#25- To get the free dessert, one must order an entree and a

[stsai]

Q25 "To get the free dessert, one must..."
I used abstraction and figured out the S-->N relations of the premise:
A-->B+C-->D; so Not D-->Not A
I eliminated answers (A), (D), and (E) for they don't have the same SN relations as above.
Yet for me, (B) and (C) seem to have the same SN relations. Though I chose (C) and got it right, I am still unsure about why (B) is wrong. Is it because it's based on abstraction? (i.e. "have something better to do" is an abstraction in comparison to "can grow blueberries)

Thanks in advance!

[Dave Killoran]

Hey Stsai,

Let's start with #25 first.

The first part of answer choice (B) matches the stimulus pretty well. So, the problem isn't there, but in the conclusion. As you correctly noted, the stimulus uses a contrapositive idea to draw the conclusion (not D --> not A). Now, take a look at the conclusion in answer choice (B). If this was going to be the correct answer, we should see a contrapositive approach similar to the following: "Therefore, anyone who does not have something better to do cannot be elected class president." Diagrammed, that would be "Not Something better to do --> Not elected class pres"

Check that conclusion against the conclusion in answer choice (B)--do you see how they subtly shifted the language (they moved the "not" around, specifically)? Try diagramming the conclusion and see what you get. What you should get if you diagram the conclusion in (B) is:

"Something better to do --> Not elected class pres"

That's different from the conclusions in the stimulus and answer choice (C).

What you've run into is a tricky re-wording by Law Services, which isn't surprising given that this is the final questions in the section.

Please let me know if that helps. Thanks!

[stsai]

And I now see the difference in the conclusion of #25. These are indeed tricky!
Thanks for the detailed explanation, Dave!

[whoonfirst]

Can someone help me understand this? I think I am wrong in how I diagram "one must."

I used the following to map the problem:

FD E &S
/E or /S FD

E or S FSD
/FSD /E or /S

/FSD /FD

Do I have this backwards?

[Jonathan Evans]

Hey Whoonfirst,

You've got the second contrapositive wrong:

E or S FSD
/FSD /E or /S

It needs to be:

E or S FSD
~FSD ~E & ~S

"One must" introduces a necessary condition. It does not appear as though you are fundamentally approaching this problem incorrectly, apart from this violation of the rule that negating "OR" becomes "AND" (and vice versa, De Morgan's laws)

[amagari]

why didn't the either/or statement in this problem generate the two normal conditional laws like

/Entree Soft Drink

/Salad Soft Drink

[Adam Tyson]

If I understand your diagram correctly, amgari, you've interpreted the stimulus to say that if you do not get an entree or you do not get a salad then you do get a free soft drink. But that's the opposite of what it says! Careful reading here is crucial. The rule is that is you DO get an entree or you DO get a salad, then you can get the free soft drink. That can be diagrammed like this:

E
OR FSD
S

You can also diagram it as Jonathan did:

E or S FSD

Same meaning, but just lumping the two sufficient conditions together. Saves space, keeps it simple, easier to type that in this forum. The contrapositive is:

FSD E AND S

That is, if you cannot get a free soft drink then you must not have gotten an entree and you must not have gotten a salad.

I hope that clears things up for you. Keep at it, and read carefully!

[amagari]

I was trying to diagram it like on page 196 in the 2016 Logical Reasoning Bible.

The book says that diagramming will allow for A or B or both. I understand why you diagrammed it as you did (I understand multiple sufficient and necessary conditions) but I thought this is that special Either/or construction not just a normal "or" choice that would be diagrammed as you did.

[Adam Tyson]

Ah, I see! That special either/or construction is used when at least one of two alternatives is required, or necessary. That would be something like "you must have either the cake or the pie". In that case, if you do not have the cake, you must have the pie, and if you do not have the pie, you must have the cake.

Here, there is nothing necessary about either the entree or the salad in the relationship with the soft drink. If you have one of those things, you can get the soda - just a normal multi-conditional if...then relationship.

Use that special either/or setup only when you know that one thing being "out" forces the other one to be "in"; when the absence of one guarantees the presence of the other.

Keep at it!

[amagari]

I thought the Either/Or implied automatically implied "at least one" condition.

So if the first sentence of the question were changed to "To get the free dessert, one must order either an entree or
a salad," would that trigger the special either/or construction that I was trying to use? Whereas the sentence I tried to use it on said "they can" as opposed to "must"? Also since the construction is in the necessary condition?