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 cross-off 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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