, we could also use the quantifier sum to make a different kind of statement. A statement to the effect that some things that fall into one category don't fall into a second category. Some humans are not female, some Coursera students are not American, some Americans are not Spanish speakers. Statements of that form will call O statements or propositions. They say that some things that fall into one category don't fall into a second category. Some F's are not G's. So those are the four kinds of statements we're going to focus on this week on categorical logic. We'll consider other kinds of statements as well and we'll mention some other kinds of quantifiers, but those are the ones that are going to be of primary interest to us. Now, I've talked about these different quantifiers and how they can be used to make statements of very different kinds, but now we can visually represent the differences in the information provided in these four different kinds of statements. So consider, how would we visually represent a statement of the form that all F's are G's? All things that fall into one category also fall into a second category. Well, if you wanted to say all F's are G's, let's see, how would you do that? You'd be saying that all of the F's that there are fall into the G circle, but if all the F's that there are fall into the G circle, then there can't be any F's outside the G circle, right? So the Venn diagram for all F's or G's would look like that. You'd be saying whatever F's there are, they've got to be in here, right? There can't be any outside there. So that would be the Venn diagram for all F's or G's or a proposition of the A form. Next, there are propositions of the E form. Now, how would we represent those? Well, let's see. Here's the Venn diagram for no F's or G's. When you say no F's or G's, you're saying if there are any F's at all, they can't be inside the G circle. So you've got to shade out that portion of the F circle that's inside the G circle, right? The shading implies that there's nothing there, right? So there are no F's inside the G circle, and that's how you represent no F's or G's. Next, there are statements of the I form. Some F's or G's. How would you represent those? Some F's or G's, you use an X mark to represent that there is a thing. So you use an X to represent that there is a thing that's in the F circle, and that's also in the G circle, right? So there's a thing right there. It's in the F circle. So it is an F, but it's also a G. So some F's are G's. That's how you'd represent, how you'd visually represent a statement of the I form. Finally, how about a statement of the O form? Some F's are not G's. How would you visually represent that? Well, you'd have to make an X to indicate that there is an F, right? There is a thing that is F, but it's not G. If it's not G, then it's got to be outside the G circle. So it's inside the F circle, but outside the G circle. And that's how you'd indicate visually some F's are not G's. Now, I'd like you to notice something. We've just gone over the Venn diagrams for propositions of the A, E, I, and O forms. But what I'd like you to notice, let's go back to those forms. Propositions of the A form, all F's are G's are negations of propositions of the O form. Some F's are not G's, right? If all F's are G's, then it's not going to be true that some F's are not G's. And if some F's are not G's, then it's not going to be true that all F's are G's. All F's are G's, in other words, is going to be true when and only when some F's are not G's is not true. And similarly, propositions of the E form are going to be negations of propositions of the I form, right? When is it going to be true that no F's are G's? It's going to be true that no F's are G's when and only when it's not true that some F's are G's, right? If some F's are G's, then it's not going to be true that no F's are G's. And if no F's are G's, then it's not going to be true that some F's are G's. So propositions of the A and O form are negations of each other, and propositions of the E and I form are negations of each other. And those relationships are represented visually on the Venn diagrams that we just drew, right? Consider again, the Venn diagram for propositions of the A form, propositions of the A form, the Venn diagram is going to have shading in here, but in propositions of the O form, the Venn diagram is going to have an X in here, in precisely the place where propositions of the A form had shading. Propositions of the E form are going to have shading in here, whereas propositions of the I form are going to have an X in here. So the way that we represent negation using Venn diagrams is one proposition is going to be the negation of another just in case the first has shading wherever the second has an X, or vice versa, right? If one proposition has an X wherever the other has shading, or it has shading wherever the other has an X, then the two propositions are going to be the negations of each other. That's how the relation of negation is visually represented in Venn diagrams. In the last lecture, we learned about propositions of the forms that we labeled A, E, I, and O. We learned how to represent these propositions verbally, and also how to represent them visually using Venn diagrams. Today, I want to apply those lessons to study a kind of argument that we're going to call an immediate categorical inference. So what's an immediate categorical inference? An immediate categorical inference is an inference with just one premise, and of course, one conclusion, where each of those two statements, both the premise and the conclusion, are of the form A, E, I, or O. They needn't be of the same form. One of them could be, let's say, of the E form and the other one of the O form or whatever. They needn't be of the same form, but they're each of one of those four forms. Now, these are very simple inferences, but we're going to study them today because we're going to see how we can use Venn diagrams to show that some of these inferences are valid and others aren't. Now, before we do that, let me say something about the kinds of propositions that are involved in these inferences, the A, E, I, and O propositions. Remember what those letters stand for. The A proposition is a proposition of the form all F's or G's, all things that fall into one category also fall into a second category, and E proposition is a proposition of the form no F's or G's, nothing that falls into the first category also falls into the second category. An I proposition is a proposition of the form some F's or G's, that's to say something that falls into the first category falls into the second category, and an O proposition is a proposition of the form some F's or not G's, something that falls into the first category does not fall into the second category. Now, notice in each of those propositions, there's one category that I'm indicating by use of the schematic letter F. There's one category that gets directly modified by the quantifier, right, the F category, right. I'm talking about in the a proposition I'm talking about all things that fall into that category. In the e proposition I'm talking about no things that fall into that category. In the i or the o proposition I'm talking about some things that fall into that category. So there's a category that gets directly modified by the quantifier and then there's another category that doesn't. Okay, let's introduce terminology to distinguish these two categories. We'll talk about the two categories as the subject term and the predicate term. So the category that gets directly modified by the quantifier, the F category is what we're going to call the subject term and the category that doesn't get directly modified by the quantifier is what we're going to call the predicate term. You'll see in our later lecture about syllogisms why this terminology is going to be useful. It's not going to be obvious today why it's useful, but it'll eventually become useful. Okay, so Fs are subject terms, Gs are predicate terms, and immediate categorical inferences are inferences that have one premise with the subject term and predicate term and one conclusion with the subject term and the predicate term. All right. What kinds of immediate categorical inferences are there? Well, there are lots, but the most common kind of immediate categorical inference is one that we'll call conversion. A conversion inference is an inference in which the conclusion just switches the subject term and predicate term as they occur in the premise. So if the premise is of the form of no Fs or Gs, let's say no Duke students are NFL football players, then the conclusion would be of the form no Gs or Fs. No NFL football players are Duke students. That's an example of an immediate categorical inference that's a conversion inference. It converts one e proposition to another e proposition. You could do it for other forms of proposition. So for instance, consider the conversion inference from an a proposition to another a proposition. Consider the inference from all Duke students are NFL football players. To the conclusion, all NFL football players are Duke students. Now, notice the first of those two conversion inferences is plausibly valid, whereas the second one isn't. Why is that? Well, we can see why that is. If we use Venn diagrams to visually represent the information contained in the premises of those inferences, more generally, I can say right now that conversion inferences are valid for propositions of the E and I form, but they're not valid for propositions of the A or O form. We can understand why that is using Venn diagrams. So why is it that conversion inferences from the A form to the A form are not valid? Why are they not valid? Well, let's consider. What are you saying when you say all Fs or Gs? Well, you're saying that whatever Fs there are, they're not outside the G circle. They're all in the G circle. So you represent that information by shading in the portion of the F circle that's outside the G circle in order to indicate that there's nothing there. Okay, but now once you've shaded in that portion of the F circle, does that tell you that all Gs are Fs? No. Of course, it leaves open that all Gs are Fs. I mean, it could be that all the Gs there are are in here, but shading in the portion of the F circle that's outside the G circle also leaves it open that there are plenty of Gs out here. So once we look at the Venn diagram from the inference for all Fs or Gs to all Gs or Fs, we understand why that inference is not valid. Shading in the portion of the F circle outside the G circle leaves open that there are plenty of Gs that are outside the F circle. Okay, what about the second kind of conversion inference for propositions of the E form from no Fs or Gs to no Gs or Fs? Well, let's see, how would we represent no Fs or Gs? Well, we do that by shading in the portion of the F circle that's inside the G circle, right? So that shows that no Fs are in the G circle, no Fs or Gs. But now look, if no Fs or Gs, then we can just read off from that that no Gs are Fs, right? If there are any Gs, they can't be in the area that's shaded in, right? Because what it means to shade in the area is that there's nothing there. So whatever Gs there are, they can't be inside the F circle. They've got to be outside the F circle. So if it's true that no Fs or Gs, then it's got to be true that no Gs are Fs. The Venn diagram for no Fs or Gs shows us that. So conversion inferences are invalid for A propositions, but they're valid for