# In West Calverton, most pet stores sell exotic birds, and most of those that sell exotic birds also sell tropical fis...

Antionette on January 21, 2014

Help

Please explain how to diagram.

7 Replies

Melody on January 22, 2014

Let's diagram:

"In West Calverton, most pet stores sell exotic birds,"

Q1: PS-most-SEB
SEB-some-PS

"and most of those that sell exotic birds also sell tropical fish."

Q2: SEB-most-STF
STF-some-SEB

"Any pet store there that sells tropical fish, but not exotic birds does sell gerbils."

P1: STF & not SEB ==> SG
not SG ==> not STF or SEB

"No independently owned pet stores in West Calverton sell gerbils."

P2: IOPS ==> not SG
SG ==> not IOPS

Answer choice (D): "No independently owned pet store in West Calverton sells tropical fish but not exotic birds."

IOPS ==> not STF or SEB
STF & not SEB ==> not IOPS

Answer choice (D) follows from connecting P2 to P1's contrapositive:

IOPS ==> not SG ==> not STF or SEB, to conclude:

IOPS ==> not STF or SEB

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

Missy on November 2, 2015

I understand why the correct answer is D but can you explain why E is incorrect.

Mehran on November 9, 2015

Thank you for your question.

None of the statements we can diagram as joint sufficient conditions (1) the quality of being independently owned and (2) the quality of not selling tropical fish. Nor are these conditions found jointly in the contrapositive form, that is, (1) the quality of not being independently owned or (2) the quality of selling tropical fish. That is why (E) is incorrect: there is no textual support in the stimulus for the proposition that IOPS & STF ==> SEB.

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

Gabby on April 29, 2020

@naz can you explain how you diagrammed D? I don't get the portion in the necessary condition.

Joshua on June 28, 2020

@Gabby, I was similarly confused by the diagramming of the necessary of (D) but maybe this will help. With the premises, we diagrammed "but not" as AND not.

"Any pet store that sells tropical fish, but not exotic birds, does sell gerbils."
STF & not SEB ==> SG
not SG ==> not STF or SEB.

However, in D, the "No Rule" kicks in. As such, we have to negate the second half of D. Therefore, the but not in the necessary now is transformed by the No at the beginning of the sentence. Therefore, we have the or, which fits with the chaining of the contrapositive outlined above.

Shunhe on July 3, 2020

Hi @Gabby_teixeira and @JoshG,

Thanks for the question! So let me try to help with the diagram of (D). (D) tells us that no independently owned pet store in West Calverton sells tropical fish but not exotic birds. As JoshG mentioned, we’re going to think of “but not” as “and not” here. Remember that when we diagram “No X are Y,” we diagram it as X—>~Y. This is because this statement is equivalent to saying “if X, then not Y.” In this example, it’s “if it’s an independently owned pet store in West Calverton, then it doesn’t sell tropical fish but not exotic birds.”

So we diagram

IOPWC —> ~(TF & ~EB)

Now, when we negate a compound statement, we can negate each term individually, and then switch the & to an v or vice versa. So we can rewrite this

IOPWC —> ~TF v EB

In other words, if it’s an independently owned pet store in West Calverton, either it doesn’t sell tropical fish, or it does sell exotic birds. In other words, if it sells tropical fish, then it doesn’t sell exotic birds; if it doesn’t sell exotic birds, it doesn’t sell tropical fish. You could, of course, also have a pet store that doesn’t sell either which is something you should keep in mind.

Hope this helps! Feel free to ask any other questions that you might have.

Ryan on November 2, 2020

@shunhe thank you!