# If a wig has any handmade components, it is more expensive than one with none. Similarly, a made–to–measure wig range...

MJA7 on November 29, 2014

Question help

I am having difficulty with the conditional statements. Please help lay out the conditional statements that allows us to conclude A as the correct answer

Replies

Naz on December 5, 2014

This is a strengthen with sufficient premise question. Remember that a sufficient premise is sufficient for a conclusion, if and only if the existence of the premise guarantees or brings about the existence of the conclusion. Therefore, we need to find the premise that 100% guarantees the conclusion. The way you want to attack these answer choices is two-pronged. Ask yourself, does it strengthen? If it doesn't, then cross it out and continue to the next answer choice. If it does strengthen, however, then ask yourself whether or not the premise guarantees the conclusion.

"If a wig has any handmade components, it is more expensive than one with none."

P1: WHC ==> METWN
not METWN ==> not WHC

"Made-to-measure wigs range from medium-priced to expensive."

P2: M2M ==> MP2E
not MP2E ==> not M2M

"Handmade foundations are never found on wigs that do not use human hair."

P3: not UHH ==> not WHC
WHC ==> UHH

"Any wig that contains human hair should be dry-cleaned."

P4: UHH ==> DC
not DC ==> not UHH

"So all made-to-measure wigs should be dry-cleaned."

C: M2M ==> DC
not DC ==> not M2M

(A): "Any wig whose price falls in the medium-priced to expensive range has a handmade foundation."

(A) MP2E ==> WHC
not WHC ==> not MP2E

Does this strengthen the argument? Yes.

We know from P2 that "M2M ==> MP2E," which we can connect to (A) "MP2E ==> WHC," which we can connect to the contrapositive of P3 "WHC ==> UHH," which we can connect to P4 "UHH ==> DC."

So it would look like this: M2M ==> MP2E ==> WHC ==> UHH ==> DC, to conclude M2M ==> DC. Therefore, answer choice (A) helps strengthen the conclusion.

Does the premise guarantee the conclusion? Yes.

As you can see from the chain we wrote down above, through the transitive property, answer choice (A) guarantees the conclusion.

Hope that clears things up! Please let us know if you have any other questions.

Anna on October 4, 2018

I am still confused on how A is correct.

Mehran on October 16, 2018

P3: "Handmade foundations are never found on wigs that do not use human hair."

not UHH ==> not WHC
WHC ==> UHH

P4: "Furthermore, any wig that contains human hair should be dry-cleaned."

UHH ==> DC
not DC ==> no UHH

C: "So all made-to-measure wigs should be dry-cleaned."

M2M ==> DC
not DC ==> M2M

Notice that we can make the following transitive chain from P3 and P4:

WHC ==> UHH ==> DC

But our conclusion is M2M ==> DC.

How can we guarantee this conclusion 100%?

P2: Similarly, a made-to-measure wig ranges from medium-priced to expensive.

M2M ==> MP2E
not MP2E ==> not M2M

So this is the jump:

M2M ==> MP2E | WHC ==> UHH ==> DC

We can guarantee this conclusion by closing this gap. How do we do that?

By making MP2E sufficient for WHC as follows:

MP2E ==> WHC

This is exactly what (A) says:

"Any wig whose price falls in the medium-priced to expensive range (MP2E) has a handmade foundation (WHC)."

So (A) would be the correct answer.

Hope that helps! Please let us know if you have any other questions.

JoshHoward504 on January 16, 2023

The condition concerning HMF not being found on HH wigs is confusing to me.

I diagrammed it as, ~HMF ---> ~UHH, but I see that you diagrammed it the other way. How did you know to put ~HMF as the necessary condition?

Emil-Kunkin on January 20, 2023

Hi, we have the statement that HMF are never found that on wigs without HH. I would try to think about this intuitively: we know that we cannot have HMF when there is HH- or that those two things are mutually exclusive. I'm other words, if one of those things happens, the other can't.

So If not human hair then not HMF
And if HMF then then we must have HH