Welcome to the third lecture. I hope you made good progress with the assignment from the last session. Our next step in becoming more precise about our use of language for use in mathematics is to take a close look at the meaning of the word implies. This turns out to be pretty tricky. Brace yourself for several days of confusion until the idea sought themselves out in your mind. Just like learning to ride a bike. Well, all the actual learning occurs before you finally get the hang of it. So too, most of the benefit from understanding the way our language is used in mathematics comes from trying to figure it out. The benefits in this case is helping to develop your mathematical thinking ability, and extra process of trying to understand the issue that goes out for you. Yeah, sure, once you've sorted this language stuff out, you'll be able to use language well actually. But by that stage, you'll just using it, the part that helps you with when and how to think mathematically is largely over. We'll need other tasks to develop your mathematical thinking ability forever. So as you get into this, bear in mind that the pay off for struggling with issues is significant in terms of being able to think like a mathematician. Here we go. In mathematics, we frequently encounter expressions of the form, Phi implies Psi. Indeed, implication is the means by which we prove results in mathematics, starting with observations or axioms. So we'd better understand how the word implies behaves. In particular, how does the truth or falsity of a statement Phi implies Psi depend upon the truth or falsity of Phi and of Psi? Well, the obvious answer is to say that Phi implies Psi if the truth of Psi follows from the truth of Phi. But is that what we want? Let me give you an example, that phi of the statement loop 2 is a rational. And that psi of the statement is 0 is less than 1. And let's ask ourselves, is the statements phi implies psi true? Well, phi is true. As I've mentioned once already, we're going to prove that later in the course. And we all know that psi is true, 0 is certainly less than 1. So we have the truth of this. We have the truth of that. Does that mean that phi implies psi? Obviously not. There's no relationship between phi and psi. This takes some effort to prove, as you'll see. We all know this one. So yes, this is true and that's true, but the truth of this doesn't follow from the truth of that. And now we realize there's a complexity with implication that we didn't meet before when we were dealing with and, with or, and with not. And the complexity is that implication Involves causality. Causality is an issue of great complexity that philosophers have been discussing for generations. Now, we're facing a problem. It didn't arise before, because when we're dealing with conjunction and disjunction, it didn't matter whether there was any kind of relationship between the two conjuncts or the two disjuncts. For example, let's look at the sentence, Julius Caesar is dead, and let's conjoin it with the sentence 1 + 1 = 3, the mathematical sentence. And let's do the same thing with disjunction. Forming a conjunction and disjunction didn't require any kind of relationship between these two. Clearly, they're independent. One's a statement about a long dead individual, and the other one is a mathematical statement. Incidentally, why do we got these in front of us? Let me just give you a quick quiz. Is the first one true, or is it false? Is the second one true, or is it false? What do you think? Well, remember what the definition was. A conjunction is true if both conjuncts are true. This one's true but this one's false, so the conjunction's false. This one's true. This one's false. All that you need for a disjunction to be true is at least one of the disjuncts to be true. So that one's true. And the fact that there's no meaningful relationship between the two conjuncts in this case, or the two disjuncts in that case, created no lull in determining what the truth value was. It was purely in terms of truth and falsity. But it's not sitting with implication, because implication involves causality. So let me just express that explicitly. Implication has a truth part and a causation part. What we're going to do is ignore that part. We're going to leave that to the philosophers if you like and we're just going to focus on the truth part, now that might sound to be a very rush thing to do. Throwing away a causation, now, we can't be left with anything useful, but it turns out yeah, it might seem a dangerous thing to do, to throw away this important causation implication, but it turns out when we focus on the truth part we are left with enough to save our leaves in mathematics. So much so that we're going to give this a name. We're going to call it the conditional. Or sometimes, the material conditional. That's the part we're going to focus on. So we're going to split implication into two parts, conditional and causation. The first part, the conditional, we're going to define entirely in terms of truth values. And the second part we're going to leave to the philosophers. The symbol that we use normally for conditional, at least the symbol I'm going to use, is a double arrow. Lots of textbooks use a single arrow, but that has many uses in mathematics, so we certainly are going to use that for something else later on in the course. So to avoid confusion, I'm going to stick to this notation for the conditional. So I'm going to write conditional expressions like