ℑ
∧
⇒ ℑ
∧ ℑ
x
x
x
P x Q x
P x
Q x
[ ( )
( )] [
( )
( )]
Solu tion
ℑ
∧
⇔
∧
∨
∧
⇔
∧
x P x Q x
P
P
P
Q
P
Q
[ ( )
( )] [( ( )
( )] [( ( )
( )]
[[ ( )
(
0
0
1
1
0
0)]
( )] [( ( )
( ))
( )]
[[ ( )
( )] [ ( )
(
∨
∧
∧
∨
⇔
∨
∧
∨
P
P
Q
Q
P
P
Q
P
1
0
0
1
0
1
0
1)] [[ ( )
( )]
[ ( )
( )]]
[[ ( )
( )] [ ( )
(
∧
∨
∧
∨
⇔
∨
∧
∨
P
Q
Q
Q
P
P
Q
Q
0
1
0
1
0
1
0
1)]]
( )
( )
⇔ ℑ
∧ ℑ
x
x
P x
Q x
(for this universe).
Hence proved.
Ì Exam ple 8.4.8: For each of the following sets of premises, list the
relevant conclusions which can be drawn and the rules of inference used in
each case.
(a) All cows are mammals. Some mammals chew their cud.
(b) All even integers are divisible by 2. The integer 4 is even but 3 is
not.
(c) What’s good for the auto industry is good for the country. What’s
good for the country is good for you. What’s good for the auto
industry is for you to buy an expensive car.
Solu tion
(a) C(x): is x is a cow.
M(x): x is a mammal.
D(x): x chew their cud
∨
→
x C x
M x
[ ( )
( )]
ℑ
→
x M x
D x
( )
( ).
(b) E(x): x is an even Integer.
D(x): x is divisible by 2.
E x
D x
( )
( )
→
.
(c) A x x
( ): is good for the auto industry
C(x): x is good for the country
Y(x): x is good for you.
b = “You buying an expensive car” (constant)
V A x
C x
x [ ( )
( )]
→
(1)
V C x
Y x
x [ ( )
( )]
→
(2)
A(b)
(3)
By US,
A b
C b
( )
( )
→
(4)
C b
Y b
( )
( )
→
(5)
Prop o si tions and Pred i cates
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