- Oct 07, 2016
- Advice on Logical Reasoning, LSAT
- Reviewed by: Matt Riley
Some? Most? All?
Quantifiers. Some LSAT students think they’re the enemy. Blueprint classes cover quantifiers (some, most, all, and the valid inferences that can be drawn from those claims) in lesson 3 and it’s a lot of new material at once. It can be scary. But it’s worth getting it down. You’re likely to see quantifiers on a small handful of questions on the LSAT. Having quantifiers down can keep those questions from tying your brain in knots. If you have to figure them out on the spot, it’s not easy. If you know what you need to know, it makes things much more straightforward.
Let’s start with a “some” statement. Some reality TV stars are narcissists. That just means there’s at least one person who is both those things. We’ll diagram it like this:
Reality TV -s- narcissist
It’s a dash, not an arrow, because it’s reversible. It’s equally true that some narcissists are reality TV stars. “Some” statements are always reversible. Also, “some” places a lower limit but not an upper limit. It could be true that most reality TV stars are narcissists or even that all reality TV stars are narcissists. All “some” means is that there’s at least one.
Now suppose I told you that all reality TV stars give on-screen interviews. That would diagram as:
Reality TV —> on screen interviews
We know that at least one reality TV star is a narcissist. OK, there’s more than that in the real world but on the LSAT “some” just means “at least one.” So let’s think of an example. We’ll call her Kim. Any resemblance to a real person is completely unintended. Since Kim is a reality TV star, Kim must also give on-screen interviews. That means there’s at least one person, namely Kim, who is a narcissist and gives onscreen interviews. Shocking, I know. The valid conclusion is:
Narcissist -s- on screen interviews.
Now suppose it were true that all politicians are narcissists:
Politician —> narcissist
That doesn’t prove anything additional. We knew only that there was at least one reality TV star narcissist. We named her Kim. Is Kim a politician? We don’t know. I know, I know, especially this year you can think of other examples. But we can’t prove anything from those statements.
Here’s the difference: In the first example, the “some” statement and the “all” statement had “reality TV” as the term in common. That term in common was on the sufficient side of the “all” statement. In the other example, they had “narcissist” in common, and that was on the necessary side of the “all” statement. The former leads to a valid conclusion but the latter doesn’t. Which side of the “some” statement is shared doesn’t matter because, once again, “some” statements are reversible.
Now let’s talk about “most.” Suppose I told you that most cats have fur. We’ll diagram that like so:
Cat -m-> fur
That means that more than half of cats have fur. It could be just barely more than half. Or it could be all of them. It happens to be true that there is such a thing as a hairless cat and it looks like an alien. But you couldn’t prove it from the claim diagrammed above.
Now let’s add a conditional statement: anything that has fur sheds.
Fur —> shed
Most cats have fur, and as soon as you know something has fur, you know that it sheds. This leads to the valid conclusion that most cats shed. Shocking, I know.
Cat -m-> shed
Now let’s stick with the premise that most cats have fur, and add another premise: all cats meow.
Cat —> meow
This is just like the first inference we made with “some” and “all.” If it works with “some,” which is weaker evidence, we can do the same thing with the stronger “most.” Name a cat with fur. That cat must meow. That leads us to the conclusion that some things with fur meow:
Fur -s- meow
In both cases, the shared term is on the sufficient side of the conditional statement. When the shared term is on the left hand side of the most statement, the conclusion is “some,” whereas when it’s on the right hand side you can draw “most” as a valid conclusion.
Let’s now say you have a friend named Zeke, and all of Zeke’s pets are cats:
Zeke’s pet —> cat
What does that tell us? Zeke could be a lover of hairless cats. Since that’s a possibility, there’s no valid conclusion to draw from this “all” statement and the claim that most cats have fur. Again, the shared term has to be on the sufficient side of the conditional statement to draw a valid conclusion.
There’s only one more valid way to combine quantifiers. Suppose that most dogs like to play with balls and most dogs will protect your home. Those claims will diagram as:
Dog -m-> play ball
Dog -m-> protect home
That’s two majorities out of the same total. Even if you tried to keep the group of dogs who play ball separate from the dogs who protect your home, each one is a majority of dogs. There has to be at least a little overlap between the groups, leading to the valid conclusion that some things that play ball will protect your home:
Play ball -m-> protect home
Let’s stick with the first premise from this last example:
Dog -m-> play ball
Now we’ll add the premise that most things that play ball are human:
Play ball -m-> human
Can we draw a valid conclusion? Nope, doesn’t work. The only way to draw a valid conclusion from two “most” statements is when the same thing is on the left hand side of both statements.
That’s it. Those are the valid inferences you can draw from quantifiers. Your task is to learn them two ways: understand why they work, but also just flat out memorize them. Understanding them will help you memorize them and help you with the harder questions involving quantifiers. But the real benefit is when you just know it, and you can take a hard question and reduce it to a pattern you’ve seen so many times before.
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