CSE451
EXERCISE 6 SOLUTIONS
QUESTION 1
Given a proof system:
S
= (
L
{∪
,
⇒}
,
E
=
F
AX
=
{
A
1
,A
2
}
,
R
=
{
(
r
)
}
)
,
where
A
1 = (
A
⇒
(
A
∪
B
))
,
A
2 = (
A
⇒
(
B
⇒
A
))
and
(
r
)
(
A
⇒
B
)
(
A
⇒
(
A
⇒
B
))
1. Solution:
Prove that
S
is
sound
under classical semantics.
Solution:
Axioms of
S
are basic classical tautologies. The proof of soundness of the rule of inference is the
following.
Assume (
A
⇒
B
) =
T
. Hence the logical value of conclusion is (
A
⇒
(
A
⇒
B
)) = (
A
⇒
T
) =
T
for all
A
.
2.
Determine whether
S
is
sound
or
not sound
under
K
semantics.
K semantics
diﬀer from ˆLukasiewicz’s semantics only in a case on implication only. This table is:
KImplication
⇒
F
⊥
T
F
T
T
T
⊥
⊥ ⊥
T
T
F
⊥
T
Solution 1:
S
is not sound under
K
semantics. Let’s take truth assignment such that
A
=
⊥
,B
=
⊥
. The
logical value of axiom A1 is as follows.
(
A
⇒
(
A
∪
B
)) = (
⊥⇒
(
⊥ ∪ ⊥
)) =
⊥
and
6 
=
K
(
A
⇒
(
A
∪
B
)).
Observe
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Bachmair,L
 Computer Science, Logic, logical value, =⊥, formal proof A1

Click to edit the document details