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 upper case F and an upper case G to represent any category at all. So in a statement of the form all Fs are Gs, 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 Fs or Gs, 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 the statement of the form no Fs are Gs. 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 we're going to call E. An 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 "some". Now, the quantifier some 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 we'll call I propositions. Finally, we can also use the quantifier some 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 we'll call O statements or O propositions. They say that some things that fall into one category don't fall into a second category. Some Fs are not Gs. So those are the four kinds of statements that we're going to focus on in 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 are Gs would look like that. You'd be saying whatever F's there are, they've got to be in here. There can't be any outside there. So that would be the Venn diagram for all F's are 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 are G's. When you say no F's are G's, you're saying if there are any F's at all, they can't be inside the G circle, so you've gotta shade out that portion of the F circle that's inside the G circle. The shading implies that there's nothing there. So, there are no F's inside the G circle. And that's how you represent no F's are G's. Next, there are statements of the I form, some F's are G's. How would you represent those? Some F's are G's. You use an X mark to represent that there is a thing. So use an X to represent that there is a thing that's in the F circle and that's also in the G circle. 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 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 gotta 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 theVenn 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. 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. If some F's are G's, then it's 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. Those relationships are represented visually on the Venn diagrams that we just drew. 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 the precisely the place where propositions of the A form had shading. Propositions of the E form are going to have shading in here, where 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's going to be the negation of another just in case the first has shading wherever the second has on X or vice versa. 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.