Welcome to Week Three. In weeks one and two, we already learned a lot about how to identify and analyze arguments. We can do close analysis, we can identify the premises and conclusions, we can put them in standard form, what's next? Well, the next step is to take those parts and put them in a certain order and fill in the missing gaps. We need to learn how to reconstruct arguments. Are you ready? Well there are lots of way to reconstruct. Think about constructing a house, or building. In order to construct a good building, you got to know what the goal is, what the standards of a good building are. The same thing goes for reconstructing arguments. In order to reconstruct an argument properly, we need to know what the standards are for reconstruction. We're trying to reconstruct it, so as to meet those standards. Because the goal is not to reconstruct the argument in order to make it look bad. The point is going to be to reconstruct arguments, so as to make them look good. because by making your opponents look bad or silly, that doesn't do anybody any good. If you want to learn about their perspective, and you want to learn from, their views. Then you need to reconstruct their argument, so as to make it look as good as possible. And to do that,you need to know about the standards for arguments, that is the standards that make arguments good or bad. So what we're going to do this week, is we're going to look first at some standards for arguments, validity and soundness in particular. And then we're going to use those standards to develop a method, called reconstruction or deep analysis. I'll explain those terms later. And then we're going to apply that method to a few concrete examples. In order to be able to take a passage, and take those premises and conclusions and fill them out and get a full fledged argument. That, if we've done it properly, will be as good as it can be and that we can learn from. That's the goal. Now, because an argument consists of premises and a conclusion. And the premises are supposed to be related in the right way to the conclusion, there can be two main ways for an argument to go wrong. Two main vices of argument you might say. The first is there might be something wrong with the premises. In particular, they might be false, or at least one of them might be false. Second, there might be something bad about the relation between the premises and the conclusion. The premises might fail to give a good reason for the conclusion. Now, each of these problems is something that we need to avoid. And when we do avoid them, we get the corresponding virtues, namely, validity and soundness. And those are the two notions that we want to discuss in this lecture, and the next. Let's begin with the relation between the premises and the conclusion. What kind of relation between the premises and the conclusion is good for an argument or makes an argument good? Well, that depends. Some arguments are deductive, and others are not. So let's focus for a moment on deductive arguments. In Deductive Arguments, the conclusion is suppose to follow from the premises. What does that mean? I mean, what does it mean for a conclusion to follow from the premises? That's a really hard notion to pin down. So what logicians usually do and what we're going to do, is focus instead on the notion of validity. And the idea is that a deductive argument is trying to structure itself, so that it's valid. And we'll explain what validity is. But for now, I want to emphasize that we're only talking about deductive arguments. There's going to be another class of arguments called inductive arguments that we'll get to later in this course. Where, they don't even pretend to be valid. They don't even pretend that the conclusion follows from the the premises. But just for simplicity, let's focus on deductive arguments now. And the idea is that the deductive argument, should be structured in such a way that it's valid. Then the next question is, what's validity? Let's start with a simple example. Suppose that you know Mary, but you don't know her children. However, you do know that she has one child who is pregnant, and you also know that only daughters can become pregnant. So, you have all that you need to know in order to draw further conclusion. Mainly, Mary has at least one daughter. So here's the argument. Mary has a child who is pregnant, only daughters can become pregnant, therefore, Mary has at least one daughter. Now, if you think about it, there's just no way, no possibility that both of those premises are true and the conclusion's false. That is the feature that we're going to call Validity. More generally, we can define validity in an argument. so that an argument is valid, if and only if it's not possible for the premises to be true and the conclusion false. That is, it's not possible for there to be a situation where both of those hold. That is, a situation where the premises are true and the conclusion is also false. Now that, might strike you as a pretty simple notion. But actually that little word possible is the problem. How do you tell what's possible or what's not possible? Well, there's not mechanical solution to that, and we'll struggle with that a little bit throughout this course. But for now, since we're right at the start, let's think of it this way. Is there any way for you to tell a coherent story, where the premises are true, and the conclusion's false. Can you describe a situation with that combination of truth values? That is, the premises being true, and the conclusion falls in the same situation. If you can tell a coherent story with that combination, then it's possible and the argument's not valid. But if there's no way to tell a coherent story. Where the premises are true and the conclusion's false, then the argument's valid. Now let's try that test on our example. Mary has a child who is pregnant, only daughters can be pregnant, therefore, Mary has a daughter. So, is there any way to tell a coherent story where the two premises are true. That is, where Marry has a child who is pregnant, and only daughters can be pregnant. But the conclusions false, Marry does not have a daughter. Well just try, suppose Marry has only one child and it is a son. There the conclusion is false, good. What about that? But then, is that son pregnant? Well, if the son is not pregnant then the first premise is false. Mary doesn't have a child who is pregnant. But if the son is pregnant somehow, don't ask me how, but if the son is pregnant, then the second premise is not true. It can't be true that only daughters can be pregnant because this child is a son. Okay, what if Mary has two children? Try that. Try to tell the story that way. Mary has a daughter and a son. Now, she's got a child who is pregnant, the daughter. And only daughters