P: B → not OP: D → not C P: ?C: B → not D

Max-Youngquist September 5, 2018

@bkatie46 start by diagramming out all contrapositives of the premises:

1) B ==> not O
O ==> not B

2) D ==> not C
C ==> not D

3) ?

C: B ==> not D
D ==> not B

So we know we need to build a bridge that gets us EITHER from B to not D, or from D to not B.

Let's say we choose B == > not D. The next step is to look at our premises/contrapositives and list anything that includes EITHER B==> or ==> not D. We'll then build a bridge to fill in any gaps.

You can see we have:

B==> not O
and
C ==> not D.

So the missing link is not O ==> C.

If we had instead started from D ==> not B:

D==> not C
and
O ==> not B.

So the missing link is not C ==> O, which can be rewritten as not O ==> C.

So to summarize:

1) list all premises/contrapositives
2) find the premises/contrapositives that involve one side of the conclusion you're trying to prove
3) fill in the gap(s)

I hope that helps!

yuetngan March 10, 2019

Can't the answer choice also be answer choice C?

yuetngan March 10, 2019

So we can pair any contrpositive with any premise? Or does the pattern have to be contrpositive with contrpositive only and a premise with a premise ? We can mix and match?

Ravi March 12, 2019

@yuetngan,

Great question. Let's take a look at the question before we go into
why the answer choice cannot also be (C).

We have

P: B - >not O
P: D - >not C
P:
C: B - >not D

We need to figure out what premise we could add to get to the conclusion.

P: B - >not O (O - >not B)
P: D - >not C (C - >not D)
P:
C: B - >not D

Notice that the conclusion has B in the sufficient condition. This
means we should probably focus on the original version of the first
premise instead of its contrapositive. However, also note that not D
is in the necessary condition of the conclusion. When we take the
contrapositive of the second premise, not D also appears in the
necessary condition. This tells us that it'll probably be easiest to
work with the contrapositive of the second premise.

P: B - >not O
P: C - >not D
P:
C: B - >not D

This makes it much easier to see what we need to add. if we connect
not O and C by adding not O - >C as the third premise, we can make a
big chain that gets us to our conclusion

B - >not O - >C - >not D

C: B - >not D

Therefore, not O - >C is the missing premise we need. (E) gives us
this, so it's the correct answer choice.

To answer your question about whether we can pair any contrapositive
with any premise: yes, we can. The reason we can is because
contrapositives are logically equivalent to their original statements.
Since they're logically equivalent, we can mix and match, and that's
exactly the approach we've taken above.

You asked about (C). (C) says O - >not C. The contrapositive of this
statement is C - >not O.

O - >not C (C - >not O) has an entirely different meaning from not
O - >C. They're fundamentally different statements. (C) doesn't help us
bridge the premises with the conclusion, so it cannot be the correct
answer.

Does this make sense? Let us know if you have any more questions!