Welcome to week five of our course. The week on categorical logic. [NOISE] 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 1, 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 biconditional, expressed in English by if and only if. We can use the truth table for the biconditional 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 either be 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 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 will 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 biconditional. Now, you see, that's an example of how we can use the truth table for propositional connective like the biconditional 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 biconditional 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 use the truth table to understand why they were invalid or valid. 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 uses 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 this 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.