this. That's the truth part of phi implies psi. When we have a conditional, we call phi the antecedent, and we call psi the consequent. And we're going to formally define the truth of phi conditional psi in terms of the truth of phi, and the truth of psi. Well, you might worry that by throwing away a causation, we're going to be left with a notion that's really of no use whatsoever. That actually is not the case. Even though we're throwing away something of great significance, hanging on to the truth part leaves us something very useful. And the reason is, whenever we have a genuine implication, which are actually the only circumstances in which we're ultimately going to be interested, whenever we have a genuine implication, the truth behavior of the conditional is the correct one. It really does capture what happens with truth and falsity, when we have a genuine implication. That probably seems a bit mysterious at this stage, but when we start to look at some examples, I think it should become clear what I mean. The advantage is that the conditional is always defined. For real implication, you've got that issue of causation, the square root of 2 and the 0 less than 1 example, the truth or falsity wasn't the issue, it was whether there was a relationship between those two statements. Now, that's a complicated issue. But because we're going to define the conditional purely in terms of the truth value of the two constituents, the antecedent and the consequent, it turns out that the conditional will always be defined. When we do have a genuine implication, the definition of the conditional will agree with the way implication behaves. And when we don't have a genuine implication, the conditional will still be defined, and so we can proceed. Again, this probably seems very mysterious when I describe it in this way. But as we develop some examples, I hope you'll be able to understand what I'm trying to get at. Okay, I think we need to catch our breaths. Let me do that by giving you a quiz. The truth of the conditional from phi to psi is defined in terms of? 1, the truth of phi and psi, or 2, whether phi causes psi, or both. Which is it? It's number 1. We define the truth of a conditional in terms of the truth and falsity of the antecedents and the consequent. And because we define the truth of the conditional in terms of truth and falsity in that way, it has a truth table. So let's see if we can figure out what it is. Well, the, I'm not giving you, this is a quiz, by the way, I'm going to work this one through. This is tricky, we're not out of the woods yet by any means, so I'm going to lead you through this one. This part we've already looked at. We define the conditional as the truth part of implication. And implication has a property that a true implication leads to a true conclusion from a true assumption. So because we take the conditional from real implication, we have to have truths all the way throughout the top level. Now the fun begins. We have to fill in these three values. And when I say fun, I mean fun. Let's look at that first row of the truth table. Suppose phi is the statement N is bigger than 7. And suppose psi is the statement N squared is bigger than 40. If phi is true, in other words, if N is bigger than 7, then N squared is certainly bigger than 40. In fact, it's bigger than 40 now. So, certainly, in this case, phi does imply psi. So that thing is true. If N is bigger than 7, then N squared is certainly bigger than 40, so that's true. We have truth everywhere. So this is consistent with the truth table. Now let's look at a different example. Let phi be the statement our old friend Julius Caesar is dead. And let psi be the statement pi is bigger than 3. Phi is true. Psi is true. According to the truth table, it follows that phi conditional psi is true. In other words, Julius Caesar is dead, conditional, pi is bigger than 3. Now, if you read this as Julius Caesar is dead implies pi bigger than 3, then you're in a nonsensical situation. But remember, this isn't implication, this is just the truth part of implication. And in terms of the truth part, there's no problem. That's true, that's true, the conditional is true. In the first example, there is a meaningful relationship between phi and psi. When we know that N is bigger than 7, then we can conclude that N squared is certainly bigger than 40. There's a connection between the two. And in that case, the behavior of the conditional is certainly consistent with what's really going on. In the second case, there's no connection between the two. The conditional is true, but it's got nothing to do with one thing following from the other. The value of doing this is, even though this has no meaning in terms of implication, its truth value is defined. In both cases, we have a well-defined truth value. In the first case, it's a meaningful truth value. That does follow from that. In the second case, it's purely a defined truth value. But that's not going to cause us any problems, because we're never going to encounter this kind of thing in mathematics. We encounter this kind of thing all the time. But we're not going to encounter this kind of thing. So all we've done is we've extended a notion to be defined under all circumstances. And we've done it in a way that's consistent with the behavior we want when something meaningful is going on. This is actually quite common in mathematics to extend the domain of definition of something so that it's always defined. So long as it has the