 Hello and welcome to this video about when are conditional statements true? This is a follow up to an important idea from the last video about whether a conditional statement is quote unquote honest or not. Let's go back to a simple example from that video which is a conditional statement or a kind of promise that I might make to my kids. If you finish your dinner then you can play outside. The hypothesis here is you finish your dinner and the conclusion is you can play outside. Now what you should notice here is that the statement is actually a statement in the mathematical sense. It's a declarative sentence that is definitely true or false but not both. It is a contract or a promise that I give to my kids and I'm either telling the truth when I make it or not. What we want to do in this video is understand how to know if I am telling the truth when I am making a claim like this. So let's first look at it from the sinister sort of back door side of everything. So under what conditions would I be lying not telling the truth or telling a falsehood when I make this statement. Let me give this to you as an early concept check. So the statement again is if you finish your dinner then you can play outside. So the question, how would you know if I am lying or not? When my kids finish dinner and are allowed to play outside? When my kids finish dinner but are not allowed to play outside? When the kids don't finish dinner but are allowed to play outside? When the kids don't finish dinner and are not allowed to play outside? Both B and C or both B and D? This is kind of a tricky one. So pause the video for a minute and think about the answer that's most correct. Okay, so the answer here is counterintuitive to a lot of people at first. But if we think about it, it will make sense. The answer here is B only. The statement, my promise, is shown to be false or a lie. Only when my kids do finish dinner but are not allowed to play outside. In all other cases, the promise is true. Now that may surprise you, so let's unpack this result. Let's go down one by one through each of these four options here at the front of this statement and see what happens. First of all, it's pretty clear that if A happens, the kids finish dinner and are allowed to play outside, then my promise is true. The kids met the hypothesis condition and the conclusion that I guarantee would follow actually does follow. So I'm telling the truth in that situation. And I think just as clearly if B happens, then my promise is false. Because I claim that if the hypothesis is met, then the conclusion is guaranteed to follow. Well, under these conditions, my kids satisfied the hypothesis, but the conclusion did not follow, so my guarantee was just a bust. So it's clear that in situation B, my conditional statement was not true. And when something's not true, we mean that it's false. So what about C and D? Let's go with D first. Now, in this situation, the hypothesis was not met and the conclusion did not happen, and that seems logically consistent. My conditional statement makes a claim as to what happens if the hypothesis is met, and so it doesn't seem wrong to tell my kids they can't play outside if they don't finish their dinner. So in this case, my original statement or promise is definitely not false. And when something's not false, we mean that it's true. Situation C is the hardest to understand. Here, the kids have failed to satisfy the condition of finishing their dinner. But I let them play outside anyway. Does this mean that my original promise was a lie? Well, actually, it doesn't mean that. My original promise was based on something happening if the hypothesis condition was met. It didn't say what would happen if the hypothesis condition was not met. So if the hypothesis condition is not met, then no matter what happens, you can't logically come back to me and say that my original statement was a lie. If the hypothesis isn't met, what I do with my kids isn't constrained by anything. I can let them play outside if I feel generous, or I can say no. Either way, my original promise stands because it tells what happens if they do eat their dinner, but leaves me completely free to do what I want if they don't. So let's do another concept check to see how well you understand this idea. Consider this statement. If it is cold outside, I will put on my gloves. So under what condition or conditions is the statement false? When it's cold outside and I don't put on my gloves? When it's not cold outside, but I put my gloves on anyway? When it's not cold outside and I do put on my gloves? Just A and B or all the above? So as we've seen in the previous example, the only situation in which a conditional statement is false is when the hypothesis is met or is true. But the conclusion is false, meaning that it doesn't follow. The hypothesis here is it is cold outside. And the conclusion is I will put on my gloves. So the only case here where the promise is false is when it's cold outside, but I do not do as I said I would do. That is I don't put on my gloves and that's option A. That's the correct answer here. In the other two options, the hypothesis is not met. So I'm not constrained by my statement. I can do whatever I want with my gloves because I didn't say what I would do if it were not cold outside. Maybe I just like wearing gloves and I can wear them whatever the temperature is. So this is a pretty major concept. And if you understand that, you're well on your way to understanding a lot of theoretical math, actually. The main takeaway from this lesson just to wrap up is that a conditional statement is always true except in one situation. And that one situation is when the hypothesis is met, the hypothesis is true, but the conclusion is not met. The conclusion is false. In the next video, we'll talk about a handy way to encapsulate the truth value of a complex statement like this called a truth table. So stay tuned and thanks for watching.