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annetr0712 on January 19, 2020

Missing Premise Drills

I am having some difficulty with the following problem: P1: X -> A P2: ? C: Not A -> B The answer I put down was the following: P1: X -> A Contrapositive: Not A -> Not X P2: Not X -> B Contrapositive: Not B -> X C: Not A -> B Contrapostive: Not B -> A But the answer for premise two is Not B -> X I am wondering if the order matters or either is okay as long as you have both of the answers?

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SamA on January 19, 2020

Hello @annetr0712,

You did not have difficulty with this problem, you answered it correctly.

Remember that a contrapositive carries the exact same logic as its counterpart. The order in which you write them does not matter. You would be correct if you only included one of the given answers. However, it is good practice to continue diagramming a premise both ways. This will help you solidify your understanding of sufficient and necessary reasoning.

annetr0712 on January 19, 2020

Also in problems like the following:
P1: A ->Y
P2: ?
P3: B -> C
C: Not Y -> Not B

Why is the answer for premise two
Not A -> Not C
instead of
Not Y -> Not C

SamA on January 19, 2020


Technically, you are correct.

not Y - - - -> not C would lead to the conclusion.

However, for the purpose of these drills, I think they want you to include all 3 premises in reaching the conclusion. With the premise that you chose, premise 1 becomes irrelevant.

If our second premise is not A - - - - - - - > not C, then we use all three premises. I will demonstrate.

P1: A - > Y
P2: ?
P3: B - > C
C: not Y - > not B

From the conclusion, we have not Y. Based on premise 1, that gives us not A. How can we connect that to premise 3? We can say not A - - > not C. This leads us back to our conclusion.