The Definitive Guide to Sufficient and Necessary Conditions: Part 2

In Part 1 of this guide, we learned about sufficient and necessary conditions, and that every sufficient condition is just the reversal of a necessary condition, and vice versa – they are essentially just two ways of stating the same logical truth.

Now, we are going to build on what we’ve learned by tying in one of the biggest buzzwords on the LSAT – contrapositives.

Don’t get scared of it if you’ve never heard of a contrapositive. I promise you that I scored my first 180 before I had any idea what a contrapositive was. However, recognizing a contrapositive can sometimes prove to be a shortcut on a tough question. Let’s talk about exactly what a contrapositive is, and everything we’ve just learned should make it very straightforward.

Recall that sufficient conditions can be stated easily in “if-then” form. The way to state that X is sufficient for Y is:

 

If X, then Y.

 

Much like sufficient conditions, there is a very simple way of stating a necessary condition in “if-then” form. The way to state that Y is necessary for X is:

 

If not Y, then not X.

 

This makes perfect sense – if I say that a good LSAT score is necessary for getting into Harvard, that’s another way of saying that if you don’t have a good LSAT score you won’t get into Harvard. If I say you need to eat vegetables in order to have a healthy immune system, that’s another way of saying that if you don’t eat vegetables you won’t have a healthy immune system.

In other words, “Y is necessary for X” = “If not Y, then not X”.

Now, once again, the simplest way of stating that X is sufficient for Y is:

 

“If X, then Y”.

 

Recall that

 

“X is sufficient for Y” is the exact same statement as “Y is necessary for X”.

 

So, it follows that since

 

“X is sufficient for Y” = “If X, then Y”

 

and

 

“Y is necessary for X” = “If not Y, then not X”

 

and

 

“X is sufficient for Y” = “Y is necessary for X”

 

therefore,

 

“If X, then Y” = “If not Y, then not X”

 

…and, these last two equal statements in bold, which we’ve just shown mean the exact same thing, are contrapositives. The contrapositive of “If X, then Y” is “If not Y, then not X”, and vice versa.

If you understand the equivalency of necessary and sufficient conditions when reversed, then this should make perfect sense.

However, like I said, it can help to navigate through tricky questions with conditional logic. Always remember that “If X, then Y” = “If not Y, then not X”. The former is wording it as a sufficient condition, and the latter is wording it as its equivalent necessary condition. Therefore, these two statements mean the exact same thing, and are just different ways of wording the same conditional truth.

So, when you hear “if it’s raining, then it’s cloudy”, you should automatically think, “if it’s not cloudy, then it’s not raining”.

When you hear “if I’m tired of conditional logic, then I’m going to strangle a puppy”, you should automatically think, “if I don’t strangle a puppy, then I must not be tired of conditional logic”. And so on…

In Part 3 of this guide, we’ll look at how to apply this knowledge of sufficient and necessary conditions to a few specific question types on the Logical Reasoning section.

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