E propositions. What about I propositions? Well, let's see, I propositions have the form some Fs or Gs. Well, how would we represent some Fs or Gs? We would indicate that there is something that's inside the F circle. It's an F, but it's also a G. It's in the G circle. We'd indicate it with an X. Okay, but notice, once we indicate by means of that X that there's something inside the F circle that's also inside the G circle, you can just read off from that, that there's something inside the G circle that's also inside the F circle. In other words, some Gs are Fs. So once again, from the Venn diagram, you can see that if some Fs or Gs is true, then some Gs or Fs has got to be true. In other words, the conversion inference for an I proposition has got to be valid. Okay, finally, what about O propositions? Some Fs are not Gs. Well, let's see, some Fs are not Gs. How would you diagram that? Well, some Fs, so you want to use an X to indicate that there is an F, but it's not a G. So it's got to be outside the G circle. So there, you draw an X that's inside the F circle to show that it is an F, but it's outside the G circle. So it's not a G. That's how you represent that some Fs are not Gs. Okay, now does that imply that some Gs are not F? No. Maybe there are no Gs at all. For all that you've just been told, right, you're told that some Fs are not G. That doesn't tell you that there are any Gs. Or if there are any Gs, maybe all the Gs are in here. You don't know whether there are any Gs in here. So from the premise that some Fs are not Gs, you can't infer that some Gs are not F. It doesn't follow that some Gs are not F. And the Venn diagram shows us why. So this last conversion inference on propositions of the O form is not valid. And once again, you can see that from the Venn diagram for propositions of the O form. So this shows how Venn diagram, simple two circle Venn diagrams can be used to establish the validity or invalidity of immediate categorical inferences. Okay, next time we'll consider inferences of a more complicated kind. See you next time. In the last class, we talked about immediate categorical inferences, which are inferences that have a single premise and a conclusion where each of those two propositions is of the A, E, I, or O form. Today, we're going to talk about a new kind of inference called a syllogism. So what's a syllogism? Here's a definition. A syllogism is an argument that has two premises and a conclusion where all three of those propositions are of the form A, E, I, or O. Now, the conclusion is going to have two categories in it. One category that's modified by a quantifier, and we're going to call that the subject category. The other category is not modified by a quantifier. We're going to call that the predicate category. Now, the subject category is what we call the subject term of the syllogism. The predicate category, the category that's not modified by a quantifier in the conclusion, that's what we're going to call the predicate term of the syllogism. Now, every syllogism is going to have a premise that includes the subject term and another premise that includes the predicate term. Now, the premise that includes the subject term is what we're going to call the minor premise of the syllogism. And the premise that includes the predicate term is what we're going to call the major premise of the syllogism. So every syllogism is going to have two premises where one premise is the minor premise and it's going to include the subject term of the syllogism, which is the category that's modified by the quantifier in the conclusion of the syllogism. And the other premise is the major premise of the syllogism. It includes the predicate term of the syllogism, which is the category that's not modified by a quantifier in the conclusion of the syllogism. Okay, now let's look at how we can use Venn diagrams to represent the information that's carried by syllogisms and so to assess whether or not a particular syllogism is valid. In order to visually represent the information that's given in the syllogism, we need to use a Venn diagram with three circles, not just two circles, because in the syllogism we have three categories. We have the subject term, the subject category, the middle term, the G's, and the predicate term, the H's. So we need to get a Venn diagram with all three of those circles to represent the information contained in the syllogism. Now, I've described all this very abstractly. Let me give some examples so you can see how this works and how we can use Venn diagrams like this to figure out whether or not syllogisms are valid and to explain why they're valid. Okay, so consider this syllogism. All Duke students are humans. All humans are animals. Therefore, all Duke students are animals. Okay, valid or not? Well, pretty obviously it is valid. But a Venn diagram could help us to understand why it's valid. So let's diagram the information contained in the premises. So the first premise recall was that all Duke students are humans. Well, if we want to show that all Duke students are humans, that's to say that whatever Duke students there are have got to be inside the circle of the humans. So this region of the Duke student circle, this region of the circle that's outside the circle of the humans, we can shade that in to indicate that there's nothing there. Right? Nothing here because whatever Duke students there are have got to be in this region. Okay, the second premise said that all humans are animals. Well, if all humans are animals, then what that tells us is that whatever humans there are have got to be inside the circle of the animals. So we can shade in the portion of the human circle that's outside the animal circle, right? Because there aren't any humans out there. So shade that in. Okay, but now we look at the diagram with those regions shaded in. And what can we conclude? Well, we can conclude that whatever Duke students there are have got to be in this region right here. That's the only region where there could be any Duke students given the two premises of our argument. In other words, all Duke students are animals. And that's just the conclusion of our syllogism. Remember, all Duke students are animals. So we just used the Venn diagram to explain why this syllogism, which is obviously valid, is valid. We explained why it is valid. It is valid because when you shade in the portion of the Duke student circle that's outside the human circle and you shade in the portion of the human circle that's outside the animal circle, the only place left over in the Duke student circle for there to be any Duke students is inside the animal circle. And so all Duke students are animals, as we all know. Now, here's a second syllogism. Let's consider this one. Some Duke students are humans. All humans are animals. Therefore, some Duke students are animals. Valid or not? Well, let's see. So, some Duke students are humans. How would we represent that information? Some Duke students are humans. What that means is that there's got to be something in this part of the circle of Duke students that's also in the circle of humans. But the first premise doesn't tell us where that thing would be. Would it be here or would it be here? Well, let's hedge our bets and draw it right here since we don't know. We'll draw it on the borderline of the animal circle since the first premise doesn't yet tell us whether it's here or here. The second premise said all humans are animals. Well, if all humans are animals, what that tells us is that there aren't any humans outside the animal circle. So, we can just shade in the part of the human circle that's outside the animal circle. Shade it in right there. But then notice, remember, we had to have an X inside our Duke student circle and our human circle inside the intersection. Well, it can't be in here because this is shaded in, which means there's nothing there. So, the only place it can be is right there. So, now we know that the X had to be over here. But once we put the X over there, which we have to, given the information in the two premises of our syllogism, what can we conclude? We can conclude that some Duke students are animals. And that's exactly what the conclusion of the syllogism is. That some Duke students are animals. Finally, oh, that should say example three, not example one. Finally, consider this syllogism. No Duke students are humans. All humans are animals. Therefore, no Duke students are animals. Okay. Now, how would we represent that using the Venn diagram? Well, no Duke students are humans. So, that means that we have to shade in the portion of the Duke student circle that's inside the human circle to show that there's nothing in there. Right? None of the Duke students are humans. Okay? All humans are animals. So, that means we have to shade in the portion of the human circle that's outside the animal circle. Right? Because there are no humans out there. All humans are animals. Right? And the conclusion was that no Duke students are animals. But wait a second. You can't read that off the diagram. There could be lots of Duke students over here who are animals. There could be all sorts of Duke students who are animals right there. They wouldn't be human. But there could still be plenty of Duke students that are animals. In other words, this third syllogism is not valid. And it's not valid for a reason that's made clear by the Venn diagram that we just looked at. The Venn diagram shows us that and explains why the inference, the syllogism, is not valid. Just because no Duke students are humans and all humans are animals, it doesn't follow that no Duke students are animals. Those premises leave it open that there are plenty of animal Duke students. They just wouldn't be the humans. So, in this lecture, I've tried to show how we can use Venn diagrams to predict that and to explain why syllogisms are valid or invalid, as the case may be. Next time, we'll apply these lessons to some examples that we've already looked at before, but that we weren't treating as syllogisms when we looked at them before. See you next time. In the last few lectures, we've seen how we can use Venn diagrams to predict that and to show why, explain why certain immediate categorical inferences and certain syllogisms are valid or invalid. Now, I'd like to make a point today about categorical logic, about categories in general. It might seem that the application of the method of Venn diagrams is really quite limited, because a lot of our statements aren't really about categories at all. They're about individuals, not categories. But I want to say that this is an example of how ordinary language can mislead us. If we think about precisely what it is that we're saying in a lot of the ordinary statements that appear to be about individuals, we'll see that, in fact, those ordinary statements really are about categories, not just about individuals. Let me give you an example to illustrate this point. Consider the statement, Mary owns a Ferrari. Now, that statement appears to be just about individuals. It doesn't appear to involve any categories, right?  