The Definitive Guide to Sufficient and Necessary Conditions: Part 1

Sufficient and necessary conditions are the crux of the LSAT – if you understand them well, it means that you have a good handle on logical reasoning and can excel on the LSAT.

After going through this guide, you should have a full understanding of sufficient and necessary conditions. It may be dense and require multiple readings to really grasp everything, but everything you need is here. In Part 2, we’ll look at contrapositives, and in Part 3, we’ll see how sufficient and necessary conditions pertain to specific types of Logical Reasoning questions.

(For the clearest video tutorial I know of on conditional logic and how it pertains to logic games specifically, check out my LSAT Course.)

Sufficient Conditions

A sufficient condition is something that guarantees a particular fact.

For example, if it’s raining, that guarantees that there must be clouds in the sky. Therefore, rain is a sufficient condition for clouds.

“Sufficient” means “enough”, as in the fact that it’s raining is enough to know that there must be clouds in the sky.

If you sleep for less than 4 hours, that guarantees that you’ll be tired the next day. Therefore, less than 4 hours of sleep is a sufficient condition for being tired.

If you get into Harvard Law School, that guarantees that you must have gotten a high score on the LSAT. Therefore, getting into Harvard Law School is a sufficient condition for having gotten a high score on the LSAT.

Now, if you’re astute, you may have noticed that these three sufficient conditions each have different chronological perspectives. Rain proves that there are clouds at the same time as the rain. Sleeping less than 4 hours guarantees fatigue the next day. Getting into Harvard Law proves that you got a high LSAT score in the past. The chronology doesn’t matter – anytime X guarantees Y, no matter the chronological sequence, X is a sufficient condition for Y.

The simplest way of stating that X is a sufficient condition for Y is:

“If X, then Y”.

In other words:

If rain, then clouds.

If less than 4 hours of sleep, then tired.

If got into Harvard, then high LSAT score.

Sufficiency only flows in one direction. In other words, just because rain is sufficient for cloudiness, doesn’t mean cloudiness is sufficient for rain – there can be times that the sky is cloudy but not raining.

Similarly, just because less than 4 hours of sleep guarantees fatigue the next day, it doesn’t mean that if you’re tired, you must have gotten less than 4 hours of sleep the night before – you could be tired for other reasons. And, just because getting into Harvard guarantees that you must have gotten a high LSAT score, it doesn’t mean that getting a high LSAT score guarantees that you will get into Harvard – maybe your application was lacking in other ways, etc.

In other words, “if X then Y” doesn’t necessarily mean that “if Y then X”.

Sufficient conditions can chain together whenever the result of one sufficient condition is itself a sufficient condition for something else. This means that:

If X is sufficient for Y,

and Y is sufficient for Z,

then X is sufficient for Z.

For example, let’s say that Pat has a bad day whenever it’s cloudy. That’s another way of saying that cloudiness guarantees Pat having a bad day, or that cloudiness is a sufficient condition for Pat having a bad day.

If we tie that into our previous sufficient condition involving cloudiness, we can make a new inference by chaining them together:

If rain, then clouds.

If clouds, then Pat has a bad day.

Therefore, if rain, then Pat has a bad day.

However, it’s important that the sufficiency is flowing in the right direction in order to be able to chain together. Here’s an example of two sufficient conditions that could not chain together:

If rain, then clouds.

If snow, then clouds.

Here I could certainly not infer that if rain then snow, or if snow then rain. Just because rain and snow are each sufficient for clouds, doesn’t mean that either one is sufficient for the other one.

Now that you understand sufficient conditions, let’s take a look at necessary conditions.

Necessary Conditions

A necessary condition is something that is required for a particular fact.

For example, I need a cup of coffee in the morning in order to have a good day. Therefore, having a cup of coffee in the morning is a necessary condition for me having a good day.

You need to eat vegetables in order to have a healthy immune system. Therefore, eating vegetables is a necessary condition for having a healthy immune system.

You need to study a lot in order to get a good score on the LSAT. Therefore, studying a lot is a necessary condition for getting a good score on the LSAT.

Now, here’s where sufficient and necessary conditions meet: every necessary condition is the reverse of a sufficient condition.

In other words, if X is sufficient for Y, that means that Y is necessary for X. The more clearly you understand that, the closer you are to a 180. Let’s look at some of our sufficiency examples from above in order to illustrate the point:

If getting admitted into Harvard is sufficient for having gotten a high score on the LSAT, that’s another way of saying that getting a high score on the LSAT is necessary for getting into Harvard – in other words, you need to get a high score in order to get in!

If rain is sufficient for cloudiness, that’s another way of saying that cloudiness is necessary for rain – in other words, there need to be clouds in the sky in order for it to be raining!

If sleeping less than 4 hours is sufficient for being tired the next day, that’s another way of saying that being tired is necessary for having slept less than 4 hours the previous night – in other words, you need to be tired if you slept less than 4 hours!

As you can see, just like sufficient conditions, necessary conditions can function in different chronological perspectives.

The point is that every sufficient condition is also a necessary condition and vice versa. They are essentially just two different ways of wording the same logical truth.

We can illustrate this with the three necessary condition examples above as well:

If a cup of coffee in the morning is necessary for me to have a good day, that’s another way of saying that having a good day is sufficient for having had a cup of coffee in the morning – in other words, me having a good day guarantees that I must have had a cup of coffee in the morning.

If eating vegetables is necessary for having a healthy immune system, that’s another way of saying that having a healthy immune system is sufficient for eating vegetables – in other words, if you have a healthy immune system that guarantees that you must be a vegetable eater.

If studying a lot is necessary for getting a good score on the LSAT, that’s another way of saying that getting a good score on the LSAT is sufficient for having studied a lot – in other words, if you got a good score that guarantees that you must have studied a lot.

Evidently, sometimes a condition is more clearly and intuitively stated as a sufficient condition and vice versa. But the point is that “X is sufficient for Y” always means the same thing as “Y is necessary for X”, and as mentioned above, the more clearly and automatically you can see this equivalency the better you’ll be at the LSAT (on all three sections).

So, now that you understand that every necessary condition is just a sufficient condition reversed, it should make sense that these two types of conditions have very similar properties.

For example, just like sufficiency, necessity only flows in one direction. Hence, just because Y is necessary for X, that doesn’t mean that X is necessary for Y. Just because a cup of coffee in the morning is necessary for me to have a good day, doesn’t mean that me having a good day is necessarily guaranteed by having a cup of coffee in the morning.

Also, necessary conditions can chain together in the same way as sufficient conditions. Thus:

If Z is necessary for Y,

And Y is necessary for X,

Then Z is necessary for X.

(…which would also mean that X is sufficient for Z.)

We can show this by fusing together two conditions stated above. One of them was originally a sufficient condition – getting into Harvard is sufficient to know that you must have gotten a good LSAT score. We also know that studying a lot for the LSAT is necessary for getting a good LSAT score.

So, if we reframe the first of these two conditions as necessary instead of sufficient, we’d see that getting a good LSAT score is necessary for getting into Harvard. And now we have a chain of necessary conditions from which we can make a new inference:

Studying a lot for the LSAT is necessary for getting a good LSAT score.

Getting a good LSAT score is necessary for getting into Harvard.

Therefore, studying a lot for the LSAT is necessary for getting into Harvard.

We could have done this the other way and turned the necessary condition into a sufficient one, in which case the chain inference would have been that getting into Harvard is sufficient to know that you must have studied a lot for the LSAT.

In Part 2, we’ll build on what we now know about sufficient and necessary conditions by discussing one of the biggest LSAT buzzwords – contrapositives.

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