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Skylar on June 28, 2020

@ash.duarte, happy to help!

In a missing premise drill we are tasked with using the information we are given to find the unlisted premise, whereas in an argument completion drill we are tasked with using the information we are given to find the unlisted conclusion. In other words, a missing premise drill will give you the conclusion and ask you to work backwards to find the missing premise, while an argument completion drill will not give you the conclusion but will give you all of the premises so that you can find it. To illustrate this, I will do a breakdown of each type of drill.

Let's start with a missing premise drill.
P: A -> C
P: ?
C: not C -> D
Our first step should be to find the contrapositives of the statements we were given. To do this, we need to reverse and negate.
P: A -> C
not C -> not A
P: ?
C: not C -> D
not D -> C
Now, we should notice that there is a new variable introduced in the conclusion- D. This tells us that we should incorporate it somehow into our missing premise. And what are we told that we want to be able to conclude about D? That not C -> D. Do we know anything about not C? Yes, the contrapositive of our first premise tells us not C -> not A. Therefore, we should make our missing premise not A -> D. This allows us to make the chain: not C -> not A -> D. This simplifies into the conclusion we want, not C -> D. Therefore, our final answer should look like:
P: A -> C
not C -> not A
P: not A -> D
not D -> A
C: not C -> D
not D -> C

Now let's try an argument completion drill.
P: X -> Y
P: Y -> Z
C: ?
Again, we should start by reversing and negating to find the contrapositives of our given statements.
P: X -> Y
not Y -> not X
P: Y -> Z
not Z -> not Y
C: ?
Now, we should look for variables that our premises have in common. If we can find a variable that is necessary in one statement and sufficient in the other, we can combine the two statements using the transitive property. Here, we have that with the variable Y. We see that Y is necessary in our first premise and sufficient in our second premise. Therefore, we can combine the two statements to get the chain: X -> Y -> Z. We can simplify this into: X -> Z, which is our missing conclusion. The contrapositive is: not Z -> not X. So, our final answer should look like:
P: X -> Y
not Y -> not X
P: Y -> Z
not Z -> not Y
C: X -> Z
not Z -> not X

Does that make sense? Hope it helps! Please let us know if you have any other questions!

ash.duarte on June 29, 2020

yes thank you! also does it matter if for the conclusion if I put the contrapositive first? For example in the argument completion drill ex:
P: not x -> not y
P: x-> not z
C:?

I got y-> not z as my answer then got z -> not y as the contrapositive but for the answer it says its z -> not y. So does the order matter?

Cade on July 3, 2020

Thank you so much for asking this Ashley!

Skylar on July 27, 2020

@ash.duarte, the order does not matter. The answer you put would be correct. Since contrapositives have the same meaning, their order is interchangeable for our purposes in the Missing Premise Drills.

Hope that helps!