After replacing his old gas water heater with a new, pilotless, gas water heater that is rated as highly efficient, J...

Skylar on June 28, 2020

@joniqueac, happy to help!

Let's break down 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. 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 that 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!