Let's do another example. Let's look at the claim all prime numbers are odd. Now we all know this is false. And we know it's false because we know that two is a prime number, which is not odd. But let me tease apart the logic that underlies that conclusion. Because in more complicated situations where we're not familiar with the domain or with the result, all we'll have to use is the logic itself. So I want to make heavy weather of this in order to understand the kind of reasoning that we'd use in situations where we don't know the answer, okay. So I'm going to let p(x) mean x is prime and I'm going to let O(x) mean this x is odd. So the sentence can be written symbolically as for all x if x is a prime, then x is odd. When I negate this. I get exists x, not be x. Here's O(x). Or another way of writing it, there is an x, so it's the P(x) and not O(x). The negation of not for x, P(x) implies O(x) is the existent x. So it's the P(x) and not O(x). In words, i.e., there is a prime. That is not odd. So in order to prove that this is false, which is the same as proving that the negation is true. The logic says, find a prime that is not odd. And we can, two is a prime that's not odd. So if I want to prove conclusively that this statement is false. Which is the same as proving that its negation is true, what I need to do is demonstrate that there is a prime that's not odd. And in this case, that's immediate. We just pulled two out of that, observe the two is prime and it's not odd. Therefore, the negation is true, the original statement is false. Well, in that simple example that was making much ado about nothing. But in more complicated situations going through the underlying logic is often the only way forward. By now you should have noticed the pattern of behavior that's happening with these symbolic negations. When we negate a for all becomes exists and the for all sort of jumps inside. And when it jumps inside, we can manipulate it and get the result into a positive form where the negation is at the most innermost point. You can actually turn these into formal symbol manipulation rules, which means you can implement them on computer systems. And there are computer systems that do these kind of reasoning for you. I'm recommending very strongly that you don't do that. Because what we're trying to get out is mathematical reasoning and moving symbols around, doing symbolic manipulation is not mathematical reasoning. It's connected with mathematical reasoning, it supports mathematical reasoning, we can use it in mathematical reasoning. But the real essence of what we're trying to do is to understand what the statements mean, understand what the symbols represent, and reason with the concepts. It's reasoning with concepts that mathematical thinking's all about, not symbolic manipulations. Okay, we showed that this was false by finding a counterexample. In this case the counterexample was two. Supposing I modify it as follows. All prime numbers bigger than 2 are odd. In symbols, for all x bigger than 2, if x is a prime, then x is odd. Okay, now I'm going to give you a quiz. What is the negation of this statement? Is it 1, there exists an x less than or equal to 2? So it's the P(x) but not O(x)? Or is it 2, there exists an x greater than 2, so it's the P(x) and not O(x)? Okay, which one of those two do you think it is? Well, the correct answer is 2. It says there exists a number bigger than 2 which is prime, it is not odd. That's a negation of that statement. Why isn't it number 1? Well, because the original statement is about numbers bigger than 2, so the negation must be about numbers bigger than 2. We don't simply negate every symbol inside and say, well, in terms of the ordering on the real line, if we negate something being bigger than 2, then it means it's less than or equal to 2. That's a symbolic negation. That's just sort of a detailed localized negation about the ordering on the real line. But the sentence as a whole means. Something about all numbers bigger than 2. So its negation means something about all numbers bigger than 2. Okay? Well, let's move on. Let x denote a person. Let P(x) mean x plays for sports team T. And let H(x) mean x is healthy. And let me look at the statement, there is an x, so it's a P(x) and not H(x). What does that mean in everyday English? There is an unhealthy player on team T. Now let me negate that. I'm going to find a negation by symbolic manipulation. Now this, remember, is the method that I'm suggesting that we don't use. I'm doing it that way in order to illustrate the mathematics known as line logic. If we negate this, we know that when we negate an exists, it turns into a for all. And then the negation moves inside if you like. And then when we move that negation inside, we get not P(x) or H(x). So the negation moves inside. We get P(x) becomes not P(x). When we negate a conjunction, we get a disjunction. And when we double negate something, we get the original thing back. Remember, this is not the way I'm recommending you do it. I'm doing this as an illustration of the underlying logic. Well, this thing looks familiar. Remember when we looked at the conditional, p conditional q means the same as not p or q. They have the same truth table. So this guy can be rewritten as P(x), Here's H(x). In English, all players on team T are healthy. Well is that the negation of that? There is an unhealthy player on team T. Well if it's not the case that there's an unhealthy player on team T, then all the players on team T must be healthy. Which is definitely the case, that is definitely the negation of that. And again let me stress, this is not the way you should do this kind of thing. I'm using this to illustrate how negation behaves with quantification. Next point, suppose I write down a sentence like the following. Flow x, if x is greater than 0 then there is a y, so should xy = 1. Is that true or is it false? Well there's no way of knowing, it depends on what the x denotes. If x denotes a motorist or an automobile or a healthy team player, it's completely nonsensical. If x denotes a natural number, it makes sense but it's false. If x denotes a rational number, it makes sense and it's true. But a point I want to make is that the quantifier only tells you something if you know what the variable denotes. Associated with any quantifier, we have what's called the demand of quantification which tells us what does the x denote? If the demand of the quantification is abusive, we've set it in advance or the context makes it clear what it is, then we can use statements like that. If there's any danger of misunderstanding, if