Q15 - At Flordyce University any

[cyruswhittaker]

Can you help me to clarify the flaw in this problem?

To me it appears that the the flaw is linking up the groups together. Just because some students who show interest in archaelogy never take a course doesn't mean that those students are actually the part of the group interested in the dig.

Likewise, in choice B, some horses that are surefooted are not gentle, but that doesn't directly mean that horses that are well schooled are contained in that group.

However, I still feel uncomfortable with the flaw, and was wondering if you could give me a hint or some help making it more explicit and clear.

Thanks!

[aileenann]

I think your description is a very good one for describing the flaw. I'll describe it in a different more formal way, so that you can say that perspective.

In very abstract terms, we could write this argument as:

Any student who is eligible to do A must have done B and C. Many students who do C do not do B. Therefore many students who want to do A are not eligible to do A.

So if you look, the flaw lies in (exactly as you said) the argument assuming that students who have done C want to do A - but this may not be true at all. The argument seems to be assuming that showing an interest in archaeology (C) amounts to wanting to go on the field trip (wanting to do A), but we have no reason to think this.

I hope this helps Thanks for posting.

[cyruswhittaker]

So basically, let's say there were a 1000 students who had shown interest in archaelogy but only ten took a class. The conclusion is saying, therefore, that many of the students who want to participate in the dig will be ineligible.

But it could be the case that only those 10 who took the class were actually interested in the first place; the others didn't even want to participate in the dig. Then it would be true that all of the students who participated are eligible. So the author is assuming that those 1000 students actually wanted to participate.

Likewise, in choice B, the argument is assuming that the sure-footed horses are actually the well-schooled horses. Maybe the only few that are well-schooled are the ones that are actually gentle, so in this case, the conclusion would not hold (in fact all of the well-schooled ones would be ideal for beginning riders).

This was a tricky argument because it is more abstract, and I appreciate your help.

[WaltGrace1983]

This one certainly had a tough flaw to fully understand but, luckily, the answer choices helped us out a bit. I am going to do a full breakdown of this with formal logic. Maybe it will help someone else down the road.

"Any student who wants to participate in a certain dig is eligible to do so but only if..."

We get some conditional language here so I am going to take a mental note of this. The "only if" signifies that we just got a sufficient condition.

"Only if the student has taken at least one course and has shown an interest..."

More conditional language! We know that two elements are necessary.

"Many students who have shown an interest...never even take one course...Therefore..."

The contrapositive! We know that (course) and (interest) are both necessary conditions. When even one of these necessary conditions fail - which we know because there is ~(course), the sufficient condition will automatically fail.

Now the "therefore" is coming so we can be fairly certain that we now have all the premises. Let's put this into symbolic logic.

(S & P & E) → (A & I)

How did I get here? Well what do we know? We know that we have the "only if" trigger, showing that everything on the right of the "only if" is a sufficient condition and everything on the left is a necessary condition (this works well in this question but don't use that as a hard-and-fast rule!!!). So as for the sufficient condition, we know you have to be student (S) who wants to participate (P) and you are eligible (E). All of these things matter! We are talking specifically about the people who are students and wanting to participate and eligible. We don't know anything about people who just want to participate or just are students - we only know about those that are S, P, and E.

The right side of the arrow is a little bit easier to see, we know that in order to be an eligible student who wants to participate, you HAVE TO have taken at least one archaeology course (A) and you HAVE TO have shown an interest (I). Just like before, both are necessary are that is why we want to express this as (A & I).

Now keep in mind the contrapositive of (A & I). What is the opposite of both A and I? It is ~A or ~I.

Many ~(A & I)

This could also be written to say that many (~A or ~I) but the logic is still the same. The point is that this gives us the contrapositive. Thus, the argument is actually valid so far. There are no problems yet!

⊢ Many (S & P & ~E)

The stuff after "therefore" is telling us that there must be some who are (students) that (want to participate) and are (ineligible). This seems great! So what is the problem? Well the problem is this: we were given the contrapositive of the necessary condition. In order to logically complete the argument, we would expect something like this:

Many ~(A & I) ⊢ Many ~(S & P & E)

This is a perfectly logical argument. All this is is simply a contrapositive of the initial premise! Super easy! It can also be expressed like this...

Many ~(A) or ~(I) ⊢ Many ~(S) or ~(P) or ~(E).

So what is the problem with the argument then? Look closely at the contraposed conclusion above (Many ~(S) or ~(P) or ~(E))

Here is the problem:

Many ~(S) or ~(P) or ~(E)


does NOT have to equal


Many (S & P & ~E)

We could conclude that many students are ~(P) just as easily (and logically) as we could conclude that many students are ~(E). We know from the premises that it must be (Many ~(S) or ~(P) or ~(E)). However, we don't know exactly which one can be negated! I hope that makes sense.

The Answer Choices

(A) (J & W) → L, ~L ⊢ ~(J & W)

This is actually a fairly valid argument. We know that in order to be both a (jar) and (worth saving) you have to have a lid! Some don't have lids. Therefore, there are some (jars) that ~(worth saving.) The problem is that the original argument gives us two necessary conditions, not just one. In addition, this appears to be logically consistent anyway.

(B) Perfect match!

(C) (R & S) → BS, many ~(R & S) ⊢ many ~(BS)

This is a different flaw. This is simply a false negation. We cannot merely negate the premises and conclude a negated conclusion!

(D) ~(RM) → (O & NR), many ~(O & NR) ⊢ many (RM)

Also a valid argument. It uses the contrapositive.

(E) (ENB & GI) → A or OS, many ~A ⊢ few (ENB & GI)

This is a different kind of flaw. It assumes that just because many are ~A means that many are also ~(OS). Maybe every building that is ~A is OS?

My head hurts.

[HongmeiH333]

Flaw - Conditional Logic (False Contrapositive)

1. False Contrapositive
A+B→C+D, (-C)+D→A+(-B)
2. Right Contrapositive
A+B→C+D, (-C)+D→(-A)or(-B)

A = student who wants to participate in a certain archaeological dig
B = eligible
C = take at least one archeology course
D = show an interest