correct behavior, the correct definition for the meaningful cases, and provided we do the definition correctly, it really doesn't cause any problems. In fact, it solves a lot of problems and eliminates a lot of difficulties if we extend the definition so that it covers all cases. Is it just something we do in mathematics all the time? May seem strange when you first meet it, but it is a part of modern advanced mathematics. Incidentally, if you think this is just playing games, Let me mention that the computer system that controls that aircraft that you'll be flying in next time depends upon the fact that expressions like this are always well defined. That software control system doesn't depend upon knowing whether Julius Caesar is dead or things like that. It doesn't depend on those kind of facts of the world. Computer systems, by and large, don't depend upon understanding causation, which is just as well, because they don't. What computer systems depend upon is that things are always accurately and precisely defined. And this expression, phi conditional psi occurs all over the place in software systems. So, quite literally, your life depends upon the fact that this is always well-defined. It doesn't depend upon the fact that the computer doesn't know whether Julius Caesar is dead or not. Okay, time to look at the second row of the truth table. What goes in here, T or F? Well we've only got two to choose from. I guess we could just make a guess but we're not going to do that, we're going to figure it out and get the right answer. Well, what would happen if we put a T here. Let me put it in very lightly, because I'm sure it's going to be a T. If this were true, when we think about it in terms of genuine implication, because we are trying to capture the truth behavior with genuine implication, remember. So if it was the case that phi really did imply psi. If that statement was true when we interpret it as real implication then the truth of psi would follow from the truth of phi. That's how we began remember. That's what real implication means, real implication means the truth of this would follow from the truth of that. So if that were a T, and it was real implication, then when we have a T here, we would have to have a T there. But we don't, we've got an F. So we can't have a T here, because if we put a T here the conditional is contrary, it contradicts real implication and we're trying to extend implication to be defined in all cases where there's no causation. So this has to be an F. If we put a T there we're in trouble. Our notion doesn't agree with real implication. In order that the conditional agrees with real implication, that has to be an F. It's if a truth, then we would have a true antecedent and a false consequence from a true implication. So we've argued backwards to conclude that this has to be F. Let me write that down just to make sure everyone's following what I'm trying to say. If there were a genuine implication phi implies psi and if that implication were true, then psi would have to be true. If phi were true. So we cannot have phi true and psi false, if phi implies psi is true. That means, that in the case where phi is true. And psi is false, we have to have a false here. Let me write that down. Are you confused? This is tricky. There's no getting around the fact that sorting out implications and then extracting conditional form implications is tricky. You're probably going to have to replay the videos several times and look at what's been written and think about it in order to sort this out. I remember when I first met this, it took me quite a while to get on top of it. It really isn't an easy thing to do. So, you certainly have my sympathy that you're going to have to struggle with this thing. It's probably one of the hardest things in the course, to really come to terms with the way we define the meaning of the conditional. Okay, final straight. Let's see if we can fill in the last two entries in the truth table for the conditional. When we've done that, we'll all go off in search of some aspirin. Now if you're like me, you have no intuitions as to what to put here. And the reason you have no intuition is that even though you're used to dealing with implication you've never dealt with an implication where the antecedent was false. You're only ever interested in drawing conclusions from true assumptions. So you never dealt with a true one of these guys. When this guy was false. You don't have any intuitions. You do however have intuitions about this guy. And the reason that's going to help us out is that negation, swaps around F and T. So corresponding to the Fs here, when we look at this guy we'll have truths. We'll have Ts for this. So you are used to having to deal with this. When this is true, and that will be equivalent to dealing with this when that's false. Because negation swaps truth and falsity. So the trick, or at least the idea by which we're going to figure out what goes here, is to stop looking at implication and look at not implication. Phi does not imply psi if even though phi is true, psi is nevertheless false. Just think about that for a minute, phi does not imply psi if even though phi is true psi is nevertheless false. That's how you know that this guy holds. You know that phi doesn't imply psi if you can check that phi is true but psi is nevertheless false. That's the only circumstance under which you can conclude this thing is true. In all other circumstances, this guy will be false. So let