ARGUMENTS INTRODUCTION
An argument, as used on the GMAT, is a presentation of facts and opinions in
order to support a position. Many arguments will be fallacious. And many co
rrect answers will be false! This often causes students much consternation;
they feel that the correct answer should be true. But the arguments are inte
nded to test your ability to think logically. Now logic is the study of the
relationships between statements, not of the truth of those statements. Bein
g overly concerned with finding the truth can be ruinous to your GMAT argume
nt score.
"2 OUT OF 5" RULE
Creating a good but incorrect answer-choice is much harder than developing t
he correct answer. For this reason, usually only one attractive wrong answer
-choice is presented. This is called the "2 out of 5" rule. That is, only tw
o of the five answer-choices will have any real merit. Hence, even if you do
n’t fully understand an argument, you probably can still eliminate the three
fluff choices, thereby greatly increasing your odds of answering the questi
on correctly.
LOGIC I
Although in theory the argument questions are designed to be answered withou
t any reference to formal logic, the section is essentially a logic test. So
me knowledge of the fundamentals of logic, therefore, will give you a defini
te advantage. Armed with this knowledge, you should quickly notice that the
arguments are fundamentally easy and that most of them fallsintosa few basic
categories. In this section, we will study the logical structure of argumen
ts. In Logic II, we will symbolize and diagram arguments in much the same wa
y as we did with games.
Conclusions
Most argument questions hinge, either directly or indirectly, on determining
the conclusion of the argument. The conclusion is the main idea of the argu
ment. It is what the writer tries to persuade the reader to believe. Most of
ten the conclusion comes at the end of the argument. The writer organizes th
e facts and his opinions so that they build up to the conclusion. Sometimes,
however, the conclusion will come at the beginning of an argument, rarely d
oes it come in the middle, and occasionally, for rhetorical effect, the conc
lusion is not even stated.
Example:
The police are the armed guardians of the social order. The blacks are the c
hief domestic victims of the American social order. A conflict of interest e
xists, therefore, between the blacks and the police.--Eldridge Cleaver, Soul
on Ice
Here the first two sentences anticipate or set up the conclusion. By changin
g the grammar slightly, the conclusion can be placed at the beginning of the
argument and still sound natural:
A conflict of interest exists between the blacks and the police because the
police are the armed guardians of the socialsgroupsand the blacks are the ch
ief domestic victims of the American social order.
The conclusion can also be forcedsintosthe middle:
The police are the armed guardians of the social order. So a conflict of int
erest exists between the blacks and the police because the blacks are the ch
ief domestic victims of the American social order.
It is generally awkward, as in the previous paragraph, to place the conclusi
on in the middle of the argument because then it cannot be fully anticipated
by what comes before nor fully explained by what comes after. On the rare o
ccasion when a conclusion comes in the middle of an argument, most often eit
her the material that comes after it or the material that comes before it is
not essential.
In summary: To find the conclusion, check the last sentence of the argument.
If that is not the conclusion, check the first sentence. Rarely does the co
nclusion come in the middle of an argument.
When determining the meaning of a conclusion, be careful not to read any mor
esintosit than what the author states. Although arguments are not worded as
precisely as games, you still need to read them with more care than you woul
d use in your everyday reading.
As with games, read the words and sentences of an argument precisely, and us
e their literal meaning.
For example, consider the meaning of some in the sentence "Some of Mary’s fr
iends went to the party." It would be unwarranted, based on this statement,
to assume that some of Mary’s friends did not go to the party. Although it m
ay seem deceiving to say that some of Mary’s friends went to the party when
in fact all of them did, it is nonetheless technically consistent with the m
eaning of some.
Some means "at least one and perhaps all."
As mentioned before, the conclusion usually comes at the end of an argument,
sometimes at the beginning, and rarely in the middle. Writers use certain w
ords to indicate that the conclusion is about to be stated. Following is a l
ist of the most common conclusion indicators:
Conclusion Indicators
hence therefore
so accordingly
thus consequently
follows that shows that
conclude that implies
as a result means that
Most often the conclusion of an argument is put in the form of a statement.
Sometimes, however, the conclusion is given as a command or obligation.
Example:
All things considered, you ought to vote.
Here, the author implies that you are obliged to vote.
The conclusion can even be put in the form of a question. This rhetorical te
chnique is quite effective in convincing people that a certain position is c
orrect. We are more likely to believe something if we feel that we concluded
it on our own, or at least if we feel that we were not told to believe it.
A conclusion put in question form can have this result.
Example:
The Nanuuts believe that they should not take from Nature anything She canno
t replenish during their lifetime. This assures that future generations can
enjoy the same riches of Nature that they have. At the current rate of destr
uction, the rain forests will disappear during our lifetime. Do we have an o
bligation to future generations to prevent this result?
Here the author trusts that the power of her argument will persuade the read
er to answer the question affirmatively.
Taking this rhetorical technique one step further, the writer may build up t
o the conclusion but leave it unstated. This allows the reader to make up hi
s own mind. If the build-up is done skillfully, the reader will be more like
ly to agree with the author, without feeling manipulated.
Example:
He who is without sin should cast the first stone. There is no one here who
does not have a skeleton in his closet.
The unstated but obvious conclusion here is that none of the people has the
right to cast the first stone.
When determining the conclusion’s scope be careful not to read any more or l
esssintosit than the author states. GMAT writers often create wrong answer-c
hoices by slightly overstating or understating the author’s claim. Certain w
ords limit the scope of a statement. These words are called quantifiers--pay
close attention to them. Following is a list of the most important quantifi
ers:
Quantifiers
all except likely
some most many
only could no
never always everywhere
probably must alone
Example:
Whether the world is Euclidean or non-Euclidean is still an open question.
However, if a star’s position is predicted based on non-Euclidean geometry,
then when a telescope is pointed toswheresthe star should be it will be ther
e. Whereas, if the star’s position is predicted based on Euclidean geometry,
then when a telescope is pointed toswheresthe star should be it won’t be th
ere. This strongly indicates that the world is non-Euclidean.
Which one of the following best expresses the main idea of the passage?
(A) The world may or may not be Euclidean.
(B) The world is probably non-Euclidean.
(C) The world is non-Euclidean.
(D) The world is Euclidean.
(E) The world is neither Euclidean nor non-Euclidean.
Choice (A) understates the main idea. Although the opening to the passage st
ates that we don’t know whether the world is non-Euclidean, the author goes
on to give evidence that it is non-Euclidean. Choice (C) overstates the main
idea. The author doesn’t say that the world is non-Euclidean, just that evi
dence strongly indicates that it is. In choice (B), the word "probably" prop
erly limits the scope of the main idea, namely, that the world is probably n
on-Euclidean, but we can’t yet state so definitively. The answer is (B).
Premises
Once you’ve found the conclusion, most often everything else in the argument
will be either premises or "noise." The premises provide evidence for the c
onclusion; they form the foundation or infrastructure upon which the conclus
ion depends. To determine whether a statement is a premise, ask yourself whe
ther it supports the conclusion. If so, it’s a premise. Earlier we saw that
writers use certain words to flag conclusions; likewise writers use certain
words to flag premises. Following is a partial list of the most common premi
se indicators:
Premise Indicators
because for
since is evidence that
if in that
as owing to
suppose inasmuch as
assume may be derived from
Example:
Since the incumbent’s views are out of step with public opinion, he probably
will not be reelected.
Here "since" is used to flag the premise that the incumbent’s positions are
unpopular.
Suppressed Premises
Most arguments depend on one or more unstated premises. Sometimes this indic
ates a weakness in the argument, an oversight by the writer. More often, how
ever, certain premises are left tacit because they are too numerous, or the
writer assumes that his audience is aware of the assumptions, or he wants th
e audience to fill in the premise themselves and therefore be more likely to
believe the conclusion.
Example:
Conclusion: I knew he did it.
Premise: Only a guilty person would accept immunity from prosecution.
The suppressed premise is that he did, in fact, accept immunity. The speaker
assumes that his audience is aware of this fact or at least is willing to b
elieve it, so to state it would be redundant and ponderous. If the unstated
premise were false (that is, he did not accept immunity), the argument would
not technically be a lie; but it would be very deceptive. The unscrupulous
writer may use this ploy if he thinks that he can get away with it. That is,
his argument has the intended effect and the false premise, though implicit
, is hard to find or is ambiguous. Politicians are not at all above using th
is tactic.
A common question on the GMAT asks you to find the suppressed premise of an
argument. Finding the suppressed premise, or assumption, of an argument can
be difficult. However, on the GMAT you have an advantage--the suppressed pre
mise is listed as one of the five answer-choices. To test whether an answer-
choice is a suppressed premise, ask yourself whether it would make the argum
ent more plausible. If so, then it is very likely a suppressed premise.
Example:
American attitudes tend to be rather insular, but there is much we can learn
from other countries. In Japan, for example, workers set aside some time ea
ch day to exercise, and many corporations provide elaborate exercise facilit
ies for their employees. Few American corporations have such exercise progra
ms. Studies have shown that the Japanese worker is more productive than the
American worker. Thus it must be concluded that the productivity of American
workers will lag behind their Japanese counterparts, until mandatory exerci
se programs are introduced.
The conclusion of the argument is valid if which one of the following is ass
umed?
(A) Even if exercise programs do not increase productivity, they will improv
e the American worker’s health.
(B) The productivity of all workers can be increased by exercise.
(C) Exercise is an essential factor in the Japanese worker’s superior produc
tivity.
(D) American workers can adapt to the longer Japanese work week.
(E) American corporations don’t have the funds to build elaborate exercise f
acilities.
The unstated essence of the argument is that exercise is an integral part of
productivity and that Japanese workers are more productive than American wo
rkers because they exercise more. The answer is (C).
Counter-Premises
When presenting a position, you obviously don’t want to argue against yourse
lf. However, it is often effective to concede certain minor points that weak
en your argument. This shows that you are open-minded and that your ideas ar
e well considered. It also disarms potential arguments against your position
.. For instance, in arguing for a strong, aggressive police department, you m
ay concede that in the past the police have at times acted too aggressively.
Of course, you will then need to state more convincing reasons to support y
our position.
Example:
I submit that the strikers should accept the management’s offer. Admittedly,
it is less than what was demanded. But it does resolve the main grievance--
inadequate health care. Furthermore, an independent study shows that a wage
increase greater than 5% would leave the company unable to compete against J
apan and Germany, forcing itsintosbankruptcy.
The conclusion, "the strikers should accept the management’s offer," is stat
ed in the first sentence. Then "Admittedly" introduces a concession; namely,
that the offer was less than what was demanded. This weakens the speaker’s
case, but it addresses a potential criticism of his position before it can b
e made. The last two sentences of the argument present more compelling reaso
ns to accept the offer and form the gist of the argument.
Following are some of the most common counter-premise indicators:
Counter-Premise Indicators
but despite
admittedly except
even though nonetheless
nevertheless although
however in spite of the fact
As you may have anticipated, the GMAT writers sometimes use counter-premises
to bait wrong answer-choices. Answer-choices that refer to counter-premises
are very tempting because they refer directly to the passage and they are i
n part true. But you must ask yourself "Is this the main point that the auth
or is trying to make?" It may merely be a minor concession.
Logic II (Diagramming)
Most arguments are based on some variation of an if-then statement. However,
the if-then statement is often embedded in other equivalent structures. Dia
gramming brings out the superstructure and the underlying simplicity of argu
ments.