Not that any of my co-workers would own a Ferrari, but suppose that Mary is a particular person, and we're talking about Mary's Ferrari, which is a particular car, and it seems that that statement is just claiming that the person, the particular person I'm talking about, owns that particular car. All right? There don't seem to be any categories involved at all. But wait a second. Compare the statement Mary owns a Ferrari to the statement, some of Mary's possessions are Ferrari cars. Now, that second statement notice is of the I form. It's of the form some F's, RGs. You're saying some things that fall into one category, the category of Mary's possessions, also fall into a second category, the category of Ferrari cars. Now, these two statements seem to amount to the same thing. When I say Mary owns a Ferrari, what I'm saying is true if and only if some of Mary's possessions are Ferrari cars. The information carried by one of those statements seems to be the same as the information carried by the other. So here's a case where to look at the language we were using when we said Mary owns a Ferrari, you would have thought, well, we're not talking about any categories there. So the method of then diagrams and all of this stuff about categorical logic is completely irrelevant to any argument we might make that uses the statement Mary owns a Ferrari either as a premise or as a conclusion. But in fact, that's not true. In fact, the statement that Mary owns a Ferrari is equivalent to the statement some of Mary's possessions are Ferrari cars, which is a statement of the I form some F surges. And so the categorical logic that we've been developing this week in the method of Venn diagrams actually can be used to study the validity of arguments where the statement Mary owns a Ferrari is included as one of the premises or as the conclusion. Because the statement Mary owns a Ferrari is just equivalent to a statement of the I form some of Mary's possessions or Ferrari cars. I want to say this phenomenon is very general. Not every statement is of the AI or O forms, but lots and lots of ordinary statements that we make are of one of those forms. Many more than ordinary language would suggest. So with that in mind, I want now to show how we can apply the categorical logic, how we can apply the lessons we learned concerning the validity of syllogisms and how we can use Venn diagrams to predict and explain the validity of syllogisms, how we can now apply those lessons to arguments that are syllogisms, but you wouldn't think that they are just to look at the language in which they're expressed. Okay, see next time. Today, we're going to talk about how you can use Venn diagrams to show that a particular inference is valid and to explain why it's valid. We're going to do that by considering the particular inferences, the particular arguments that we looked at at the beginning of last week when we were starting our unit on deductive logic. So let's look at those inferences right now. Remember the first one was the example that Walter gave back in week three, our argument about Mary. Premise one was Mary has a child who is pregnant. Premise two was only daughters can become pregnant. And so the conclusion of those two premises was therefore Mary has at least one daughter. Okay, now how do we represent the information contained in the premises of this inference using a Venn diagram? Well, let's begin by asking what are the categories that this inference that this argument brings into relation with each other is Mary one of the categories? Well, no, because Mary's not a category, Mary's an object, Mary's a thing. She's a particular thing. She's not a category. What is a category though, that's mentioned in this argument is Mary's children. And the information in the first premise is that the category of Mary's children includes something that is also in the category of pregnant people or pregnant beings. So we can visually represent that information as follows. The category of Mary's children includes something that's in the category of pregnant people. Right, so we don't know whether or not the category of Mary's children includes anything out here. But whether or not it includes anything out here, it's got to include something that's in this region. So do we draw an X? Well, let's wait a minute. Let's ask, where would we draw that X? Would we draw it here? Or would we draw it here? That's going to make a difference. That's going to make a difference where we draw it. So we don't yet know whether to draw the X in here or in here. We know though that the X is supposed to go somewhere in this football shaped region that's in the intersection of the category of Mary's children and the category of pregnant people. Maybe premise two will help us figure out where to draw that X. Let's see. So premise two says only daughters can become pregnant. So how do we visually represent the information that only daughters can become pregnant? Well, only daughters can become pregnant means that there can't be any pregnant people who are not daughters. So we have to shade out the circle, the portion of the circle of pregnant people, people that's outside the circle of daughters. Right? There can't be any pregnant people who are outside this circle. All the pregnant people are inside this circle. Okay, but remember, what did premise one tell us? Premise one told us that Mary has a child who is inside this circle. So here's what we know. Mary has a child who's inside the circle of pregnant people and anyone who's inside the circle of pregnant people must also be inside the circle of daughters. Well, so Mary's child who's inside the pregnant people circle must be right here. So that's where the X goes. So what we can figure out from just the first two premises, remember, I haven't relied on any information here other than the information given in premises one and two. What premises one and two tell us is that there's an X and it goes right here. Okay, but if the X goes right there, then what does that tell us? That tells us that at least one of Mary's children is a daughter. Mary has at least one daughter. Maybe she has more than one daughter, but we know based on the information contained in premises one and two, we can conclude that she has at least one daughter. We know that for sure based on the information contained in premises one and two. So if the information contained in premises one and two is correct, then Mary must have at least one daughter. And so the argument is valid. And the Venn diagram shows us that the argument is valid because once we visually represent the information contained in premises one and two, then we can simply read off the diagram that the conclusion is true. We can read off the diagram that Mary has at least one daughter. Okay, now let's see if we can apply this technique to the other arguments of the same form that we considered at the beginning of last week. Remember, we considered an argument concerning Terry. Premise one was Terry has a job in which she arrests people. Premise two was only police officers can arrest people. And the conclusion of those two premises was therefore at least one of Terry's jobs is as a police officer. Now, how would we visually represent the information contained in the premises? Well, let's consider premise one tells us that Terry's jobs includes a job in which she arrests people, right? So at least one of Terry's jobs has to be in this circle. Because whatever else it is that Terry does, she has a job in which she arrests people. So at least one of her jobs must be in this circle. And of course, it has to be in this circle because that's the circle of Terry's job. So at least one of Terry's jobs must be in this football shaped region right here in this intersection of the category of Terry's jobs and professionals who arrest people. That's what premise one tells us. But again, it doesn't tell us where that particular job of Terry's would be. Is it here? Or is it here? Right? Premise one doesn't give us that information. But what does premise two tell us? Well, remember premise two tells us that only police officers can arrest people. So how would we visually represent that? Only police officers can arrest people. That means that there's in the circle of professionals who arrest people, we have to shade in this part because there are no professionals who arrest people other than police officers. Only police officers arrest people. So we shade in that portion of the professionals who arrest people circle. And I remember what we learned from premise one was that Terry has a job in which she arrests people. So somewhere in this football shaped region, there has to be an X. But we know now that it can't be here. So the X must be here. So that's the information that we get from premises one and two. But notice, once we visually represent that information in our Venn diagram, then we can simply read off the Venn diagram that Terry has a job, at least one job in which she's a police officer. So at least one of Terry's jobs is as a police officer. And we can just read that off the Venn diagram once we've used the Venn diagram to represent the information contained in premises one and two. So the Venn diagram, again, can be used to represent the validity of the inference. And not only that, notice also that the Venn diagram for this inference is the same in the placement of its shading in its X's as the Venn diagram for the previous inference about Mary. Well, what does that tell us? That tells us that the two inferences, the two arguments are of the same form. They're of the same form. And you can see that by seeing that their Venn diagrams look the same. The only difference between the two Venn diagrams is the labeling of the circles is the categories, the specific categories involved in the arguments. Other than the specific categories involved in the arguments, the labeling of the circles, everything else is the same. The shading is in the same place, the X is in the same place. Okay. Now, can we apply this to our third argument about Robert? You'll remember this third argument that we considered of the same form at the beginning of week four at the beginning of our unit on deductive logic. Remember, we considered an argument that went like this. Premise one, Robert has a pet who is canine. Premise two, only mammals are canine. Conclusion, therefore at least one of Robert's pets is mammal. Okay, well, how would we visually represent the information contained in premises one and two of this argument? Here's how. At least one of Robert's pets is canine. So that means that whatever other pets Robert has, right, maybe Robert has some non canine pets, but at least one of Robert's pets must be in this circle. Well, that particular pet that we're talking about has to be in this circle. But of course, it also has to be in this circle because this is the circle of Robert's pets. So the pet must be in this football shaped region right here, the intersection of Robert's pets and canines. But where is it? Is it here or is it here? Well, now let's look at premise two. What do we find out from premise two? What we find out from premise two is that only mammals are canine. Okay, well, if only mammals are canine, then that means that there can't be any canines outside the circle of mammals, right? All the canines, whatever canines there are, are in this region right here. There aren't any canines outside the mammal circle. So we know that whatever canines there are have to be in this intersection right here of the circle of canines in the circle of mammals. And we also know that whatever that Robert has at least one pet in the circle of canines. Okay, well, if Robert has a pet in the circle of canines, and we also know that there's nothing in the circle of canines that's over here, then we know that Robert's pet, which has to be in this circle, right, this is the circle of Robert's pets, Robert's pet has got to be in here, right? The pet that