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 are 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 with inferences, the A, E, I, and O propositions. Remember what those letter stand for. The A proposition is a propositions in the form all F's are Gs. All things that fall into one category also fall into a second category. An E proposition is a proposition of the form no Fs are Gs. Nothing that falls into the first category also falls into the second category. An I proposition is a proposition of the form some Fs are Gs. That's to say some thing that falls into the first category falls into the second category. And an O proposition is a proposition of the form some Fs are not Gs, right? Some thing 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 that 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 a subject term and a predicate term, and one conclusion with a subject term and a 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 no Fs are Gs, let's say no Duke students are NFL football players. Then, the conclusion would be in the form of no Gs are 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 propositions. 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 influences 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, right? 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 are 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. All right, 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 F's are G's to no G's are F's? Well, let's see, how would we represent no F's are G's? Well, we do that by shading in the portion of the F circle that's inside the G circle. All right? So, that shows that no F's are in G the circle. No F's are G's. But now look, if no F's are G's, then we can just read off from that that no G's are F's. Right? If there are any G's, 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 G's 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 are Gs, then it's got to be true that no Gs are Fs. The Venn diagram for no Fs are Gs shows us that. So, conversion inferences are invalid for A propositions, but they're valid for E propositions. What about I propositions? Well, lets see. I propositions have the form some Fs are Gs. Well, how would we represent some Fs are 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 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 G's are F's. So, once again, from the Venn Diagram, you can see that if some Fs are Gs is true, then some Gs are Fs has gotta 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 gotta 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 G's at all for all that you've just been told. Right? You're told that some F's are not G. That doesn't tell you that there are any G's. Or if there are any G's, maybe all the G's are in here. You don't know whether there are any G's 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 diagrams, 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.