If-Then
A-->B
By now you should be well aware that if the premise of an if-then statement
is true then the conclusion must be true as well. This is the defining chara
cteristic of a conditional statement; it can be illustrated as follows:
A-->B
A
Therefore, B
This diagram displays the if-then statement "A-->B," the affirmed premise "A
," and the necessary conclusion "B." Such a diagram can be very helpful in s
howing the logical structure of an argument.
Example: (If-then)
If Jane does not study for the GMAT, then she will not score well. Jane, in
fact, did not study for the GMAT; therefore she scored poorly on the test.
When symbolizing games, we let a letter stand for an element. When symbolizi
ng arguments, however, we may let a letter stand for an element, a phrase, a
clause, or even an entire sentence. The clause "Jane does not study for the
GMAT" can be symbolized as ~S, and the clause "she will not score well" can
be symbolized as ~W. Substituting these symbolssintosthe argument yields th
e following diagram:
~S-->~W
~S
Therefore, ~W
This diagram shows that the argument has a valid if-then structure. A condit
ional statement is presented, ~S-->~W; its premise affirmed, ~S; and then th
e conclusion that necessarily follows, ~W, is stated.
Embedded If-Then Statements
Usually, arguments involve an if-then statement. Unfortunately, the if-then
thought is often embedded in other equivalent structures. In this section, w
e study how to spot these structures.
Example: (Embedded If-then)
John and Ken cannot both go to the party.
At first glance, this sentence does not appear to contain an if-then stateme
nt. But it essentially says: "if John goes to the party, then Ken does not."
Example: (Embedded If-then)
Danielle will be accepted to graduate school only if she does well on the GR
E.
Given this statement, we know that if Danielle is accepted to graduate schoo
l, then she must have done well on the GRE. Note: Students often wrongly int
erpret this statement to mean:
"If Danielle does well on the GRE, then she will be accepted to graduate sch
ool."
There is no such guarantee. The only guarantee is that if she does not do we
ll on the GRE, then she will not be accepted to graduate school.
"A only if B" is logically equivalent to "if A, then B."
Affirming the Conclusion Fallacy
A-->B
B
Therefore, A
Remember that an if-then statement, A-->B, tells us only two things: (1) If
A is true, then B is true as well. (2) If B is false, then A is false as wel
l (contrapositive). If, however, we know the conclusion is true, the if-then
statement tells us nothing about the premise. And if we know that the premi
se is false (we will consider this next), then the if-then statement tells u
s nothing about the conclusion.
Example: (Affirming the Conclusion Fallacy)
If he is innocent, then when we hold him under water for sixty seconds he wi
ll not drown. Since he did not die when we dunked him in the water, he must
be innocent.
The logical structure of the argument above is most similar to which one of
the following?
(A) To insure that the remaining wetlands survive, they must be protected by
the government. This particular wetland is being neglected. Therefore, it w
ill soon perish.
(B) There were nuts in that pie I just ate. There had to be, because when I
eat nuts I break out in hives, and I just noticed a blemish on my hand.
(C) The president will be reelected unless a third candidate enters the race
.. A third candidate has entered the race, so the president will not be reele
cted.
(D) Every time Melinda has submitted her book for publication it has been re
jected. So she should not bother with another rewrite.
(E) When the government loses the power to tax one area of the economy, it j
ust taxes another. The Supreme Court just overturned the sales tax, so we ca
n expect an increase in the income tax.
To symbolize this argument, let the clause "he is innocent" be denoted by I,
and let the clause "when we hold him under water for sixty seconds he will
not drown" be denoted by ~D. Then the argument can be symbolized as
I-->~D
~D
Therefore, I
Notice that this argument is fallacious: the conclusion "he is innocent" is
also a premise of the argument. Hence the argument is circular--it proves wh
at was already assumed. The argument affirms the conclusion then invalidly u
ses it to deduce the premise. The answer will likewise be fallacious.
We start with answer-choice (A). The sentence
"To insure that the remaining wetlands survive, they must be protected by th
e government"
contains an embedded if-then statement:
"If the remaining wetlands are to survive, then they must be protected by th
e government."
This can be symbolized as S-->P. Next, the sentence "This particular wetland
is being neglected" can be symbolized as ~P. Finally, the sentence "It will
soon perish" can be symbolized as ~S. Using these symbols to translate the
argument gives the following diagram:
S-->P
~P
Therefore, ~S
The diagram clearly shows that this argument does not have the same structur
e as the given argument. In fact, it is a valid argument by contraposition.
Turning to (B), we reword the statement "when I eat nuts, I break out in hiv
es" as
"If I eat nuts, then I break out in hives." This in turn can be symbolized a
s N-->H.
Next, we interpret the clause "there is a blemish on my hand" to mean "hives
," which we symbolize as H. Substituting these symbolssintosthe argument yie
lds the following diagram:
N-->H
H
Therefore, N
The diagram clearly shows that this argument has the same structure as the g
iven argument. The answer, therefore, is (B).
Denying the Premise Fallacy
A-->B
~A
Therefore, ~B
The fallacy of denying the premise occurs when an if-then statement is prese
nted, its premise denied, and then its conclusion wrongly negated.
Example: (Denying the Premise Fallacy)
The senator will be reelected only if he opposes the new tax bill. But he wa
s defeated. So he must have supported the new tax bill.
The sentence "The senator will be reelected only if he opposes the new tax b
ill" contains an embedded if-then statement: "If the senator is reelected, t
hen he opposes the new tax bill." (Remember: "A only if B" is equivalent to
"If A, then B.") This in turn can be symbolized as R-->~T. The sentence "But
the senator was defeated" can be reworded as "He was not reelected," which
in turn can be symbolized as ~R. Finally, the sentence "He must have support
ed the new tax bill" can be symbolized as T. Using these symbols the argumen
t can be diagrammed as follows:
R-->~T
~R
Therefore, T
[Note: Two negatives make a positive, so the conclusion ~(~T) was reduced to
T.] This diagram clearly shows that the argument is committing the fallacy
of denying the premise. An if-then statement is made; its premise is negated
; then its conclusion is negated.
Transitive Property
A-->B
B-->C
Therefore, A-->C
These arguments are rarely difficult, provided you step back and take a bir
d’s-eye view. It may be helpful to view this structure as an inequality in m
athematics. For example, 5 > 4 and 4 > 3, so 5 > 3.
Notice that the conclusion in the transitive property is also an if-then sta
tement. So we don’t know that C is true unless we know that A is true. Howev
er, if we add the premise "A is true" to the diagram, then we can conclude t
hat C is true:
A-->B
B-->C
A
Therefore, C
As you may have anticipated, the contrapositive can be generalized to the tr
ansitive property:
A-->B
B-->C
~C
Therefore, ~A
Example: (Transitive Property)
If you work hard, you will be successful in America. If you are successful i
n America, you can lead a life of leisure. So if you work hard in America, y
ou can live a life of leisure.
Let W stand for "you work hard," S stand for "you will be successful in Amer
ica," and L stand for "you can lead a life of leisure." Now the first senten
ce translates as W-->S, the second sentence as S-->L, and the conclusion as
W-->L. Combining these symbol statements yields the following diagram:
W-->S
S-->L
Therefore, W-->L
The diagram clearly displays the transitive property.
DeMorgan’s Laws
~(A & B) = ~A or ~B
~(A or B) = ~A & ~B
If you have taken a course in logic, you are probably familiar with these fo
rmulas. Their validity is intuitively clear: The conjunction A&B is false wh
en either, or both, of its parts are false. This is precisely what ~A or ~B
says. And the disjunction A or B is false only when both A and B are false,
which is precisely what ~A and ~B says.
You will rarely get an argument whose main structure is based on these rules
--they are too mechanical. Nevertheless, DeMorgan’s laws often help simplify
, clarify, or transform parts of an argument. They are also useful with game
s.
Example: (DeMorgan’s Law)
It is not the case that either Bill or Jane is going to the party.
This argument can be diagrammed as ~(B or J), which by the second of DeMorga
n’s laws simplifies to (~B and ~J). This diagram tells us that neither of th
em is going to the party.
A unless B
~B-->A
"A unless B" is a rather complex structure. Though surprisingly we use it wi
th little thought or confusion in our day-to-day speech.
To see that "A unless B" is equivalent to "~B-->A," consider the following s
ituation:
Biff is at the beach unless it is raining.
Given this statement, we know that if it is not raining, then Biff is at the
beach. Now if we symbolize "Biff is at the beach" as B, and "it is raining"
as R, then the statement can be diagrammed as ~R-->B.
CLASSIFICATION
In Logic II, we studied deductive arguments. However, the bulk of arguments
on the GMAT are inductive. In this section we will classify and study the ma
jor types of inductive arguments.
An argument is deductive if its conclusion necessarily follows from its prem
ises--otherwise it is inductive. In an inductive argument, the author presen
ts the premises as evidence or reasons for the conclusion. The validity of t
he conclusion depends on how compelling the premises are. Unlike deductive a
rguments, the conclusion of an inductive argument is never certain. The trut
h of the conclusion can range from highly likely to highly unlikely. In reas
onable arguments, the conclusion is likely. In fallacious arguments, it is i
mprobable. We will study both reasonable and fallacious arguments.
We will classify the three major types of inductive reasoning--generalizatio
n, analogy, and causal--and their associated fallacies.
Generalization
Generalization and analogy, which we consider in the next section, are the m
ain tools by which we accumulate knowledge and analyze our world. Many peopl
e define generalization as "inductive reasoning." In colloquial speech, the
phrase "to generalize" carries a negative connotation. To argue by generaliz
ation, however, is neither inherently good nor bad. The relative validity of
a generalization depends on both the context of the argument and the likeli
hood that its conclusion is true. Polling organizations make predictions by
generalizing information from a small sample of the population, which hopefu
lly represents the general population. The soundness of their predictions (a
rguments) depends on how representative the sample is and on its size. Clear
ly, the less comprehensive a conclusion is the more likely it is to be true.
Example:
During the late seventies when Japan was rapidly expanding its share of the
American auto market, GM surveyed owners of GM cars and asked them whether t
hey would be more willing to buy a large, powerful car or a small, economica
l car. Seventy percent of those who responded said that they would prefer a
large car. On the basis of this survey, GM decided to continue building larg
e cars. Yet during the’80s, GM lost even more of the market to the Japanese
..
Which one of the following, if it were determined to be true, would best exp
lain this discrepancy.
(A) Only 10 percent of those who were polled replied.
(B) Ford which conducted a similar survey with similar results continued to
build large cars and also lost more of their market to the Japanese.
(C) The surveyed owners who preferred big cars also preferred big homes.
(D) GM determined that it would be more profitable to make big cars.
(E) Eighty percent of the owners who wanted big cars and only 40 percent of
the owners who wanted small cars replied to the survey.
The argument generalizes from the survey to the general car-buying populatio
n, so the reliability of the projection depends on how representative the sa
mple is. At first glance, choice (A) seems rather good, because 10 percent d
oes not seem large enough. However, political opinion polls are typically ba
sed on only .001 percent of the population. More importantly, we don’t know
what percentage of GM car owners received the survey. Choice (B) simply stat
es that Ford made the same mistake that GM did. Choice (C) is irrelevant. Ch
oice (D), rather than explaining the discrepancy, gives even more reason for
GM to continue making large cars. Finally, choice (E) points out that part
of the survey did not represent the entire public, so (E) is the answer.