there's any potential ambiguity, we can make the quantifier more explicit, by stating the domain of quantification. To make this more explicit, I could write it as. For all x in the set of rational numbers, if x is positive and there is a y, just said xy = 1. And now I've got a true statement, that's not ambiguous. It tells us that for every positive rational, well wait a minute, wait a minute. I haven't actually been explicit as to what the y is, have I? Now arguably, having made it clear that the x denotes a rational, it's reasonable to assume that the y denotes a rational. But if I want to, I could be even more explicit and say for all x in Q, if x is greater than 0 then there is a y in Q. So it so it said xy = 1. I think most mathematicians would agree that if you're explicit about the quantifier at the beginning of a statement, then absent any indication to the contrary, the other quantifiers denote the same thing. But if there's any possibility of misunderstanding or if there's any potential ambiguity, it's better to be explicit everywhere as to what the quantifiers denote. Something else you need to be aware of with regards to quantification is that mathematicians sometimes omit the quantifier. We write things like if x is greater than or equal to 0 then the square root of x is greater than or equal to 0. If you see an expression like this What that means is the following. For all x, perhaps in R or perhaps in something else. For all x in R, x greater than or equal to 0 apply square root of x is greater than 0 or something like that, depending on what the x denotes. I've assume from the context and this is a reasonable assumption, I've assume that the variable is meant to range over the real numbers. Even though there is no explicit quantify here, this would be read as meaning something like that. This is what's known as implicit quantification. The mathematician is simply leaving out of the formal expression any explicit mention of the quantifier. It's left implicit in the way this is written. This is a fairly advanced point, I'm certainly not recommending you do this. But the professionals do it all the time, so you need to be aware of it in case, you come across it in something that you read. But please avoid doing this. I realized by saying this, that I'm saying do as I say not as I do. Because like all professional mathematicians I do like things this way all the time. But while you're still learning about these ideas, it's best to avoid doing it. Because if you don't use this in the right kind of context, all kind of difficulties can arise. And having given you one caution, let me give you another one. This is about combating quantifiers with conjunction and disjunction. Let me just do this by way of an example. Let N, the set of natural numbers, be the domain of quantification. Let E(x) mean x is even and that O(x) mean x is odd. Look at the formulas for all x E(x) or O(x), and for all x E(x) or for all of x O(x). I guess I don't need that last bracket. For all x E(x) or o(x) and for all x E(x) or all of x of O(x). Notice that the first one is true. It says, for every natural number it's either even or it's odd. What does the second one say? It says, every natural number is even or every natural number is odd. Well that's false. The point I'm trying to make here is you can't just take a for all and take it inside a bracketed expression. If you do that you're likely to end up turning a true statement into a false statement or vice versa. Similarly, If I took if exists x, E(x) and O(x). We do it with an un this time. And I'll take existing x E(x) and existing x O(x). Again, I've put that extra bracket in there. I can't stop myself doing that. What have I got? This says, the reason x. Which is both even and odd, which is false. What does this one say? There is an x that's even and there is an x that's odd, that's true. So the same thing happens with exists and conjunctions there. You can take an existence statement like this, and in this case it's false. But if you pair them up in this way, it's true. In other words, be very careful when you're reasoning with quantifiers, conjunctions, and disjunctions. You can't simply take things inside the way you sometimes do in arithmetic. If you think about what these things mean, you're not likely to run into that difficulty. But if you start treating these as symbolic expressions to manipulate, then things can go badly wrong. Time for a quiz. Okay, well we've just seen that this expression, something like for all of x, A(x) or B(x) is not equivalent to for all x A(x) or for all x B(x). The example we looked at was where the numbers are even or odd. And it was a case that for all natural numbers, the numbers are either even or odd. But it's not the same as saying that all natural numbers are even or all natural numbers are odd. In the case of even or odd, for natural numbers, this is true. But both of these disjuncts are false, so the disjunction is false. Okay, what about this? Instead of having disjunction, we've got conjunction. Are these equivalent or not, what do you think? Well, the answer is yes. They are equivalent. Or you could argue this in an abstract form, but let's just look at an example that's pretty illustrative. Let's take something like, All athletes, Are big and strong. Okay, so x denotes athletes, A means big, B means strong. So that says all athletes are big and strong. This would say, All athletes are big. And then we'd have all athletes are strong. It's pretty clear that in the case of the example, that is equivalent to the conjunction of those two. If all athletes are both big and strong, then in particular they're all big and they're all strong. And conversely, if they're all big and they're all strong, then they're all big and strong. In other words, when you've got universal quantification, if it's combined with a conjunction, then you get the equivalents of these two forms. But if universal quantification is combined with a disjunction, they're not necessarily equivalent. This isn't a proof, this is just an example. But if you take that example, you should be able to come up with a simple little logical argument showing that that implies that. And conversely that implies that. So the answer is yes in this case. Let's look at a another variant. Well again in this case, when we had even and odd numbers, that showed that these are not necessarily equivalents. But what about this, where we have an existential quantifier combined not with a conjunction, but with a disjunction? Do we have equivalency in this case? What do you think? Again, the answer