Sufficient question, question

[tungj]

In a sufficient question, if we are given:
A->B
Therefore C

One correct answer would be B->C

But if we were to look at the contrapositive of the premise,
~B->~A
Therefore C

Would another correct answer be ~A->~C so we have:
~B->~A->C

So in essence, we could have 4 possible correct answers (including the contra positives) to fill in the gap:
B->C (and ~C->~B)
~A->~C (and C->A)

I have a feeling I've made a mistake somewhere but I can't figure it out.

Thanks!

[ohthatpatrick]

I'm confused by aspects of your question, so you're probably going to have to clarify.

You initially said that if we have
A -> B
and conclude C
then one Suff Assump would be
B -> C.

That's not technically right (and although I'll explain why, it ultimately has little to do with your question).

If I have
prem: A -> B
+
Suff Assump: B ->C

Can I conclude C?

No. I can only conclude A -> B -> C

The difference here is the difference between a statement of fact and a conditional phrase.

For example, consider this argument:
(A -> B)
If Amanda comes to the party, Bob will be happy.
(conc: C)
Thus, Carmen will be happy.

Suff. Assump (B->C)
If Bob is happy, Carmen will be happy.

If you combine the premise and the SA, all you get is
If Amanda comes -> Bob is happy -> Carmen is happy.

Have you proven that Carmen will be happy?

No. You're still missing something: you need to establish that Amanda is coming to the party.

This whole conditional chain might exist, but until I know Amanda is coming, I don't know if Bob or Carmen will be happy.

However, I think this technical error had nothing to do with your real question.

Let's change the argument to this:
If Amanda comes to the party, Bob will be happy. Thus, Carmen will be happy, since Amanda will come to the party.

A -> B
A.
=====
C

The sufficient assumption here would normally be
B -> C
and the contrapositive would also be valid
~C -> ~B

(Very often the correct answer to Suff Assump is written in contrapositive form, so you should be looking out for it)

Another sufficient assumption that would work here is
A --> C
as well as its contrapositive
~C --> ~A

However, that's it.

~A --> ~C (C --> A)
would not be a sufficient assumption.

This would only help us prove that A is true, but the argument is trying to prove that C is true.

You never want to see a conditional set up as
Conclusion --> Premise

Let me know if you have questions about any of this or want to clarify what you were originally asking.