Analogy
To argue by analogy is to claim that because two things are similar in some
respects, they will be similar in others. Medical experimentation on animals
is predicated on such reasoning. The argument goes like this: the metabolis
m of pigs, for example, is similar to that of humans, and high doses of sacc
harine cause cancer in pigs. Therefore, high doses of saccharine probably ca
use cancer in humans.
Clearly, the greater the similarity between the two things being compared th
e stronger the argument will be. Also the less ambitious the conclusion the
stronger the argument will be. The argument above would be strengthened by c
hanging "probably" to "may." It can be weakened by pointing out the dissimil
arities between pigs and people.
Example:
Just as the fishing line becomes too taut, so too the trials and tribulation
s of life in the city can become so stressful that one’s mind can snap.
Which one of the following most closely parallels the reasoning used in the
argument above?
(A) Just as the bow may be drawn too taut, so too may one’s life be wasted p
ursuing self-gratification.
(B) Just as a gambler’s fortunes change unpredictably, so too do one’s caree
r opportunities come unexpectedly.
(C) Just as a plant can be killed by over watering it, so too can drinking t
oo much water lead to lethargy.
(D) Just as the engine may race too quickly, so too may life in the fast lan
e lead to an early death.
(E) Just as an actor may become stressed before a performance, so too may dw
elling on the negative cause depression.
The argument compares the tautness in a fishing line to the stress of city l
ife; it then concludes that the mind can snap just as the fishing line can.
So we are looking for an answer-choice that compares two things and draws a
conclusion based on their similarity. Notice that we are looking for an argu
ment that uses similar reasoning, but not necessarily similar concepts. In f
act, an answer-choice that mentions either tautness or stress will probably
be a same-language trap.
Choice (A) uses the same-language trap--notice "too taut." The analogy betwe
en a taut bow and self-gratification is weak, if existent. Choice (B) offers
a good analogy but no conclusion. Choice (C) offers both a good analogy and
a conclusion; however, the conclusion, "leads to lethargy," understates the
scope of what the analogy implies. Choice (D) offers a strong analogy and a
conclusion with the same scope found in the original: "the engine blows, th
e person dies"; "the line snaps, the mind snaps." This is probably the best
answer, but still we should check every choice. The last choice, (E), uses l
anguage from the original, "stressful," to make its weak analogy more tempti
ng. The best answer, therefore, is (D).
Causal Reasoning
Of the three types of inductive reasoning we will discuss, causal reasoning
is both the weakest and the most prone to fallacy. Nevertheless, it is a us
eful and common method of thought.
To argue by causation is to claim that one thing causes another. A causal ar
gument can be either weak or strong depending on the context. For example, t
o claim that you won the lottery because you saw a shooting star the night b
efore is clearly fallacious. However, most people believe that smoking cause
s cancer because cancer often strikes those with a history of cigarette use.
Although the connection between smoking and cancer is virtually certain, as
with all inductive arguments it can never be 100 percent certain. Cigarette
companies have claimed that there may be a genetic predisposition in some p
eople to both develop cancer and crave nicotine. Although this claim is high
ly improbable, it is conceivable.
There are two common fallacies associated with causal reasoning:
1. Confusing Correlation with Causation.
To claim that A caused B merely because A occurred immediately before B is c
learly questionable. It may be only coincidental that they occurred together
, or something else may have caused them to occur together. For example, the
fact that insomnia and lack of appetite often occur together does not mean
that one necessarily causes the other. They may both be symptoms of an under
lying condition.
2. Confusing Necessary Conditions with Sufficient Conditions.
A is necessary for B means "B cannot occur without A." A is sufficient for B
means "A causes B to occur, but B can still occur without A." For example,
a small tax base is sufficient to cause a budget deficit, but excessive spen
ding can cause a deficit even with a large tax base. A common fallacy is to
assume that a necessary condition is sufficient to cause a situation. For ex
ample, to win a modern war it is necessary to have modern, high-tech equipme
nt, but it is not sufficient, as Iraq discovered in the Persian Gulf War.
SEVEN COMMON FALLACIES
Contradiction
A Contradiction is committed when two opposing statements are simultaneously
asserted. For example, saying "it is raining and it is not raining" is a co
ntradiction. Typically, however, the arguer obscures the contradiction to th
e point that the argument can be quite compelling. Take, for instance, the f
ollowing argument:
"We cannot know anything, because we intuitively realize that our thoughts a
re unreliable."
This argument has an air of reasonableness to it. But "intuitively realize"
means "to know." Thus the arguer is in essence saying that we know that we d
on’t know anything. This is self-contradictory.
Equivocation
Equivocation is the use of a word in more than one sense during an argument.
This technique is often used by politicians to leave themselves an "out." I
f someone objects to a particular statement, the politician can simply claim
the other meaning.
Example:
Individual rights must be championed by the government. It is right for one
to believe in God. So government should promote the belief in God.
In this argument, right is used ambiguously. In the phrase "individual right
s" it is used in the sense of a privilege, whereas in the second sentence ri
ght is used to mean proper or moral. The questionable conclusion is possible
only if the arguer is allowed to play with the meaning of the critical word
right.
Circular Reasoning
Circular reasoning involves assuming as a premise that which you are trying
to prove. Intuitively, it may seem that no one would fall for such an argume
nt. However, the conclusion may appear to state something additional, or the
argument may be so long that the reader may forget that the conclusion was
stated as a premise.
Example:
The death penalty is appropriate for traitors because it is right to execute
those who betray their own country and thereby risk the lives of millions.
This argument is circular because "right" means essentially the same thing a
s "appropriate." In effect, the writer is saying that the death penalty is a
ppropriate because it is appropriate.
Shifting The Burden Of Proof
It is incumbent on the writer to provide evidence or support for her positio
n. To imply that a position is true merely because no one has disproved it i
s to shift the burden of proof to others.
Example:
Since no one has been able to prove God’s existence, there must not be a God
..
There are two major weaknesses in this argument. First, the fact that God’s
existence has yet to be proven does not preclude any future proof of existen
ce. Second, if there is a God, one would expect that his existence is indepe
ndent of any proof by man.
Unwarranted Assumptions
The fallacy of unwarranted assumption is committed when the conclusion of an
argument is based on a premise (implicit or explicit) that is false or unwa
rranted. An assumption is unwarranted when it is false--these premises are u
sually suppressed or vaguely written. An assumption is also unwarranted when
it is true but does not apply in the given context--these premises are usua
lly explicit.
Example: (False Dichotomy)
Either restrictions must be placed on freedom of speech or certain subversiv
e elements in society will use it to destroy this country. Since to allow th
e latter to occur is unconscionable, we must restrict freedom of speech.
The conclusion above is unsound because
(A) subversives do not in fact want to destroy the country
(B) the author places too much importance on the freedom of speech
(C) the author fails to consider an accommodation between the two alternativ
es
(D) the meaning of "freedom of speech" has not been defined
(E) subversives are a true threat to our way of life
The arguer offers two options: either restrict freedom of speech, or lose th
e country. He hopes the reader will assume that these are the only options a
vailable. This is unwarranted. He does not state how the so-called "subversi
ve elements" would destroy the country, nor for that matter, why they would
want to destroy it. There may be a third option that the author did not ment
ion; namely, that society may be able to tolerate the "subversives" and it m
ay even be improved by the diversity of opinion they offer. The answer is (C
).
Appeal To Authority
To appeal to authority is to cite an expert’s opinion as support for one’s o
wn opinion. This method of thought is not necessarily fallacious. Clearly, t
he reasonableness of the argument depends on the "expertise" of the person b
eing cited and whether she is an expert in a field relevant to the argument.
Appealing to a doctor’s authority on a medical issue, for example, would be
reasonable; but if the issue is about dermatology and the doctor is an orth
opedist, then the argument would be questionable.
Personal Attack
In a personal attack (ad hominem), a person’s character is challenged instea
d of her opinions.
Example:
Politician: How can we trust my opponent to be true to the voters? He isn’t
true to his wife!
This argument is weak because it attacks the opponent’s character, not his p
ositions. Some people may consider fidelity a prerequisite for public office
.. History, however, shows no correlation between fidelity and great politica
l leadership.
--
I would fly you to the moon and back
If you’ll be if you’ll be my baby
Got a ticket for a worldswhereswe belong
So would you be my baby
Testprep充分性精解轉(zhuǎn)載smth 2001-10-14 10:51:58發(fā)信人: ykk (我不說話并不代表我不在乎),信區(qū): EnglishTest
標(biāo)題: (GMAT)Testprep充分性精解
發(fā)信站: BBS水木清華站(Fri Oct 12 16:07:05 2001)
Data Sufficiency
----------------------------------------------------------------------------
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INTRODUCTION DATA SUFFICIENCY
Most people have much more difficulty with the Data Sufficiency problems tha
n with the Standard Math problems. However, the mathematical knowledge and s
kill required to solve Data Sufficiency problems is no greater than that req
uired to solve standard math problems. What makes Data Sufficiency problems
appear harder at first is the complicated directions. But once you become fa
miliar with the directions, you’ll find these problems no harder than standa
rd math problems. In fact, people usually become proficient more quickly on
Data Sufficiency problems.
THE DIRECTIONS
The directions for Data Sufficiency questions are rather complicated. Before
reading any further, take some time to learn the directions cold. Some of t
he wording in the directions below has been changed from the GMAT to make it
clearer. You should never have to look at the instructions during the test.
Directions: Each of the following Data Sufficiency problems contains a quest
ion followed by two statements, numbered (1) and (2). You need not solve the
problem; rather you must decide whether the information given is sufficient
to solve the problem.
The correct answer to a question is
A if statement (1) ALONE is sufficient to answer the question but statement
(2) alone is not sufficient;
B if statement (2) ALONE is sufficient to answer the question but statement
(1) alone is not sufficient;
C if the two statements TAKEN TOGETHER are sufficient to answer the question
, but NEITHER statement ALONE is sufficient;
D if EACH statement ALONE is sufficient to answer the question;
E if the two statements TAKEN TOGETHER are still NOT sufficient to answer th
e question.
Numbers: Only real numbers are used. That is, there are no complex numbers.
Drawings: The drawings are drawn to scale according to the information given
in the question, but may conflict with the information given in statements
(1) and (2).
You can assume that a line that appears straight is straight and that angle
measures cannot be zero.
You can assume that the relative positions of points, angles, and objects ar
e as shown.
All drawings lie in a plane unless stated otherwise.
Example:
In triangle ABC to the right, what is the value of y?
(1) AB = AC
(2) x = 30
Explanation: By statement (1), triangle ABC is isosceles. Hence, its base an
gles are equal: y = z. Since the angle sum of a triangle is 180 degrees, we
get x + y + z = 180. Replacing z with y in this equation and then simplifyin
g yields x + 2y = 180. Since statement (1) does not give a value for x, we c
annot determine the value of y from statement (1) alone. By statement (2), x
= 30. Hence, x + y + z = 180 becomes 30 + y + z = 180, or y + z = 150. Sinc
e statement (2) does not give a value for z, we cannot determine the value o
f y from statement (2) alone. However, using both statements in combination,
we can find both x and z and therefore y. Hence, the answer is C.
Notice in the above example that the triangle appears to be a right triangle
.. However, that cannot be assumed: angle A may be 89 degrees or 91 degrees,
we can’t tell from the drawing. You must be very careful not to assume any m
ore than what is explicitly given in a Data Sufficiency problem.
