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

Dealing With Conditionals: Formalizing What you Already Know

There comes a time in every LSAT student’s life when they must learn stop worrying and love the conditional. Conditionals are just those little funny guys that we use to talk about, well, conditions. If….then, When….then… Unless …, then …. and so on.

Take a classic conditional:

1) If you have a cat, then fur will get all over.

We can make our lives a little easier by symbolizing the conditional like this:

1) If X, then Y.

Here, we just replace “you have a cat” with X, and “fur will get all over” with Y. We can do this for similar reasons as we can replace numbers with variables in math: what matters for logic is just the structure of what’s being said. This can make our lives much easier on the test, as we won’t have to continually rewrite what was relevant.

We can even make this shorter and more concise by using arrow “—>.” to represent the logic that if…then represents. So we can just say

1) X —> Y

This is useful because it is easier to manipulate and also it gives a regular form to the conditional.

Now, it’s not uncommon to see conditionals written like

2) I only go out if I want a sandwich.

But we can’t just rewrite this as:

3) If I want a sandwich, I can go out.

This is because only if is actually a different logic than just if…then. Whenever we see “only if,” we treat that part of the sentence as the second part of the arrow. So really, (2) is written like

2) If I go out, then I want a sandwich.

The big thing to remember then is that “if…” by itself represents what goes in the first part of a condition, and “only if” represents the second part.

Because order matters when we put things into a conditional, we’ve given special names to the first and second piece. The sentence before the arrow is the antecedent. The sentence after the arrow is the consequent.

So

4) If I want to pet a cat, I have to be quiet.

Can be symbolized like

P —> Q

where P stands for “I want to pet a cat” and Q stands for “I have to be quiet.”

I want to pet the cat is the antecedent, and I have to be quiet is the consequent.

So, P is the antecedent and Q is the consequent.

With that, we have the language and symbolization to work with LSAT questions that require conditionals, like inference questions or parallel structure questions.

Try a few more symbolizations!

5) Only when I have coffee can I face the world.

6) If a tree gets big enough, it will only fall if it is cut down.

7) I’m usually angry, if I’m hungry.

8) The house is happy only if mom is happy.

—Ryan Born, 2/5/2025