can be pregnant, but she has a son. Wait a minute, she's got a son and a daughter. So now the conclusion's true. Because she does have a daughter even though she also has a son. Well, oh wait, how about this one? What if Mary has a child who is biologically female, but sees himself as a male. And so she sees that child as a male, but that child is pregnant, because after all, they're biologically female. Now, are the premises true and the conclusion false? Does that story make sense? Wait a minute. Either, her child is a daughter or her child is a son. Now, if it's a daughter and it's pregnant, no problem,the conclusion's true. If it's a son, because that child sees himself as a male, then you've got a choice. Well, what about the first premise? The first premise is going to be true, she does have a child who is pregnant. But what about the second premise, only daughters can be pregnant. Wait a minute, if that really is a son, if we're going to call that a son, then, it's no longer true that only daughters can be pregnant. So, now the second premise is false. So, try it again. Try it with, you know, sex changes and try it with hermaphrodites, tell the story any way you want about Mary's children. And there's no way that both premises come out true when the conclusion's false. That shows that the argument is valid. It might be just that we can't imagine the coherent story which makes it invalid. But the fact that we've tried hard and looked at all the possibilities we can think of. At least gives us a good reason to think, that this argument is valid. Now, some people like to think of it in the reverse direction. They say, let's imagine that the conclusion's false, and then, if it has to be the case that at least one of the premises is false, the argument's valid. Then you could define validity as, it's necessarily the case that if the conclusion is false, one of the premises is false. Or, in every possible situation, if the conclusion's false, one of the premises is false. We can apply this new account of validity to the same old example. It's got to be the case. That if Mary didn't have a daughter, then she doesn't have a child who was pregnant or else, there are at least some children who are pregnant who are not daughters. So in other words, this case you're reasoning back from the falsehood of the conclusion to at least one of the premises has to be false. Where as in the earlier definition, you were saying it's not possible in the situations where the premises are true for the conclusion to be false. You can look at it either way, either direction. Just pick the one that works for you and go with that definition. Because in the end, the two definitions are equivalent. It's just a matter of what's going to help you understand which arguments are valid and which ones are not. In addition to understanding what validity is, it's also very important to understand what validity is not. A lot of people get confused by the notion of validity in this context, because they're thinking that, to call an argument valid, must be to call it good, right? You call a driver's license valid. When its good in the eyes of the law, but that's not what we're talking about here. The notion of validity is getting used by logitions here as a technical notion. And its very, very, very important to remember, that to call an argument valid is not to call it good. For some arguments like deductive arguments, being valid might be necessary for them to be good, but it's not enough. And we'll see a lot of examples of that later on. The second point about what validity is not, is that validity does not depend on whether the premises and the conclusion are actually true or false. Instead, it depends on what's possible. Whether there's a certain combination, true premises and a false conclusion, that's even possible. So, whether the premise is actually true in the actual world is not what's at issue. And we can see this by seeing that some arguments with false premises can still be valid. And some arguments with true conclusions can be invalid. So let's look at some examples of that. Indeed, there are four possibilities, because remember, the conclusion could be true or false, and premises, could be all true or at least one false. So, we've got four possibilities, and all of those are possible, except for one. The one combination that's not possible for valid arguments is True Premises and a False Conclusion. But if you've got True Premises and a True Conclusion, it might be valid, it might not. If you've got false premises in a true conclusion, it might be valid it might not. If you've got false premises, and a false conclusion, it might be valid it might not. So, let's look at some examples of each of those possibilities in order to better understand the relation between premises and conclusion, that exists when the argument is valid. It's hard to give examples with true premises and false conclusion, or any of these other combinations. When the truth is controversial. So we're going to have a really simple example, and we're going to start just by stipulating what the facts are. We're going to assume that all Ford cars have four tires, but some Ford cars do not have four doors. We're also going to assume that Henry's car is a Ford that has four doors. And Jane's car is a Chrysler, it has only two doors, not four doors. And we're just going to take those facts for granted and assume that that's the situation we're talking about. And then, we can give examples of all the combinations that we discussed before. Let's begin with true premises and a true conclusion. Here's an example, all Ford cars have four ties. Henry's car is a Ford. So Henry's car had, has four tires. Are the premises true? Well yeah, our assumptions tell us that all Ford cars have four tires. And then Henry's car is a Ford. What about the conclusion, is that true? Sure. That's another one of our assumptions. Henry's car has four tires. But, the fact that the premises are true and conclusion's also true, doesn't yet tell us it's valid. Because, to be valid, it has to be impossible for the premises to be true and the conclusion false. There has to be no coherent story where the premises are true and the conclusions false. So what about this case? Is it valid? Well, is it possible? Is there any way to tell a coherent story where, all Ford cars have four tires? Henry's car is a Ford, but Henry's car does not have four tires just try. One way to tell not going to able to do it, is to reason backwards. Assume that the conclusion falls, that Henry's car does not have four tires, may be it got six tires. Well, then, how could both premises be true? If Henry's car is still a Ford and it has six tires, then the first premise is not true. Because the first premise says that all Ford cars have