ELIMINATION
Data Sufficiency questions provide fertile ground for elimination. In fact,
it is rare that you won’t be able to eliminate some answer-choices. Remember
, if you can eliminate at least one answer choice, the odds of gaining point
s by guessing are in your favor.
The following table summarizes how elimination functions with Data Sufficien
cy problems.
Statement Choices Eliminated
(1) is sufficient B, C, E
(1) is not sufficient A, D
(2) is sufficient A, C, E
(2) is not sufficient B, D
(1) is not sufficient and (2) is not sufficient A, B, D
Example 1: What is the 1st term in sequence S?
(1) The 3rd term of S is 4.
(2) The 2nd term of S is three times the 1st, and the 3rd term is four times
the 2nd.
(1) is no help in finding the first term of S. For example, the following se
quences each have 4 as their third term, yet they have different first terms
:
0, 2, 4
-4, 0, 4
This eliminates choices A and D. Now, even if we are unable to solve this pr
oblem, we have significantly increased our chances of guessing correctly--fr
om 1 in 5 to 1 in 3.
Turning to (2), we completely ignore the information in (1). Although (2) co
ntains a lot of information, it also is not sufficient. For example, the fol
lowing sequences each satisfy (2), yet they have different first terms:
1, 3, 12
3, 9, 36
This eliminates B, and our chances of guessing correctly have increased to 1
in 2.
Next, we consider (1) and (2) together. From (1), we know "the 3rd term of S
is 4." From (2), we know "the 3rd term is four times the 2nd." This is equi
valent to saying the 2nd term is 1/4 the 3rd term: (1/4)4 = 1. Further, from
(2), we know "the 2nd term is three times the 1st." This is equivalent to s
aying the 1st term is 1/3 the 2nd term: (1/3)1 = 1/3. Hence, the first term
of the sequence is fully determined: 1/3, 1, 4. The answer is C.
Example 2: In the figure to the right, what is the area of the triangle?
(1)
(2) x = 90
Recall that a triangle is a right triangle if and only if the square of the
longest side is equal to the sum of the squares of the shorter sides (Pythag
orean Theorem). Hence, (1) implies that the triangle is a right triangle. So
the area of the triangle is (6)(8)/2. Note, there is no need to calculate t
he area--we just need to know that the area can be calculated. Hence, the an
swer is either A or D.
Turning to (2), we see immediately that we have a right triangle. Hence, aga
in the area can be calculated. The answer is D.
Example 3: Is p < q ?
(1) p/3 < q/3
(2) -p + x > -q + x
Multiplying both sides of p/3 < q/3 by 3 yields p < q.
Hence, (1) is sufficient. As to (2), subtract x from both sides of -p + x >
-q + x, which yields -p > -q.
Multiplying both sides of this inequality by -1, and recalling that multiply
ing both sides of an inequality by a negative number reverses the inequality
, yields p < q.
Hence, (2) is also sufficient. The answer is D.
Example 4: If x is both the cube of an integer and between 2 and 200, what i
s the value of x?
(1) x is odd.
(2) x is the square of an integer.
Since x is both a cube and between 2 and 200, we are looking at the integers
:
which reduce to
8, 27, 64, 125
Since there are two odd integers in this set, (1) is not sufficient to uniqu
ely determine the value of x. This eliminates choices A and D.
Next, there is only one perfect square, 64, in the set. Hence, (2) is suffic
ient to determine the value of x. The answer is B.
Example 5: Is CAB a code word in language Q?
(1) ABC is the base word.
(2) If C immediately follows B, then C can be moved to the front of the code
word to generate another word.
From (1), we cannot determine whether CAB is a code word since (1) gives no
rule for generating another word from the base word. This eliminates A and D
..
Turning to (2), we still cannot determine whether CAB is a code word since n
ow we have no word to apply this rule to. This eliminates B.
However, if we consider (1) and (2) together, then we can determine whether
CAB is a code word:
From (1), ABC is a code word.
From (2), the C in the code word ABC can be moved to the front of the word:
CAB.
Hence, CAB is a code word and the answer is C.
UNWARRANTED ASSUMPTIONS
Be extra careful not to read any moresintosa statement than what is given.
?The main purpose of some difficult problems is to lure yousintosmaking an u
nwarranted assumption.
If you avoid the temptation, these problems can become routine.
Example 6: Did Incumbent I get over 50% of the vote?
(1) Challenger C got 49% of the vote.
(2) Incumbent I got 25,000 of the 100,000 votes cast.
If you did not make any unwarranted assumptions, you probably did not find t
his to be a hard problem. What makes a problem difficult is not necessarily
its underlying complexity; rather a problem is classified as difficult if ma
ny people miss it. A problem may be simple yet contain a psychological trap
that causes people to answer it incorrectly.
The above problem is difficult because many people subconsciously assume tha
t there are only two candidates. They then figure that since the challenger
received 49% of the vote the incumbent received 51% of the vote. This would
be a valid deduction if C were the only challenger (You might ask, "What if
some people voted for none-of-the-above?" But don’t get carried away with fi
nding exceptions. The writers of the GMAT would not set a trap that subtle).
But we cannot assume that. There may be two or more challengers. Hence, (1)
is insufficient.
Now, consider (2) alone. Since Incumbent I received 25,000 of the 100,000 vo
tes cast, I necessarily received 25% of the vote. Hence, the answer to the q
uestion is "No, the incumbent did not receive over 50% of the vote." Therefo
re, (2) is sufficient to answer the question. The answer is B.
Note, some people have trouble with (2) because they feel that the question
asks for a "yes" answer. But on Data Sufficiency questions, a "no" answer is
just as valid as a "yes" answer. What we’re looking for is a definite answe
r.
CHECKING EXTREME CASES
?When drawing a geometric figure or checking a given one, be sure to include
drawings of extreme cases as well as ordinary ones.
Example 1: In the figure to the right, AC is a chord and B is a point on the
circle. What is the measure of angle x?
Although in the drawing AC looks to be a diameter, that cannot be assumed. A
ll we know is that AC is a chord. Hence, numerous cases are possible, three
of which are illustrated below:
In Case I, x is greater than 45 degrees; in Case II, x equals 45 degrees; in
Case III, x is less than 45 degrees. Hence, the given information is not su
fficient to answer the question.
Example 2: Three rays emanate from a common point and form three angles with
measures p, q, and r. What is the measure of q + r ?
It is natural to make the drawing symmetric as follows:
In this case, p = q = r = 120, so q + r = 240. However, there are other draw
ings possible. For example:
In this case, q + r = 180. Hence, the given information is not sufficient to
answer the question.
Problems:
1. Suppose 3p + 4q = 11. Then what is the value of q?
(1) p is prime.
(2) q = -2p
(1) is insufficient. For example, if p = 3 and q = 1/2, then 3p + 4q = 3(3)
+ 4(1/2) = 11. However, if p = 5 and q = -1, then 3p + 4q = 3(5) + 4(-1) = 1
1. Since the value of q is not unique, (1) is insufficient.
Turning to (2), we now have a system of two equations in two unknowns. Hence
, the system can be solved to determine the value of q. Thus, (2) is suffici
ent, and the answer is B.
2. What is the perimeter of triangle ABC above?
(1) The ratio of DE to BF is 1: 3.
(2) D and E are midpoints of sides AB and CB, respectively.
Since we do not even know whether BF is an altitude, nothing can be determin
ed from (1). More importantly, there is no information telling us the absolu
te size of the triangle.
As to (2), although from geometry we know that DE = AC/2, this relationship
holds for any size triangle. Hence, (2) is also insufficient.
Together, (1) and (2) are also insufficient since we still don’t have inform
ation about the size of the triangle, so we can’t determine the perimeter. T
he answer is E.
3. A dress was initially listed at a price that would have given the store a
profit of 20 percent of the wholesale cost. What was the wholesale cost of
the dress?
(1) After reducing the asking price by 10 percent, the dress sold for a net
profit of 10 dollars.
(2) The dress sold for 50 dollars.
Consider just the question setup. Since the store would have made a profit o
f 20 percent on the wholesale cost, the original price P of the dress was 12
0 percent of the cost: P = 1.2C. Now, translating (1)sintosan equation yield
s:
P - .1P = C + 10
Simplifying gives
..9P = C + 10
Solving for P yields
P = (C + 10)/.9
Plugging this expression for PsintosP = 1.2C gives
(C + 10)/.9 = 1.2C
Since we now have only one equation involving the cost, we can determine the
cost by solving for C. Hence, the answer is A or D.
(2) is insufficient since it does not relate the selling price to any other
information. Note, the phrase "initially listed" implies that there was more
than one asking price. If it wasn’t for that phrase, (2) would be sufficien
t. The answer is A.
4. What is the value of the two-digit number x?
(1) The sum of its digits is 4.
(2) The difference of its digits is 4.
Considering (1) only, x must be 13, 22, 31, or 40. Hence, (1) is not suffici
ent to determine the value of x.
Considering (2) only, x must be 40, 51, 15, 62, 26, 73, 37, 84, 48, 95, or 5
9. Hence, (2) is not sufficient to determine the value of x.
Considering (1) and (2) together, we see that 40 and only 40 is common to th
e two sets of choices for x. Hence, x must be 40. Thus, together (1) and (2)
are sufficient to uniquely determine the value of x. The answer is C.
5. If x and y do not equal 0, is x/y an integer?
(1) x is prime.
(2) y is even.
(1) is not sufficient since we don’t know the value of y. Similarly, (2) is
not sufficient. Furthermore, (1) and (2) together are still insufficient sin
ce there is an even prime number--2. For example, let x be the prime number
2, and let y be the even number 2 (don’t forget that different variables can
stand for the same number). Then x/y = 2/2 = 1, which is an integer. For al
l other values of x and y, x/y is not an integer. (Plug in a few values to v
erify this.) The answer is E.
6. Is 500 the average (arithmetic mean) score on the GMAT?
(1) Half of the people who take the GMAT score above 500 and half of the peo
ple score below 500.
(2) The highest GMAT score is 800 and the lowest score is 200.
Many students mistakenly think that (1) implies the average is 500. Suppose
just 2 people take the test and one scores 700 (above 500) and the other sco
res 400 (below 500). Clearly, the average score for the two test-takers is n
ot 500. (2) is less tempting. Knowing the highest and lowest scores tells us
nothing about the other scores. Finally, (1) and (2) together do not determ
ine the average since together they still don’t tell us the distribution of
most of the scores. The answer is E.
7. The set S of numbers has the following properties:
I) If x is in S, then 1/x is in S.
II) If both x and y are in S, then so is x + y.
Is 3 in S?
(1) 1/3 is in S.
(2) 1 is in S.
Consider (1) alone. Since 1/3 is in S, we know from Property I that 1/(1/3)
= 3 is in S. Hence, (1) is sufficient.
Consider (2) alone. Since 1 is in S, we know from Property II that 1 + 1 = 2
(Note, nothing in Property II prevents x and y from standing for the same n
umber. In this case both stand for 1.) is in S. Applying Property II again s
hows that 1 + 2 = 3 is in S. Hence, (2) is also sufficient. The answer is D.
8. What is the area of the triangle above?
(1) a = x, b = 2x, and c = 3x.
(2) The side opposite a is 4 and the side opposite b is 3.
