Solu tion
1.
C D
∨
P
2.
C D
H
∨ → ¬
P
3.
¬ H
(1), (2), Modus Ponens
4.
¬ →
∧ ¬
H
A
B
(
)
P
5.
A
B
∧ ¬
(3), (4), Modus Ponens
6.
(
) (
)
A
B
R S
∧ ¬ →
∨
P
7.
R S
∨
(5), (6), Modus Ponens
Ì Exam ple 8.2.6: State whether the following argument is valid or not. If
valid, give proof. If not valid, give counter example.
If a baby is hungry the baby cries.
If the baby is not mad, then he does not cry.
If a baby is mad, then he has a red face.
Therefore, if a baby is hungry, then he has a red face.
Solu tion
H : Baby is hungry
C : Baby cries
M : Baby is mad
R : Baby has a red face.
Given:
H C
→
¬ → ¬
M
C
M
R
→
∴ →
H
R
Verification:
1.
H C
→
Prem ise
2.
¬ → ¬
M
C
Prem ise
3.
C
M
→
(2), Contrapositive
4.
H
M
→
(1), (3), Hyp. Syll.
5.
M
R
→
Prem ise
6.
H
R
→
(4), (5), Hyp. Syll.
Ì Exam ple 8.2.7: State whether the following argument is valied or not.
“If Nixon is not re-elected, then Tulsa will lose its air base. Nixon will
be re-elected if and only if Tulsa votes for him. If Tulsa keeps its air base,
Nixon will be re-elected. Therefore, Nixon will be re-elected”.
Prop o si tions and Pred i cates
269