is yes. These are in fact equivalent. What would be a good example? Well let's take something like, There is a player, let's keep it with athletic examples. A player who, let's say, I know, who is, A good attacker, Or a good defender. Okay, so there's a player who is a good attacker or a good defender. Is that the same as saying, Let's just simplify it a little bit. There's a good attacker, Or there is a good defender. Okay, and we're going to disjoin those. So if there was a player who was a good attacker or a good defender, one or the other, then whichever one it is, will be here. If the player who is referred to here is going to either be a good attacker or a good defender, so one of those two is going to be true. Which means that the disjunction is true. So if this is true, then that's true. And conversely, if there's a player who's a good attacker, then that player is a good attacker or a good defender, in fact a good attacker. Or if it's this one, then that's true. So we have an implication that way with a disjunction. And we have an implication that way with a disjunction. They're equivalent. Again, this isn't a proof. This is just an example to illustrate the fact. But you shouldn't have any difficulty taking that example and turning that into a simple little argument. At least I hope you won't have any difficulty doing that, turning that into a simple argument. To show that that implies that, and that implies that. So in the case of existential quantification, if we have it with a conjunction, then we don't get equivalence. But if we have it with a disjunction, then we do have equivalence. And if you think about what's going on here, the fact is that for all, Is like, and let me put quotes on here because this is a sort of specialized use. For all is like conjunction, and exists, Is like disjunction. Because for all says something's true for all, and conjunction means that all of the conjuncts have to be true. This simply says there's at least one, and this says there's at least one. So that's all about all things, and that's about at least one. So we get good behaviour, whether you call this good or not, you get this nice behaviour. If we have exists with disjunction because exists is essentially a disjunction sort of thing. And for all, that was on the previous slide, for all is really a conjunction sort of thing. And it's when they're mixed up when you get the disjunctive thing here and the conjunctive thing here that things fall apart, okay. Well, there you go, how did you get on with those two parts of the quiz? And this is the last point I want to make about quantifiers. Suppose we're having a discussion about, let's say, the real numbers. So, if we have variables x, y and z, they're assumed to denote real numbers. The domain of quantification will be the set of real numbers. And in the course of the discussion about real numbers, we want to talk about rational numbers. Or particular rational numbers. We might want to mention that there's a rational number x. Well, there's one way we can do that. We can talk about the rational number in this context. We can say there is an x in Q, or for we can say for all y in Q. Or if I want then to talk about a natural number, I can say for all N in the set of natural numbers. So by introducing explicit demands of quantification in this way, in the course of discussing real numbers, I can restrict attention to particular rationals, or natural numbers, or whatever. And so in the course of an argument, I can actually use multiple domains of quantification. Well, in examples like this where we're talking about different sets of numbers, this is fine. But in other situations, it really doesn't make sense to introduce multiple domains of quantification. For example, suppose I'm going to be talking about, let's say animals. So suppose the domain of quantification is the set of animals. And then I might want to say something like, every leopard has spots. Well, how would I do that? Well, I suppose I could say for all x in the set of leopards, x have spots. But in the next sentence, I might want to talk about tigers, and then giraffes, and then who knows what. And very soon, I could have a whole range of different domains of quantification floating around. And that's not very clean. It's not very nice, and it's certainly not very sensible. Because the discussion isn't really about leopards, or tigers, or giraffes, or whatever. It's about animals. And if the discussion's about animals, then the set of animals should be the domain of quantification. So instead of writing something like that, what I should write is for all animals, if that animal is a leopard then it has spots. That allows me to say something like, there is an animal which is a horse and has spots. There's a spotted horse. Or I could say for all x, if x is a tiger, then x doesn't have spots. The point is that I'm talking about animals. Then when I refer to specific animals, I'm not changing the domain of quantification. I'm still talking about animals. I'm just describing properties of particular kinds of animals. The discussion is still about animals, so the domain of quantification should be animals. It's different here, because the domain of quantification arguably is the set of real numbers or the set of all numbers. And then because we have these standard subsets, it's okay to introduce things like this. You don't have to, you could actually use something analogous to this. But in this kind of situation, yes, if you're talking just about real numbers, then you really don't want to be introducing particular subdomains. You should use the analog of these kinds of expressions, and talk about if it's a rational, then this. If it's a natural number, then this. But if this is really a discussion about all numbers, not just the real numbers, then these are just particular categories of real numbers and natural numbers, and so forth. Then you can have subdomains. You can have different domains of quantification. This isn't an issue of right or wrong. It's an issue of whether it's sensible and helpful to work in a certain way. And using separate subdomains of quantification is fine if they are natural domains, and it makes sense to talk about it that way. But the idea of a domain of quantification is that that tells us what it is we're talking about. If we're talking about animals, then don't introduce different domains of quantification that are not animals. Even if they're subdomains, it gets complicated. Okay, that's enough about quantifiers for now. I suggest you go away and see how you get along with assignment number six.
