1.8. Implication 5
Probably th e easies t wa y t o giv e a n exampl e o f a n irrationa l i s just t o
write dow n a nonrepeating , nonterminatin g decima l lik e
.01 001 0001 00001 000001....
although that' s n o numbe r you'v e eve r hear d of .
1.7. Logi c an d ambiguou s English . Englis h i s a treacherou s lan -
guage, an d w e have to b e very careful . Fo r example , ther e ar e tw o meaning s
of "or " i n English :
(1) A craz y "exclusive " or , meanin g "on e o r th e other , bu t no t both. "
I ha d lunc h i n th e Frenc h Quarter , an d the y offere d m e a choic e of
black-eyed pea s or greens , bu t the y wouldn' t giv e m e both .
(2) Th e sensible , mathematica l "inclusive " or , meanin g "on e o r th e
other o r both. " Yo u ca n wor k o n you r rea l analysi s homewor k da y
or night .
1.8. Implication . Ther e ar e man y way s t o sa y tha t on e statemen t A
implies anothe r statemen t B. Th e followin g al l mea n exactl y th e sam e
thing:
I f A, the n B.
A implie s B , writte n A = B.
A onl y i f B.
B i f A , writte n B ^ A.
Here ar e som e example s o f suc h equivalen t statements :
I f a numbe r n i s divisible b y four , the n i t i s even .
n divisibl e b y fou r implie s n even .
n i s divisibl e b y fou r onl y i f i t i s eve n (neve r i f it i s odd) .
n i s even i f divisibl e b y four .
Such a n implicatio n i s true i f B i s true o r i f A i s false (i n whic h cas e we
say tha t th e implicatio n i s "vacuousl y true") . Fo r example , "I f 5 i s even ,
then 15 i s prime " i s vacuousl y true . Thi s i s pur e logi c an d ha s nothin g t o
do wit h causation . (Indeed , fro m th e poin t o f vie w o f causation , i f 5 wer e
even, the n 15, as a multipl e o f 5 , would b e eve n an d henc e not prime. )
Such a n implicatio n i s false onl y i f A i s true an d B i s false .
Such a n implicatio n i s logicall y equivalen t t o it s contrapositive : no t B
implies no t A. Thi s i s the basi s fo r proo f b y contradiction . W e assum e th e
negation o f th e B w e ar e tryin g t o prove , an d arriv e a t a contradictio n o f
the hypothesi s A.
Previous Page Next Page