# The author uses the word "immediacy" (line 39) most likely in order to express

on April 15, 2020

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?

Skylar on April 15, 2020

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

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.

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!