Welcome to this video for Chapter One of Basic Sentential Logic, Informal Fallacies, and Cognitive Biases. In this video I'll be talking about material that shows up near the beginning of Chapter One and Section 1.1 of that text. The topics will include arguments, premises and conclusions, the distinction between induction and deduction, and discussion of what "validity," "soundness," "strength," and "cogency" mean. But overall the topic is arguments. What is an argument? An argument, as we're going to be using it, is a verbal or written or propositional or sentential representation of some episode of reasoning. So in the last video I gave an example of reasoning: the Pizza Argument. Now this was a piece of reasoning that the girl at LAX went through in her head. She wasn't talking out loud as she was deciding or realized that the pizza in the box she was holding was a cheese pizza. This was all going on in her head, but if we're going to study reasoning, we want to be able to represent these processes that are sometimes purely internal, psychological processes so that we can study them. And an argument is how we're going to do that. So it's a way of verbally representing it or representing it in written form. Now as I mentioned last time, reasoning is the process of determining what follows from certain facts or assumptions. And there's two elements to this. There's the thing in green here - the certain facts or assumptions that you're starting with - and then the element in blue - what follows from these facts or assumptions. So reasoning has always these two components. When we put these two things together, we see that our written representation of a piece of reasoning is going to have two parts. There's going to be a part where we're representing the information that we're starting with or that the person going through the piece of reasoning is starting with, and we're going to call those "premises." So there'll be sentences or statements that we'll write down - those are premises, and they represent the initial information. And there will be a conclusion, and that's the piece of information that is taken to follow from those premises. So if we go back to the Pizza Argument, what I've done here is I've taken that argument, and I've represented the information - the premises and conclusion - in sentences. And I've done some other things too. I've given them numbers. So Sentences 1 and 2-- or I'll use the word "statements" from here on out. Statements 1 and 2 represent the information that she was starting with. 3 represents the new piece of information that she realized followed from 1 and 2. But I've separated these out, and I've given these different things numbers. I've also put a line here to separate the premises from the conclusion, and I've got this triple dot, which is the first piece of notation that we're getting. And this triple dot basically means "therefore," or it's a way of marking off the conclusion. Now in any argument, there's always going to be exactly one conclusion, but there could be any number of premises. There can be one, three, a hundred. In Chapter Four we'll see some weird cases where there's zero premises. Those are kind of odd. Don't worry about those. But just keep in mind, there can be any number of premises, but there's always one conclusion. Now arguments come in a couple of varieties, and the two varieties that we're going to look at are, first, deductive arguments. And a deductive argument is-- this is just how we're going to define it, and it's rough but don't worry about it. It's enough for our purposes. An argument is deductive if the premises are taken to provide complete, watertight support for the conclusion. Now it might not be a good deductive argument. It might be that the person going through the piece of reasoning or presenting the reasoning thinks that it provides watertight support, but it really doesn't. But the key is - do they think it does? Is the argument taken to provide watertight support? And if that's the case, then that's a deductive argument. An inductive argument is one where the premises are taken to provide probable support for the conclusion, not watertight. So the person going through a piece of inductive reasoning realizes that even if the reasoning is good, the premise isn't a hundred percent guaranteed. But they do think that the premises make the conclusion very likely, very probable. Don't worry about how probable is probable enough. For purposes of this class we can be very rough and intuitive about that. So that's the distinction between these two main types of arguments. Let's see some examples. Here's an example of a deductive argument that has two premises and a conclusion. Premise 1: If I file my taxes, I will get a refund. Premise 2: I will file my taxes. Therefore, I will get a refund. Now this is a deductive argument. It's also a good deductive argument. We'll get to what it means to be good in a minute, but if I'm going through this piece of reasoning... Let's say this is a representation of the reasoning that I go through when I'm thinking about my taxes. I might think that these premises provide watertight support for the conclusion, and as long as I think that, it's a deductive argument even if I'm