me write that down. Okay? This guy is true if you have phi true and psi false. In all other circumstances, this guy will be false. You still with me? Because we're only one line away from the conclusion now. Let me just, Rewrite that as follows. Because negation swaps F and T, this guy will be false, and this guy is true. So if we have T and F, which we have here. Then negation is false, which means this guy is true. In all other circumstances, this guy is true, so those are true. And we're done. Phew, that was difficult. In fact, in my mind, this is the most difficult part of the entire course. This is not easy stuff, it's going to take you some time before you sort it out in your mind. You're going to have to keep working at this. Look through the lecture several times and think about it. It will take effort, but you'll get there. We all get there in the end, but it's just not easy. In the meantime, you've got this far. We should start thinking about opening a bottle of champagne. But before you crack open the bottle, let me leave you with one more little quiz. Which of the following are true? Phi conditional psi is true whenever, (1) Phi and psi are both true. (2) Phi is false and psi is true. (3) Phi and psi are both false. And (4), phi is true and psi is false. And I want you to check all that are true. Well, which of these four, Are the case? Which of these four conditions tell you when phi, psi is true? The answer is 1, 2, and 3. Phi conditional psi is true either when 1 is true, or when 2 is true, or when 3 is true. The only time when phi conditional psi is false is when 4 holds. So the correct answer is 1, 2, and 3, and 4 is not. Okay, now you can open that champagne. Let me sum up what we've done. We've defined a notion, the conditional, that captures only part of what implies means. To avoid difficulties, we base our definition solely on the notion of truth and falsity. Our definition agrees with our intuition concerning implication in all meaningful cases. The definition for a true antecedent is based on an analysis of the truth values of genuine implication. The definition for our false antecedents is based on a truth value analysis of the notion does not imply. In defining the conditional the way we do, we do not end up with a notion that contradicts a notion of genuine implication. Rather, we obtain a notion that extends genuine implication to cover those cases where the claim of implication is irrelevant, because the antecedent is false or meaningless when there's no real connection between the antecedent and the consequence. In the meaningful case where there is a relationship between phi and psi, and in addition, where phi is true, namely, the cases covered by the first two rows of the truth table, the truth value of the conditional will be the same as the truth value of the actual implication. Remember, it's the fact that the conditional always has a well-defined truth value that makes this notion important in mathematics since in mathematics, we can't afford to have statements with undefined truth values floating around. Let's finish this lecture with a short quiz. When you've done that, you should complete assignment three. I've kept assignment three fairly short since I expect you'll need most of your time simply understanding our analysis of implication and the definition of the conditional. Remember to discuss these assignments with other students. Don't struggle for too long on your own. On this kind of material, it's much more effective to work with others. Here's the quiz. Well, the answer to the first one is that it's true. The antecedent is true, and the consequence is true, so the conditional is true. In fact, there's a deeper result going on here. Providing you take a positive number, instead of pi, any positive number, then if the square of that positive number is bigger than 2, that number must be bigger than 1.2. Because the square root of 2 is 1.4, etc, etc, etc. So for positive numbers, it doesn't have to be pi, it can be anything, any positive number whose square is bigger than 2, it must be bigger than 1.2. And that would be a case of genuine causation, genuine implication. But in terms of the conditional, it's enough that the antecedent is true, then the consequence is true. For this one, it's also true. Now the consequence is false. Pi certainly does not equal 3. But the antecedent is false. Add if you have a false antecedent, the conditional is always true. Pi squared is most certainly not less than 0. So, you've got false, false, that makes the conditional true. Number three, That one's false. The antecedent is true, and the consequence is false. And you cannot obtain a false conclusion from a true assumption. What about the next one? Well, that one's true. The antecedent is true, and the consequence is true. What about this one? Do triangles have four sides? No. Do squares have five sides? No. But anything with a false antecedent is true, so that's true. You've got false, conditional, false, and that's always true. What about this one? Well, we don't know when Euclid's birthday was. At least, I don't know when Euclid's birthday was. I suspect you don't either. Rectangles certainly have four sides, however. Either this is true, or it's false. Either way, since the consequence is true, the thing is true. We've going to run down using the two consequence. So either we have true, true, in which case, it's true, or we have false, true, in which case, it's true. Okay, how did you get on with that?