we're talking about, the pet that's canine has got to be in there. Okay, so that's a way of visually representing the information contained in the in the two premises of our argument. But once we visually represent that information, we can just look at the diagram and read off the fact that at least one of Robert's pets is mammal. Because at least one of Robert's pets is inside the circle of mammals, right, inside the circle, which means that at least one of Robert's pets is mammal. But that's exactly what the conclusion says. And so this Venn diagram shows us that this argument about Robert is valid just as the arguments about Mary and Terry were valid. And furthermore, this argument about Robert is valid for the same reason as the arguments about Mary and Terry were valid. They're valid not because of their subject matter, right, the three arguments have completely different subject matters. They're valid because of their form. And their form is what's revealed by the Venn diagram. The form is what's in common to the three Venn diagrams. The placement of shading and X's in the Venn diagrams is their common form. And that's what explains why all three of the arguments are valid. Okay, see you next time. In the last few lectures, we've seen how we can use Venn diagrams to visually represent the information that's conveyed in A, E, I or O propositions. And we've also seen how there are different ways in ordinary language of expressing such propositions. Sometimes such propositions are expressed even by sentences that use individuals' names and don't explicitly mention categories. Today, I want to talk a bit more about that. I want to talk about the various ways in which, in English at least, and in many other natural languages, we can represent the information conveyed by A, E, I or O propositions using forms of language that we haven't yet discussed. So let's consider some examples. So consider propositions of the form not all F's or G's. We haven't talked about not all before. Well, when you say not all F's or G's, what exactly are you saying? Usually, when you say not all F's or G's, what you mean to be saying is that there are some F's that are not G's. In short, what you mean to be saying when you say not all F's or G's is usually a proposition of the O form. So how would you represent not all F's or G's? Well, all F's or G's, remember, would be represented by shading this out. That would be all F's or G's, but you're saying not all F's or G's. In other words, some F's are not G's. So instead of shade there, what you would do is put an X right there. There's some F, something out there that's not a G. It's outside the category of the G's. And for some examples, you might say not all geniuses take Corsera courses. In other words, some geniuses don't take Corsera courses. Or you might say not everything that Pat does is intended to annoy Chris. In other words, some of the things that Pat does, some of the things that fall into the category, things that Pat does, are intended to annoy Chris. That is to say they do fall into the category, things that are intended to annoy Chris. So those are both examples of statements of the form not all F's or G's. And they're both equivalent to a statement of the form some F's or not G's. Now consider statements of the form all F's are not G's. Well, when you say something of the form all F's or not G's, that's usually intended to convey that no F's are G's. In other words, it's usually intended to convey a proposition of the E form. So when you say no F's or G's, how do you represent that? Well, you represent it by shading in the intersection of the F's and the G's. And that's just to say that all F's are not G's. Right? If there are any F's at all, they've got to be out here. They've got to be outside the category of the G's. Some examples of statements like that. All Nobel Prize winners are not alcoholics. Right? You're saying if there are any Nobel Prize winners, they're outside the category of the alcoholics. Everything that Pat does is not designed to achieve victory. Again, you're saying that whatever falls into the category of things that Pat does, it is outside the category of things designed to achieve victory. Right? So if this is the category of Pat's actions, and this is the category of things designed to achieve victory, then when you say everything that Pat does is not designed to achieve victory, you're saying there's nothing in this intersection. Whatever Pat's actions are, they've got to be out here. Okay, finally, let's consider a statement of the form. Some F's are both G's and H's. If you say some F's are both G's and H's, what are you saying? Well, are you saying that there are some F's that are G's and some F's that are H's? Well, that's something you could say, of course. You could say some F's are G's and some F's are H's. You could say some men are tall and some men are short. But when you say some F's are both G's and H's, you're not saying that. You're saying that some F's are in this intersection of G's and H's. So some F's have got to be here. That's what you're saying. When you say some F's are both G's and H's. That's why you couldn't say, at least you couldn't truthfully say some men are both tall and short. There's no man who is both tall and short. But you could say there are some men who are both wealthy and happy. They're a man who is wealthy and who is happy. Other examples, some philosophers are both robotic and monotone. It's not just that there are some philosophers who are robotic and some philosophers who are monotone. There are actually some philosophers who are both robotic and monotone. Some of the things that Pat does are both intended to amuse and to provoke. It's not just that some of the things she does are intended to amuse and some of the things she does are intended to provoke. It's that some of the things she does are intended to do both amuse and to provoke. So if these are Pat's actions right here, and this is the category of things intended to amuse, and this is the category of things intended to provoke, then this last statement, some of the things that Pat does are both intended to amuse and to provoke, is saying that some of Pat's actions fall into this area right here, this intersection of things designed to amuse and things designed to provoke. But of course, since they're Pat's actions, they have to fall into this portion of the area right here. Okay, I hope I've given you a sense of the various ways in which ordinary language can express ideas that we can represent visually using Venn diagrams. Over the past couple of weeks, we've seen how we can use truth tables in Venn diagrams to predict and explain the validity of arguments. But the arguments that we've looked at are all arguments that are given in ordinary language that we can understand. But of course, not all arguments are like that. Some arguments are given in foreign languages that we don't understand, or they're given in technical languages that we don't understand. Could you use truth tables or Venn diagrams to predict and explain the validity of those arguments? Even those arguments that are given in languages that you don't yet understand? Well, today, I'm gonna show that you can. You can do that. And I'm gonna show it by giving some examples. So let's start with a simple but fanciful example.   And you've translated most of their words, but there's this word they use, spook, and you haven't yet figured out what that means. And so you're observing their behavior.  And you're pretty confident of this hypothesis. Your hypothesis is that they use spook as a truth functional connective. And it's a truth functional connective that has this truth table. It connects two propositions to make a larger proposition. And that resulting proposition, p spook q, for whatever propositions p and q spook is connecting, that resulting proposition is gonna be true whenever p is true or whenever both p and q are false. So the only scenario in which p spook q is gonna be false is where p is false and q is true. So that's your hypothesis about what they mean by spook. Okay, now one day you hear one of the members of this foreign culture give an argument. And using the translation manual that you've developed, you translate their argument as follows. John is riding his bicycle. Spook, Jill is walking to the park. Premise to Jill is walking to the park. Therefore, John is riding his bicycle. Now, is this argument valid or invalid? Well, you can use the truth table for spook to figure out whether the argument is valid or invalid and also to see why. So, let's consider how the truth table would go, right? This is the truth table again for spook applied to the case that we're considering. Well, premise one tells us that John is riding his bicycle, spook, Jill is walking to the park. So, if premise one is true, then we're either in this scenario, in this scenario, or in this scenario. Premise two tells us that Jill is walking to the park. So, if premise two is true, then we're either in this scenario or in this scenario. Okay, so what do premises one and two put together? What do they tell us? Well, premises one and two tell us that, first of all, we're not in this scenario because in this scenario, it's not gonna be true that Jill is walking to the park. Again, premises one and two put together tell us that we're not in this scenario because in this scenario, it's not gonna be true that John is riding his bicycle, spook, Jill is walking to the park. Again, premises one and two tell us that we're not in this scenario, because in this scenario, it's not gonna be true that Jill is walking to the park. So if premises one and two are both true, that tells us that we've gotta be in this scenario. But in this scenario, it's gotta be true that John is riding his bicycle. And so using the truth table for spook, you can figure out that this argument is valid. And you can explain why it's valid. See? So you can use the truth table for a truth functional connective that you don't have any understanding of independently of the truth table for it, right? It's not a connective that you use in your language. You can just use the truth table to figure out that a particular argument is valid. Let's try that with a more complicated case. Okay. So now suppose you hear some member of this foreign culture arguing as follows, at least according to your translation. John is riding his bicycle spook. Jill is walking to the park or Frank is sick. Frank is not sick. John is not riding his bicycle. Therefore, Jill is walking to the park. Okay. Now is that argument valid or not? Okay. So let's look at this more complicated truth table. Okay. So premise three tells us that John is not riding his bicycle. So that tells us that we've got to be in one of these four scenarios. Premise two tells us that Frank is not sick. So that tells us that we've got to be in one of these four scenarios. And premise one tells us that John is riding his bicycle spook. Jill is walking to the park or Frank is sick. So that tells us that we've got to be in one of these scenarios right here. Okay. So based on that information, what can we figure out? Well, we can immediately figure out that we're not in one of these top four scenarios because in all of those top four scenarios, Frank is riding his bicycle, right? So premises one, two and three together rule out the top four scenarios. We can also figure out that we're not in scenarios five or seven because in scenarios five or seven, Frank is sick. So we can rule out those scenarios. And