wrong. Now in this case I'm actually right if I think that, but just me thinking it - that it's watertight - is enough to make the argument deductive. Compare that to this argument: If my plants were watered, the hose would be moved. The hose was moved. Therefore my plants were watered. Now I might go through this reasoning and think that likewise it's a hundred percent watertight. But in this case you can see it's not. In fact, it could just be that someone tripped over the hose and moved the hose, so even if those premises are true, the conclusion doesn't have to be true. The conclusion doesn't follow with a hundred percent, but that doesn't matter. As long as the person going through the reasoning thinks that it does - thinks that it's watertight - that makes the argument deductive. Let's see some examples of inductive arguments. Again I'll show you an example of a good one and a bad one, but they're both inductive. South Park has always been on Wednesday at 10 pm. It's now Wednesday at 10 pm. Therefore, South Park is probably on now. If I'm going through this reasoning, I realize that the conclusion doesn't follow with a hundred percent certainty. This is what I'm indicating here with the "probably" in brackets. It's just a way of indicating that the person going through the reasoning realizes that it doesn't nail the conclusion down with complete certainty. But even so the person might think that it makes the conclusion very likely. They'd be willing to bet a lot of money on it. And in this case they would be right. If these premises are true, then the conclusion is probably true. Here would be a bad-- it's still an inductive argument, but one that doesn't really render the conclusion very likely. The last time I watched South Park the first commercial was for Old Navy. Therefore, the first commercial on tonight's episode of South Park will probably be for Old Navy. That's probably not worth betting a lot of money on. Commercial order and even what commercials appear on various TV shows changes all the time and so just because that premise is true, that doesn't render the conclusion very likely to be true. It might be, but I wouldn't bet a lot of money on it. The difference between deductive and inductive arguments isn't how good they are. And that's what I was trying to illustrate by giving an example of a good and a bad deductive argument and a good and a bad inductive argument. It's just a difference in how strongly the person going through the reasoning thinks that the conclusion is supported by the premises. Now it turns out there's going to be two different ways of measuring an argument's goodness. I've been talking about a good argument and a bad argument, and we'll see that's very rough. We need to sort of come up with a better way of discussing how arguments can succeed or fail. So we're going to distinguish a couple of different ways, and these ways will apply both to inductive and deductive arguments. One is the inferential relationship between the premises and the conclusion. So that's one way in which an argument can succeed or fail. This inferential relationship can be good or it can not be good. The second way has to do with the truth of the premises. The premises might all be true or they might not be true, but that's independent of the first type of goodness. And I'll give you some examples of this shortly. So first let's talk about this first measure of goodness, this inferential relationship between the premises and the conclusion. And here's a way to think about it. Ask yourself this question: if the premises were true-- now for any given piece of reasoning, the premises might be false, they might be true, you don't always know, and for purpose of this question we don't care - what you want to ask yourself for purposes of this first question is just assume they were true - maybe they actually are, maybe they aren't - but if they were true, would the conclusion necessarily be true? And that's how you would ask the question if it's a deductive argument. If it's an inductive argument, you would ask the same question with a slight variation. You would say, if the premises were true, would the conclusion probably be true? And if the answer is yes, then the inferential relationship between the premises and conclusion is good. And note - I'm gonna return to this point a bunch - you can assess that question, you can answer yes or no, even if the premises are in fact false because the question is if they were true, would the conclusion have to be true. So let's go back to this argument. This is the same example I used earlier. Premise 1: If I file my taxes, I will get a refund. Premise 2: I will file my taxes. Therefore, I will get a refund. Now you can ask yourself, "If these premises were true (in this case it's Statement 1 and 2; these are the premises), then would the conclusion have to be true?" And the answer is yes. If those premises were true, the conclusion would have to be true. Now some people watching this video might question that. I'll get back to this in a minute, but for current purposes just take my word for it. If those premises were true, the conclusion would have to be true. Another way of putting it is this: if the conclusion were false, at least