four tires, and Henry's car would then under that assumption be a Ford car without four tires, but with six tires instead. So the first premise would be false. Well, another possibility is that Henry's car has six tires and it's not a Ford. And then you can have the first premise true, all Ford's have four tires. But you couldn't have the second premise true, namely that Henry's car is a Ford. So there's no way when the conclusion is false, for both premises to be true and that shows you that the argument is valid. Nonetheless, there are other examples where the premises are true and the conclusion's true, but the argument is not valid. Instead it's invalid. Here's an example of that combination. All Ford cars have four tires. Henry's car has four tires. Therefore Henry's car is a Ford. Now in this new argument, are all the premises true? Yes, the first premises says, all Ford cars have four tires. And that's true by our assumption, the second premise says Henry's car has four tires. And that's also true by our assumptions. And is the conclusion true? Yes, our assumptions also tell us that Henry's car is a Ford. But, is it possible, is there any way to tell a coherent story where, those premises are true and the conclusion is false. Yes, absolutely. All that has to happen is Jane and Henry switch cars. Then, the first premises can be true, because all Ford cars have four tires, and the second premise is going to be true. Because Henry's car has four tires, of course now it's a Chrysler, because he got it from Jane. But the conclusion can be false Henry's car is not a Ford, because Ford and Chrysler are different companies. So if he switches cars with Jane and he has a Chrysler, then he doesn't have a Ford. He's car is not a Ford, okay? So now you've got a situation, where the premises are true, and the conclusion false. It's not an actual situation, but it's a possible situation. You can tell a coherent story, where the premises are true, and the conclusion's false. And that tells you that the argument is invalid. Next, let's consider an example with false premises and a true conclusion. Premise one, all Fords have four doors. Premise two, Henry's car is a Ford. Conclusion, Henry's car has four doors. Is the first premise true? No, it's not true that all Fords have four doors. Our assumptions tell us that. Second, is Henry's car a Ford? That's true. So one of the premises is false and the other one is true. That means they're not all true, and the conclusion, is that true? Yes. It is true that Henry's car has four doors. But remember, the fact that, that's actually the case doesn't tell us whether or not it's valid. So is it valid? That depends on when it's whether it's possible for the premises to be true and the conclusion false. Premises aren't actually true, but is there a possible story that you could tell that would be coherent. Where the premises are true but the conclusion, false. That's the test of validity, so lets apply it to this case. Well, just imagine that the conclusion is false. That, Henry's car does not have four doors, it's only got two doors. Then, there is really only two possibilities, Its either a Ford or it is not a Ford. If it is a Ford then the first premise is false. It is not true that all Fords have four doors. But if Henry's car is not a Ford, then the second premise is false, because it says that Henry's car is a Ford. So there's no coherent way in which it could possibly be true. That both of these premises are true and the conclusion's false. So this argument's valid, and notice that that shows, that an argument can be valid, even though it's got a false premise. Now, you might be thinking to yourself, this is crazy. How can an argument be valid, when one of its premises is false? I mean, an argument is not good when it's premises are false. But notice what that does. That confuses the notion of valid, like in a valid drivers license, where to be valid is good, with the technical notion of validity that we're using here. The technical notion of validity that we're using here, has to do with the relation between the premises and the conclusion. And in particular it has to do with possibilities. And not with the actual falsehood of the premise. So what we have to ask ourselves is, what would happen if it really were true? That all Fords have four doors. It's not true, in the actual world, but we're concerned with possibility. And if all Fords did have four doors, and, if Henry's car was a Ford, then it would have to have four doors. So that possibility of the premise being true, even though it's not, is what's crucial for determining validity. Because it's not possible for the premises to be true and the conclusion false. That makes it valid in our technical sense even if it's not valid in the common sense notion of validity as goodness. We're not saying the arguments are good argument, we're saying that is meets this technical definition of validity that logicians use. Now the only combination of truth values in premise and conclusion, that you cannot get with a valid argument. Is to have true premises and a false conclusion. So here's an example of that. Premise one, some Ford cars do not have four doors. Premise two, Henry's car is a Ford. Conclusion, Henry's car does not have four doors. The premises by our assumptions are both true, and the conclusions false. And it's not valid, because it's easy to see how it might be possible for the premises to be true and the conclusion false. It's simple. Even if some Fords don't have four doors, Henry's car is one of the Fords that does have four doors. And then, both the premises can be true, and the conclusions false. So that's how you can get an invalid argument with true premises and a false conclusion. But you don't even really need that. Look, every argument that has true premises and a false conclusion has to be invalid. Because if it does in fact actually have true premises and a false conclusion, then it's possible for it to have true premises and a false conclusion. So you can know right off the bat, that every argument with true premises and a false conclusion is invalid. What you can't know is for the other combinations, then you have to think through what's possible, instead of simply, what's actual. We haven't been through all of the possibilities, but we have seen that you can have invalid arguments with true premises and true conclusions. And you can have valid arguments, with false premises and true conclusions. And we've got a little table that shows us the other possibilities. Instead of going through all of those possibilities myself. I think it would be better if you did a few excercises, and that'll make sure that you understand this notion of validity, before we go on and try to show how validity is related to soundness.
