Difference b/t sufficient and necessary assumption questions (formal logic)

Peter

It really helps me understand the concepts of assumption questions by looking at it formally. Could someone check my logic?

So say we have a premise (B->C) and we have the conclusion (A->C) and say we have the two question stems

Question stem 1: Which one of the following, if assumed, ALLOWS the conclusion to be properly drawn?
Question stem 2: Which one of the following assumptions is REQUIRED for the conclusion to be properly drawn?

For question stem 1, we're looking for a sufficient assumption. Is that just saying something like
___ (and) (B->C) -> (A->C)
so in this case (A->B) would be the obvious sufficient assumption that fills the blank?

For question stem 2, we're looking for the necessary assumption. Is that saying
(B->C) (and) (A->C) -> ___
where the necessary assumption fills in the blank? Something about this doesn't seem quite right.

J.Y. Ping

"Valid" = "properly drawn"

You're on the money for the sufficient assumption question (stem 1). I'd emphasize "if assumed" as opposed to "allows".

You're also right to feel some unease about stem 2, the necessary assumption question stem. Necessary assumptions are assumptions that must be true if the argument is valid. Please note that "the argument is valid" is different from "the premise is true and the conclusion is true." The difference is that for the argument to be valid the conclusion must derive its truth from the premises. In LSAT world, anyway, that's the case. So necessary assumption questions are asking us to find an assumption that must be true so that the existing premises still have a shot of forming (with other assumptions and the existing conclusion) a valid argument. In other words, if the necessary assumption is false, then existing premises will not be able to support ("lend truth to") the conclusion.

Instead of (B-->C) (and) (A-->C) --> necessary assumption answer choice
It should be: [(___) (and) (B-->C) --> (A-->C)] --> necessary assumption answer choice

As it turns out, sometimes, the sufficient assumption is also the necessary assumption. Often, that happens in simple arguments like the one you gave as an example.

B-->C
_______
A-->C

It's evident that A-->B and B-->C implies A-->C. That A-->B is a sufficient assumption is therefore evident.

It's harder to see why A-->B is also a necessary assumption. Consider if A wasn't sufficient for B. In other words, A and not B. Then, B-->C becomes useless to support A-->C. Sure, it could be that A-->X and X-->C, therefore, A-->C. But that's a brand new argument, an argument in which B-->C played no role in supporting A-->C. So if we deny A-->B, then there's no way for B-->C to get us to A-->C. That's why A-->B is also a necessary assumption for that particular argument.

For enrolled students, relevant lessons below:
http://7sage.com/lesson/how-to-find-the-sufficient-assumption/
http://7sage.com/lesson/how-to-find-necessary-assumptions/
http://7sage.com/lesson/advanced-negate-all-statements/

Peter

Again, thanks for such a detailed response. I really appreciate you taking the time JY. Could you give the formula for an argument where one of the conditions is necessary but not sufficient?

Would it be something like {A (and) (A->B) (and) (B->C) (and) C} -> (A->C)
where A is sufficient but not necessary, since (not) A wouldn't wreck the argument?
(A->B), (B->C) are both sufficient and necessary
C is not sufficient but necessary, (not) C would wreck the argument?

J.Y. Ping

I'm having a hard time remember a formal argument where the necessary assumption isn't also a sufficient assumption.

I suppose in the example we had, you could say "Well, A-->B isn't really necessary. All we need is just "It's possible that 'A-->B'" I don't think the LSAT gets that deep into what is "really necessary."

In the example you gave
{A (and) (A->B) (and) (B->C) (and) C} -> (A->C)
A and C in the premises are superfluous premises, neither necessary nor sufficient. Your conclusion is conditional, i.e., if A then C. So we don't need to "trigger" A or C. All men are mortal, all mortals are silly sometimes. Therefore, all men are silly sometimes. We don't need to talk about a particular man.

Most Necessary Assumption questions are not formal, I believe.

jnewton0828

JY is it fail-safe to treat the answer choices in NA questions like MBT questions?