we can rule out the sixth scenario because in the sixth scenario, it's not true that John is riding his bicycle spook. Jill is walking to the park or Frank is sick. So based on the information that we get from premises one, two and three of the argument, we can deduce that we're in this last scenario. We've got to be in this last scenario. And in this last scenario, it's false that Jill is walking to the park. In other words, Jill is not walking to the park. But that's not what the conclusion of our argument says. So this argument, this more complex argument is not valid. And we can prove that as we just did using the truth table for spook and disjunction. Okay. Now, let me show how we can do the same thing with quantifiers. So imagine as you're translating this foreign language, you hear a word. It's always said in a high-pitched tone, an excited tone, jid. And you wonder, what does jid mean? Well, after watching the way they use that word, you come up with a hypothesis. Your hypothesis is that when they use the word jid, they're using a quantifier. And the quantifier works like this. When you say jid f are g, you're saying that there aren't any f that are outside the g category, but there is an f that's inside the g category. So this diagram represents, according to your hypothesis, this diagram represents what the members of this foreign culture mean by jid. Okay. So now, suppose you hear one of the members of this foreign culture give the following argument. Jid, giraffes or herbivores, you translate. Then they say jid, herbivores are mammals. And then they draw the conclusion there are some giraffes and all of them are mammals. Now, is that argument valid? Well, let's use the Venn diagram for jid to figure out whether or not it's valid. So the first premise says, jid, giraffes or herbivores. So, how do we represent that? Well, remember, we have to shade out the portion of the giraffe circle that's outside the herbivore circle, and we have to draw an x that's in the giraffe circle and in the herbivore circle. We don't know quite where to draw that x, so let's hedge our bets right now and draw it over here. Okay. The second premise, though, tells us that jid, herbivores are mammals. Now, how do we represent that? Well, remember, by the Venn diagram, we have to shade out that part of the herbivore circle that's outside the mammal circle and draw an x inside the herbivore circle that's also inside the mammal circle. Right, now we know that this x that we drew on the edge has to be inside the mammal circle, not outside the mammal circle. Okay, so we're gonna put this x right here on the edge of the mammal circle and the giraffe circle because we don't know specifically which one it's supposed to go into. Oh, but wait, we do, and here's why we do. Remember that when we drew the x on the border of the mammal circle and the herbivore circle right here inside the giraffe circle, we weren't sure if that x was supposed to go in here or in here. Now we know that this x is supposed to go in here. So maybe there are also herbivore mammals that are not giraffes, but what we can definitely conclude from premises one and two is that there are herbivore mammals that are giraffes. So we know that there's an x that's supposed to go in this region right here. There's an x in that region right here, and then this whole region is shaded out. There's nothing in this whole region. That's what we know from premises one and two using the Venn diagram for jid. Excuse me, for jid. Okay, now let's apply that to our argument. Well, what does our argument say? It says there are some giraffes and all of them are mammals. But that's exactly what we can read off from the Venn diagram that we just constructed, right? If you look at the Venn diagram that we just constructed, there's an x that's inside the giraffe circle and inside the mammal circle and every part of the giraffe circle that's outside the mammal circle is shaded out. So there aren't any giraffes that are not mammals. Okay, so we just figured out that this argument is valid and we figured it out even though our only understanding of jid was from the Venn diagram that we constructed. Now, I have another question. Given this Venn diagram, how would you translate jid into English? Well, often people use the word all to mean the same thing that jid means according to this Venn diagram. People speaking ordinary English use the word all to mean precisely this quantifier, right? When they use the word all, they often mean that there are things in the category that they're modifying by all. Like if I say all ravens are birds, often what I'm understood to mean is there actually are some ravens and all the ravens there are are birds. So there are some things in the F circle but they're all inside the G circle. There are some things in the circle of ravens, let's say, but they're all also inside the circle of birds. There are no ravens outside the bird circle but there are ravens inside the bird circle. That's often what people would understand me to mean if I said all ravens are birds. Now, we've been using the quantifier all, we've been understanding the quantifier all in a different way so that it doesn't imply that there actually are members of the category that's being modified by all. So the way we've been using the quantifier all, if you say all ravens are birds, all you mean is there aren't any ravens that are not birds. But that's not the same as saying there are ravens and all of them are birds. So it looks like we can use the Venn diagram for JIT to translate that quantifier in a foreign language into a familiar quantifier from ordinary English. Well, welcome back. I've really missed you over the last couple of weeks when you've been off with raw meta learning about propositional logic and then categorical logic. And I hope that you enjoyed that, it's very important. But in the next few weeks, we're gonna turn to a different kind of argument because that type of logic is appropriate for deductive arguments. And what we're going to study instead is inductive arguments. Now, deductive arguments are given quite often throughout life, but inductive arguments are probably even more common. Whenever you wanna figure out who committed a crime, we were talking about an interest of the best explanation. Then there are arguments from analogy which happened quite often just think of all the analogies that you run into in life. And then we'll see statistical generalizations. Well, just think about all the polls and politics these days and statistical applications.  And then we'll turn to causal reasoning. Well, causes are crucial to everything from science to figuring out what made your car stop and decision-making and probability. So we got a lot of things to cover. And the first thing that we need to do is to see the difference between inductive and deductive arguments. Because you've been focused for a couple of weeks with Rahm about deductive arguments. You need to understand what's the difference between those arguments and the ones that we're gonna study over the next couple of weeks. In order to see that difference, I'm gonna show you a little video. It's a video I made a few years ago. So you might notice that I was a little younger back then. Okay, fair enough. But I think this video will help you see what the difference is between inductive arguments and deductive arguments. Rainy days and Mondays always get me down. The sun will come out tomorrow But your bottom dollar that tomorrow will be sun tomorrow, tomorrow Wait a minute, wait a minute. How do you know that? I heard a weather report. So what? They get it wrong all the time. Well, maybe this one won't come out tomorrow but at least it'll come up tomorrow. So you say, but is that really true? How many of you agree that the sun will come up tomorrow morning? Of course, definitely. Absolutely, yes. Everybody knows that. Okay, so you all agree with her but do you really know that it's true? How could you show that the sun will come up tomorrow? I don't know, I'm just a kid. Wait, I didn't mean to get you all upset but there really is an issue here. The question is, how can you prove that the sun will come up tomorrow morning? Well, here's one way that some people use. The conclusion is that the sun will come up tomorrow. The premise is yesterday the sun came up. The second premise, the day before that the sun came up. The third premise, the day before that the sun came up. Then the day before that, then the day before that, and the day before that, and the day before that, and the day before that, and the day before that. Well, that can get awfully long so we'll shorten the argument. The conclusion is still that the sun will come up tomorrow but the premises are just the sun came up yesterday and the sun came up every day before that for an awfully long time. Now is the argument valid? Yes or no? Right. It's not valid. An argument is valid if and only if it's not possible for the premises to be true and the conclusion false. But it is possible for these premises to be true and this conclusion to be false. Can you imagine how that would happen? One way would be if a meteor struck the earth in the middle of the night and stopped it from spinning. Then the sun wouldn't rise tomorrow. Now you're really freaking me out. But don't worry, that's extremely unlikely. However, it's possible and that shows that the argument's not valid. The next question is, is this argument any good? An argument's good if it serves its purpose. The purpose of this argument and many other arguments is to provide reasons for its conclusion. So, do the premises in this argument provide reasons for its conclusion? Yes, some philosophers deny this, but most people think that these premises provide good reasons for its conclusion. So let's assume that for now. What does that show? That shows that some arguments can be good even if they're not valid. But don't generalize too quickly. There are other arguments that are no good because they're invalid. Take Bub, for example. Is it true that every sophomore is a student? Yeah. Are you a sophomore? No. So you're not a student, right? Wrong. I am a student. Well, then what's wrong with the argument? Every sophomore is a student. You're not a sophomore, so you're not a student. Well, both the premises are true, but the conclusion's false, so the argument must be invalid. So you're telling me that what makes that argument bad is that it's invalid. Yep. Good job. Now, we've got a puzzle. Some arguments are bad because they're invalid, whereas other arguments are good even though they're invalid. How can that be? Simple. There are two kinds of arguments, deductive and inductive. Deductive arguments are bad when they're invalid. Inductive arguments can be good even though they're invalid. Why? Because deductive arguments are intended to be valid, whereas inductive arguments are not intended to be valid. The crucial point is that there are different kinds of standards for evaluating arguments. Deductive standards ask whether an argument is valid or invalid. Inductive standards ask whether an argument is strong or weak. There's several important differences between these standards. First, deductive validity is all or nothing. An argument is either valid or not.  