These questions require us to make logical inferences from a given set of premises. The contrapositive is one type of valid logical inference, but we can also use other rules of inference to combine premises to reach a conclusion. Let's look at this problem:
(1): A- > ~B (2) C -> X (3) ~ C -> B
Let's start by identifying common elements. (1) & (3) have B & ~ B. How can we convert both to B?
Let's take a contrapositive of (1):
(4) B -> ~A
Now, we can combine (3) and (4) into a following logical chain:
~C -> B -> ~A , which is equivalent to:
(5) ~C -> ~ A
Now let's look at the resulting premise (5) and the remaining premise (2), note the common element C & ~ C. Let's take a contrapositive of (5):
(6) A -> C
Combine (6) and (2):
A -> C -> X, which is equivalent to:
(7) A -> X
This statement is logically equivalent to its contrapositive:
~ X -> ~A
The properly inferred conclusion is, therefore, either:
A -> X or ~ X -> ~A
Does this make sense?
Let me know if you have any further questions.
NicoleBanksAugust 22, 2019
Ok I'm still a little confused on how do you make up these chains? Is it like alphabetical order or the order they are in the problem?
