 Continuing with lecture one in our series, we're now gonna talk about, well, the ideas of logic, in particular, what is a statement in the logical sense? The previous video was a lot more verbose than this one's gonna be, and that's mostly just because as we were introducing ourselves to the lecture series, there's a lot of things we had to say, like what is advanced mathematics? We had to define what that meant, while we were defining what definition meant, there's a lot of confusion going on there, I'll give you that. This one's gonna be a little bit more precise when we're looking for, because as I mentioned earlier, our lecture's always gonna be broke up into three parts, one about mathematical content, one about logical content, that's the one we're looking at right now, and then the third one about communication. All right, so this is our first, our very first video about logic, so where do we begin our discussion of logic? Well, logic is basically the study of valid reasoning. Logic professes to give us inferences for which are gonna be absolutely irrefutable because of the laws of logic. Logic lets us know when things are true or they're false, but when I say when things are true or false, what can be true or false? That's exactly what a statement is. A statement in the sense of logic is a declarative sentence. Sentence is a grammatical thing, I'm not gonna define what that is, but it's a declarative sentence. A statement is a sentence that is either true or false, and I should also mention that a statement cannot be true and false, it's one or the other, there's an exclusive or here. It's either true or it's either false. Now, as we develop what is so called Boolean logic, sometimes called Boolean algebra, the things are related to each other, one calls it Boolean Algebra because it's kind of like an algebra, but you work with statements as your quantities, like in a previous algebra class that you've probably taken, like called algebra, the variables you consider are typically going to be real numbers, or at least they're numbers, and you use things like x, y, z to represent a generic number. In Boolean algebra, which is a type of logic, our quantities are actually not numbers, our quantities are statements for which they could be true or false. And it's actually very common when one represents a statement to use variables like PQR. We typically reserve x, y, and z for a variable real number, a variable statement could be PQR. We'll use some of these in the future, don't worry about that so much right now. So Boolean logic, you're gonna combine multiple statements, which each statement could be true or false into an expression, and then decide, is that expression itself true or false? We often use capital T to abbreviate true, and we abbreviate false by capital F. And so if you think of these as variables for statements, what that means is the variable could take on one of two quantities. Like our variable P could equal T, or it could equal F, right? So unlike college algebra, where there's infinite possibilities, maybe for your real numbers, in Boolean algebra, you only have two possibilities. And this is something we're gonna do in the future, we'll be able to brute force problems that is we try every combination to see whether it works or not. Is the statement true or false, depending upon what these primitive statements take on. That's why we often use P. P is a primitive statement. It's the most simple kind. We'll talk about compound statements another time. But right now, I just wanna focus on what is a statement. A statement is a sentence that is either true or false. Now, to be true or false, we do not necessarily need to know whether it's true or false. To be a statement just means it's one or the other, not both, even if its truthfulness is unknown to us. Take for example here, the sentence, cancer is the leading killer of women. This is a statement because it is either true or false. Either cancer is the leading killer of women or it's not because maybe some other cause is the leading killer of women like, I don't know, anvil's falling on their head. I don't claim that's the case, but this is a statement because it's either true or it's false. But I myself don't actually have the statistical data to know whether this is true or not. And maybe it depends, like, are we talking about women in the United States versus women in the United Kingdom or women in Japan? Maybe it changes based upon what set of women are we considering? But despite there are some ambiguities here, right? Cause which women are we referring to? Every woman that's ever existed. Are we talking about the women who live in Cedar City, Utah, where SU is located? We might have to clarify that, okay? But this is a statement, despite some of the shortcomings I mentioned here. I don't know if it's true or false, but it's a statement. Another example of a statement. If you eat less and exercise more, then you will lose weight. This is a statement. This is either true or it's false. It's one of those two things, right? Now, is it true or false? Well, a lot of fitness people will tell you it is. A lot of nutritionists will tell you it is. And for the most part, I think most of us believe it, but I don't have to prove whether it's true or false for it to be a statement. Another example, in the last 10 years, we have reduced by 25% the amount of greenhouse gases in the atmosphere that produce global warming. I have no idea if that's true or not, but I do know it is a statement, all right? Let's focus on something more mathematical now. If a circle has a radius of r, then its area is pi r squared, all right? That's a statement. It's in fact a true statement. We often typically will abbreviate this as a equals pi r squared. Oftentimes, mathematical statements are abbreviated using formulas and mathematical notation of some kind. That is quite commonplace. This is a statement. It's true, in fact. Here's another statement. Every even number is divisible by two. This is another statement. In fact, it's a true statement. I know that every even number is divisible by two because that's the definition of what it means to be even, right? We saw this in the previous video. You're even if you can factor it as two times a number, okay? Another statement here. Some right triangles are esosceles. Yep, that's a true statement. Look at this one right here. All right triangles are esosceles. Now, this is where you have to be very careful. This last example is the one that students often make the most mistakes on. All right triangles are esosceles. I should mention that this is a statement. It is a statement, but the important thing to notice here, it is a false statement. It is not true that all right triangles are esosceles. You can come up with this right triangle, for example. Right, to be a right triangle means that one of your angles is 90 degrees. To be esosceles means that two sides are congruent to each other. And while this is not a perfectly drawn triangle, I hope my diagram is enough to illustrate that. This is a right triangle that is not esosceles. So this is a statement, but it is a false statement. Statements do not have to be true. And just because I know a statement is false does not remove the fact that it is