Inductive strength, in contrast, comes in degrees. An argument is stronger when it gives more and better reasons for its conclusion. So an argument can be very weak when the reasons it gives for its conclusion are very weak or it can be moderately strong when it gives moderately strong reasons for its conclusion or it can be very strong when it gives very strong reasons for its conclusion. Since inductive strength comes in degrees, we can't simply ask whether an argument is strong. We need to ask whether it's strong enough. That depends on the context and the values at stake. It's like you've got your cooking lesson? Yeah! One of the most important things in cooking is to make sure that whatever it is you're cooking is done. When people cook cakes the way they normally test it for being done is they take a straw or a piece of bamboo and they stick it in the middle of the cake. It comes up with raw dough, it's not done, but if it comes up clean like this, that means it's done. Well, at least in that spot, but if you want to make sure that it's done throughout the whole cake, then you have to test other spots as well. So much for that. Now, when you're cooking turkey, you could use the same method. You could stick in a straw or a piece of bamboo and see whether any pink juice comes out. If it comes out pink, then it's not done. But if it comes out clean, then the turkey's done. See any pink juice? No. Then it must be done. Let's try a piece. But you wouldn't want to do that with turkey because you can get very sick from uncooked turkey. That's why most of the time they build a pop up meat thermometer right into the turkey to make sure that nobody eats uncooked turkey. But this is not a cooking class. The point here is simply that whether an argument is strong enough depends on what's at stake. When there's a lot to lose, we demand better reasons and stronger arguments. Context is also important in another way. Additional information from the context can weaken an inductive argument. That is, it can change it from strong to weak. In technical language, the inductive standard of strength is called defeasible or non-monotonic. In contrast, the deductive standard of validity is indefensible or monotonic. That means that no matter what premises you add to a valid argument, it will still be valid. Go ahead. Try. Here's an example. If Joe is a sophomore, then Joe is a student. Joe is not a student, so Joe is not a sophomore. And any premise you want to that argument, and it will still be valid, which shows that the deductive standard of validity is indefensible. In contrast, the inductive standard of strength is defeasible. If you add additional premises or additional information, it can make a strong argument weak. Consider a courtroom. I'm sure it was him. He was only 10 feet away from me. I saw him do it. I'm sure it was him. I saw him with my own eyes. I'd like to cross-examine this witness. Are you positive that it was the defendant sitting right there, or could it have been his identical twin brother who is entering the courtroom right now? It could have been. I can't tell him apart. Now I would like to recall the first witness. Let me remind you that you're still under oath. Can you tell the difference between those two gentlemen? No, they look like the same person to me. Now there's at least a reasonable doubt, isn't there? What was very strong evidence becomes much weaker when we add the new evidence about the identical twin. That is an example of the defeasibility of inductive standards. Although tricky, it's important to classify arguments as inductive or deductive because it affects whether they're good or bad. An inductive argument might not be valid, but it can still be good, for example. The sum came up yesterday, it came up every day for thousands of years, therefore it will come up tomorrow. That's a good argument, even though it's invalid, because it's not intended to be valid. This is just one instance of a general rule that you shouldn't criticize something for not being what it's not supposed to be. This book can be a good book, even though it's no good as a frisbee. Similarly, an inductive argument can be a good argument, even if it's not valid, if it's not intended to be valid. That is, if it's an inductive argument. This point undermines many common mistakes about inductive arguments. Many people think that inductive arguments are somehow inferior to deductive arguments because they're not valid. But that can't be right, because it's easy to take any inductive argument and turn it into a deductive argument. For example, most of Joe's friends are seniors, so Joe must be a senior too. Well, we can make that argument valid simply by adding a conditional premise. If most of Joe's friends are seniors, then Joe is a senior too. But that just shifts all the doubts about the inductive argument into doubts about the conditional premise. So it hasn't really made us more sure of whether Joe is a senior. Our definition of induction also undermines a second common mistake. Many people believe that induction always takes us from the particular to the general. But no matter how many people say that, it's just plain wrong. Inductive generalization is one kind of inductive argument, but it's not the only kind. In fact, we are going to study five different types of inductive arguments. To see the differences between these arguments, let's do a simple experiment. Excuse me, we're testing a new type of lemonade here. Could you help us out? Sure. Thank you. First, could you try one from that cup? That's pretty good. Great. Now, could you try this second cup, please? Oh, that's terrible. Your reaction suggests that you don't like that. Well, obviously, you put dishwashing liquid in there. How do you know that? Well, it tastes like soap. Is that why you spit it out? Well, I didn't spit out the first one, did I? You might be interested to know that you agree with over 90% of the people we've tested so far. They all like our lemonade better than dishwashing liquid. That's great.  Well, of course they would.  I'm out of here. This experiment would be stupid if it were intended to show something about lemonade. But all it's supposed to show is the differences among the different kinds of inductive arguments. Our next step will be to look at each of these kinds of induction in more detail. Now that we've seen which features distinguish inductive arguments from deductive arguments, we want to look at the different kinds of inductive arguments one by one. And the first two we're going to look at are moving up from a sample to a generalization and then moving back down from the generalization to some kind of prediction in a particular case. Generalizations are all around us. Almost all movies have credits. Most popular bands have drummers. Many restaurants are closed on Mondays. Three quarters of police officers are very nice people. Two-thirds of books have chapters in them. Half of the people I know like to play sports. And my favorite of all, 87.2% of statistics are made up on the spot. It should come as no surprise at all that there's so many generalizations filling our lives because generalizations can be extremely useful, especially when you need to make a decision.  And they also need to know whether doctors can usually help people who have symptoms of that sort. All of those generalizations are relevant to whether or not you want to go to the doctor to see if he can get some help with your sick stomach. But then the problem is how can we decide which generalizations to believe? Since they're generalizations and they apply to all instances of a certain sort, you can't check them all out. You have to take a sample of some sort.  You've made hundreds of jelly donuts. You're not going to tear them all apart and check everyone for how much jelly they have in the middle because then you have no donuts left to sell to your customers. Or imagine that you want to buy a car. This time you're the customer. You want to buy a car and you need to know how often these cars had problems in the first year because if they have problems a lot during the first year you don't want to buy that kind of car. But you don't want every car of that sort to have been tested for the first year because then you can't buy a new car. You're only going to have a used car since they've all been used in the tests. Or imagine that you want to know what types of trees grew during a certain period of history. So you look at the soil and you dig down and you start looking for how many pollen grains there are at a certain level that indicates a certain age. Well you can't check every spot in the field because if you did that you'd just be destroying the entire field. So in these cases and many other cases you just have to take samples. You can't test the whole class and then you have to generalize from the sample to the larger class. All of these generalizations from samples share a certain form. First we look at one instance of the sample. We say that first F is G. That is the thing that fits in the class F and has the property G. Then we check another. The second F is also G and the third F is also G and all the rest of the F's in the sample are G. So we conclude that all F's are G. Now notice that all in the conclusion means everything in the class of F's. It doesn't only mean the ones in our sample. So we've started from premises that are only about the sample which is only a part of the general class and we've reached a conclusion about the whole class when we say that all F's are G. Now other arguments of the same general type are a little bit different because you don't always get complete uniformity. You don't always get all F's are G.  So you get like two-thirds of the F's are G and then you reach the conclusion that two-thirds of all F's are G. Again though you've only looked at the F's in the sample. You've only observed a small part of that total class of all F's and you draw a conclusion that two-thirds of the overall class the whole class of F's has that property G. So that's why you have a general that's why it's called a generalization. You start from a smaller sample and reach a conclusion about a much larger class. You generalize to the whole class. The fact that the argument moves from a small part of the class to the whole class shows that it's inductive and what does that mean? Well first is the argument valid. Think back to one of the examples. Almost all bands that I know of have drummers therefore almost all bands have drummers. Well is it possible that the premises are true and the conclusion false?  