 In this video, I want to introduce to you the notion of a conditional statement. This is a very important type of compound statement in logic. One of the perhaps the most important ones I can say. So let's first define it and we'll give you some explanation of what they're all about. So a conditional statement, sometimes it's called conditional for short, expresses the notion of if and then. So as we previously learned in logic, a conjunction gives us the notion of and, a disjunction gives us the notion of or, and a negation gives us the notion of not. A very important thing when we consider in logic is this idea of if and then. If this happens, then something else happens. This is exactly what a conditional statement is. And so you would put in other primitive statements right here, if P, then Q. That's the structure of a conditional statement. We'll often denote it using an arrow. So we would write something like P arrow Q and we would just read that as if P, then Q. When considering a conditional statement, it does connect together to other statements P and Q. But unlike the conjunctions and disjunctions we saw before, the order does matter. So if if P, then Q is not the same thing as Q, if Q, then P. Another way we read this is we might read it as like P implies Q. That way if you don't want to say if and then, those are synonyms in that situation. But P implies Q is not the same thing as Q implies P. The order of the elements makes a big difference. Conjunctions and disjunctions, that is, ands and ors are commutative operations. The order didn't matter, but first conditionals, it makes a big, big deal. And as such, we might need to distinguish between the first statement and the second statement. This first statement in a conditional is often referred to as the hypothesis. Some people like to call it a premise. And then for the second one, this is typically referred to as the conclusion. Some people might call a result. That seems to be less common though. So we have our hypothesis that comes first and then our conclusion. And the idea is if we assume this is true, hence our hypothesis, then we conclude that this one must be true. And so let's make sure we understand symbolically how we can work with conditionals. Suppose we have two statements here, two primitive statements. P will be the statement, I go to the store. And Q will be the statement, I will buy an apple. So then if you look at the conditional, P implies Q, you could read that in regular English as, if I go to the store, then I will buy the apple. You can see the primitive P right here, I go to the store. You can see the primitive Q right here, I will buy an apple. And it's connected by these connecting words, if and then. And again, the order does matter. P is the hypothesis of this conditional and Q is the conclusion. If you go to the store, then you will buy an apple. That's what this conditional gives us. But of course we have to be able to go the other way around as well. Sometimes we have the statement in plain English and we have to represent it mathematically or logically into symbols here. So let's read this one. If I do not go to the store, then I will not buy an apple. So we can clearly see the if and then structure, although it looks like someone misspelled it. Sorry about that. We see the if then structure going on there. But what's the rest of it? I will, I do not go to the store that sounds like P, but there's this not inside of there. Same thing with the other one. I will not buy an apple. There's a not going on there. Going to the store is P going to buy an apple is Q. So we then end up with the conditional not P implies not Q. Remember, we can use these little twiddles for not. We might also write this as not P implies not Q. Those are both ways we can write negations here. But I should mention that this is not the same thing as saying not P implies Q. Those are different statements. Not P implies not Q is not the same thing as not P implies Q, which of course is also not the same thing as not P implies Q. You have to be careful as you read those because in English, we might describe them using the same words, but they are different statements. They're not logically equivalent. Something will explore more in the future here. This is why the notation is very important as we work with these logical statements because while in spoken language, we can cause ambiguities, the notation removes any confusion so that we know that these three things all give us different statements. But with that, we have to be able to make sure we can express conditional statements in this if then form. To write a conditional in the if then format means we have correctly identified who is the hypothesis and who is the conclusion. Because if we don't know who the hypothesis and conclusion are if you're conditional, we don't really know the conditional. So let's actually take a moment and look at a few examples of conditional statements which are not written in the if then form, but we can translate into it. So take our first sentence A right here. If your driver's license, your driver's license will be suspended if you are convicted of driving under the influence of alcohol right there. So there is a conditional happening here. There's two primitives. There's one primitive statement, your driver's license will be suspended. And