one of the premises would have to be false. Now I'm gonna return to this question. Some of you might question what I just said, might think, "Well but wait a minute, Rick, couldn't the conclusion be false," and of course the conclusion could be false. So here's one way it could be false. Maybe there's a legal lien on my income such that I don't get a refund even if I've overpaid my taxes, and normally I would get them back, but because there's a lien, I don't actually get any refund. That's certainly possible, but notice something. If there is a lien on my income such that I don't get a refund and so the conclusion is false, that also means Premise 1 is now false. So the situation that we imagined that makes the conclusion false is a situation that also makes Premise 1 false. Because if there's a lien on my income, then it's not true that if I file my taxes I get a refund. The government just takes it and gives it to whoever legally has a right to that money. So what valid means is that if the premises are true, the conclusion must be true. The conclusion could be false, but only if one or more of the premises is also false. Now compare that to this argument: If my plants were watered, the hose would be moved. The hose was moved. Therefore, my plants were watered. Notice even if those premises were true, the conclusion could be false. So here's the situation. This is true. Premise 1 is true. If my plants were watered, the hose wiII be moved because the only way to water them is with the hose, and where the hose is now you can't reach the plants so maybe Premise 1 is true. Premise 2 could be true. The hose was moved. And yet it's obviously the case that the conclusion could still be false. Here's the situation. Someone just walked around and tripped over the hose, and they move the hose, but they didn't water the plants. So you can easily imagine a scenario where the premises are true, but the conclusion is still false at the same time. So this argument, it fails that test. It doesn't have this good inferential relationship between the premises and conclusion. If a deductive argument has this good inferential relationship between the premises and the conclusion, then that argument is "valid." That's the word we're going to use for it. And here's the definition: an argument is valid, a deductive argument is valid if and only if - that's not a typo; it's a way of stating definitions (don't worry about it) - if all the premises of the argument were true, the conclusion would have to be true. Another way of saying the same thing: If the conclusion were false, then one or more of the premises would have to be false. Now a point I really want to make clear is that an argument can be valid even if the premises are in fact false because that's not the important issue. The issue for validity is 'if they were true, would the conclusion have to be true?' So I want to illustrate that with this argument. Premise 1: If I'm President of the United States, then I get all the free BMWs I want. Premise 2: I am President of the United States. Therefore, 3, I get all the free BMWs I want. Now notice, if you ask that diagnostic question I gave you earlier, ask yourself 'If those premises were true, would the conclusion have to be true?' The answer is yes. If Premise 1 were true, if there was some provision in the Constitution such that whoever's president gets all the free BMWs I want, and if it was also the case that Premise 2 was true, if I was President of the United States, then 3 would follow with 100% certainty. Now it doesn't matter that Premise 1 and 2 are actually false and the conclusion's actually false. What matters for validity is if the premises were true, would that conclusion have to be true, and you can see here that the answer is yes. So even an argument where everything's false can be valid. The next point I want to make is that even if everything's true the argument could be invalid so consider this argument. If I have a mass greater than 0 kilograms, then I am subject to gravitational forces. That's true actually. I like pizza. That's also true. Therefore I am explaining validity. That conclusion is true, but you'll notice that this is a horrible argument. Just intuitively you can see that this is a bad piece of reasoning, even though the premises are true and the conclusion is true. Why? Because, again, go back to that diagnostic question. If the premises were true, would the conclusion have to be true? Hopefully you can see the answer is no, and in fact a little bit later on, after I've stopped explaining validity, Premise 1 will still be true, Premise 2 will still be true, but the conclusion will be false. So it's super possible for the premises to be true, but the conclusion to be false. Even though they are in fact all true right now, there isn't that relationship between them. Now why does validity matter? Here's why. Validity is what let LAX Girl skip opening the box. So remember that example. She knew that it was either cheese or pepperoni. She learned it wasn't pepperoni, so she then determined that the pizza in the box she was holding had to be cheese, and she went straight back to her chair without opening it up. The reason she could do that is because the reasoning in her