In the last class we talked about immediate categorical inferences, which are inference 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 a syllogism we have three categories. We have the subject category, the middle term, the Gs, and the predicate term, the Hs. 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 there is 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 gotta be inside the circle of the animals. So we can shade in the portion of the humans circle that's outside the animals 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 students circle that's outside the humans circle and you shade in the portion of the humans circle that's outside the animals circle, the only place left over in the Duke students circle for there to be any Duke students is inside the animals 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 animals circle since the first premise doesn't tell us whether it's here or here, okay. The second premise said, all humans are animals. Okay, well, if all humans are animals, what that tells us is that there aren't any humans outside the animals circle. So we can just shade in the part of the humans circle that's outside the animals circle. All right, shade it in right there. But then notice, remember we have to have an x inside our Duke students circle and our humans circle inside the intersection. Well, it can't be in here because this is shaded in, which means there's nothing there. So this only place it can be is right there. So now we know that the x had to be over here. But once we've 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, is that some Duke students are animals. Finally, 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 students circle that's inside the humans 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 humans circle that's outside the animals circle, all right? Because there are no humans out there. All humans are animals, all 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 reasons 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 human and all the 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.