- Fri Oct 16, 2015 5:02 pm
#20229
Hi jonwg5121,
The flaw in this question is a numbers and percentages error, and it’s an important one to be alert to, since it appears relatively frequently in a number of variations. The argument tells us that an indeterminate percentage of weaklings (“all too many”) are cowards, which lets us know that at least one weakling must be a coward, allowing us to say more technically that “some weaklings are cowards,” and to infer that therefore at least one coward must be a weakling (or again that “some cowards are weaklings”).
We also learn that few cowards fail to be fools, meaning that there must be at least one coward who is a fool--which, again, we can state as “some cowards are fools.” (Even if you want to say, instead, that this tells us that most cowards must be fools, note that this is information about the percentage of cowards who are fools, which would not give us the information we need to draw the author’s conclusion without knowing something about the relative sizes of the two groups of weaklings and cowards (see more on this below).)
So all we know about cowards is that some of them are weaklings and some of them (possibly most of them) are fools. But we have no knowledge about the sizes of these subgroups within the entire group of cowards, and thus have no way to know whether the two subgroups overlap with one another. By contrast, if we knew that “most” cowards (51% or more) were weaklings and “most” cowards (51% or more) were fools, then it would be impossible for these two subgroups of cowards not to overlap, since each would constitute more than half of the entire group. But based on what we actually do know, there could be 100 cowards out there, 10 of whom are weaklings and 60 of whom are fools, but none would need to be both--so what we are actually told about cowards doesn’t numerically entitle us to infer that there is anyone among the cowards who is both a weakling and a fool. We would need different information about the sizes of these two subgroups to reach that conclusion.
The stimulus author tries to circumvent this weakness by erroneously suggesting that, if “all too many” weaklings are cowards, we can infer that this makes up a large enough percentage of the total group of cowards that it must overlap with at least one instance of those cowards who are fools. But there are two things wrong with this reasoning. First of all, we don’t really know what “all too many” means as a percentage of the group of weaklings--“too many” is really so subjective that it might only mean one. Second, and far more importantly, we have absolutely no information about the relative sizes of the groups of weaklings and cowards. So even if we knew that every weakling was a coward, and most cowards were fools, we still couldn’t infer, as the stimulus does, that there was necessarily an overlap between those groups. If, for example, there were 40 weaklings, every one of whom was a coward, and a total group of 100 cowards, most of whom (e.g., 51) were fools, we would have no way of knowing whether there was any overlap between these groups of 40 and 51 within the total group of 100.
Without that missing vital information about the relative group sizes of weaklings and cowards, even knowing something so extreme about percentages as that every weakling (100%) is a coward and that most cowards (51% or more) are fools would not allow us to know whether the groups of cowards who are weaklings and cowards who are fools actually overlap. The stimulus doesn’t get us that far regarding percentages, but it is important to see that even if it did, that would not be a basis for drawing the author’s conclusion about an actual overlap. So the primary error here is in assuming that simply knowing something about percentages can tell you something definite about actual numbers.
In abstract terms, the stimulus tells us this:
Some (or many) As are Bs
Some (or most) Bs are Cs
Thus, there must be at least one A who is a C
Answer choice (C) says the same thing, just rearranging the terms, and thus makes the same error as the stimulus. But answer choice (E) says something crucially different. If you slightly adjust your diagram for (E) to present things more numerically, it makes it easier to see why the reasoning in (E) is actually correct:
P1: Majority voting population-->favor total ban [in other words, MOST voters favor the total ban, or most As are Bs]
P2: Favor total ban-->not opposed to stiffer tariff [in other words, EVERY person who favors the total ban is not opposed to stiffer tariffs, or all Bs are Cs]
C: At least one person-->Not opposed to stiffer tariff [thus, in other words, at least one one voter must be not opposed to stiffer tariffs, or at least one A must be a C]
Knowing that the whole group of people who favor the total ban also are not opposed to stiffer tariffs allows us to conclude that anyone who falls into that group has the characteristic that is shared by everyone in the group. In other words, because every B has to be a C, if there is at least one A that is a B, that A must also be a C (since all Bs are Cs), so (E) is correct in concluding that at least one voter must not be opposed to the ban. Knowing something absolute about the group of people who favor the total ban takes us out of the uncertain situation set up by the stimulus, where we don't know anything about the entire group of Bs and thus can't extrapolate what we do know to As, just because some (or even many) As happen to be part of the group of Bs.
I hope this helps!
Laura
/CA (courage to act...)