CSE451
EXERCISE 7
SOLUTIONS
Problem 1
Given a proof system:
S
= (
L
{¬
,
⇒}
,
E
=
F
AX
=
{
(
A
⇒
A
)
,
(
A
⇒
(
¬
A
⇒
B
))
}
,
(
r
)
(
A
⇒
B
)
(
B
⇒
(
A
⇒
B
))
)
.
1.
Prove that
S
is
sound
under classical semantics.
2.
Prove that
S
is
not sound
under
K
semantics.
3.
Write a formal proof in
S
with 2 applications of the rule (
r
).
Solution of 1.
Definition:
System
S
is sound if and only if
(i)
Axioms are tautologies and
(ii)
rules of inference are sound, i.e lead from all true premisses to a true conclusion.
We verify the conditions (i), (ii) of the definition as follows.
(i)
Both axioms of
S
are basic classical tautologies.
(ii)
Consider the rule of inference of
S
.
(
r
)
(
A
⇒
B
)
(
B
⇒
(
A
⇒
B
))
)
.
Assume that its premise (the only premise) is True, i.e. let
v
be any truth assignment, such that
v
*
(
A
⇒
B
) =
T
. We evaluate logical value of the conclusion under the truth assignment
v
as follows.
v
*
(
B
⇒
(
A
⇒
B
)) =
v
*
(
B
)
⇒
T
=
T
for any
B
and any value of
v
*
(
B
).
Solution of 2.
System
S
is
not sound
under
K
semantics because axiom (
A
⇒
A
) is not a
K
semantics
tautology.
Solution of 3.
There are many solutions. Here is one of them.
Required formal proof is a sequence
A
1
, A
2
, A
3
,
where
A
1
= (
A
⇒
A
)
(Axiom)
A
2
= (
A
⇒
(
A
⇒
A
))
Rule (
r
) application 1 for
A
=
A, B
=
A
.
A
3
= ((
A
⇒
A
)
⇒
(
A
⇒
(
A
⇒
A
)))
Rule (
r
) application 2 for
A
=
A, B
= (
A
⇒
A
).
1
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Problem 2
Prove, by constructing a formal proof that
‘
S
((
¬
A
⇒
B
)
⇒
(
A
⇒
(
¬
A
⇒
B
)))
,
where
S
is the proof system from Problem 1.
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 Spring '08
 Bachmair,L
 Computer Science, Logic, formal proof

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