the second one is you're convicted of a DUI. And then there is this magic word if that sits in between them. We don't have the if then but what happens here is actually our conditional is written backwards in this language. The hypothesis, the premise is actually over here. Hypothesis. And then the conclusion actually showed up first. And try to work that in your mind for a second right when will you get suspended your driver's license after you have the DUI. This is the hypothesis that implies the conclusion over here. So this thing is actually a way of writing a hypothesis conditional backwards. So the original form actually looks like Q comes from P so that arrows going from right to left instead of left to right. So we can rewrite that as if you are convicted of driving with a DUI then you your license will be suspended. So now we've rewritten this in the form P implies Q. Okay, that one wasn't so bad. So typically if you have a statement and then an if and then another statement it's typically just the conditional backwards if then puts it in the right order. But you have to be careful about that because there is a way that that can get a little bit frustrating the following is such an example. You will graduate only if you have a 2.5 grade point average. So how does the conditional work here. If you ignore the word only here like if it wasn't in the sentence, this actually would feel like the previous example you will graduate if you have a 2.5 GPA that would say something like oh if your GPA is 2.5, then you will graduate. This only if actually changes the direction of the if it changes the meaning of the if so that this actually is in the correct order now. This only if is suggesting that the direction goes this way this if before actually has it going backwards. So this statement is already in the correct order. We can just rewrite it with if and then. So you will only graduate if you will you will graduate only if you have a 2.5 GPA could be written as you will graduate if you graduate then you have a 2.5 GPA. So notice how what's saying here because after all there might be other conditions to graduation other than just having a GPA if you said it the other way around. If your GPA is 2.5 then you will graduate then that's the only condition necessary for graduation but there might be other things to do like you have to take certain classes maybe have a capstone project what have you. And so if you will graduate then you can imply there's this 2.5 GPA the only if goes in one direction the if goes in another. And so we'll talk about this more later on in our lecture series but this has come up already before we've used this language if and only if. For which we commonly abbreviate this as IFF it's a phrase we use all the time which is why it deserves a abbreviation. It comes from this phenomenon that the word if by itself gives you one direction and the phrase only if gives you the other directions so if you suggested the phrase if and only if you're saying you want arrows to go in both directions. And so if and only if is a way of saying that two statements are logically equivalent to each other again we'll we'll talk about this more in the next lecture. But let's say some more about conditionals right here to hold your reservation it is sufficient to give us your credit card number this word sufficient is related to this if and only if business we were talking about before. What is the direction of the implication here to order to hold your reservation it is sufficient to give you are for you to give us the credit card. What this means is if you give us the credit card then we will hold your reservation right that giving the credit card causes something else to happen the reservation there. Now there could be other ways to hold a reservation so it might not be necessary to give us your credit card but if you do give us a credit card number that will get you a reservation. And so that is a sufficient condition. Now this word sufficient is important and I've already given away its counterpart the word necessary. And so a last example of this to qualify for a discount on your airline tickets it is necessary to pay for them two weeks in advance necessary implies an implication but it gives us the other direction. You could rewrite this as an if then statement as if you qualify for a discount on your airline tickets then you will pay for them two weeks in advance. Okay, so while the sufficiency before switched the direction. This one actually kept them in the same direction they were so this is like the if and only if we saw before. There's this notion of necessity and there's this notion of sufficiency that we that there are there are opposites of each other and so if you have both sufficiency and necessity, then you actually have the implication going in both directions. These by conditional statements will explore in the next video. So try to be cautious of these things. These words sufficient and necessary mean opposite things from each other. Now to end this video on conditionals I do want to talk about their truth of values. Okay, so in Boolean logic, what does it mean if you like your hypothesis is true and then your conclusion is false or something like that. So when you read a conditional let's suppose we know that the primitive is the hypothesis is true and let's suppose