If/Then Statements and Contrapositives

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All right, in the next couple of lessons, we are going to discuss formal logic. Formal logic is very important in the LSAT. It comes up a lot in the logical reasoning sections and it also comes up a lot in logic games. Fortunately, you only really need to know the basics of formal logic for the logic games and that's what we'll cover over the next few lessons here.

In this video, we're gonna learn about the if then statement and how to form the contra positive of an if then statement. If then statements appear in logic games in conditional rules. The standard form of an if then statement is the simple if a then b format so an example might be if runner A is chosen for the race, then runner B is also chosen for the race.

You've probably heard in the past of people using the terms necessary versus sufficient to describe this type of relationship and while that's fine, I would encourage you to think of it rather as sufficient. Leads to the necessary result, because that mimics the if A then B. So if we look at, in the best way to think of this is to just go to a real world example, so we're going to look at for example, a rainy.

So on a rainy day there's two components, rain and cloud. What we want to do is take those two components and ask ourselves two questions about them. The first one If it's raining outside, are there necessarily clouds in the sky? And if you just think about that for a second, the answer is yes. If it's raining out, there are definitely clouds somewhere in the sky.

You can't have rain on a perfectly clear day, right? Therefore, rain is sufficient to know that there are clouds in the sky. Clouds are necessary for rain so the sufficient term leads to the necessary term, okay. The second question we want to ask ourselves is if there are clouds in the sky is it raining?

And the answer to this question is maybe. We don't know for sure. There could be clouds in the sky and it's just a gloomy, overcast day with no rain. Therefore we can't create an if then statement going in that direction. We can't go from the necessary term to the sufficient term. Now let's look at how you can form the contra positive o a statement like that.

So again we'll take our basic phrase here if it's raining then there are clouds in the sky and what we're going to do with it is negate it and flip both terms. This is going to give us the contrapositive which is also a true statement, and is giving to be very helpful for making inferences in logic games. So here's what we mean by negating and flipping both terms.

We're going to take our clouds and our rain We're going to switch their positions, and we're going to add not into each one. Therefore, our correct contra-positive becomes, if there are not clouds in the sky, then it's not raining. And again, if you think about that logically for a minute, it makes perfect sense.

If there is no clouds anywhere to be seen, there is no way it's going to be raining out. With contrapositives there's a couple of things to look out for, very common mistakes people make. The first one is to flip the terms, but not to negate them. That would lead to a result like, if there are clouds in the sky It's raining, and we've already talked about the fact that that's not necessarily true, so that's not a correct contrapositive.

Another mistake is to negate the terms but not to flip them. So it's to say, if it's not raining, there are not clouds in the sky. And again, if you think about that for a minute, you can tell that that's not necessarily true. So Let's take a minute here to practice if/then statements and contrapositives. Go ahead and pause the video and fill in the contrapositives or the if/then forms for each of the sentences on the page here.

Okay great, let's walk through these one at a time. So for this top one, if x then z. Forming the contra positive first what excuse me sorry about that. First we're gonna flip the terms. And then we're just gonna put the no in front to negate them. So our contra positive of if x then z Is if no z then no x.

Moving on to the next one if it's snowing then it's cold outside. Again let's flip the terms and say cold on one side and snowing on the other side. And we're gonna say to negate them if it's not cold Then it's not snowing. Great. Next one.

Whenever I work in the office, I drink coffee. Now, this one was a little bit trickier, because we didn't have it set up in the typical if, then form right off the bat, so hopefully, when you were thinking through this one, you were asking yourself which of these terms is sufficient, and which one's necessary. And the answer is that working in the office.

Is sufficient to know that I'm drinking coffee because I say whenever. In other words, every time I work in the office, I drink coffee. So if I'm working in the office, then I'm drinking coffee. And from that point the contrapositive is pretty straightforward. Not drinking coffee. Then not working.

Great and now we move on to the last one. We go to the beach on every holiday. Again this is not in the standard if/then form so you might have had to think about this one for just a minute. And hopefully what you were able to realize is that if it's a holiday, then we go to the beach.

Because they say, we go to the beach every holiday, so if it's a holiday come out we're definitely at the beach, right. Then the contrapositive once again is pretty straightforward. No beach, then we know it not a holiday. So to recap formal logic basics. We've got our basic statement, if it's raining, then there are clouds in the sky.

Another way of writing that is, if rain then clouds in the sky, that arrow is a very important symbol that we're gonna use a lot in logic games, it's just a quicker way to write this phrase. And, in fact, an even faster way to write it is simply Rain, then clouds. Next we have the contrapositive of that statement. If there are no clouds in the sky, then it's not raining.

So again, a slightly shorter way of writing it. And the shortest way of writing it of all and this is the one that I'd really recommend getting used to because it'll help you speed up your process of diagramming and inferencing during the logic games. Great.

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