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

jakennedy January 15, 2022

Hi,

This is an argument completion drill. These are examples of what you may find on a must be true question if it happens to contain conditional logic.

On such a question, you first want to work with your conditional statements to combine them and make inferences.

Beginning with the first premise, we have:

X ? Y
X ? not Z

I separated the "and" statement so that we can add the remaining premises separately.

Next, we want to add premise two, A ? Z. The problem is, our current diagram has not Z rather than Z. To solve this problem, we have to take the contrapositive:

not Z ? not A

Combining that with our original diagram, we get:

X ? Y
X ? not Z ? not A

We then repeat the same process for premise 3, B ? A. Since we need not A, and the premise has A, we need the contrapositive:

not A ? not B

Then combine with our previous diagram:

X ? Y
X ? not Z ? not A ? not B

With that, we have completed the combining step. Let’s add the contrapositives as well:

X ? Y
not Y ? not X

B ? A ? Z ? not X
X ? not Z ? not A ? not B

On a must be true question, there are two things that they can do from here.

They could just leave it at this and go into the answer choices. If this occurs, you will look for a conditional statement that matches one of the diagrams that we made. For example, the following would all be acceptable answers because they are proven by our diagrams:
“If X then not B”
"if A then not X”
“if not Y then not X''

or

They can trigger something on the diagram for you. Consider the following conditional statement:

To take the LSAT, one must register weeks in advance: (Take LSAT ? Register)

This in itself does not tell us that anyone IS taking the LSAT. It just tells us what is required if someone happens to take it. Now imagine that the stimulus added another sentence:

To take the LSAT, one must register weeks in advance. Roma is taking the LSAT today.

This would trigger our diagram: Take LSAT ? Register. From this, we would know that Roma must have registered weeks in advance.


When these drills say that something exists, we are in this second option. They are triggering those elements on your diagram. It is your job to go to your diagram and see what follows from those letters.

In this case, they said that Y and A exist, so we have to see what those elements trigger on our diagram:

Our diagrams:

X ? Y
not Y ? not X

B ? A ? Z ? not X
X ? not Z ? not A ? not B


First, we go to the letter Y and see that no arrows come from Y, so we learned nothing new.
Then we go to A and see the following:

B ? A ? Z ? not X

We follow the arrows from A and get to Z and not X. Therefore, we know that Z and not X must be true.

That is how we get to answer choice D.

Please let me know if I can clarify in any way.

Hope this helps!

Lamont February 13, 2022

Got it. Thaks

Lamont February 13, 2022

Sorry, with the contra positive explanation I am very confused.

Ross-Rinehart February 14, 2022

Whenever you have two conditional statements with the same term, on the same side of the arrow, but one is negated and the other is not negated, you’ll need to take the contrapositive of one of the two conditional statements to make a deduction.

Let’s say we had:

1. Register for LSAT ? Sign Up for LSAT Weeks in Advance
2. NOT Registered for the LSAT ? CAN’T Apply to Law School

Notice how “Register for the LSAT” is in both 1 and 2, and it’s on the same side of the arrow in both, but it’s negated in 2 and not in 1? That means we have to take the contrapositive of one of them to make a transitive deduction. If we take the contrapositive of 2, we’d get:

CP of 2: Apply to Law School ? Register for LSAT
1. Register for LSAT ? Sign Up for LSAT Weeks in Advance

From that, we can make the conclusion:

Apply to Law School ? Sign Up for LSAT Weeks in Advance

Hope that helps!

Lamont March 4, 2022

Thanks Ross. Your explanation helped me to understand how to link the statement to answer D.

Alex-Hoston May 10, 2022

Thanks, Ross the contrapositive explanation helped tremondously.

Mariyam May 30, 2024

I've never seen anyone in the last month solve one of these problems using question marks. What section should I study to understand what you just explained?

Mariyam May 30, 2024

Also, I could have sworn that I saw in one of our lessons that IF X THEN Y (X --> Y) "meant" All Xs are Ys. Am I mistaken about this because since Y exists & All Xs are Ys, that wouldn't mean that all Xs exist as well?

Emil-Kunkin June 2, 2024

I think that the question marks are some sort of rendering error. I've never seen it happen that way but I would treat those as arrows like we normally see for diagramming conditional logic.

And yes I think you're right that if x then y means that if all y has a quality then all X also have that quality.

Mariyam June 2, 2024

Thank you so much, I though that after all this time I had completed missed a critical element, I feel much better equipped to deal with this now :-)