From (1) we can determine the measures of the angles: a + b + c = x + 2x + 3
x = 6x = 180
Dividing the last equation by 6 gives: x = 30
Hence, a = 30, b = 60, and c = 90. However, different size triangles can hav
e these angle measures, as the diagram below illus
An argument, as used on the GMAT, is a presentation of facts and opinions in
order to support a position. Many arguments will be fallacious. And many co
rrect answers will be false! This often causes students much consternation;
they feel that the correct answer should be true. But the arguments are inte
nded to test your ability to think logically. Now logic is the study of the
relationships between statements, not of the truth of those statements. Bein
g overly concerned with finding the truth can be ruinous to your GMAT argume
nt score.
"2 OUT OF 5" RULE
Creating a good but incorrect answer-choice is much harder than developing t
he correct answer. For this reason, usually only one attractive wrong answer
-choice is presented. This is called the "2 out of 5" rule. That is, only tw
o of the five answer-choices will have any real merit. Hence, even if you do
n’t fully understand an argument, you probably can still eliminate the three
fluff choices, thereby greatly increasing your odds of answering the questi
on correctly.
LOGIC I
Although in theory the argument questions are designed to be answered withou
t any reference to formal logic, the section is essentially a logic test. So
me knowledge of the fundamentals of logic, therefore, will give you a defini
te advantage. Armed with this knowledge, you should quickly notice that the
arguments are fundamentally easy and that most of them fallsintosa few basic
categories. In this section, we will study the logical structure of argumen
ts. In Logic II, we will symbolize and diagram arguments in much the same wa
y as we did with games.
Conclusions
Most argument questions hinge, either directly or indirectly, on determining
the conclusion of the argument. The conclusion is the main idea of the argu
ment. It is what the writer tries to persuade the reader to believe. Most of
ten the conclusion comes at the end of the argument. The writer organizes th
e facts and his opinions so that they build up to the conclusion. Sometimes,
however, the conclusion will come at the beginning of an argument, rarely d
oes it come in the middle, and occasionally, for rhetorical effect, the conc
lusion is not even stated.
Example:
The police are the armed guardians of the social order. The blacks are the c
hief domestic victims of the American social order. A conflict of interest e
xists, therefore, between the blacks and the police.--Eldridge Cleaver, Soul
on Ice
Here the first two sentences anticipate or set up the conclusion. By changin
g the grammar slightly, the conclusion can be placed at the beginning of the
argument and still sound natural:
A conflict of interest exists between the blacks and the police because the
police are the armed guardians of the socialsgroupsand the blacks are the ch
ief domestic victims of the American social order.
The conclusion can also be forcedsintosthe middle:
The police are the armed guardians of the social order. So a conflict of int
erest exists between the blacks and the police because the blacks are the ch
ief domestic victims of the American social order.
It is generally awkward, as in the previous paragraph, to place the conclusi
on in the middle of the argument because then it cannot be fully anticipated
by what comes before nor fully explained by what comes after. On the rare o
ccasion when a conclusion comes in the middle of an argument, most often eit
her the material that comes after it or the material that comes before it is
not essential.
In summary: To find the conclusion, check the last sentence of the argument.
If that is not the conclusion, check the first sentence. Rarely does the co
nclusion come in the middle of an argument.
When determining the meaning of a conclusion, be careful not to read any mor
esintosit than what the author states. Although arguments are not worded as
precisely as games, you still need to read them with more care than you woul
d use in your everyday reading.
As with games, read the words and sentences of an argument precisely, and us
e their literal meaning.
For example, consider the meaning of some in the sentence "Some of Mary’s fr
iends went to the party." It would be unwarranted, based on this statement,
to assume that some of Mary’s friends did not go to the party. Although it m
ay seem deceiving to say that some of Mary’s friends went to the party when
in fact all of them did, it is nonetheless technically consistent with the m
eaning of some.
Some means "at least one and perhaps all."
As mentioned before, the conclusion usually comes at the end of an argument,
sometimes at the beginning, and rarely in the middle. Writers use certain w
ords to indicate that the conclusion is about to be stated. Following is a l
ist of the most common conclusion indicators:
Conclusion Indicators
hence therefore
so accordingly
thus consequently
follows that shows that
conclude that implies
as a result means that
Most often the conclusion of an argument is put in the form of a statement.
Sometimes, however, the conclusion is given as a command or obligation.
Example:
All things considered, you ought to vote.
Here, the author implies that you are obliged to vote.
The conclusion can even be put in the form of a question. This rhetorical te
chnique is quite effective in convincing people that a certain position is c
orrect. We are more likely to believe something if we feel that we concluded
it on our own, or at least if we feel that we were not told to believe it.
A conclusion put in question form can have this result.
Example:
The Nanuuts believe that they should not take from Nature anything She canno
t replenish during their lifetime. This assures that future generations can
enjoy the same riches of Nature that they have. At the current rate of destr
uction, the rain forests will disappear during our lifetime. Do we have an o
bligation to future generations to prevent this result?
Here the author trusts that the power of her argument will persuade the read
er to answer the question affirmatively.
Taking this rhetorical technique one step further, the writer may build up t
o the conclusion but leave it unstated. This allows the reader to make up hi
s own mind. If the build-up is done skillfully, the reader will be more like
ly to agree with the author, without feeling manipulated.
Example:
He who is without sin should cast the first stone. There is no one here who
does not have a skeleton in his closet.
The unstated but obvious conclusion here is that none of the people has the
right to cast the first stone.
When determining the conclusion’s scope be careful not to read any more or l
esssintosit than the author states. GMAT writers often create wrong answer-c
hoices by slightly overstating or understating the author’s claim. Certain w
ords limit the scope of a statement. These words are called quantifiers--pay
close attention to them. Following is a list of the most important quantifi
ers:
Quantifiers
all except likely
some most many
only could no
never always everywhere
probably must alone
Example:
Whether the world is Euclidean or non-Euclidean is still an open question.
However, if a star’s position is predicted based on non-Euclidean geometry,
then when a telescope is pointed toswheresthe star should be it will be ther
e. Whereas, if the star’s position is predicted based on Euclidean geometry,
then when a telescope is pointed toswheresthe star should be it won’t be th
ere. This strongly indicates that the world is non-Euclidean.
Which one of the following best expresses the main idea of the passage?
(A) The world may or may not be Euclidean.
(B) The world is probably non-Euclidean.
(C) The world is non-Euclidean.
(D) The world is Euclidean.
(E) The world is neither Euclidean nor non-Euclidean.
Choice (A) understates the main idea. Although the opening to the passage st
ates that we don’t know whether the world is non-Euclidean, the author goes
on to give evidence that it is non-Euclidean. Choice (C) overstates the main
idea. The author doesn’t say that the world is non-Euclidean, just that evi
dence strongly indicates that it is. In choice (B), the word "probably" prop
erly limits the scope of the main idea, namely, that the world is probably n
on-Euclidean, but we can’t yet state so definitively. The answer is (B).
Premises
Once you’ve found the conclusion, most often everything else in the argument
will be either premises or "noise." The premises provide evidence for the c
onclusion; they form the foundation or infrastructure upon which the conclus
ion depends. To determine whether a statement is a premise, ask yourself whe
ther it supports the conclusion. If so, it’s a premise. Earlier we saw that
writers use certain words to flag conclusions; likewise writers use certain
words to flag premises. Following is a partial list of the most common premi
se indicators:
Premise Indicators
because for
since is evidence that
if in that
as owing to
suppose inasmuch as
assume may be derived from
Example:
Since the incumbent’s views are out of step with public opinion, he probably
will not be reelected.
Here "since" is used to flag the premise that the incumbent’s positions are
unpopular.
Suppressed Premises
Most arguments depend on one or more unstated premises. Sometimes this indic
ates a weakness in the argument, an oversight by the writer. More often, how
ever, certain premises are left tacit because they are too numerous, or the
writer assumes that his audience is aware of the assumptions, or he wants th
e audience to fill in the premise themselves and therefore be more likely to
believe the conclusion.
Example:
Conclusion: I knew he did it.
Premise: Only a guilty person would accept immunity from prosecution.
The suppressed premise is that he did, in fact, accept immunity. The speaker
assumes that his audience is aware of this fact or at least is willing to b
elieve it, so to state it would be redundant and ponderous. If the unstated
premise were false (that is, he did not accept immunity), the argument would
not technically be a lie; but it would be very deceptive. The unscrupulous
writer may use this ploy if he thinks that he can get away with it. That is,
his argument has the intended effect and the false premise, though implicit
, is hard to find or is ambiguous. Politicians are not at all above using th
is tactic.
A common question on the GMAT asks you to find the suppressed premise of an
argument. Finding the suppressed premise, or assumption, of an argument can
be difficult. However, on the GMAT you have an advantage--the suppressed pre
mise is listed as one of the five answer-choices. To test whether an answer-
choice is a suppressed premise, ask yourself whether it would make the argum
ent more plausible. If so, then it is very likely a suppressed premise.
Example:
American attitudes tend to be rather insular, but there is much we can learn
from other countries. In Japan, for example, workers set aside some time ea
ch day to exercise, and many corporations provide elaborate exercise facilit
ies for their employees. Few American corporations have such exercise progra
ms. Studies have shown that the Japanese worker is more productive than the
American worker. Thus it must be concluded that the productivity of American
workers will lag behind their Japanese counterparts, until mandatory exerci
se programs are introduced.
The conclusion of the argument is valid if which one of the following is ass
umed?
(A) Even if exercise programs do not increase productivity, they will improv
e the American worker’s health.
(B) The productivity of all workers can be increased by exercise.
(C) Exercise is an essential factor in the Japanese worker’s superior produc
tivity.
(D) American workers can adapt to the longer Japanese work week.
(E) American corporations don’t have the funds to build elaborate exercise f
acilities.
The unstated essence of the argument is that exercise is an integral part of
productivity and that Japanese workers are more productive than American wo
rkers because they exercise more. The answer is (C).
Counter-Premises
When presenting a position, you obviously don’t want to argue against yourse
lf. However, it is often effective to concede certain minor points that weak
en your argument. This shows that you are open-minded and that your ideas ar
e well considered. It also disarms potential arguments against your position
.. For instance, in arguing for a strong, aggressive police department, you m
ay concede that in the past the police have at times acted too aggressively.
Of course, you will then need to state more convincing reasons to support y
our position.
Example:
I submit that the strikers should accept the management’s offer. Admittedly,
it is less than what was demanded. But it does resolve the main grievance--
inadequate health care. Furthermore, an independent study shows that a wage
increase greater than 5% would leave the company unable to compete against J
apan and Germany, forcing itsintosbankruptcy.
The conclusion, "the strikers should accept the management’s offer," is stat
ed in the first sentence. Then "Admittedly" introduces a concession; namely,
that the offer was less than what was demanded. This weakens the speaker’s
case, but it addresses a potential criticism of his position before it can b
e made. The last two sentences of the argument present more compelling reaso
ns to accept the offer and form the gist of the argument.
Following are some of the most common counter-premise indicators:
Counter-Premise Indicators
but despite
admittedly except
even though nonetheless
nevertheless although
however in spite of the fact
As you may have anticipated, the GMAT writers sometimes use counter-premises
to bait wrong answer-choices. Answer-choices that refer to counter-premises
are very tempting because they refer directly to the passage and they are i
n part true. But you must ask yourself "Is this the main point that the auth
or is trying to make?" It may merely be a minor concession.
