Supply the missing premise that makes the conclusion follow logically:P: A → not BP: ?C: B → C

Mehran July 5, 2018

Hi @sean18, thanks for your post.

Let's take a close look.

Premise #1: A ==> not B. The contrapositive is B ==> not A.
Premise #2: missing
Conclusion: B ==> C

Notice that the sufficient condition of the contrapositive of the first premise and the sufficient condition of the conclusion are the same: B. So how do we get from the first premise to the given conclusion?

The correct answer, (E), bridges this gap. If you add this premise to the contrapositive of the first premise, you can construct a transitive argument:
B ==> not A ==> C

therefore: B ==> C

Let's look at answer choice (C):
P1: A ==> not B; cp: B ==> not A
P2: C ==> not A

You cannot link up the positive or contrapositive of the given premise with the premise stated in answer choice (C) and reach the given conclusion.

Hope this helps! Please let us know if you have any additional questions.

alp2115 April 1, 2019

Would b-> not a be the same as not a -> b?

Jacob-R April 3, 2019

No. The direction of logical conditions is absolutely crucial! Here is one example: If your flowers grow, there weren't pests in the garden. We cannot say that therefore if there aren't pests in the garden, your flowers will grow! (There may be, for example, a lot of other requirements!) The only way we can flip a logical statement is with a contrapositive: Where b -> not a becomes a (negated!) -> not b (also negated!)

I hope that helps. Please let us know if you have further questions.