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“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:
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充分性精解转载smth 2001-10-14 10:51:58发信人: ykk (我不说话并不代表我不在乎),信区: EnglishTest
标题: (GMAT)Testprep充分性精解
发信站: BBS水木清华站(Fri Oct 12 16:07:05 2001)
Data Sufficiency
----------------------------------------------------------------------------
----
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)。
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