That is they weren't part of my sample. Okay and is the argument defeasible? That means that you can add further information to the premises and it'll make the argument much less strong. It might even undermine the argument totally. And of course it could because you could give me all kinds of examples of bands that don't have drummers and then I would have to change my conclusion that almost all bands have drummers. So this argument is defeasible. Does that mean that it can't be strong? No. It could still be strong. It can come in different types of strength. How would you get a stronger argument or a weaker argument in this case? Well you can have a larger or a smaller sample. If I take a very large sample the argument's going to be stronger. If I take a very small sample the argument's going to be weaker. So it's not like validity which is on or off. It's either valid or not. Instead the strength of the argument comes in degrees depending on how big the sample is. Now does that mean that since it can't be valid it can only be strong to a certain degree? That it's no good? No. Because it's an inductive argument. Inductive arguments don't even try to be valid. They don't even pretend to be valid. That's not what they're supposed to be. And so you can't criticize this argument by saying it's not valid. It's doing everything that it's supposed to do if it provides you with a strong reason and a strong enough reason for the conclusion. That's when an inductive argument is supposed to do. So if this argument does it it's a good argument. But even if inductive arguments including generalizations from samples don't have to be valid they still do need to be strong. And so we need to figure out how to tell when a generalization from a sample is a strong argument. When it provides a strong reason for the conclusion. I don't really answer that question. You got to turn to statistics and the area of mathematics and learn how to analyze the data much more carefully than we'll be able to do here. But even mentioning the name statistics raises fear in some people. Mark Twain is famous for having said there are three kinds of lies that are lies, they're damn lies, and they're statistics. He actually gives credit to Disraeli for having said it first. But the point is that statistics are even worse than damn lies. You can't trust him at all. We're still Mark Twain says. But on the other hand some people say statistics that's math. It's all about numbers. Can't question that. Here's one example of somebody who suggests such a position. There's still one thing that's irrefutable and that is the numbers. Numbers don't lie. They can't parse poll numbers. They're infallible. Okay, fine. He knows that statistics aren't infallible. He's just being sarcastic. But there are many people who put a lot of faith in the numbers and in statistics. They seem to think that you always have to trust it when a statistician comes up with an answer. And we're going to learn that neither of these views is correct. It's not true the statistics are always worse than damn lies and it's also not true that the numbers are irrefutable. The truth lies somewhere in the middle. So now we really got to face that crucial question. How can we tell when a generalization from a sample is a good argument? We're not going to be able to do a lot of mathematical statistics here. That would take a whole course and you ought to take one that's useful, it's important, it's interesting. Definitely worth taking a course on statistics. But all we're going to be able to do here is look at some really common errors that people make. So you won't be misled in these obvious ways. It won't so much help you do statistical studies of your own but it will help you avoid getting fooled by people who cite statistical studies for conclusions that you might not want to believe. To illustrate these problems I did a little survey. I happen to love chocolate chip cookies. Now not all chocolate chip cookies are equally good. Some of them I think have too few chips. You need a lot of chocolate chips.  That stuff's good too. So you want just the right balance here. You want I would say in a say a three inch diameter cookie you want about 10 to 12 chocolate chips. And there happened to be five bakeries in my hometown that serve chocolate chip cookies. So I went and got 10 chocolate chip cookies from each of the five bakeries and counted the chocolate chips in the cookies. And I found out that 80 percent of the chocolate chip cookies that I bought from Bakery A had 10 to 12 chips in them. So I conclude that 80 percent of the chocolate chip cookies from Bakery A have 10 to 12 chocolate chips in them. Now that sounds like a pretty good argument, doesn't it? I mean I cut a sample 20 or 30 but turns a pretty good number for a sample. So you ought to believe that conclusion, right? There's nothing wrong with that argument, is there? Well what's the problem? The problem is I'm lying. I didn't buy a single cookie. I didn't do this survey. What does that show you? That shows you that one problem for statistical generalizations from samples is that the premises have to be true. Kind of obvious. And what it shows is that just like a valid argument's no good unless it's also sound when it's a deductive argument, similarly what would be a strong argument if the premises were true is no good if the premises aren't true. But also like deductive arguments it's not enough for the premises to be true. What if I counted, I really did buy the cookies and I really did count, but I missed a bunch of them because I was going so fast or I couldn't count the chocolate chips very well because they were all like all melted together or maybe I got the cookies from Bakery A mixed up with the cookies from Bakery B or Bakery C or Bakery D then I just made a mistake when I was counting the cookies. Then it turns out the premises of the argument of the generalization from the sample those premises are false but not because I'm lying rather because I didn't count well. So I'm unjustified and of course it doesn't help if I'm justified if you're not justified you're not going to have any reason to believe this about the cookies unless not only did I count them accurately and carefully and reliably but you have reason to believe that I did so. So the general point is simply that when you face a generalization from a sample then the premises have to be true and justified so the first question you ought to ask about any generalization from a sample is are the premises true and are they justified? Next let's assume that I'm honest I'm not lying and I count carefully and thoroughly so I don't make a mistake and I'm just if I'm believing that I haven't made a mistake I go to all five of the bakeries and I buy a cookie from each bakery and I count the chips in the cookies that I bought and it turns out that the cookie from Bakery A has 11 chips in it and the cookies from Bakery B, Bakery C, Bakery D, Bakery E they all have less than 10 chips in them so now I can do two of these generalizations from samples right? I can say well 100% of the cookies that I sampled from Bakery A have between 10 and 12 chips therefore 100% of the cookies from Bakery A have between 10 and 12 chips but 0% of the cookies I sampled from Bakery B have 10 to 12 chips so 0% of the cookies from Bakery B have 10 to 12 chips. What's wrong with that argument? I hope it's pretty clear the problem is you can't generalize from just one cookie. The cookies aren't made in a totally mechanical way where they count the chips before they put in the cookies then you can't know that every cookie is the same and if you can't know that every cookie is the same then you can't generalize from one cookie alone to all the cookies in the bakery. You might have gotten one that happened to have 11 chips when all the others cookies in the bakery had less than 10 or for the other bakeries for BCDE you might have gotten a cookie that had less than 10 when actually they all all the rest of the cookies in the store had between 10 and 12. So when you generalize from such a small sample from a sample that's too small then it's called the fallacy of hasty generalization and it's so obvious that it's hard to believe but it's actually often committed. It's a very common fallacy you know your next door neighbor will buy a new car and it breaks down and you say ah kind of car is no good or you meet somebody from Sweden and this person from Sweden likes football and so you say ah people from Sweden like football and people just constantly generalize from extremely small samples in order to form generalizations that guide their behavior in everyday life. Sometimes they're right but a lot of times they're wrong. That's when you have the fallacy of hasty generalization. To avoid that fallacy we need to ask a second question about generalizations from samples namely we have to ask is the sample large enough? I notice the samples come in varying sizes from just one item to almost the whole set and so the question that we have to ask first is is it large but it's not clear what that means but we really want to know is is it large enough because sometimes a very small sample can be plenty big just imagine that you come across an apple tree and you want to find out whether the apples off that tree float in the water so you bring in a tub of water you pull off an apple off the tree and you put it in the tub and it floats. Now you can generalize that all the apples or maybe almost all the apples off that tree will float in water. One is a big enough sample in that case now why is that? It's because you have background information from biology that the apples on that tree are going to be very similar to each other because of how they arose and so sometimes a single instance is going to be enough even if it's not enough when we're counting chips and chocolate chip cookies. The other point about the sample being large enough that you need to keep in mind is that whether it's large enough depends on what the stakes are. If you're testing a bunch of parachutes to see if they work you better not just check a few even 10 is not enough like we use for the chocolate chip cookies you want to check every parachute to make sure that it's packed properly and it's going to work or you're going to have disaster when they fail. But what about chocolate chip cookies? Suppose you take a sample of 10 and it turns out that that sample is not really representative. Big deal! It's other nine chips or 13 chips you know it's just not that serious an issue. So a sample can be large enough for something that doesn't matter like chocolate chip cookies without being large enough for something that really does matter like whether parachutes are packed properly. So whether the sample is large enough depends on the background information that's what the apple case showed us and also on what's at stake that's what the parachute case showed us. Next let's assume that I'm not lying the count's accurate you're justified in believing it's accurate and the sample's big enough all that is settled right so what I did was I went into the five bakeries and I turned to the person behind the counter and I said I'm doing this little survey because I got to figure out which place I want to buy my chocolate chip cookies in town and and I want to find out whether how many of your cookies have between 10 and 12 chocolate chips so could you sell me 10 chocolate cookies and I'm going to do my survey so then I take 10 cookies from each of the five bakeries and I bring them home and I count them and sure enough from bakery A we found that 80 percent of the cookies that I bought from bakery A have 10 to 12 chips and so I conclude that 80 percent of the cookies that are made in bakery A have 10 to 12 chips and that seems like a pretty good argument doesn't it it's got a big enough sample if you don't believe that let's say it's 20 cookies there's still something wrong there's something wrong with that argument what's wrong with that argument well I told the guy behind the counter that I was doing this survey to figure out where I was going to buy my chocolate chip cookies so if he wants me to buy chocolate chip cookies from his cookie shop or his bakery then he's going to look in the counter for the 10 cookies that look like they have about 10 to 12 chips in them so it might be no surprise that 80 percent of the cookies that I sampled have 10 to 12 chips because he picked out the ones that did this is called the problem of bias sampling sometimes the sample that you take is not representative of the whole because it's biased in a certain way in this case it was biased by the person behind the counter and what he knew about what I was doing and