head that she was going through was valid. That was the key bit, and the premises were true. We'll get to that later. Remember valid means if the premises are true or were true, the conclusion must be true. So if I know or if LAX girl knows that the reasoning is valid and the premises are true, then you just know the conclusion is true. You don't have to check that independently. You don't have to look in the box or however else you would check on that piece of information. So that's why validity is important. If an argument isn't valid, meaning it's such that even if the premises were true the conclusion could still be false, then you'd have to look in the box. Even if you knew the premises were true you'd still have to check on that conclusion independently. So that's why validity is important. What the definition of validity rules out is the possibility of a certain combination, namely, all true premises and a false conclusion. Now one way you can look at that is that this combination then, the one that's ruled out - all true premises and a false conclusion - is diagnostic for validity. If that combination's possible, that argument's not valid. If it's impossible, the argument's valid. So this combination is going to be important for us because we're going to use this combination, especially in Chapter Two, as a way of diagnosing whether an argument is valid or not. So we're gonna have a name for this combination - all true premises and a false conclusion - we're going to call it a counter-example. Now this sounds pretty fancy, the way I've defined it and everything, but notice that people have an intuitive grasp of this. It's very common for people to point out that someone else's reasoning is flawed by producing a counter-example. So for example suppose Roberto says, "I know that Juan won lotto." That's his conclusion. Now we're gonna get his reasoning. "I saw Juan in a Tesla this morning, and Juan always said that if he won lotto, he'd buy a Tesla." So that's Roberto's reasoning for his conclusion, and we can make it an argument like this. If Juan one lotto, he would have a Tesla. He does have a Tesla. Therefore, he won lotto. Now it turns out that's a bad piece of reasoning. It's not valid, but how could we show that? Well Maria, maybe that's who Roberto's talking to, maybe Maria realizes that's a bad piece of reasoning so how does she respond? She says, "Well, maybe he inherited a lot of money." Notice this is a very intuitive way. Maria's not a logician. You could have come up with this yourself, but look at what's being described here. What Maria is doing is she's producing a counter-example. She's describing a situation where all the premises are true. Of course if Juan won lotto, he'd have a Tesla. He says it all the time. Let's believe it, and Juan does have a Tesla. We saw him this morning driving around in it. Those are all true, but he didn't win lotto possibly. I'm going to tell you a situation where those premises are still true, but the conclusion is false. So Maria is producing a counter-example in order to show that Roberto's reasoning is bad. So it's a very intuitive thing. Now of course most people don't think of it in terms of validity and counter-examples and things like that. They just have an intuitive grasp of it, but roughly that's what's going on. Now let's turn to inductive arguments. Remember the example. South Park has always been on Wednesday at 10:00 p.m. It's now Wednesday at 10:00 p.m. Therefore, South Park is probably on now. In this argument there is a good inferential relationship between the premises and conclusion. If those premises are true, the conclusion is probably true. I would bet a decent amount of money on it. Again don't worry about how probable is probable enough, but this was an example of one where that reasoning wasn't so great. I wouldn't really bet much money on this or maybe any money. If the starting information was just the last time I watched South Park, the first commercial was for Old Navy doesn't really say much about whether that will still be the first commercial on tonight's episode. So there's not a good inferential relationship. This inductive argument fails that test. So here's the definition. Again, we're looking at the first type of goodness as applied to deductive and now inductive arguments. An inductive argument has a good inferential relationship between the premises and conclusion. If it does, then it's "strong." That's going to be our word for that. So here's a definition of a strong argument. An inductive argument is strong, if and only if, if all the premises of the argument were true, the conclusion would probably be true. So the first kind of goodness: I said there were two types. This inferential relationship - and I've just talked quite a bit about how to assess that. 'If the premises were true, would the conclusion necessarily be true' for deductive argument. 