a statement. Okay? Now, to talk about statements, we have to also talk about things that are not statements. The following are not statements. I'll give you reasons why. So the first sentence here, please open your textbooks. This is not a statement because it's not a sentence that is either true or false. This is actually a command. It's not declaring anything. It's telling you what to do. Open your books. That can't be a true or false thing. All right, the next one here. The number 42. This is not a statement because it's what we actually call a fragment sentence. It's not even a complete sentence, right? We have a noun, the number 42, but there's no verb going on here. It's a fragment. And as such, I can't tell you whether it's true or false because it didn't declare anything, okay? So that's not a statement. The next one, when did the dinosaurs become extinct? This is not a statement either. It's not a true or false statement. It's a question. I don't know. I mean, maybe scientists know, but it can't be true. It can't be false, all right? It's an example of a question. It's an inquisition. It's not a statement. Now, this last one is the most curious of all of the examples of non-statements. This is not a statement. And look at this sentence here. This statement is false. Now, it's tempting to think this is a statement because it refers to itself as a statement. And it does look like a sentence. It's a sentence, so it's not a fragment. It didn't ask anything. It doesn't tell you to do anything. So what's left when it comes to grammar? It feels like it should be a declarative sentence. It's declaring something. It's like, hey, this sentence is false. That sounds like a declaration. So why isn't it a statement? The problem is that a statement has to be either true or false. And this is an example of a paradox. A paradox is a sentence that does declare something, but it can't take on the assignment of true or false because of the following situation. That let's call this sentence P, okay? So P is the sentence. This statement is false. Now, if P is true, that means we have to then accept what this says as fact, in which case then it would say that the statement is false. So it's like, okay, if the statement is true, then we believe what it says, which tells it's false. So P is true and false. That's not possible. But on the other hand, if P is false, that means the thing it tells us is not true. And so it tells me that the statement is false. So if that's not true, then the opposite would have to be true because you do have that dichotomy there, which means the statement is actually true and we get the same problem again. Yikes. You know, we can't get both. We have these logical conflict right here. The logical arrows are crashing into each other like cars. The statement is the statement, if it were a statement, I should say, the sentence, if it were a statement would have to be both true and false, which that's not possible for a statement. So we don't get a statement, we actually get a paradox. One has to be very, very cautious about that in mathematics. Now, like I mentioned earlier, it's quite common to describe statements using mathematical notation. Now, the whole statement itself could be a mathematical notation like the equation. This is very common for the equations. Two plus three equals five, that is a true statement. But you also could just combine regular language with mathematical notation and you can make statements that way. So like the function f of x equals one over x is continuous. Well, it depends on how you define continuity, right? But it's a statement. Don't worry about whether it's true or false. If an integer n is a multiple of six, then n is an even number, right? That is a statement. It uses mathematical notation to describe it. That's okay. You use some variables in this case of what we're doing. Now, I should mention that this is an example, excuse me, the next example here, let's also take the integer n is even. This is an example we call an open statement. It's an open statement because it contains variables and depending upon how you choose those variables will change whether it's true or false, okay? And so in this situation, you have this unspecified integer n. n could be even, but n could be odd. I don't know without more information about n, okay? So in this open statement that it is a statement, but you do need further information before you can decide whether it's true or false. We use those in mathematics all the time. Now, by contrast, when you look at this sentence right here, if n is an integer, if an integer n is a multiple of six, then n is even. This is not an example of an open statement. This is actually an example of a true statement. This is a true statement because if n is a multiple of six, then it's even. How do I know that? Because I can actually prove it. There is a valid proof in this situation. Scooch up a little bit here. We'll put the proof right here. And so we might write something like the following, that if, you know, since n is a multiple of six, we take that assumption there. Since, since six divides n, that's what it means for n to be a multiple of six, there is, oops, there is a number. Let's call that number. Sorry, I ran off the screen there. We'll call it a, there's a number a such that, such that n equals six times a, right? So that, that's what it means to be. n is a multiple of six there. So we might even throw in something like by definition, by definition, okay? And so then we also make the observation. We'd say something like note, note that n equals six a is equal to two times three times a, because you can factor six as an integer. And then by properties of integer multiplication, we can write this as two times three a. So we get that and be like, oh, since three a is an integer, we have that six, that's not what we're trying to do. We're trying to show that two divides n like so. And that then proves the statement. So n is even. So we were able to prove this statement. There is a proof, which then provides it. So it doesn't matter what n is, if it satisfies the hypothesis of this conditional statement, then it'll satisfy the conclusion. And so as we end this video, I wanna point out here that there's an important dichotomy when it comes to statements, there's the so-called simple statements that contain just a single idea. And then there's a compound statement, which will contain several simple statements together, right? You connect simple statements together using connectives that form then a compound statement. So when you look at this one right here, and the integer n is even, that's a simple statement. There's only one idea present, the number n being even. But then we look at this one right here, this would be a compound statement because there's two ideas in play here. n is a multiple of six, and n is a multiple of two. And these connecting words, if and then, allow us to change two open statements actually into a true statement. And we can then provide a proof on why that is. And so don't worry about how I wrote the proof right now. This is something we'll practice in the future. We just started beginning with this. We'll learn more about connectives and compound statements in future videos about logic.