Necessary vs. Sufficient Conditions (LSAT)

I won't go as far as to say that any one concept is more important than all others on the LSAT, but I do think an understanding of necessary vs sufficient conditions is one of the more important concepts to know well.

Take the following statement (which is not true, by the way!):

"If you are to get accepted into Yale Law School then you must score above a 175 on the LSAT."

If we take "x" to be acceptance into Yale Law School and "y" to be scoring above a 175 on the LSAT then the above statement is equivalent to"

if x then y.

 or 

x --> y 

The "x" above in the "x-->y" is the sufficient condition and the "y" is the necessary condition.

Let's say that a candidate scores a 176 on the LSAT. We cannot conclude that the candidate will get accepted into Yale Law School. This is because scoring above a 175 on the LSAT is a necessary condition for admission. And satisfying the necessary condition does not allow us to conclude that the sufficient condition is satisfied.

Knowing that y is satisfied does not mean that x is satisfied.

On the other hand, assume that the candidate gets accepted into Yale Law School. We now can also conclude that the candidate scored above a 175 on the LSAT. Getting into Yale Law School is the sufficient condition and that condition has been satisfied. Satisfying the sufficient condition allows us to conclude that the necessary condition is satisfied.

Knowing that x is satisfied means that y is satisfied.

Again, the statement: 

"If you are to get accepted into Yale Law School then you must score above a 175 on the LSAT."

Let's negate the sufficient condition and say that the student does not get accepted into Yale Law School. Can we conclude that the candidate did not score above a 175 on the LSAT?

We can not. Merely knowing that the sufficient condition has not been satisfied does not allow us to conclude anything about the necessary condition.

Knowing that x is not satisfied does not mean that y is not satisfied.

But what if we negate the necessary condition:

Let's negate the necessary condition and say that the candidate did not score above a 175 on the LSAT. Do we know that the candidate did not get accepted into Yale Law School? We do, and that's because here I've negated the necessary condition. And negating the necessary condition will always negate the sufficient condition. When you negate the necessary condition, the sufficient condition is negated as well and you get the contrapositive of the original statement. 

~y --> ~ x is the contrapositive of x --> y

Knowing y is not satisfied means that x is not satisfied.

In short:

Satisfying the sufficient condition satisfies the necessary condition.

Satisfying the necessary condition does not satisfy the sufficient condition.

Negating the sufficient condition does not negate the necessary condition.

Negating the necessary condition does negate the sufficient condition.