#14 - Joseph: My encyclopedia says that the mathematician

[smile22]

For question 14, I realize that correct answer C is describing a mistaken reversal. Could you please provide a diagram to help me visualize the reasoning error that Laura is making?

Thank you.

[KelseyWoods]

Hi smile22!

So Joseph's argument is basically that since no one can prove Fermat's theorem, then Fermat was probably lying or mistaken. We can diagram it like this:

• Someone proved Fermat's theorem Fermat lying OR Fermat mistaken

Laura says that someone has proved Fermat's theorem so therefore, Joseph is wrong that Fermat was lying or mistaken. We can diagram her reasoning like this:

• Someone proved Fermat's theorem Fermat lying AND Fermat mistaken

Diagramming it that way, we can see that Laura has made a mistaken negation. Joseph said that the fact that no one had proved Fermat's theorem was sufficient to prove that Fermat was lying or mistaken. Laura negated the sufficient condition by saying that someone had proved the theorem but negating the sufficient condition is not enough to prove that Fermat was definitely not lying and definitely not mistaken.

Hope that helps!

Best,
Kelsey

[PB410]

I was able to recognize the mistaken negation in the stimulus, yet I got tripped up with the wording of the answer choices. I'm having trouble understanding how answer choice c, It mistakes something that is necessary for its conclusion to follow for something that ensures that the conclusion follows, applies to
NOT proven --> Fermat lying or mistaken
Provable ----> NOT lying or mistaken
I don't understand what C is stating when it says it mistakes something that is necessary- which would be lying and mistaken- for the conclusion to follow for something that ensures (sufficient?) that the conclusion follows. But the second half of the answer seems to describe a mistaken reversal, not a mistaken negation. Can someone help explain how to read this.

Also, I am not sure what Answer choice E means or looks like.

[AthenaDalton]

Hi PB410,

Thanks for your question!

In the context of this argument, here's what necessary vs sufficient would look like:

Necessary to support the claim that Fermat was lying: Fermat's theorem was capable of being proved, but Fermat did not personally complete the proof in his lifetime

Sufficient to support the claim that Fermat was lying: Fermat's theorem is not capable of being proved, by anyone, so obviously Fermat was lying when he claimed he had solved it

The issue with the argument in the stimulus is that Joseph starts out arguing that Fermat was lying, and offers for support the claim that Fermat's theory has never been solved and is perhaps unsolvable (if true, this is sufficient to show Fermat was lying). Laura responds with proof that the theorem is solvable. Just because the theorem can be solved by someone doesn't mean that Fermat solved it himself. Laura still needs to prove that Fermat was in fact mistaken or lying (necessary conditions) to support her argument.

I hope that distinction makes sense. Good luck studying!

[Hershel]

Hi PB410,

Thanks for your question!

In the context of this argument, here's what necessary vs sufficient would look like:

Necessary to support the claim that Fermat was lying: Fermat's theorem was capable of being proved, but Fermat did not personally complete the proof in his lifetime

Sufficient to support the claim that Fermat was lying: Fermat's theorem is not capable of being proved, by anyone, so obviously Fermat was lying when he claimed he had solved it

The issue with the argument in the stimulus is that Joseph starts out arguing that Fermat was lying, and offers for support the claim that Fermat's theory has never been solved and is perhaps unsolvable (if true, this is sufficient to show Fermat was lying). Laura responds with proof that the theorem is solvable. Just because the theorem can be solved by someone doesn't mean that Fermat solved it himself. Laura still needs to prove that Fermat was in fact mistaken or lying (necessary conditions) to support her argument.

I hope that distinction makes sense. Good luck studying!

Sorry, but I still do not understand how this explains the wording of answer C.

[Dave Killoran]

Sorry, but I still do not understand how this explains the wording of answer C.

Hi Hershel,

Thanks for the question! The mistake made in this argument is a Mistaken Negation. I altered the formatting of Kelsey's explanation above to more clearly show how that negation works but the original explanation was correct and hasn't been changed. Basically, Joseph makes a conditional comment, and then Laura found that the sufficient condition did not occur as stated, and then she concluded that the necessary conditions did not occur. That's a classic logic flaw based on mixing up conditions that is covered extensively in our books and courses.

Answer choice (C) is an abstract description of a conditional flaw based on mixing up conditions. If it looks like that is a description of a Reversal and not a Negation, remember that that is okay! The MN and MR of a statement are CPs of each other, so you can describe this conditional flaw in any way that reflects confusion between sufficient and necessary.

Please let me know if that helps. Thanks!

[caroline222]

How did you know to diagram the since... therefore... construction as conditional reasoning (not proven --> mistake OR lying)? Can "since... therefore..." usually be diagramed as conditional reasoning?

[Adam Tyson]

Speaking just for myself, Carolina, I did NOT at first see this in a conditional way. "Since" is a premise indicator that may or may not trigger a conditional analysis, and in this case nothing about it set off any conditional alarms for me. My prephrase was just "Fermat could still have been lying - maybe he didn't prove it." In my first look I classified this one as a "some evidence" flaw, in that Laura had some evidence that Joseph might be wrong, which only means she weakened his position, and then she acted as if she had completely destroyed his argument. I was actually a little surprised to see the conditional answer and almost didn't select it!

But in a larger sense, every argument is in some way conditional. Every author thinks that their premises are sufficient for their conclusion, and that their conclusion necessarily follows from those premises. Not finding any "some evidence" type of answer choice here made me give it another look through that lens, and sure enough, it fit.

I suspect I may be in the minority here and that most of my colleagues at least "felt" the conditional nature of this argument, but it's still possible to get to the right answer when you don't at first see that, largely through process of elimination. With practice and patience you will be able to adjust your view and correct your course mid-stream to accept an answer that at first bore little resemblance to your prephrase.

[menkenj]

Speaking just for myself, Carolina, I did NOT at first see this in a conditional way. "Since" is a premise indicator that may or may not trigger a conditional analysis, and in this case nothing about it set off any conditional alarms for me. My prephrase was just "Fermat could still have been lying - maybe he didn't prove it." In my first look I classified this one as a "some evidence" flaw, in that Laura had some evidence that Joseph might be wrong, which only means she weakened his position, and then she acted as if she had completely destroyed his argument. I was actually a little surprised to see the conditional answer and almost didn't select it!

But in a larger sense, every argument is in some way conditional. Every author thinks that their premises are sufficient for their conclusion, and that their conclusion necessarily follows from those premises. Not finding any "some evidence" type of answer choice here made me give it another look through that lens, and sure enough, it fit.

I suspect I may be in the minority here and that most of my colleagues at least "felt" the conditional nature of this argument, but it's still possible to get to the right answer when you don't at first see that, largely through process of elimination. With practice and patience you will be able to adjust your view and correct your course mid-stream to accept an answer that at first bore little resemblance to your prephrase.

I did the same in my rephrase! I felt that E was closest to this prephrase but to be honest, I don't quite understand what E is saying. Do you mind helping me understand the flaw that E is describing and why it's wrong please?

Thanks!

[menkenj]

I think I see it now.

E is saying that there was a mistake between mixing up an untrue statement believed to be true and a true statement believed to be false. This didn't happen in the stimulus because we don't know whether or not Fermat proved his theorem because there is no written proof.