E 15 R P
P
F
∧ ∧ ¬
⇔
(
)
E 16 P Q
P Q
→ ⇔ ¬ ∨
E 17 ¬ →
⇔ ∧ ¬
(
)
P Q
P
Q
E 18 P Q
Q
P
→ ⇔ ¬ → ¬
E 19 P
Q R
P Q
R
→
→
⇔ ∧ →
(
)
(
)
E 20 ¬ ⇔
⇔ ⇔ ¬
(
)
P
Q
P
Q
E 21 P
Q
P Q
Q P
⇔ ⇔ →
∧ →
(
) (
)
E 22 (
)
(
) (
)
P
Q
P Q
P
Q
⇔
⇔ ∧ ∨ ¬ ∧ ¬
Ì Exam ple 8.2.1: Show that R ∨ S follows logically from the premises
C D C D
H
∨
∨
→ ¬
, (
)
, ¬ →
∧ ¬
H
A
B
(
) and (
) (
)
A
B
R S
∧ ¬ →
∨ .
Solu tion
1.
C D
H
∨ → ¬
P
2.
¬ →
∧ ¬
H
A
B
(
)
P
3.
C D
A
B
∨ →
∧ ¬
(
)
Using Hyp. Syll in (1), (2)
4.
(
) (
)
A
B
R S
∧ ¬ →
∨
P
5.
(
) (
)
C D
R S
∨
→
∨
Using Hyp. Syll. in (3), (4)
6.
C D
∨
P
7.
R S
∨
Modus Ponens
Ì Exam ple 8.2.2: Show that S R
∨ is tautologically implied by
(
) (
) (
).
P Q
P
R
Q S
∨ ∧ →
∧ →
Solu tion
1.
P Q
∨
P
2.
¬ →
P Q
(
(
))
Q P
P
P Q
→ ⇔ ¬ ∨
3.
Q S
→
P
4.
¬ →
P
S
(2), (3), Hyp. Syll.
5.
¬ →
S
P
(From (4), using (
)
P Q
Q
P
→ ⇔ ¬ → ¬ )
6.
P
R
→
P
7.
¬ →
S
R
(5), (6) & Hyp. Syll.
8.
S R
∨
Ì Exam ple 8.2.3: Show that R P Q
∧ ∨
(
) is a valid conclusion from the
premises P Q Q R P
M
∨
→
→
,
,
and ¬ M.
Prop o si tions and Pred i cates
267