what his motives were in trying to get me to buy from his shop but the fallacy of bias sampling is something we have to watch out for and what we need to do then to avoid this fallacy is to ask a third question about all generalizations from samples we want to know whether premises are true and justified we want to know whether the sample is large enough we also have to ask is the sample biased in any way that's going to weaken the argument it might seem that the fallacy of bias sampling is so obvious that nobody who was careful and any good at what they were doing would ever commit that mistake but actually people do it all the time some of the top pollsters in history have done it the most famous example was Franklin Delano Roosevelt who was running for president in 1936 against Alf Landon and the literary digest did a poll that took just tons of data I can't remember how many tens of thousands of letters they sent out in this poll and they reached the conclusion from their poll that Alf Landon was going to win 56 to 44 but that's not the way it turned out turned out that Roosevelt won 62 to 38 they were way off they weren't even close so what was going on with that well they needed to get addresses of people to send the survey to and what did they use for that they used a phone book but back then remember this is 1936 a lot of people didn't have phones and in particular poor people didn't have phones and people who lived in rural areas didn't have phones and it was those poor people and people who lived in rural areas that love Franklin Roosevelt because of the new deal and all the policies that he had put in that helped them out so they voted for him and he won by a landslide and the prediction had gone the other way and it's all because of biosampling and the same problem continues today many pollsters especially when they want to do a really quick poll will call up people on their phones but there are lots of restrictions and lots of problems with that for one thing you're not supposed to do polls on cell phones but a lot of young people only have cell phones and don't have landlines so that means that young people are going to be underrepresented in the sample and even if they have a landline well they often have caller ID and they know it's a pollster so they don't answer so you get very low response rates and some studies have found that women tend to answer the phone more than men so you get samples skewed in that direction so what do you do when you get a biased sample like this well if you know it's been biased in these ways then you try to correct for that that's where the pollsters don't all agree they correct in different ways because think about it suppose at this time you got a lot more people from the liberal party than you did last time that might mean that your sample is skewed but it might mean that there's just more people that have moved over to the liberal party or to the conservative party either way how do you know the proper way to correct so as to get an accurate answer and different polling organizations use different techniques and that explains why very often the polls reach quite different results about who's ahead in an election another reason my polls often reach different results is that some pollsters are dishonest I know it comes as a shock but it's true and pollsters can sometimes reach the conclusions that they want to reach by slanting their questions so the fourth question that we want to ask about any generalization from samples is was the question slanted in some way which means was phrased in a way that made it more likely to reach one result rather than a conflicting result so how does this happen well a simple way to do it is to word the question in such a way that people are going to feel bad about giving a certain answer so for example if you want to find out how many people in a certain society think that it's it's wrong to experiment in scientific experiments on animals then you can always ask one question if you want one answer and another question if you want another answer so what about this one suppose they say is it okay to kill a mouse in order to save a human I think most people are going to say yes and then they're going to say ah so they those people who gave that answer they actually support scientific experiments on mice to save human lives therefore they support animal experimentation if you want to reach the other result in your poll then you can say should scientists torture animals in their experiments nobody wants to say they're gonna be for torture so they're going to say no scientists shouldn't do that and then you can conclude ah so most people are against animal experimentation because animal experimentation is torturing animals in fact you can go a little further you could say should scientists stop torturing animals in their experiments it's like should when did you stop beating your wife it presupposes that they are torturing animals and then people who say of course they should stop it I didn't even realize they were doing it well you're just guaranteeing the result in your poll by the way you ask the question another way to slant your questions in a poll is to give the survey participants limited options so in one example the New York Times magazine reported a poll by the Doris Day Animal League where they said 51 percent of the people we surveyed think that chimpanzees should be treated about the same as children but what happened in the survey was they were only given four options they could say chimpanzees should be treated like property or they should be treated a lot like children or they should be treated like adult humans or you're not sure well notice that given those options you don't want to say chimps are like property because you can just destroy your property if you want if you've got an old car and you want to tear it to bits with a sledgehammer that's up to you and they're not like adult humans because we don't think they should be given schooling and so on or votes or so on so you're pretty much left with either saying you're not sure and you feel like well I should have thought about this a little bit see most people were saying 51 percent were saying that chimpanzees ought to be treated similar to children so by limiting the options to a list where all the other options were undesirable for one reason or another the pollsters can get you to pick the option that they want you to pick and then the next trick is you report the conclusion right they actually didn't say the 51 percent said that chimpanzees should be treated similar to children they said that the survey showed that primates should be treated the same as children they changed chimpanzees to primates which includes a lot of very small primates that aren't nearly as intelligent and close to us as chimps are and they also said it's not similar to children the same as children so you can also slap the poll not by playing around with the question but by playing around with the way you report the conclusion of the survey so those are some of the tricks that people use in order to reach a predetermined result and to fool you into believing that their poll results are reliable and we saw other mistakes that people make even when they're not trying to fool you so what you need to do is to keep your eye on all those you need to ask these questions that i've emphasized throughout this lecture namely you guys see are the primates true and justified is the sample large enough is the sample biased in some way and have the questions or the conclusion been reported in a slanted way and there's a lot more to learn about statistics and again i want to emphasize to be useful to take a statistics course people ought to know more about statistics but we can't go into all of those details here still if you can just avoid these few simple mistakes that i've been talking about then you can avoid being misled in a number of situations in everyday life generalizations are not much use if you can't apply them back to particular cases it rains about 35 percent of the days in north carolina great but is it going to rain tomorrow that's what i want to know i don't want to know the general statistic i want to know about tomorrow because i got to decide where they're going to picnic 65 percent of the snakes in this area are poisonous great but i want to know whether the one i'm stepping on right now is poisonous i don't want to know the general statistic so we need some way to take these generalizations and apply them down to particular cases and that's the job of a form of argument we're going to call application of a generalization not a great name but they're usually called statistical syllogisms and that's an even worse name because they're not really necessarily statistical in the mathematical sense because they don't have to have numbers in them and they're also not syllogisms like the categorical syllogisms that you studied back in the section on deductive arguments so we're going to call them applications of generalizations and they just happen all the time they really do you might say for example i almost never like horror movies and that's a horror movie so i don't want to go to it because i'm not going to like it or you might say i don't want to invest in that restaurant because 80 percent of restaurants fail during the first two or three years and so this is a restaurant that's new it's probably going to fail too or how about this one most people who are on the track team are pretty thin now you know there's some exceptions to that the people who throw shot put that's what i used to do but still it's true that most of them do and so if you know that sally is on the track team and that's all you know about her then you've got some reason to believe that sally is probably thin she might throw the shot put but there are many more people on the track team who run and jump and they're going to be thin if they're any good at it so although it might be wrong it's a pretty good bet that sally's thin if she's on the track team and that's the kind of argument that we're going to call an application of a generalization here's my favorite am i wearing shoes right now can't see can you maybe i maybe i'm not but most professors wear shoes when they're teaching a class walter is a professor who's teaching a class so walter is probably wearing shoes right now that's an application of a generalization and it has the same form as the other examples that we saw before to see that form we can substitute variables for the terms in the english argument so we can substitute the letter f for the reference class which is the set of professors who are teaching then we can substitute the letter g for what's called the attribute class which is the people who are wearing shoes then we can substitute the letter a usually in lower case for the individual that we're talking about in this case that individual is me walter and we can substitute x percent for the quantifier if it's most or almost all or something like that then x percent takes its place and then the form of the argument that i just gave about wearing shoes is simply that x percent of f are g a is f therefore a is probably g notice that this application of a generalization moves in the opposite direction from the generalization from a sample we might start off saying x percent of the f's in the sample are g so x percent of f's are g and that's generalization from a sample but then we take the generalization in that conclusion and use it as a premise