Logical reasoning PrepTest 122 · Section 2 · Question 22

Incorrect. The argument is over the acceptance of theories, not the ease of the work done to prove them, so this answer is out of scope.

Incorrect. It doesn't matter how frequently mathematicians succeed at proving theorems, just whether computer-assisted programs can lead to verification. Whether that almost always, almost never, or usually happens, what matters is whether it can happen at all.

Incorrect. The conclusion of the argument is specifically about computer-assisted proofs "involving astronomically many types of instances," so this answer about a computer-assisted proof that has only a very limited number of steps is out of scope.

Incorrect. What is true of non-computer-assisted proofs is out of scope of this argument about whether computer-assisted proofs can serve to verify a theorem.

Correct. Argument or Facts:
Argument

Valid or Flawed:
Flawed

Question Type:
Strengthen with Necessary Premise

Stimulus Summary:
Principle - not Each step of math proof verified → not Accept theorem proved by one mathematician
Computer-assisted proofs - Run a ton of calculations that no human could ever review
Conclusion - Computer assisted proofs not Accept

Answer Anticipation:
This stimulus falls into a common category of Strengthen with Necessary Premise questions—it raises a principle, and then it applies that principle to a specific case. When that's the case, the argument usually fails to fully establish the sufficient condition of the principle is met, and the correct answer connects a detail from the argument to that sufficient condition.

Here, the opening line is the principle. Until functions the same as unless, so we treated it here as "if not" and diagrammed out the rule. The argument then goes on to describe the situation to which it's attempting to apply the rule before reaching a conclusion that is potentially supportable by the principle—that a theory (in this case, a class of theories) should not be accepted. That class of theories is those that are proven using computer assistance.

In order for that conclusion to be properly drawn, it would need to be shown that each step of the proof hasn't been independently verified. What is established instead is that each step in the proof in a computer-assisted proof can't be reviewed by a human. The argument is assuming, then, that a computer verifying the proof triggers that principle—that such a proof doesn't count as independent verification.

Answer Explanation:
While this answer uses some fancy language at the end, it starts off exactly matching our anticipation—using an independent computer program doesn't count as independent verification. When it starts to get unnecessarily wordy, it just means the proof is long enough that it can't be verified by humans (i.e., otherwise, not using a computer). If we negate this answer, then the computer-assisted proofs don't trigger the conditional principle and the conclusion based on the application of that principle can't be met, killing the argument.

Key Takeaway:
When a Strengthen with Necessary Premise question tries to apply a premise to a specific instance, it's usually flawed in that it fails to fully establish the sufficient condition is present in the case. When this happens, the answer connects the details of the situation to that sufficient condition.