Each of the candidates in this year's mayoral election is a small-business owner. Most small-business owners are comp...

Theresa on October 1, 2015

Help

Can you please diagram this question and explain why B is correct.

2 Replies

Melody on October 15, 2015

Let's diagram:

"Each of the candidates in this year's mayoral election is a small-business owner."

P1: CYME ==> SBO
not SBO ==> not CYME

Most small-business owners are competent managers.

Q1: SBO-most-CM
CM-some-SBO

"no competent manager lacks the skills necessary to be a good mayor."

P2: CM ==> SNGM
not SNGM ==> not CCM

"So, most of the candidates in this year's mayoral election have the skills necessary to be a good mayor."

C: CYME-most-SNGM
SNGM-some-CYME


As you can see, the argument is incorrectly connecting P1 to Q1. Remember, the correct way to connect a Sufficient & Necessary statement to a Quantifier statement is to connect them through the right-hand side variable of the Quantifier Statement and the sufficient condition of the Sufficient & Necessary statement (which have to be the same variable). Thus, we cannot viably connect P1 and Q1. Further, the argument continues to then incorrectly connect P1 to Q1 to P2 to somehow come up with CYME-most-SNGM, which is in no way correct.

Answer choice (B) has the same reasoning:

"Everything on the menu at Maddy's Shake Shop is fat-free."

P1: MMSS ==> FF
not FF ==> not MMSS

"Most fat-free foods and drinks are sugar-free."

Q1: FF-most-SF
SF-some-FF

"And all sugar-free foods and drinks are low in calories."

P2: SF ==> LC
not LC ==> not SF

"Hence, most items on the menu at Maddy's are low in calories."

C: MMSS-most-LC
LC-some-MMSS

Similarly, answer choice (C) incorrectly connects P1 to Q1, which we know we cannot do, and then continues across the chain to connect P2 to Q1 like so: P1 to Q1 to P2.

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

Andrew on August 15, 2018

For questions such as this in which you have to diagram the statements to see if you have the same structure, I have been finding it easier to diagram the statements in an abstract form instead of using the form presented in the question. ( A->B instead of CYME->SBO) Do you advise against this?