Hi @nsmith45, thanks for your post. To be sure we are clear, we understand your question to be why is the answer not "X ==> not D"? In other words, why is (E) incorrect. (Please let us know if you meant instead to ask about answer choice (D).)
Let's look at the setup first.
P: X ==> B [in plain English: X is sufficient to guarantee B; If X, then B] contrapositive: not B ==> not X
P: not A ==> not B [in plain English: if you don't have A, you don't have B] contrapositive: B ==> A
P: [missing premise]
C: not D ==> A [if not D, then A] contrapositive: not A ==> D
We need to select the answer choice that provides the missing premise.
Answer choice (C) is correct: not X ==> D. The contrapositive is "not D ==> X." Take this contrapositive, add the positive of the first premise, and the contrapositive of the second premise: not D ==> X ==> B ==> A
Via transitive, this yields "not D ==> A," which is the given conclusion.
Answer choice (E) is incorrect; it is essentially an incorrect negation of the correct answer choice. You cannot add "X ==> not D" or its contrapositive "D ==> not X" to the given premises (whether in their positive or contrapositive forms) and get to the given conclusion.
Hope this helps. Please let us know if you have any additional questions.
nsmith45November 8, 2018
Gracias parce
yuetnganFebruary 6, 2019
How would you know to select 'not D is X' in the answer choice in the first place out of all the answers?
RaviFebruary 6, 2019
@yuetngan,
The key to being able to know what to select in the answer choices is to first have a great grasp of what's going on in the stimulus.
We're told that
P: X - >B (which also means not B - >not X)
P: not A - >not B (which also means B - >A)
P: ?
C: not D - >A (which also means not A - >D)
See the term that's in the conclusion but missing in the premises? It's D.
This means D has to be in the answer choice we pick. As it turns out, D is in all of them.
Let's now look to see how we can link up the premises together to see what's missing for us to get to the conclusion
The first two premises can be combined to make
X - >B - >A which is also
not A - >not B - >not X
The conclusion says not D - >A which also means not A - >D
Looking at our premises combined, in the one that has the contrapositives (not A - >not B - >not X) we have the same not A term that appears in the conclusion.
If we put not X - >D at the end of this chain, we would be able to conclude not A - >D, which is the contrapositive of the conclusion we're given in the stimulus (not D - >A).
not A - >not B - >not X - >D
not A - >D, which means not D - >A
From this, we know the right answer is not X - >D or its contrapositive, which is not D - >X
(C) says not X - >D, and this is exactly what we're looking for, so it's the correct choice.
Does this help? Let us know if you have any more questions!
