If you’ve already started your LSAT prep journey, you’ve likely stumbled across necessary and sufficient conditions.  Just what the heck do these terms actually mean?

Necessary and sufficient conditions are the cornerstone of formal logic and conditional reasoning.  Don’t worry, it’s not as complicated as it sounds: you encounter necessary and sufficient conditions all the time in real life!  Consider the following:

“If Jack drinks a six pack, then he will get drunk.”

Let’s break that down.  We know that if Jack drinks a six pack, then he will get drunk.  Let’s represent the first part (“drinks a six pack”) with “A,” and the second part (“get drunk”) with “B.”  Now we can write this out as:

If A is true, then B must be true.

or

A → B

So if A is true (Jack chugged down that six pack), then B must also be true (Jack’s drunk). In other words, when A is true, it’s necessary that B must also be true.  So in this example B (“getting drunk”) is the necessary condition.  When A is true, so is B.

So A (Jack drinks a six pack) is the sufficient condition.  That means that drinking a six pack is enough to guarantee that Jack will become drunk.  Think of the sufficient condition like a trigger, switch, or activator.  If we pull the trigger or flip the switch (make the sufficient condition true), then we "activate" the necessary condition.

But is drinking a six pack the only way for Jack to get drunk?  What if he downs that bottle of tequila?  Or knocks back a few White Russians?  Drinking a six pack is just one of the many ways Jack could end up inebriated.  

If we only know that Jack is drunk (that the necessary condition is true), then we don’t know anything about whether the sufficient condition is true or not.  In other words, while drinking that six pack is enough to get Jack drunk, it’s not the only way; it’s just one of many.  It’s merely a sufficient condition.

In sum, when the sufficient condition occurs, you know that the necessary condition also occurs (the sufficient condition ‘activates’ the necessary condition). But just because the necessary condition occurs doesn’t mean that the sufficient condition occurs too.

Let’s recap:

Necessary Condition:  Our necessary condition is the element that must occur, when the sufficient condition is true.  We know that if Jack drinks that six pack, he will get drunk.  If A is true then B must be true.  A → B.  If we know that Jack drinks a six pack, then we know he’s going to get drunk.  So getting drunk (B) is the necessary condition in this example.  

Sufficient Condition: Our sufficient condition, on the other hand, is merely enough to make the other element true.  Just because we know Jack got drunk, that doesn’t mean we know he drank a six pack.  He could have gobbled up a tray of jello shots, or downed a handle of whiskey.  Drinking a six pack is just one of the various things that could get Jack drunk.  Sure, drinking that six pack will get Jack drunk, but so could lots of other things too.

Let’s try another example, and see if you can identify the necessary and sufficient conditions:
 

“To get into Harvard, then you must score above a 170 on the LSAT.”

Let’s reason through this one.  If A is true (I got into Harvard), then what do we know?  Well, we know that I must have gotten above a 170 on the LSAT.  So B must be true too.  But what if I only know that B is true (I got above a 170 on the LSAT)?  Do we know anything else?  No, we don’t!  Just because I got a 170 on the LSAT doesn’t mean I got into Harvard.  What if my GPA was terrible, or I never sent in a personal statement?  There are lots of things I also need to do to get into Harvard, so just because I scored above a 170 doesn’t mean I’ll get in.

So, our necessary condition is: “you must score above a 170 on the LSAT”

And our sufficient condition is: “Get into Harvard”

If we want to write this out in formal logic, we can show that:

if A is true then B must be true

if A then B

A → B

But the following are wrong since the sufficient condition doesn’t have to occur just because the necessary condition occurs:

WRONG: if B is true then A is true

WRONG: if B then A

WRONG: B → A

Now, take a look at this statement:

“If you don’t score above a 170 on the LSAT, then you won’t get into Harvard.”

Looks pretty similar to our statement above, right?  In fact, the meaning is exactly the same! From this statement, we know that if you fail to get above a 170 on the LSAT, then you won’t be able to get into Harvard.  Which is just another way of stating, “If you get into Harvard, you must have scored a 170 on the LSAT.”  Structurally, all we’ve done is negate the sufficient and necessary conditions and then flip them.  This is called the contrapositive.  In formal logic, that means:

if A then B   

(A → B)

Has exactly the same meaning as:

if not B then not A   

(~B → ~A)

(in formal logic a ‘~’ symbol means ‘negate’ or ‘not’)

So, the statement:

“To get into Harvard, then you must score above a 170 on the LSAT.”

Is logically equivalent to:

“If you don’t score above a 170 on the LSAT, then you won’t get into Harvard.”

Whenever you have A → B, you can automatically infer the contrapositive: ~B → ~A.

Let’s break down one more statement:

“If Sarah went to the party, then we ordered pizza.”

Necessary condition: “we ordered pizza”

Sufficient condition: “Sarah went to the party”

Contrapositive: “If we didn’t order pizza, then Sarah didn’t come to the party”

In formal logic notation:

A (Sarah went to the party) → B (we ordered pizza)

~B (we didn’t order pizza) → ~A (Sarah didn’t come to the party)

Have you noticed a pattern with the notation? The sufficient condition (A) is always on the left-hand side of the arrow, while the necessary condition (B) is always on the right-hand side of the arrow! You don’t have to use A and B when notating conditional statements. In fact, we suggest that you use letters that relate to the content of the conditional statement. So for the above statement, you might want to write that as:

S → P

Where S = Sarah and P = Pizza.

Correctly identifying the necessary and sufficient conditions, along with the contrapositive, is the core of formal logic!

Next time, we’ll tackle a few more complicated formal logic constructions.  Still trying to wrap your head around it?  Get in touch with us at contact@blackacreprep.com or schedule a free consultation online!