P: X → Y & not ZP: A → ZP: B → AP: Y and A exist.C: ?

shunhe August 1, 2020

Hi @Angeni-Wang,

Thanks for the question! So as for your first question, let’s think about this with a simple example. Let’s say that all Xs are Ys and Ys exist; in other words

X —> Y
Y

Can we then know that Xs must exist? Well, let’s take this following example. Unicorns (if they exist) are mammals (since they’re just horses with horns). So all unicorns are mammals. And mammals exist.

Unicorn —> Mammal
Mammal

Can we then conclude that unicorns exist? No, we can’t. Sure, if they did exist, they would be mammals, but that doesn’t mean they exist. And so that’s the reasoning behind why we can’t conclude “X” from “X —> Y.”

As to your second question: So let’s say that Z exists and no Zs are Xs, in other words

Z
Z —> ~X

Now we want to know if we can necessarily deduce that X doesn’t exist. It really depends on what you mean by this. What we CAN say is that if something has the characteristic of Z, then it cannot also have the characteristic of X. For example, No unicorns are alligators. And unicorns exist now. Then yes, we can conclude of the unicorn that it is not an alligator.

However, we can’t say that alligators don’t exist at all, because obviously, they do. This starts to get into predicate logic vs. propositional logic which is not something that you need to know for the LSAT. All you need to know is that we only care about the situations in which we want to say that the unicorns aren’t alligators.

Hope this helps! Feel free to ask any other questions that you might have.

Angeni-Wang August 2, 2020

Thank you shunhe, that helps a lot! The only thing I don't understand is how we can prove that X doesn't exist. I now get how we cannot prove that X does exist, but can't seem to make the leap as to how we can prove that X doesn't exist.

Thank you!!

Angeni-Wang August 2, 2020

Sorry I actually had one other question too. Regarding the statements X --> Y and Y exists, I had thought to understand it as "all Ys exist". If we apply it to the unicorn/mammal example, would this translate to "all mammals exist"? And if all mammals included unicorns, then wouldn't unicorns have to exist if they were mammals? Otherwise I seem to understand that the previous statement must be "some mammals exist", since if all mammals exist and unicorns are mammals, by not existing they would have to lose their definition of being a mammal (or mammals would have to lose their all-encompassing definition of existing)

Can you help me to see where I am going wrong? I can't seem to figure this one out.

Thanks!!

Victoria August 5, 2020

Hi @Angeni-Wang,

Happy to help!

As a general rule, you can never conclude the sufficient condition based on the necessary condition i.e. you can never work backwards over the arrow - the logic flows in the direction of the arrow.

Using the above example, we can conclude that mammals exist if we have proof that unicorns exist, but we cannot conclude that unicorns exist solely based on the proof that mammals exist.

I think reading the example as "all mammals exist" complicates things a bit.

If we have proof that unicorns exist, then we know that mammals exist because all unicorns are mammals and unicorns exist.

If we have proof that mammals exist, then we cannot conclude that unicorns exist even if we understand it as "all mammals exist." Just because we know that mammals exist, this does not mean that unicorns are necessarily part of this group of existing mammals. What we know is that, if unicorns did exist, then they would be mammals.

We can't prove that unicorns don't exist, but we also can't prove that they do. Overall, we cannot draw any conclusions about unicorns solely from the existence of the necessary condition i.e. mammals.

Let's try an example using different language to help clear things up a bit.

"All lawyers went to law school." And we know that an individual went to law school (i.e. law school "exists" in a sense).

Lawyer --> Law School

Law School "exists"

From this, we cannot conclude that this person is a lawyer. All we know is that they went to law school. They could have taken an entirely different career path after going to law school.

As with the above example, we cannot prove that this person is not a lawyer, but we also can't prove that they are a lawyer. We cannot draw any conclusions about this person's career solely from the existence of the necessary condition i.e. that they went to law school.

On the other hand, if we know that someone is a lawyer, then we can conclude that they must have gone to law school because all lawyers went to law school.

These concepts can definitely be difficult to grasp when you are first learning them, but always remember that you can never conclude the sufficient condition from the existence of the necessary condition.

Hope this helps clear things up a bit! Keep up the good work and please let us know if you have any further questions.

Angeni-Wang August 10, 2020

Thank you for taking the time to explain! I understand it now!

ohanamgt July 28, 2021

Cannot conclude from a necessary condition, in your example Y

Mariyam May 30, 2024

That was great Victoria, thank you.