Welcome to the second unit of our course. In the first unit of the course, you learned how to listen to what someone was saying, or read what they wrote and separate out the arguments that they were giving from the rest of their words. You learned what arguments are, what their parts are, and how those parts work together to achieve the various purposes of argument. Now in this second unit of the course, and the third unit that follows it, you'll learn some rules for evaluating the success of arguments. In particular in the second unit of the course, we'll focus on rules for evaluating the validity of deductive arguments. And then in the third unit of the course, we'll focus on rules for evaluating the strength of inductive arguments. Recall that deductive arguments are arguments that are presented as being valid, and they are successful only if they're valid. Inductive arguments, in contrast, are arguments that are not presented as being valid. They're presented as being strong, and they're successful only to the extent that they are strong. Okay, now since in this second unit of the course we're focusing on rules for evaluating the validity of deductive arguments, let me say a bit more now about validity and about the rules for assessing validity. See, I just got a new pet clownfish, Nimo. Now maybe you don't know much about clown fish anatomy, but I'm going to try to persuade you right now that clown fish have gills. Here's an argument that I could give you for the conclusion that clownfish have gills. Well, catfish have gills, goldfish have gills and sharks have gills. Therefore, clownfish have gills. Now, is that argument valid? No, it's not. It's not valid because it's possible for the premises to be true even when the conclusion is false. It could be that catfish and sharks and goldfish all have gills, even though clownfish don't. But now suppose I give you a different argument for the conclusion that clown fish have gills. Here's how this different argument goes. All fish have gills. Clown fish are a kind of fish. Therefore, clownfish have gills. Now, that argument is valid. There's no possible way for the premises of that argument to be true if the conclusion is false. Recall that in the first unit of the course, Walter defined validity as follows. He said that an argument is valid if there is no possible way for all the premises of the argument to be true while the conclusion is false. And he gave an example of an argument that was valid. The example, recall, went like this. Premise one, Mary has a child who is pregnant. Premise two, only daughters can become pregnant. Therefore, conclusion, Mary has at least one daughter. Now remember, as Walter pointed out, you can imagine situations where the first premise is false. Maybe Mary doesn't have a child who was pregnant. You can imagine situations in which the second premise is false. Maybe not only daughters can become pregnant, maybe there's some way for sons to become pregnant, or maybe there's some way for children from some third gender or some unspecified gender to become pregnant. And you can imagine a situation which the conclusion is false, in which Mary doesn't have at least one daughter. But what you can't imagine, what's completely incoherent, what's completely impossible is for there to be a situation in which both of the premises are true, and the conclusion is false. In any possible situation, either the premises, in any possible situation in which the premises are true, the conclusion has to be true. There's no possible situation for the premises to be true and the conclusion false. So that's what makes that argument valid. Now what I'd like to do right now is show that there are other arguments that have the same form as this first argument about Mary and are valid for what seems to be the same reason as the argument about Mary is valid, even though they have a completely different subject matter. So here's another example, consider the following argument. Premise one, Terry has a job in which she arrests people. Premise two, only police officers can arrest people. Therefore, conclusion, at least one of Terry's jobs is as a police officer. Now again. Maybe, premise one of this argument is false. You can certainly imagine that it's false, right? Maybe Terry doesn't have a job in which she arrests people. Maybe premise two of the argument is false, you can imagine a situation which not only police officers can arrest people. Maybe there's a world in which police officers, and judges, and plumbers, and actors can all arrest people. And, of course, you can imagine a situation which the conclusion is false, in which it's not true that at least one of Terry's jobs is as a police officer. But what you can't imagine, what's completely impossible, is that both of the premises of that argument are true and the conclusion is false. If both of the premises of that argument about Terry are true, the conclusion has to be true, and so the argument is valid. The argument about Terry is valid, just as the argument about Mary is valid. And notice the two arguments have completely different subject matters. The first is about Mary's children, the second is about Terry's jobs. But even though they have completely different subject matters, they have something in common. There seems to be something in common to their form. Something signaled by their use of the terms only and at least. Something in common to the two arguments that explains why they're valid. To get at what this common feature is a bit more clearly, let me give a third and final example of this form. Let's see if you get the idea. So consider the following argument. Premise one, Robert has a pet who is canine. Premise two, only mammals can be canine. So conclusion, Robert has at least one pet who is mammal. Now, of course you can imagine a situation in which premise one is false. Maybe Robert doesn't have any canine pets. You can imagine a situation which premise two is false. In which not only mammals are canine, maybe some canines are reptiles, or some canines are robots, or who knows. And of course you can imagine a situation where the conclusion is false, in which Robert doesn't have any mammal pets. Maybe all of his pets are birds or reptiles, or maybe he doesn't have any pets. But in any case, you can imagine these possibilities. But what you cannot imagine, what's completely ruled out, is the possibility that the premises of that argument are true and the conclusion is false. If the premises of the argument are true, the conclusion has got to be true. And so this argument about Robert is also valid, just like the arguments about Mary and Terry were valid. But of course this argument about Robert has a completely different subject matter than the arguments about Mary or Terry. One was an argument about Mary's children, the other was an argument about Terry's jobs and this is an argument about Robert's pets. Completely different subject matters, but they're all valid, and it seems they're all valid for the same reason. They have the same form, and it's a form that's signaled by their use of the phrases only and at least. So what we'll be doing in the second unit of the course is studying the forms of valid argument. The forms that explain why valid arguments are valid. Now in the arguments that we just looked at, the arguments about Mary, Terry, and Robert, their common form, I said, involved the use of the word only and it also involved the use of the phrase at least. Only and at least are both phrases that we're going to call quantifiers. And next week, in week five of the course, when we study the subject that we're going to call categorical logic, we're going to be studying in particular quantifiers like only, at least, some, all, none, and phrases like that. And we're going to be studying how the use of phrases like that, the use of quantifiers, can make arguments valid no matter what those arguments are about. If the arguments use quantifiers like only, at least, some, all and none in certain ways, then those arguments are going to be valid regardless of their subject matter. But this week, in week four, we're not going to be studying quantifiers like only or at least, instead we're going to be studying a kind of phrase that we're going to call a propositional connective. In particular, a kind of propositional connective that we'll call a truth-functional connective. So what are propositional connectives? What are truth-functional connectives? How do they work to make arguments valid? Those questions and more, we'll address in the next lecture.