Logic II (Diagramming)
Most arguments are based on some variation of an if-then statement. However,
the if-then statement is often embedded in other equivalent structures. Dia
gramming brings out the superstructure and the underlying simplicity of argu
ments.
If-Then
A-->B
By now you should be well aware that if the premise of an if-then statement
is true then the conclusion must be true as well. This is the defining chara
cteristic of a conditional statement; it can be illustrated as follows:
A-->B
A
Therefore, B
This diagram displays the if-then statement "A-->B," the affirmed premise "A
," and the necessary conclusion "B." Such a diagram can be very helpful in s
howing the logical structure of an argument.
Example: (If-then)
If Jane does not study for the GMAT, then she will not score well. Jane, in
fact, did not study for the GMAT; therefore she scored poorly on the test.
When symbolizing games, we let a letter stand for an element. When symbolizi
ng arguments, however, we may let a letter stand for an element, a phrase, a
clause, or even an entire sentence. The clause "Jane does not study for the
GMAT" can be symbolized as ~S, and the clause "she will not score well" can
be symbolized as ~W. Substituting these symbolssintosthe argument yields th
e following diagram:
~S-->~W
~S
Therefore, ~W
This diagram shows that the argument has a valid if-then structure. A condit
ional statement is presented, ~S-->~W; its premise affirmed, ~S; and then th
e conclusion that necessarily follows, ~W, is stated.
Embedded If-Then Statements
Usually, arguments involve an if-then statement. Unfortunately, the if-then
thought is often embedded in other equivalent structures. In this section, w
e study how to spot these structures.
Example: (Embedded If-then)
John and Ken cannot both go to the party.
At first glance, this sentence does not appear to contain an if-then stateme
nt. But it essentially says: "if John goes to the party, then Ken does not."
Example: (Embedded If-then)
Danielle will be accepted to graduate school only if she does well on the GR
E.
Given this statement, we know that if Danielle is accepted to graduate schoo
l, then she must have done well on the GRE. Note: Students often wrongly int
erpret this statement to mean:
"If Danielle does well on the GRE, then she will be accepted to graduate sch
ool."
There is no such guarantee. The only guarantee is that if she does not do we
ll on the GRE, then she will not be accepted to graduate school.
"A only if B" is logically equivalent to "if A, then B."
Affirming the Conclusion Fallacy
A-->B
B
Therefore, A
Remember that an if-then statement, A-->B, tells us only two things: (1) If
A is true, then B is true as well. (2) If B is false, then A is false as wel
l (contrapositive). If, however, we know the conclusion is true, the if-then
statement tells us nothing about the premise. And if we know that the premi
se is false (we will consider this next), then the if-then statement tells u
s nothing about the conclusion.
Example: (Affirming the Conclusion Fallacy)
If he is innocent, then when we hold him under water for sixty seconds he wi
ll not drown. Since he did not die when we dunked him in the water, he must
be innocent.
The logical structure of the argument above is most similar to which one of
the following?
(A) To insure that the remaining wetlands survive, they must be protected by
the government. This particular wetland is being neglected. Therefore, it w
ill soon perish.
(B) There were nuts in that pie I just ate. There had to be, because when I
eat nuts I break out in hives, and I just noticed a blemish on my hand.
(C) The president will be reelected unless a third candidate enters the race
.. A third candidate has entered the race, so the president will not be reele
cted.
(D) Every time Melinda has submitted her book for publication it has been re
jected. So she should not bother with another rewrite.
(E) When the government loses the power to tax one area of the economy, it j
ust taxes another. The Supreme Court just overturned the sales tax, so we ca
n expect an increase in the income tax.
To symbolize this argument, let the clause "he is innocent" be denoted by I,
and let the clause "when we hold him under water for sixty seconds he will
not drown" be denoted by ~D. Then the argument can be symbolized as
I-->~D
~D
Therefore, I
Notice that this argument is fallacious: the conclusion "he is innocent" is
also a premise of the argument. Hence the argument is circular--it proves wh
at was already assumed. The argument affirms the conclusion then invalidly u
ses it to deduce the premise. The answer will likewise be fallacious.
We start with answer-choice (A). The sentence
"To insure that the remaining wetlands survive, they must be protected by th
e government"
contains an embedded if-then statement:
"If the remaining wetlands are to survive, then they must be protected by th
e government."
This can be symbolized as S-->P. Next, the sentence "This particular wetland
is being neglected" can be symbolized as ~P. Finally, the sentence "It will
soon perish" can be symbolized as ~S. Using these symbols to translate the
argument gives the following diagram:
S-->P
~P
Therefore, ~S
The diagram clearly shows that this argument does not have the same structur
e as the given argument. In fact, it is a valid argument by contraposition.
Turning to (B), we reword the statement "when I eat nuts, I break out in hiv
es" as
"If I eat nuts, then I break out in hives." This in turn can be symbolized a
s N-->H.
Next, we interpret the clause "there is a blemish on my hand" to mean "hives
," which we symbolize as H. Substituting these symbolssintosthe argument yie
lds the following diagram:
N-->H
H
Therefore, N
The diagram clearly shows that this argument has the same structure as the g
iven argument. The answer, therefore, is (B).
Denying the Premise Fallacy
A-->B
~A
Therefore, ~B
The fallacy of denying the premise occurs when an if-then statement is prese
nted, its premise denied, and then its conclusion wrongly negated.
Example: (Denying the Premise Fallacy)
The senator will be reelected only if he opposes the new tax bill. But he wa
s defeated. So he must have supported the new tax bill.
The sentence "The senator will be reelected only if he opposes the new tax b
ill" contains an embedded if-then statement: "If the senator is reelected, t
hen he opposes the new tax bill." (Remember: "A only if B" is equivalent to
"If A, then B.") This in turn can be symbolized as R-->~T. The sentence "But
the senator was defeated" can be reworded as "He was not reelected," which
in turn can be symbolized as ~R. Finally, the sentence "He must have support
ed the new tax bill" can be symbolized as T. Using these symbols the argumen
t can be diagrammed as follows:
R-->~T
~R
Therefore, T
[Note: Two negatives make a positive, so the conclusion ~(~T) was reduced to
T.] This diagram clearly shows that the argument is committing the fallacy
of denying the premise. An if-then statement is made; its premise is negated
; then its conclusion is negated.
Transitive Property
A-->B
B-->C
Therefore, A-->C
These arguments are rarely difficult, provided you step back and take a bir
d’s-eye view. It may be helpful to view this structure as an inequality in m
athematics. For example, 5 > 4 and 4 > 3, so 5 > 3.
Notice that the conclusion in the transitive property is also an if-then sta
tement. So we don’t know that C is true unless we know that A is true. Howev
er, if we add the premise "A is true" to the diagram, then we can conclude t
hat C is true:
A-->B
B-->C
A
Therefore, C
As you may have anticipated, the contrapositive can be generalized to the tr
ansitive property:
A-->B
B-->C
~C
Therefore, ~A
Example: (Transitive Property)
If you work hard, you will be successful in America. If you are successful i
n America, you can lead a life of leisure. So if you work hard in America, y
ou can live a life of leisure.
Let W stand for "you work hard," S stand for "you will be successful in Amer
ica," and L stand for "you can lead a life of leisure." Now the first senten
ce translates as W-->S, the second sentence as S-->L, and the conclusion as
W-->L. Combining these symbol statements yields the following diagram:
W-->S
S-->L
Therefore, W-->L
The diagram clearly displays the transitive property.
DeMorgan’s Laws
~(A & B) = ~A or ~B
~(A or B) = ~A & ~B
If you have taken a course in logic, you are probably familiar with these fo
rmulas. Their validity is intuitively clear: The conjunction A&B is false wh
en either, or both, of its parts are false. This is precisely what ~A or ~B
says. And the disjunction A or B is false only when both A and B are false,
which is precisely what ~A and ~B says.
You will rarely get an argument whose main structure is based on these rules
--they are too mechanical. Nevertheless, DeMorgan’s laws often help simplify
, clarify, or transform parts of an argument. They are also useful with game
s.
Example: (DeMorgan’s Law)
It is not the case that either Bill or Jane is going to the party.
This argument can be diagrammed as ~(B or J), which by the second of DeMorga
n’s laws simplifies to (~B and ~J). This diagram tells us that neither of th
em is going to the party.
A unless B
~B-->A
"A unless B" is a rather complex structure. Though surprisingly we use it wi
th little thought or confusion in our day-to-day speech.
To see that "A unless B" is equivalent to "~B-->A," consider the following s
ituation:
Biff is at the beach unless it is raining.
Given this statement, we know that if it is not raining, then Biff is at the
beach. Now if we symbolize "Biff is at the beach" as B, and "it is raining"
as R, then the statement can be diagrammed as ~R-->B.
CLASSIFICATION
In Logic II, we studied deductive arguments. However, the bulk of arguments
on the GMAT are inductive. In this section we will classify and study the ma
jor types of inductive arguments.
An argument is deductive if its conclusion necessarily follows from its prem
ises--otherwise it is inductive. In an inductive argument, the author presen
ts the premises as evidence or reasons for the conclusion. The validity of t
he conclusion depends on how compelling the premises are. Unlike deductive a
rguments, the conclusion of an inductive argument is never certain. The trut
h of the conclusion can range from highly likely to highly unlikely. In reas
onable arguments, the conclusion is likely. In fallacious arguments, it is i
mprobable. We will study both reasonable and fallacious arguments.
We will classify the three major types of inductive reasoning--generalizatio
n, analogy, and causal--and their associated fallacies.
Generalization
Generalization and analogy, which we consider in the next section, are the m
ain tools by which we accumulate knowledge and analyze our world. Many peopl
e define generalization as "inductive reasoning." In colloquial speech, the
phrase "to generalize" carries a negative connotation. To argue by generaliz
ation, however, is neither inherently good nor bad. The relative validity of
a generalization depends on both the context of the argument and the likeli
hood that its conclusion is true. Polling organizations make predictions by
generalizing information from a small sample of the population, which hopefu
lly represents the general population. The soundness of their predictions (a
rguments) depends on how representative the sample is and on its size. Clear
ly, the less comprehensive a conclusion is the more likely it is to be true.
Example:
During the late seventies when Japan was rapidly expanding its share of the
American auto market, GM surveyed owners of GM cars and asked them whether t
hey would be more willing to buy a large, powerful car or a small, economica
l car. Seventy percent of those who responded said that they would prefer a
large car. On the basis of this survey, GM decided to continue building larg
e cars. Yet during the’80s, GM lost even more of the market to the Japanese
..
Which one of the following, if it were determined to be true, would best exp
lain this discrepancy.
(A) Only 10 percent of those who were polled replied.
(B) Ford which conducted a similar survey with similar results continued to
build large cars and also lost more of their market to the Japanese.
(C) The surveyed owners who preferred big cars also preferred big homes.
(D) GM determined that it would be more profitable to make big cars.
(E) Eighty percent of the owners who wanted big cars and only 40 percent of
the owners who wanted small cars replied to the survey.
The argument generalizes from the survey to the general car-buying populatio
n, so the reliability of the projection depends on how representative the sa
mple is. At first glance, choice (A) seems rather good, because 10 percent d
oes not seem large enough. However, political opinion polls are typically ba
sed on only .001 percent of the population. More importantly, we don’t know
what percentage of GM car owners received the survey. Choice (B) simply stat
es that Ford made the same mistake that GM did. Choice (C) is irrelevant. Ch
oice (D), rather than explaining the discrepancy, gives even more reason for
GM to continue making large cars. Finally, choice (E) points out that part
of the survey did not represent the entire public, so (E) is the answer.
