missing premise drills

hi, i've been going through the missing premise drills and i definitely have a better understanding so thank you! i was wondering though, on the card that says P: X -> A P: ? C: not A -> B should the missing premise be not X -> B? the card says that is the contrapositive of the missing premise but i don't understand how. i was also wondering, on the card that says P: X -> C P: not X -> A P: ? C: not A -> Z the answer is C -> Z, but I was wondering why it couldn't be A -> Z?

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Skylar April 15, 2020

@cholman, happy to help! I'm glad the drills are becoming clearer for you.

Let's take a look at the first card you asked about:
P: X -> A
P: ?
C: not A -> B

Let's find the contrapositives first:
P: X -> A
not A -> not X
P: ?
C: not A -> B
not B -> A

Now, we have two ways to find the missing premise:

(#1) Let's start at the original conclusion: not A -> B. Where else do we see the Sufficient variable of this conclusion, "not A"? We see "not A" in the contrapositive of our first premise, which tells us: not A -> not X. The conclusion that we are trying to reach ends in "B," so we should make our missing premise: not X -> B. This allows us to put together our larger chain of: not A -> not X -> B, which can be simplified into: not A -> B, our conclusion. So, our missing premise is: not X -> B and the contrapositive is: not B -> X.

(#2) We can also start at the contrapositive of our conclusion: not B -> A. Where else do we see the Sufficient condition, "not B"? We don't, so let's look at the Necessary condition, "A." We see "A" in our first premise, which says: X -> A. We know that we want our conclusion to be: not B -> A, so we should make our missing premise: not B -> X. This allows us to make a larger chain of: not B -> X -> A, which can be simplified into: not B -> A. So, our missing premise is: not B -> X and the contrapositive is: not X -> B.

Both "not B -> X" and "not X -> B" are equally valid answers. Either one could be listed as the missing premise, and the other should be listed as its contrapositive.

Let's take a look at the second card you asked about:
P: X -> C
P: not X -> A
P: ?
C: not A -> Z

Again, we should first find the contrapositives of each statement we have been given:
P: X -> C
not C -> not X
P: not X -> A
not A -> X
P: ?
C: not A -> Z
not Z -> A

Now, let's take a look at our conclusion: not A -> Z. Where do we see "not A"? Well, the contrapositive of our second premise tells us that: not A -> X. Where do we see "X"? Our first premise tells us that: X -> C. Where else do we see "C"? We don't. We know that we want our conclusion to end in "Z," so we should make our missing premise: C -> Z. This allows us to make the chain: not A -> X -> C -> Z. This can be simplified into our conclusion, not A -> Z. So, our missing premise is: C -> Z and its contrapositive is: not Z -> not C.

You ask why the missing premise can't be "A -> Z." This is not the correct answer because it does not complete the chain above. Moreover, the contrapositive of "A -> Z" is "not Z -> not A." The contrapositive of the original conclusion we were given is "not Z -> A." We cannot have "not Z" indicating both "not A" and "A."

Does that make sense? Please let us know if you have any other questions and best of luck studying!