we know the the conclusion is true in this situation if your premise is true and your conclusion is true. Then we say that the conditional is true that in fact the the premise and pull the truthfulness of the premise imply the truthfulness of the conclusion. Conversely, if your premise is true, but your conclusion is false, that actually makes the conditional a false statement, because you're like, oh, let's use some examples I often talk about promises I make to my children, right. If I tell my children, oh, if you do good on your math test, I will bring home ice cream for dinner tonight, right. So let's say the kids do well on their test. I bring home ice cream I told the truth Mike if then statement was true if you do this then I will do this. Now I was telling the truth. On the other hand, if the kids do well on their math test, but I don't bring home ice cream. That makes me a liar because they they did what they were supposed to but I didn't do what I said I would do if they had done it the if part happened but the end didn't. So if the hypothesis is true but the conclusion false, that makes the conditional false. But if the hypothesis is true and the conclusion is true, that makes the conditional true. All right, people usually understand those two very well. The second conditions we have to be a little bit more careful about what happens, what happens if the hypothesis is false. Let's begin with this last one here. If the hypothesis is false, and the conclusion is likewise false, that actually makes it a true statement. Okay, so again using the analogy of the children the math test, let's say my my kids don't do well on their math test bump bump bump. And then I don't bring home ice cream. Well, I told the truth. I said, if they do well on their math tests, I'll bring home ice cream. They didn't do well. I didn't bring home ice cream. I kept my promise. I was telling the truth. But looking at this one right here, this is this is the one where people sometimes get a little bit vexed. If the hypothesis is false, but the conclusion is nonetheless true, that actually makes the conditional true as well. So let's use that analogy here. If my kids do bad on their math test. So that's what happens, but then I still bring home ice cream. That means I told the truth. Why what I thought you said you would bring home ice cream if they did well on the test. That's the thing is that the conclusion is only under any obligation when the hypothesis is satisfied. My promise to the kids was if they do well on the on the math test, then I will get them ice cream. But if they do bad on the math test, I'm under no obligation to do anything. Because the promise is only contingent upon what they do. If they do well on the test, then I will do something. But if they don't do well on the test, I'm not obligated to do anything. If they don't do well on the test, then I'd be like stinks to be you. You don't get ice cream. Life is full of disappointments, right? But I might also be like, well, you didn't do well on the test, but I still saw that you did the best you could. I'm going to reward you with ice cream because of your efforts, even if you didn't get the grade you wanted. The promise, the conditional is only contingent upon when the statement when the hypothesis is true. When the hypothesis is false, there is no obligation. There is no promise. These two examples are examples of what we mean by something being vacuously true. Vacuously true here, meaning that the hypothesis fails. So therefore the hypothesis is in a vacuum, but the conditional is likewise true. If the hypothesis fails, then the conditional is automatically true. Of course, if the hypothesis is true and the conclusion is true, then the conditional is true. And then the last situation right here, this is the one you have to be worried about. This is the only time a conditional is false. If the hypothesis is true and the conclusion is false, then the conditional is false. That's when you're lying here. And so this notion of being vacuously true needs to be something that is remembered and understood by mathematical students. It can be difficult to understand at first, but as we go more and more into these things, we'll see better examples of what it means to be vacuously true. But for our purpose understanding here, that for a conditional statement, if the hypothesis fails, if the hypothesis is a vacuum, then the conditional statement is true. The issue is that some people conflate the truthfulness of the conclusion with the truthfulness of the conditional. The conditional is a new statement. Remember, we have a primitive statement P, which could be true or false. We have a primitive statement Q, which could be true or false. And then we make a compounded statement P implies Q, which could be true or false. Now, the truthfulness, the truth value of P and Q influence the truth value of the conditional, but they are not the same thing. The truth value of a conditional is not necessarily the truth value of the hypothesis. And definitely the truth value of a conditional is not necessarily the truth value of the conclusion. They're related, but they are very different. Like we said here, if the hypothesis fails, then this conditional is automatically true. The only time a conditional is true is when hypothesis is true, but conclusion is false.