GMT考试--Testprep数学精解（2）

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