Joseph: My encyclopedia says that the mathematician Pierre de Fermat died in 1665 without leaving behind any written proof for a theorem that he claimed nonetheless to have proved. Probably this alleged theorem simply cannot be proved, since---as the article points out---no one else has been able to prove it. Therefore it is likely that Fermat was either lying or else mistaken when he made his claim.
Laura: Your encyclopedia is out of date. Recently someone has in fact proved Fermat’s theorem. And since the theorem is provable, your claim---that Fermat was lying or mistaken---clearly is wrong.
Laura: Your encyclopedia is out of date. Recently someone has in fact proved Fermat’s theorem. And since the theorem is provable, your claim---that Fermat was lying or mistaken---clearly is wrong.
49. Joseph’s statement that “this alleged theorem simply cannot be proved” plays which one of the following roles in his argument?
(A) an assumption for which no support is offered
(B) a subsidiary conclusion on which his argument’s main conclusion is based
(C) a potential objection that his argument anticipates and attempts to answer before it is raised A
(D) the principle claim that his argument is structured to refute
(E) background information that neither supports nor undermines his argument’s conclusion
50. Which one of the following most accurately describes a reasoning error in Laura’s argument?
(A) It purports to establish its conclusion by making a claim that, if true, would actually contradict that conclusion.
(B) It mistakenly assumes that the quality of a person’s character can legitimately be taken to guarantee the accuracy of the claims that person has made.
(C) It mistakes something that is necessary for its conclusion to follow for something that ensures that the conclusion follows.
(D) It uses the term “provable” without defining it.
(E) It fails to distinguish between a true claim that has mistakenly between believed to be false and a false claim that has mistakenly been believed to be true.
答案分别是：B,C。题目不是很清楚，两个问题也看不大懂，还请大虾指教，不胜感激！！
Joseph's reasoning follows as:
Since the encyclopedia says..., 'this alleged theorem simply cannot be proved.' Therefore, either lying or mistaken.
So, B is the key of Q49. Because B says the sentence in question is a conclusion and used to support the main conclusion - 'either lying or mistaken'.
As to Q50, i think there is exactly a logic error in Laura's reasoning. That is, there is a possibility that Fermat did in fact not prove the the theorem when he claimed that he proved it. So, Laura's argument - 'someone has in fact proved Fermat’s theorem'- does not suffice to show that Joseph's main conclusion is wrong.
Further, if the theorem is exactly unprovable, then Joseph's conclusion is right; in other words, Laura is wrong. The hypothetical sentence is the negative of Laura's argument that the theorem is provable. So Laura's argument, in fact, is necessary to her conclusion.
Therefore, C is the key of Q50.
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