Sufficient and Necessary Conditions

In the first blog post, we talked about just formalizing sentences. But conditions play an essential role in our logic and reasoning, even if we don’t always acknowledge them. Today, we’ll talk about the relationship that the pieces of the conditional have with one another: sufficiency and necessity.

Think about the United States (U.S) . We all know that the capital of the United States is Washington D.C (D.C) And I can tell you that any capital of a country is inside that country. So, D.C must be inside the United States.

Not the most impressive argument, but since it’s all true, it is a good example. See, if you have the conditional that

1) If A city is the capital of a country, than it must be inside that country.

And I tell you

2) Ottawa is the capital of Canada.

You have sufficient information to say that

3) Ottawa is in Canada.

This kind of inference is super familiar to us, which makes it convenient for these examples. It also makes it convenient for thinking about LSAT questions, as the LSAT is very strict about what you are allowed to infer. In general, if the LSAT does not explicitly give you information, or give you information for a formal deduction, you cannot use that information — even if it it is true in the real world. The LSAT doesn’t mind if you use entirely uncontroversial facts (like, the moon comes out at night), but they have to be genuinely uncontroversial.

So, Austin is in Texas. And Texas is in the United States. So I can also say:

4) If I am in Austin, I am in the United States.

Now imagine I tell you that I’m actually not in the United States — I’m in Ireland. Then

5) I’m not in the United States.

So, can I be in Austin? Obviously not. But why?

Well, imagine I also said I am in Austin.

But if I am in Austin, I must be in the United States. Because being in Austin is sufficient for being in the United States.

So, it just can’t be true that if I am in Austin I am also not in the United States.

Thus, we can say it is necessary for me to be in the United States for me to be in Austin.

That was easy enough. What LSAT students often miss out on when preparing for the LSAT at the beginning (or even later in their LSAT prep) is that sufficient and necessary conditions do not imply one another. Being in Austin is sufficient to be in the United States, but it’s not necessary (I could be in D.C). And being in the United States is necessary to be in Austin, but it doesn’t guarantee that I am in Austin (I could be in D.C). It is essential to never confuse neccessity and sufficiency.

Building on what I said in the first post, remember that
P —> Q
is a good symbolization for a conditional. P, the antecedent, will always be sufficient for Q. Q, the consequent, will always be necessary for P. Important: conditionals are really two conditions in one:

P—> Q

and

not Q —> not P.

We call this second equivilent conditional the “contrapositive” of the first. The LSAT loves using contrapositives in questions, so it’s good practice to simply write them everytime you see any conditional.

Not is a pretty important logical word, so we should give it a symbol as well! Typing-wise, it’s easiest to use tilde (~).

So

P—>Q implies ~Q —> ~P

and vice-versa.

Try some yourself!

1) What is the contrapositive of Q—>P?

2) Translate “Only if I want a cat will my partner want a cat.” and write it’s contrapositive

3) Suppose you notice that anytime it is raining, the street is wet. How can we write this as a conditional? What is the sufficient condition? What is the necessary condition? How can you show that the necessary condition is necessary?

—Ryan Born 2/5/2025