we may do it as an intermediate step. However, we do need
to have the conditional we wish to prove as a SHOW line.
We are allowed to introduce new SHOW lines whenever we
wish.
Once we crossoff the “SHOW” of the newly introduced
“SHOW” line we can then use it as though it were a normal
line.
However,
once a box has been drawn around some lines, we
cannot use those lines again
in our derivation. Examples:
1.
2.
3.
4.
5.
6.
7.
[
P
!
(
P
_
Q
)]
!
R
⇠
⇠
⇠
SHOW:
R
⇠
⇠
⇠
SHOW:
P
!
(
P
_
Q
)
P
⇠
⇠
⇠
SHOW:
P
_
Q
P
_
Q
R
Pr
DD
CD
Ass
DD
4
_
I
1,3
!
O
Here, we used CD in proving the SHOW line at line 3, but the
ultimate conclusion, the SHOW line at line 2, was not proven
by CD, but by
!
O, which is DD rule, so we write “DD” at 2,
and “CD” at line 3.
This basic procedure allows us to use multiple CDs, combined
with
$
I, to prove biconditional (iff) statements.
We Frst prove the conditional one way, and then the other.
Then we put them together.
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 Spring '08
 BOHN
 Logic, Proof theory

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