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 and 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 as saying that some Brazilians speak Portuguese, certainly that's true. Maybe not all Brazilians speak Portuguese, and maybe there are some Brazilian citizens should'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 or in Portugal not in Brazil. Or in any case are in some other part of the world than Brazil. And that none of the Brazilian's 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 it's 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 Brazilian speak Portuguese and Portuguese speakers understand Spanish. Do Brazilians understand Spanish? Maybe there's a different way to understand the argument. Maybe the right way to understand the argument is by 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 gathered 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 speaker 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 the Portuguese speaker, and the category of a person who understands Spanish. Then we saw that by the use of certain modifiers, the modifier some, 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. 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 argument 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 are say that all Brazilians speak Portuguese? Well, you're saying that there is no Brazilian who falls outside the category of Portuguese speakers. 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 gotta 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 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 speakers circle. And to represent some Brazilians speak Portuguese, we put an X in the Brazilian circle that's also inside the Portuguese speakers circle. Now we used a Venn diagram to represent the information contained in the first premise of the argument about Brazilians but how could we use the 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. 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. So that represents that there is something that is both a Portuguese speaker and an understand 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, 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 understands 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, while that argument is not valid. 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 and 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. So we can shade out the portion of the Brazilian 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. According to premise one all Brazilians are Portuguese speakers, according to that premise all the Brazilians there are have to be in here. Premise two says, all Portuguese speakers understand Spanish. Well, if all Portuguese speakers understand Spanish then there cannot be any Portuguese speakers who are outside the category of those who understand Spanish. 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 understands Spanish circle. There are no Portuguese speakers that fall outside the category of understanders of Spanish according to premise two of this argument. That all Portuguese speakers understand Spanish. So now we can represent that information that way. 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 if 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 shade it in, the circles that we had to shade in to represent the first two premises of the argument. 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 understanders 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.
