#22 - In Sheldon most bicyclists aged 18 and over have

[Administrator]

Complete Question Explanation

Parallel Flaw. The correct answer choice is (C)

This is a fantastic question for paralleling key terms in the premise(s) and conclusion. The premise uses the term “most” twice, and the conclusion uses “most” once. Hence, the correct answer choice must mirror this logical structure/language by containing three “most” (or synonymous terms, such as “almost all,” “more than half,” etc.) terms.

The flaw here is a numbers and percentages error: you cannot conclude that most bicyclists that have lights on their bicycles are at least 18 simply because most bicyclists who are at least 18 have lights, whereas most under 18 do not. Consider the following example that meets the premises but yields a different conclusion:

• There are 400 bicyclists in Sheldon: 100 are at least 18, and 300 are under 18.
  
  60 out of 100 bicyclists 18+ yrs old have lights (most have lights).
  
  100 out of 300 bicyclists under 18 yrs have lights (most do not have lights).
  
  In this case, most bicyclists with lights are under 18 (100 vs. 60).

Answer choice (A): The conclusion uses the phrase “fewer than half,” which is not the same as “most.”

Answer choice (B): The conclusion in this answer choice is causal, so it does not match the stimulus.

Answer choice (C): This is the correct answer choice. Note that it also contains three “most” statements, and it contains the same error as the one in the stimulus (numbers and percentages, as described above).

Answer choice (D): The first premise says “every,” which does not parallel the “most” premises in the stimulus.

Answer choice (E): The first premise says “everyone,” which does not parallel the “most” premises in the stimulus.

[taxstonefromthefeds]

When I diagrammed this question, I also noticed a mistaken reversal flaw. I was wondering if my thought process was right.

1) B18+ lights on bike
2) not B18+ not lights on bike
3) Conclusion: lights on bike B18+

Depending on how this is analyzed, the conclusion is a mistaken reversal of statement number one. This flaw is captured in AC C.

[Jonathan Evans]

Hi, Taxstone!

You're right that this is very similar to a Mistaken Reversal flaw! With these "most" statements you cannot travel backwards against the direction of the arrow, with the following exception:

• Most children like ice cream. Most children like pizza. Therefore, there is at least one child who likes both ice cream and pizza.
  
  Children like ice cream
  Children like pizza
  Children like ice cream and pizza

Strictly speaking, Mistaken Reversal refers to the formal conditional fallacy of the converse (or affirming the consequent) and not to quantity statements such as these, but your reasoning is valid.

Good job!

[jennifersuh]

Hi!
I'd like some help understanding this problem. I wrote out the contrapositives like so:

B18+ --> lights
contrapositive: no lights --> not 18+

not 18+ --> no lights
contrapositive: lights --> 18+

Doesn't that second contrapositive make this conclusion valid? I did also see the mistaken reversal issue, but I also saw this and am rather confused.

Additionally, could you explain why D is incorrect? Does it have to do with the way the conclusion is worded?

[Robert Carroll]

Jennifer,

The statements in the stimulus are not conditionals at all, but "most" statements. "Most" statements have some similarities to conditionals, but also some differences:

Similarities:

-"Most" statements have a direction. "A B" does not entail "B A"
-Because of that, reversing a "most" statement is a mistake, similar to Mistaken Reversal for conditionals
-Negating the terms of a "most" statement without changing the direction is a mistake, too. "A B" does not entail "A B".
-Because of that, "most" statements don't allow Mistaken Negations, like conditionals

Difference:

-Reversing the order and negating each element of a "most" statement is not a valid inference. "A B" does not entail "B A"
-Because of that, there is no such thing as a contrapositive of a "most" statement, unlike with conditionals, which are all equivalent to their own contrapositives

Because the premises are "most" statements, it's not valid to infer the reversal, or the contrapositive, of any of them.

As far as answer choice (D), one of the premises of that answer choice is a conditional, and the conclusion lacks a quantitative "most" statement. The stimulus has only "most" statements in each of its two premises and its conclusion. Answer choice (D) therefore fails the "Double the Premises" and "Double the Conclusion" elements for Parallel Reasoning/Parallel Flaw questions.

Robert Carroll

[ashpine17]

How would I go about effectively diagramming (aged 18 and over)? I don't think 18+ is enough because that implies 18 and over but not exactly 18, right? Is this basically an illegal reversal of a most statement? And is the second half of the first sentence useless? (whereas...do not)

[Rosepose24]

I just did a straight match of terms and occurrences:

stimulus:
most over 18 group —> lights on
most under 18 group —> lights off
most with lights on are over 18.

C:
most who voted —> on the list
most who not voted > not on the list
most on the conservative list are residents who voted.

Is this method to match an answer choice appropriate.... I can see analogous terms more quickly than I can analyze or create diagrams with Most --> some statements. Although, I have a firm grasp on both. Is it risky to just jump to "abstraction" as a method for solving before trying the reasoning/ validity/ conclusion/ premise test? Noticing when I look at discussion threads, I'm not immediately breaking it down into diagrams like discussed but still getting the right answer.

[Adam Tyson]

Ashpine, I would definitely use "18+" for shorthand because that does include those who are 18, while "-18" with the minus sign before the number would, for me, clearly indicate those under 18.

And Rosepose, be careful about that exact issue, because in your diagram you DID exclude people who are exactly 18! Not that it mattered in this case, but it might in some other instance. Other than that, great job matching terms here!

Two other ways of looking at this argument without a diagram:

1) Match the strength of the language in the premises and conclusion. In the stimulus, the argument is "most...most...therefore most," and only answer C matches that strength of language in both premises and in the conclusion.

2) Note that this argument is about comparing two distinct groups whose relative sizes are unknown, and then drawing a numeric conclusion that requires knowing the relative sizes. Again, only answer C is comparing two groups in that way and then making that kind of numeric conclusion. Answer A looks at one group doing two different things; answer B does the same (one group - the library - doing two different things); answer D doesn't compare the two types in the conclusion; answer E is again just one group and we are looking at two different activities that some members of the group do.