Q22 - A mathematical theorem proved

[noah]

22. (E)
Question type: Assumption
To boil down this rather long-winded argument, since people could never verify the huge number of calculations that a computer computes in the process of checking a mathematical proof, computer-verified proofs cannot be accepted. Why shouldn’t these proofs be accepted? Because they can’t be verified by a human. Thus the argument assumes that human verification is required and computer verification is insufficient. Answer choice (E) spells out this assumption, and if we were to negate it, the argument would fall apart _ the sign of a necessary assumption _ since we would have two contradictory premises. One saying that computer verification is insufficient, and the other saying it is.

(A) is irrelevant to the argument.
(B) is irrelevant to the argument.
(C) is irrelevant to the argument.
(D) is attractive, however it is irrelevant to the argument how proofs that are not verified by computer are processed.

[cyruswhittaker]

I chose E after POE and using the negation technique, but am still slightly confused by the ending of choice E: "...that is extended enough.." Any thoughts?

[noah]

I chose E after POE and using the negation technique, but am still slightly confused by the ending of choice E: "...that is extended enough.." Any thoughts?

That's just a fancy way of referring to the fact that humans couldn't verify each step.

extended enough (so long) to be otherwise unverifiable (that humans couldn't do it).

Make sense?

[cyruswhittaker]

Okay, so would a sufficient assumption for this argument be:

Whenever a computer-assisted proof involves astronomically many types of instances, there will be certain steps that cannot be independently verified. Hence...

?

The reason I ask is because I'm trying to better understand the relationship between necessary and sufficient assumptions. The above would be a sufficient assumption because it relates the necessary condition of being accepted (independent verification) to the collective characteristic being analyzed (astonomically-many instances), to the sufficient condition "should not be accepted" by means of the contrapositive.

So in other words, it's providing a link to the necessary condition.

Choice E, on the hand, which is a necessary assumption, merely shows that the collective characteristic (astronomically-many instances) doesn't satisfy the necessary condition, but this might be related to only a specific instance.

So for a simple argument:

"All dogs are green. James is green."

Sufficient Assumption: James is a dog.

Now let's say we added in an additional piece of information:

"All dogs are green. All humans are red. James is green."

The sufficient assumption could still be "James is a dog," but a necessary assumption would be "James is not a human."

I guess what I'm trying to say is that it seems like necessary assumptions can be predicated on additional information in an argument that result in additional necessary conditions that need to be met but are not required by the conclusion. On the other hand, many wrong answer choices will provide information irrelevant to the information that is in the argument.

For example, the necessary assumption above "John is not a human" would no longer be necessary, but rather irrelevant, unless there was a statement specifically relating a human to the necessary condition.

And ultimately it seems like this "lack" of information is the only difference seperating an air-tight argument from one that is not; and this is why weaken, strengthen and justify questions all revolve around assumptions, but the difficulty is in trying to isolate the individual components embedded in all of the details.

[noah]

Okay, so would a sufficient assumption for this argument be:

Whenever a computer-assisted proof involves astronomically many types of instances, there will be certain steps that cannot be independently verified. Hence...

Thoughtful question. I'm not sure I followed it all the way, but I think I did enough to respond:

First of all, yes, your new answer to the problem would be a sufficient assumption - but I think it's also necessary. The argument conclude we should not accept these proofs, i.e. never, so we need them to never be verifiable.

A more typical sufficient (and not necessary) assumption for this sort of argument would be "nothing but a human can verify a step." - and the easiest way to think about that is that we don't need that to be the case - it's more than we need. In this case, it isn't hanging on some extraneous piece of information. However, you're right that necessary assumptions don't have to relate to the core of an argument in the same strict manner that a sufficient assumption or a sufficient and necessary assumption generally does. This is particularly true in cases when the necessary assumption is eliminating alternate possibilities.

For example:Tim is sick therefore he must have played in the rain.

A necessary assumption doesn't even need to relate to something in this argument, it could simply be: Tim is not sick because he licked a germ-infested lollipop. Where's that lollipop coming from?!

A sufficient one would be: The only way Tim gets sick is by playing in the rain.

I wasn't exactly catching your drift about the linking, and perhaps it's me as I shy away from formalized approaches. But, for this following part, you're a bit off:

So for a simple argument:

"All dogs are green. James is green."

Sufficient Assumption: James is a dog.

Now let's say we added in an additional piece of information:

"All dogs are green. All humans are red. James is green."

The sufficient assumption could still be "James is a dog," but a necessary assumption would be "James is not a human."

"James is not a human" is necessary in the first argument as well! For the argument to work, meaning for the conclusion to flow from the premise, we need James not to be human.

I guess what I'm trying to say is that it seems like necessary assumptions can be predicated on additional information in an argument that result in additional necessary conditions that need to be met but are not required by the conclusion. On the other hand, many wrong answer choices will provide information irrelevant to the information that is in the argument.

For example, the necessary assumption above "John is not a human" would no longer be necessary, but rather irrelevant, unless there was a statement specifically relating a human to the necessary condition.

And ultimately it seems like this "lack" of information is the only difference separating an air-tight argument from one that is not; and this is why weaken, strengthen and justify questions all revolve around assumptions, but the difficulty is in trying to isolate the individual components embedded in all of the details.

I think you're talking about the issue that necessary assumptions can be an almost annoying fact that must be eliminated in order for the argument to make sense. I agree. But it doesn't have to link to some side premise mentioned in the argument.

By the way, the Assumption Family of questions is a bit larger than you mentioned. It also includes flaw and principle questions (the ones where you make the argument work, not where you apply the principle), as well as assumption questions (of course).

