EXERCISE 8 SOLUTIONS
Problem 1
H
is the following proof system:
H
= (
L
{⇒
,
¬}
,
F
, AX
=
{
A
1
,A
2
,A
3
}
, MP
)
A1
(
A
⇒
(
B
⇒
A
))
,
A2
((
A
⇒
(
B
⇒
C
))
⇒
((
A
⇒
B
)
⇒
(
A
⇒
C
)))
,
A3
((
¬
B
⇒ ¬
A
)
⇒
((
¬
B
⇒
A
)
⇒
B
)))
A4
(((
A
⇒
B
)
⇒
A
)
⇒
A
)
MP
(Rule of inference)
(
MP
)
A
; (
A
⇒
B
)
B
(1)
Prove that
H
is SOUND under classical semantics.
Solution:
Soundness Theorem holds because all axioms of
H
are tautologies and MP leads from tautologies
to a tautology.
(2)
Why Deduction Theorem holds for
H
?
Solution:
System
H
extends by one extra axiom
A
3 the proof system
H
1
for which we have proved the
deduction theorem.
(3)
Is
H
COMPLETE?
Solution:
YES. Axioms A1, A2, A3 of
H
are axioms of the system
H
2
from Chapter 8. It is stated in Chapter
8 and proved in Chapter 9 that Completeness Theorem holds for
H
2
.
Problem 2
S
is the following (sound) proof system:
S
= (
L
{⇒
,
∩}
,
F
, AX
=
{
A
1
} R
=
{
(
r
1
)
,
(
r
2
)
}
)
,
where
Axiom:
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 Spring '08
 Bachmair,L
 Computer Science, Logic, Proof theory, Completeness Theorem, Axiom A1

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