'If the premises were true, would the conclusion probably be true' for inductive argument. And we introduced two words for that: "valid" - that's if a deductive argument has a good inferential relationship and "strong" - that's if an inductive argument has this inferential relationship. Let's move on to the second type of goodness which is just whether the argument has all true premises or not. So here are two arguments. This is the BMW argument I talked about earlier. Compare that to this argument. If I'm over 21, then I can legally drink beer at the pub. I am over 21. Therefore, I can legally drink beer at the pub. Now both of these arguments are valid in that if those premises are true, then the conclusion is certainly true. But only the second one is sound because, notice, even though this first argument - the BMW argument - is valid, the premises are both false. On the other hand, the second argument - the pub argument - is also valid, but in addition all of its premises are true. So the second kind of goodness that an argument can have is the truth of its premises, and if a deductive argument is valid - so it passes that first test - and it also passes the second test - it has all true premises - then it's "sound." That's going to be our name for that status. Turning to inductive arguments, if an inductive argument is strong - so it passes the first test - and it also has all true premises, then that argument is "cogent." So that'll be our word for that category of argument. Now I've just run through a whole bunch of terminology so let me summarize these in a little diagram. So we have arguments overall. This is the category of all arguments, and there's two varieties. There's deductive and inductive. And recall this has to do with the degree of support that the person presenting or going through the reasoning thinks that the premises provide to the conclusion. If they think it's complete, watertight support, then it's a deductive argument. If they think it's probable support, then it's inductive. Looking first at deductive arguments, if it passes that first test - if the premises do provide complete, watertight support - if the premises were true, the conclusion would have to be true, then it's valid. If they don't - even if the premises were true, the conclusion could still be false - then it's invalid. So that's the first type of goodness. When we move to inductive arguments, if an inductive argument has that first type of goodness, it's strong. (If the premises were true, the conclusion would probably be true.) If it doesn't, then it's weak. (Even if the premises were true, the conclusion wouldn't probably be true.) What about the second type of goodness? Well if a deductive argument is valid and it has all true premises, then it's sound. If it's valid but doesn't have all true premises-- so this would be like the pub argument was here. That was valid and all the premises were true so it was sound. The BMW argument, that argument was valid but it wasn't sound. So we don't really have a snappy, single word for this type of argument. If it's valid but one or more of the premises are false, then it's just valid but not sound. And when we move to inductive arguments, if an inductive argument is strong and it has all true premises, it's cogent. If it's strong but one or more of the premises is false, then it's strong but not cogent. Now this is quite a bit of stuff to look at, but the good news is that really what we're interested in for the first four chapters is just validity. So we're not going to be concerned about inductive arguments for the first four chapters. When we get to Chapters Five and Six, we'll turn to inductive arguments. And we're not concerned with soundness. What we're concerned about is, given a deductive argument, is it valid or invalid? How do we verify that it's valid? Now to recap, I said a while ago that validity - this was in the last lecture - is often a function of the formal features of an argument. Whether a piece of reasoning is good is a function of formal features, and validity in particular, that's what is a function of formal features. So this was the Pizza Argument, and it has this form. I talked about this in the last video. Either X or Y. Not Y. Therefore, X. That form is a valid form. Any argument that has that form will be valid. It will be such that if those premises were true, the conclusion would be true. For instance either the rabbit ran down the left trail or the rabbit ran down the right trail. The rabbit did not run down the right trail. Therefore the rabbit ran down the left trail. That argument is valid, and the reason it's valid is because of that form. Well what does that mean? That means in order for us to assess validity, what we want to do is first be able to take an argument and determine its form because whether it's valid or not is a function of its form not what it's about, not the content, but the form of the argument. So the first thing we need to do is figure out, if I give you an argument, what is the form of that argument? And that's going to be Chapter One - the rest of Chapter One. Then in Chapters Two, Three, & Four, we'll develop tools for determining, of some given formal representation of an argument after we've already got the form, whether that argument is valid or not, whether that is a valid form. That'll be Chapters-- we'll do it in Chapter Two and then we'll do it in a different way in Chapters Three & Four. So that's a preview of what's coming up in later lectures and the rest of the Chapters Two, Three, & Four, and that's it for this video.