THE IF-THEN STATEMENT IS OFTEN EMBEDDED1 IN OTHER EQUIVALENT STRUCTURES. DIA
GRAMMING BRINGS OUT THE SUPERSTRUCTURE AND THE UNDERLYING2 SIMPLICITY3 OF ARGU
MENTS.
IF-THEN
A--B
BY NOW YOU SHOULD BE WELL AWARE THAT IF THE PREMISE4 OF AN IF-THEN STATEMENT
IS TRUE THEN THE ConCLUSION MUST BE TRUE AS WELL. THIS IS THE DEFINING CHARA
CTERISTIC OF A CONDITIONAL5 STATEMENT; IT CAN BE ILLUSTRATED6 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 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 SYMBOLIZING7 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 SYMBOLIZED9 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 VALID10 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:
JOHN AND KEN11 CANNOT BOTH GO TO THE PARTY.
AT FIRST GLANCE, THIS SENTENCE DOES NOT APPEAR TO CONT人工智能N AN IF-THEN STATEME
NT. BUT IT ESSENTIALLY12 SAYS: IF JOHN GOES TO THE PARTY, THEN KEN DOES NOT.
EXAMPLE:
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
进销存RET 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: IF
A IS TRUE, THEN B IS TRUE AS WELL. IF B IS FALSE, THEN A IS FALSE AS WEL
L . 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 , THEN THE IF-THEN STATEMENT TELLS U
S NOTHING about THE CONCLUSION.
EXAMPLE:
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?
TO INSURE THAT THE REM人工智能NING WETLANDS SURVIVE, THEY MUST BE PROTECTED BY
THE GOVERNMENT. THIS PARTICULAR WETLAND IS BEING NEGLECTED. THEREFORE, IT W
ILL SOON PERISH.
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 BLEMISH13 ON MY HAND.
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.
EVERY TIME MELINDA HAS SUBMITTED HER BOOK FOR PUBLICATION IT HAS BEEN RE
JECTED. SO SHE SHOULD NOT BOTHER WITH ANOTHER REWRITE.
WHEN THE GOVERNMENT LOSES THE POWER TO TAX ONE AREA OF THE ECONOMY, IT J
UST TAXES ANOTHER. THE SUPREME14 COURT JUST OVERTURNED THE SALES TAX, SO WE CA
N EXPECT AN INCREASE IN THE INCOME TAX.
TO SYMBOLIZE8 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 INVALIDLY15 U
SES IT TO DEDUCE THE PREMISE. THE ANSWER WILL LIKEWISE BE FALLACIOUS.
WE START WITH ANSWER-CHOICE . THE SENTENCE
TO INSURE THAT THE REM人工智能NING WETLANDS SURVIVE, THEY MUST BE PROTECTED BY TH
E GOVERNMENT
CONT人工智能NS AN EMBEDDED IF-THEN STATEMENT:
IF THE REM人工智能NING 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 , 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 INT进销存RET 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 .
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 NEGATED16.
EXAMPLE:
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 CONT人工智能NS AN EMBEDDED IF-THEN STATEMENT: IF THE SENATOR IS REELECTED, T
HEN HE OPPOSES THE NEW TAX BILL. 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 ~ 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
DS-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 DONT 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