I hope that helps - I imagine you still have some questions about this rather large topic! We should probably move this to the general LR questions section...

[cyruswhittaker]

Thank you for the response. It definately helps me to make better sense out of all the question types and the roles of assumptions.

Let me just try to clarify where I went wrong with my example.

If the argument is:

"All dogs are green. Thus, James is green."

and we consider the statement "James is not a human."

I was intially saying it was not a necessary assumption because we don't have any explicit information from the argument that denies the truth of the conclusion "James is green" IF "James is a human." Afterall, if we negate it and say "James is a human," it seems that from the argument, it still allows both facts--James is a human and green--to be consistent, but the argument itself wouldn't work anymore. In other words, I would have more instantly classified it as necessary if the premise was "Only dogs are green." With this, "James is a human" instantly contradicts the argument.

But I think this is where I went wrong: it's in the relationship that is stated that makes this a necessary assumption. The argument's conclusion "James is green" is suggested to follow, using the word "thus," from the statement "All dogs are green." So if we were to say that James is a human, the argument would be dis-jointed.

[noah]

You got it!

[lwilliams25619]

Will someone tell me if my thinking is correct. I was able to get to the right answer, but it took a looong time! This is how I thought this through:

The conclusions says: computer assisted proofs with a lot of calculations should not be accepted.

From the premises I took that:1. mathematicians theorems should not be accepted until all of the steps were indep. verified, 2. computer assisted proofs make a lot of calculations, 3. so many calculations that they, in turn, can not be verified.

This seems to be a necessary assumption question since the stem says the argument relies on the assumption. Therefore, to connect the conclusion soundly to the stated premises, one can not accept the computer either when there are a lot of calcuations because it would be too difficult to verify.

[cyruswhittaker]

I think you're on the right track but there's a little more subtle detail with the author's premises/conclusion.

Notice that the first sentence provides the necessary condition for the a proof being accepted: that every steps needs to be independently verified. Then, the author claims that these computer-assisted proofs require so many steps that no human can verify them. So, the author is using the contrapositive of his first statement (first sentence) to illustrate that a computer proof should not be accepted because it doesn't meet the necessary condition of independent verification.

But notice that the author only says "independently verified" as the necessary condition. There is no qualification that stipulates this to be contained to human verification. So the question could be posed: what if there is another computer, rather than a human, that can independently verify the steps?

For this author's conclusion to hold, of course, the answer to this must be "no" and this leads us precisely to choice (E).

As always, the assumption is a gap in the author's argument, and in this case, the gap came in the form of a subtle lack of qualification to the necessary condition.

[bbirdwell]

cyrus! Good explanation. I can tell you've improved quite a bit since you first started posting.

I'll add my two cents, because, while the language of this argument is intimidating, it's all smoke and mirrors. The correct answer is a piece of cake to find if you stick to good analysis and good, aggressive elimination.

Important premises:
1. theorem should not be accepted until independently verified
2. no human could ever review every step in a computer proof

Conclusion:
No computer proof can ever be accepted.

Break the conclusion down into it's basic elements and see how they are tied (or not tied) to the evidence. The main concept in the conclusion is "acceptance." What do we know about acceptance? That we cannot accept until we have independent verification. Therefore, this author is assuming that we cannot independently verify the computer proof. Blammo. The LSAT can be so simple if we just let it.

Now take a look at the choices and pretend you only have 20 seconds to choose. Eliminate the ones that are nowhere close to the basic elements of the argument:
(A) "greatly simplified?" Conclusion has nothing to do with simplicity. It's about acceptance. Easy elimination.
(B) "most attempts?" Classic wrong answer choice. We have no idea whether most or many or several or none or all attempts do this.
(C) who cares? The conclusion is about tasks for which there are "astronomical instances."
(D) On a quick pass, we might leave this one just on the basis that it's intimidating and sounds like it might be relevant somehow. It does seem oddly confusing with its conditional language, though -- the argument didn't really have any conditional language. Let's leave it for now.
(E) Bam. Verification, acceptance, computers. This is it. We could almost have only been told the conclusion and then found the right answer, simply applying a good process of elimination.

[shirando21]

22. (E)
Question type: Assumption
To boil down this rather long-winded argument, since people could never verify the huge number of calculations that a computer computes in the process of checking a mathematical proof, computer-verified proofs cannot be accepted. Why shouldn’t these proofs be accepted? Because they can’t be verified by a human. Thus the argument assumes that human verification is required and computer verification is insufficient. Answer choice (E) spells out this assumption, and if we were to negate it, the argument would fall apart _ the sign of a necessary assumption _ since we would have two contradictory premises. One saying that computer verification is insufficient, and the other saying it is.

(A) is irrelevant to the argument.
(B) is irrelevant to the argument.
(C) is irrelevant to the argument.
(D) is attractive, however it is irrelevant to the argument how proofs that are not verified by computer are processed.

after re-read this argument, I found the condition"independently verified" in the first sentence is more important than " no human being could review" in working out this question. only E has this expression "independent verification" among the choices.

[timmydoeslsat]

I skipped through the posts in this thread so that I could give, what is hopefully, a different perspective.

We have evidence that:

Should be accepted ---> IV

Then we are told about how computer-assisted proofs do all of these calculations that no human could review every step.

We then have a conclusion of ~Should be accepted.

This keys us in on the reasoning of the argument. The arguer has assumed that ~IV is occurring in the argument to allow for a conclusion of ~Should be accepted.

But we do not have, explicitly, ~IV occurring. We want to protect the idea that ~IV is occurring. Answer choice E does just that.