Analogy
To argue by analogy is to claim that because two things are similar in some
respects, they will be similar in others. Medical experimentation on animals
is predicated on such reasoning. The argument goes like this: the metabolis
m of pigs, for example, is similar to that of humans, and high doses of sacc
harine cause cancer in pigs. Therefore, high doses of saccharine probably ca
use cancer in humans.
Clearly, the greater the similarity between the two things being compared th
e stronger the argument will be. Also the less ambitious the conclusion the
stronger the argument will be. The argument above would be strengthened by c
hanging "probably" to "may." It can be weakened by pointing out the dissimil
arities between pigs and people.
Example:
Just as the fishing line becomes too taut, so too the trials and tribulation
s of life in the city can become so stressful that one’s mind can snap.
Which one of the following most closely parallels the reasoning used in the
argument above?
(A) Just as the bow may be drawn too taut, so too may one’s life be wasted p
ursuing self-gratification.
(B) Just as a gambler’s fortunes change unpredictably, so too do one’s caree
r opportunities come unexpectedly.
(C) Just as a plant can be killed by over watering it, so too can drinking t
oo much water lead to lethargy.
(D) Just as the engine may race too quickly, so too may life in the fast lan
e lead to an early death.
(E) Just as an actor may become stressed before a performance, so too may dw
elling on the negative cause depression.
The argument compares the tautness in a fishing line to the stress of city l
ife; it then concludes that the mind can snap just as the fishing line can.
So we are looking for an answer-choice that compares two things and draws a
conclusion based on their similarity. Notice that we are looking for an argu
ment that uses similar reasoning, but not necessarily similar concepts. In f
act, an answer-choice that mentions either tautness or stress will probably
be a same-language trap.
Choice (A) uses the same-language trap--notice "too taut." The analogy betwe
en a taut bow and self-gratification is weak, if existent. Choice (B) offers
a good analogy but no conclusion. Choice (C) offers both a good analogy and
a conclusion; however, the conclusion, "leads to lethargy," understates the
scope of what the analogy implies. Choice (D) offers a strong analogy and a
conclusion with the same scope found in the original: "the engine blows, th
e person dies"; "the line snaps, the mind snaps." This is probably the best
answer, but still we should check every choice. The last choice, (E), uses l
anguage from the original, "stressful," to make its weak analogy more tempti
ng. The best answer, therefore, is (D).
Causal Reasoning
Of the three types of inductive reasoning we will discuss, causal reasoning
is both the weakest and the most prone to fallacy. Nevertheless, it is a us
eful and common method of thought.
To argue by causation is to claim that one thing causes another. A causal ar
gument can be either weak or strong depending on the context. For example, t
o claim that you won the lottery because you saw a shooting star the night b
efore is clearly fallacious. However, most people believe that smoking cause
s cancer because cancer often strikes those with a history of cigarette use.
Although the connection between smoking and cancer is virtually certain, as
with all inductive arguments it can never be 100 percent certain. Cigarette
companies have claimed that there may be a genetic predisposition in some p
eople to both develop cancer and crave nicotine. Although this claim is high
ly improbable, it is conceivable.
There are two common fallacies associated with causal reasoning:
1. Confusing Correlation with Causation.
To claim that A caused B merely because A occurred immediately before B is c
learly questionable. It may be only coincidental that they occurred together
, or something else may have caused them to occur together. For example, the
fact that insomnia and lack of appetite often occur together does not mean
that one necessarily causes the other. They may both be symptoms of an under
lying condition.
2. Confusing Necessary Conditions with Sufficient Conditions.
A is necessary for B means "B cannot occur without A." A is sufficient for B
means "A causes B to occur, but B can still occur without A." For example,
a small tax base is sufficient to cause a budget deficit, but excessive spen
ding can cause a deficit even with a large tax base. A common fallacy is to
assume that a necessary condition is sufficient to cause a situation. For ex
ample, to win a modern war it is necessary to have modern, high-tech equipme
nt, but it is not sufficient, as Iraq discovered in the Persian Gulf War.
SEVEN COMMON FALLACIES
Contradiction
A Contradiction is committed when two opposing statements are simultaneously
asserted. For example, saying "it is raining and it is not raining" is a co
ntradiction. Typically, however, the arguer obscures the contradiction to th
e point that the argument can be quite compelling. Take, for instance, the f
ollowing argument:
"We cannot know anything, because we intuitively realize that our thoughts a
re unreliable."
This argument has an air of reasonableness to it. But "intuitively realize"
means "to know." Thus the arguer is in essence saying that we know that we d
on’t know anything. This is self-contradictory.
Equivocation
Equivocation is the use of a word in more than one sense during an argument.
This technique is often used by politicians to leave themselves an "out." I
f someone objects to a particular statement, the politician can simply claim
the other meaning.
Example:
Individual rights must be championed by the government. It is right for one
to believe in God. So government should promote the belief in God.
In this argument, right is used ambiguously. In the phrase "individual right
s" it is used in the sense of a privilege, whereas in the second sentence ri
ght is used to mean proper or moral. The questionable conclusion is possible
only if the arguer is allowed to play with the meaning of the critical word
right.
Circular Reasoning
Circular reasoning involves assuming as a premise that which you are trying
to prove. Intuitively, it may seem that no one would fall for such an argume
nt. However, the conclusion may appear to state something additional, or the
argument may be so long that the reader may forget that the conclusion was
stated as a premise.
Example:
The death penalty is appropriate for traitors because it is right to execute
those who betray their own country and thereby risk the lives of millions.
This argument is circular because "right" means essentially the same thing a
s "appropriate." In effect, the writer is saying that the death penalty is a
ppropriate because it is appropriate.
Shifting The Burden Of Proof
It is incumbent on the writer to provide evidence or support for her positio
n. To imply that a position is true merely because no one has disproved it i
s to shift the burden of proof to others.
Example:
Since no one has been able to prove God’s existence, there must not be a God
..
There are two major weaknesses in this argument. First, the fact that God’s
existence has yet to be proven does not preclude any future proof of existen
ce. Second, if there is a God, one would expect that his existence is indepe
ndent of any proof by man.
Unwarranted Assumptions
The fallacy of unwarranted assumption is committed when the conclusion of an
argument is based on a premise (implicit or explicit) that is false or unwa
rranted. An assumption is unwarranted when it is false--these premises are u
sually suppressed or vaguely written. An assumption is also unwarranted when
it is true but does not apply in the given context--these premises are usua
lly explicit.
Example: (False Dichotomy)
Either restrictions must be placed on freedom of speech or certain subversiv
e elements in society will use it to destroy this country. Since to allow th
e latter to occur is unconscionable, we must restrict freedom of speech.
The conclusion above is unsound because
(A) subversives do not in fact want to destroy the country
(B) the author places too much importance on the freedom of speech
(C) the author fails to consider an accommodation between the two alternativ
es
(D) the meaning of "freedom of speech" has not been defined
(E) subversives are a true threat to our way of life
The arguer offers two options: either restrict freedom of speech, or lose th
e country. He hopes the reader will assume that these are the only options a
vailable. This is unwarranted. He does not state how the so-called "subversi
ve elements" would destroy the country, nor for that matter, why they would
want to destroy it. There may be a third option that the author did not ment
ion; namely, that society may be able to tolerate the "subversives" and it m
ay even be improved by the diversity of opinion they offer. The answer is (C
).
Appeal To Authority
To appeal to authority is to cite an expert’s opinion as support for one’s o
wn opinion. This method of thought is not necessarily fallacious. Clearly, t
he reasonableness of the argument depends on the "expertise" of the person b
eing cited and whether she is an expert in a field relevant to the argument.
Appealing to a doctor’s authority on a medical issue, for example, would be
reasonable; but if the issue is about dermatology and the doctor is an orth
opedist, then the argument would be questionable.
Personal Attack
In a personal attack (ad hominem), a person’s character is challenged instea
d of her opinions.
Example:
Politician: How can we trust my opponent to be true to the voters? He isn’t
true to his wife!
This argument is weak because it attacks the opponent’s character, not his p
ositions. Some people may consider fidelity a prerequisite for public office
.. History, however, shows no correlation between fidelity and great politica
l leadership.
--
I would fly you to the moon and back
If you’ll be if you’ll be my baby
Got a ticket for a worldswhereswe belong
So would you be my baby
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Data Sufficiency
----------------------------------------------------------------------------
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INTRODUCTION DATA SUFFICIENCY
Most people have much more difficulty with the Data Sufficiency problems tha
n with the Standard Math problems. However, the mathematical knowledge and s
kill required to solve Data Sufficiency problems is no greater than that req
uired to solve standard math problems. What makes Data Sufficiency problems
appear harder at first is the complicated directions. But once you become fa
miliar with the directions, you’ll find these problems no harder than standa
rd math problems. In fact, people usually become proficient more quickly on
Data Sufficiency problems.
THE DIRECTIONS
The directions for Data Sufficiency questions are rather complicated. Before
reading any further, take some time to learn the directions cold. Some of t
he wording in the directions below has been changed from the GMAT to make it
clearer. You should never have to look at the instructions during the test.
Directions: Each of the following Data Sufficiency problems contains a quest
ion followed by two statements, numbered (1) and (2). You need not solve the
problem; rather you must decide whether the information given is sufficient
to solve the problem.
The correct answer to a question is
A if statement (1) ALONE is sufficient to answer the question but statement
(2) alone is not sufficient;
B if statement (2) ALONE is sufficient to answer the question but statement
(1) alone is not sufficient;
C if the two statements TAKEN TOGETHER are sufficient to answer the question
, but NEITHER statement ALONE is sufficient;
D if EACH statement ALONE is sufficient to answer the question;
E if the two statements TAKEN TOGETHER are still NOT sufficient to answer th
e question.
Numbers: Only real numbers are used. That is, there are no complex numbers.
Drawings: The drawings are drawn to scale according to the information given
in the question, but may conflict with the information given in statements
(1) and (2).
You can assume that a line that appears straight is straight and that angle
measures cannot be zero.
You can assume that the relative positions of points, angles, and objects ar
e as shown.
All drawings lie in a plane unless stated otherwise.
Example:
In triangle ABC to the right, what is the value of y?
(1) AB = AC
(2) x = 30
Explanation: By statement (1), triangle ABC is isosceles. Hence, its base an
gles are equal: y = z. Since the angle sum of a triangle is 180 degrees, we
get x + y + z = 180. Replacing z with y in this equation and then simplifyin
g yields x + 2y = 180. Since statement (1) does not give a value for x, we c
annot determine the value of y from statement (1) alone. By statement (2), x
= 30. Hence, x + y + z = 180 becomes 30 + y + z = 180, or y + z = 150. Sinc
e statement (2) does not give a value for z, we cannot determine the value o
f y from statement (2) alone. However, using both statements in combination,
we can find both x and z and therefore y. Hence, the answer is C.
Notice in the above example that the triangle appears to be a right triangle
.. However, that cannot be assumed: angle A may be 89 degrees or 91 degrees,
we can’t tell from the drawing. You must be very careful not to assume any m
ore than what is explicitly given in a Data Sufficiency problem.
ELIMINATION
Data Sufficiency questions provide fertile ground for elimination. In fact,
it is rare that you won’t be able to eliminate some answer-choices. Remember
, if you can eliminate at least one answer choice, the odds of gaining point
s by guessing are in your favor.
The following table summarizes how elimination functions with Data Sufficien
cy problems.
Statement Choices Eliminated
(1) is sufficient B, C, E
(1) is not sufficient A, D
(2) is sufficient A, C, E
(2) is not sufficient B, D
(1) is not sufficient and (2) is not sufficient A, B, D
Example 1: What is the 1st term in sequence S?
(1) The 3rd term of S is 4.
(2) The 2nd term of S is three times the 1st, and the 3rd term is four times
the 2nd.
(1) is no help in finding the first term of S. For example, the following se
quences each have 4 as their third term, yet they have different first terms
:
0, 2, 4
-4, 0, 4
This eliminates choices A and D. Now, even if we are unable to solve this pr
oblem, we have significantly increased our chances of guessing correctly--fr
om 1 in 5 to 1 in 3.
Turning to (2), we completely ignore the information in (1). Although (2) co
ntains a lot of information, it also is not sufficient. For example, the fol
lowing sequences each satisfy (2), yet they have different first terms:
1, 3, 12
3, 9, 36
This eliminates B, and our chances of guessing correctly have increased to 1
in 2.
Next, we consider (1) and (2) together. From (1), we know "the 3rd term of S
is 4." From (2), we know "the 3rd term is four times the 2nd." This is equi
valent to saying the 2nd term is 1/4 the 3rd term: (1/4)4 = 1. Further, from
(2), we know "the 2nd term is three times the 1st." This is equivalent to s
aying the 1st term is 1/3 the 2nd term: (1/3)1 = 1/3. Hence, the first term
of the sequence is fully determined: 1/3, 1, 4. The answer is C.
Example 2: In the figure to the right, what is the area of the triangle?
(1)
(2) x = 90
Recall that a triangle is a right triangle if and only if the square of the
longest side is equal to the sum of the squares of the shorter sides (Pythag
orean Theorem). Hence, (1) implies that the triangle is a right triangle. So
the area of the triangle is (6)(8)/2. Note, there is no need to calculate t
he area--we just need to know that the area can be calculated. Hence, the an
swer is either A or D.
Turning to (2), we see immediately that we have a right triangle. Hence, aga
in the area can be calculated. The answer is D.
Example 3: Is p < q ?
(1) p/3 < q/3
(2) -p + x > -q + x
Multiplying both sides of p/3 < q/3 by 3 yields p < q.
Hence, (1) is sufficient. As to (2), subtract x from both sides of -p + x >
-q + x, which yields -p > -q.
Multiplying both sides of this inequality by -1, and recalling that multiply
ing both sides of an inequality by a negative number reverses the inequality
, yields p < q.
Hence, (2) is also sufficient. The answer is D.
Example 4: If x is both the cube of an integer and between 2 and 200, what i
s the value of x?
(1) x is odd.
(2) x is the square of an integer.
Since x is both a cube and between 2 and 200, we are looking at the integers
:
which reduce to
8, 27, 64, 125
Since there are two odd integers in this set, (1) is not sufficient to uniqu
ely determine the value of x. This eliminates choices A and D.
Next, there is only one perfect square, 64, in the set. Hence, (2) is suffic
ient to determine the value of x. The answer is B.
Example 5: Is CAB a code word in language Q?
(1) ABC is the base word.
(2) If C immediately follows B, then C can be moved to the front of the code
word to generate another word.
From (1), we cannot determine whether CAB is a code word since (1) gives no
rule for generating another word from the base word. This eliminates A and D
..
Turning to (2), we still cannot determine whether CAB is a code word since n
ow we have no word to apply this rule to. This eliminates B.
However, if we consider (1) and (2) together, then we can determine whether
CAB is a code word:
From (1), ABC is a code word.
From (2), the C in the code word ABC can be moved to the front of the word:
CAB.
Hence, CAB is a code word and the answer is C.
UNWARRANTED ASSUMPTIONS
Be extra careful not to read any moresintosa statement than what is given.
?The main purpose of some difficult problems is to lure yousintosmaking an u
nwarranted assumption.
If you avoid the temptation, these problems can become routine.
Example 6: Did Incumbent I get over 50% of the vote?
(1) Challenger C got 49% of the vote.
(2) Incumbent I got 25,000 of the 100,000 votes cast.
If you did not make any unwarranted assumptions, you probably did not find t
his to be a hard problem. What makes a problem difficult is not necessarily
its underlying complexity; rather a problem is classified as difficult if ma
ny people miss it. A problem may be simple yet contain a psychological trap
that causes people to answer it incorrectly.
The above problem is difficult because many people subconsciously assume tha
t there are only two candidates. They then figure that since the challenger
received 49% of the vote the incumbent received 51% of the vote. This would
be a valid deduction if C were the only challenger (You might ask, "What if
some people voted for none-of-the-above?" But don’t get carried away with fi
nding exceptions. The writers of the GMAT would not set a trap that subtle).
But we cannot assume that. There may be two or more challengers. Hence, (1)
is insufficient.
Now, consider (2) alone. Since Incumbent I received 25,000 of the 100,000 vo
tes cast, I necessarily received 25% of the vote. Hence, the answer to the q
uestion is "No, the incumbent did not receive over 50% of the vote." Therefo
re, (2) is sufficient to answer the question. The answer is B.
Note, some people have trouble with (2) because they feel that the question
asks for a "yes" answer. But on Data Sufficiency questions, a "no" answer is
just as valid as a "yes" answer. What we’re looking for is a definite answe
r.
CHECKING EXTREME CASES
?When drawing a geometric figure or checking a given one, be sure to include
drawings of extreme cases as well as ordinary ones.
Example 1: In the figure to the right, AC is a chord and B is a point on the
circle. What is the measure of angle x?
Although in the drawing AC looks to be a diameter, that cannot be assumed. A
ll we know is that AC is a chord. Hence, numerous cases are possible, three
of which are illustrated below:
In Case I, x is greater than 45 degrees; in Case II, x equals 45 degrees; in
Case III, x is less than 45 degrees. Hence, the given information is not su
fficient to answer the question.
Example 2: Three rays emanate from a common point and form three angles with
measures p, q, and r. What is the measure of q + r ?
It is natural to make the drawing symmetric as follows:
In this case, p = q = r = 120, so q + r = 240. However, there are other draw
ings possible. For example:
In this case, q + r = 180. Hence, the given information is not sufficient to
answer the question.
Problems:
1. Suppose 3p + 4q = 11. Then what is the value of q?
(1) p is prime.
(2) q = -2p
(1) is insufficient. For example, if p = 3 and q = 1/2, then 3p + 4q = 3(3)
+ 4(1/2) = 11. However, if p = 5 and q = -1, then 3p + 4q = 3(5) + 4(-1) = 1
1. Since the value of q is not unique, (1) is insufficient.
Turning to (2), we now have a system of two equations in two unknowns. Hence
, the system can be solved to determine the value of q. Thus, (2) is suffici
ent, and the answer is B.
2. What is the perimeter of triangle ABC above?
(1) The ratio of DE to BF is 1: 3.
(2) D and E are midpoints of sides AB and CB, respectively.
Since we do not even know whether BF is an altitude, nothing can be determin
ed from (1). More importantly, there is no information telling us the absolu
te size of the triangle.
As to (2), although from geometry we know that DE = AC/2, this relationship
holds for any size triangle. Hence, (2) is also insufficient.
Together, (1) and (2) are also insufficient since we still don’t have inform
ation about the size of the triangle, so we can’t determine the perimeter. T
he answer is E.
3. A dress was initially listed at a price that would have given the store a
profit of 20 percent of the wholesale cost. What was the wholesale cost of
the dress?
(1) After reducing the asking price by 10 percent, the dress sold for a net
profit of 10 dollars.
(2) The dress sold for 50 dollars.
Consider just the question setup. Since the store would have made a profit o
f 20 percent on the wholesale cost, the original price P of the dress was 12
0 percent of the cost: P = 1.2C. Now, translating (1)sintosan equation yield
s:
P - .1P = C + 10
Simplifying gives
..9P = C + 10
Solving for P yields
P = (C + 10)/.9
Plugging this expression for PsintosP = 1.2C gives
(C + 10)/.9 = 1.2C
Since we now have only one equation involving the cost, we can determine the
cost by solving for C. Hence, the answer is A or D.
(2) is insufficient since it does not relate the selling price to any other
information. Note, the phrase "initially listed" implies that there was more
than one asking price. If it wasn’t for that phrase, (2) would be sufficien
t. The answer is A.
4. What is the value of the two-digit number x?
(1) The sum of its digits is 4.
(2) The difference of its digits is 4.
Considering (1) only, x must be 13, 22, 31, or 40. Hence, (1) is not suffici
ent to determine the value of x.
Considering (2) only, x must be 40, 51, 15, 62, 26, 73, 37, 84, 48, 95, or 5
9. Hence, (2) is not sufficient to determine the value of x.
Considering (1) and (2) together, we see that 40 and only 40 is common to th
e two sets of choices for x. Hence, x must be 40. Thus, together (1) and (2)
are sufficient to uniquely determine the value of x. The answer is C.
5. If x and y do not equal 0, is x/y an integer?
(1) x is prime.
(2) y is even.
(1) is not sufficient since we don’t know the value of y. Similarly, (2) is
not sufficient. Furthermore, (1) and (2) together are still insufficient sin
ce there is an even prime number--2. For example, let x be the prime number
2, and let y be the even number 2 (don’t forget that different variables can
stand for the same number). Then x/y = 2/2 = 1, which is an integer. For al
l other values of x and y, x/y is not an integer. (Plug in a few values to v
erify this.) The answer is E.
6. Is 500 the average (arithmetic mean) score on the GMAT?
(1) Half of the people who take the GMAT score above 500 and half of the peo
ple score below 500.
(2) The highest GMAT score is 800 and the lowest score is 200.
Many students mistakenly think that (1) implies the average is 500. Suppose
just 2 people take the test and one scores 700 (above 500) and the other sco
res 400 (below 500). Clearly, the average score for the two test-takers is n
ot 500. (2) is less tempting. Knowing the highest and lowest scores tells us
nothing about the other scores. Finally, (1) and (2) together do not determ
ine the average since together they still don’t tell us the distribution of
most of the scores. The answer is E.
7. The set S of numbers has the following properties:
I) If x is in S, then 1/x is in S.
II) If both x and y are in S, then so is x + y.
Is 3 in S?
(1) 1/3 is in S.
(2) 1 is in S.
Consider (1) alone. Since 1/3 is in S, we know from Property I that 1/(1/3)
= 3 is in S. Hence, (1) is sufficient.
Consider (2) alone. Since 1 is in S, we know from Property II that 1 + 1 = 2
(Note, nothing in Property II prevents x and y from standing for the same n
umber. In this case both stand for 1.) is in S. Applying Property II again s
hows that 1 + 2 = 3 is in S. Hence, (2) is also sufficient. The answer is D.
8. What is the area of the triangle above?
(1) a = x, b = 2x, and c = 3x.
(2) The side opposite a is 4 and the side opposite b is 3.
From (1) we can determine the measures of the angles: a + b + c = x + 2x + 3
x = 6x = 180
Dividing the last equation by 6 gives: x = 30
Hence, a = 30, b = 60, and c = 90. However, different size triangles can hav
e these angle measures, as the diagram below illus