 #1
VeraMason
 1
 0
 Homework Statement:

Prove that the following sentences are theorems (Truths of Logic):
A iff ~~A *Do not use Double Negation*
 Relevant Equations:
 NA
My thought was to break up the sentence into its equivalent form: (A >~~A) & (~~A > A)
From there I assumed the premise of both sides to use indirect proofs, so:
1. ~(A > ~~A) AP
2. ~(~A or ~~A) 1 Implication
3. ~~A & ~~~A 2 DeMorgan's
4. A > ~~A 13 Indirect Proof
5. ~(~~A > A) AP
6. ~(~~~A or A) 5 Implication
7. ~~~~A & ~A 6 DeMorgan's
8. ~~A > A 57 Indirect Proof
9. (A >~~A) & (~~A > A) 4,8 Conjunction
10. A iff ~~A 9 Equivalence
To me, this looks like it would be correct. Obviously, lines 3 and 7 would look a lot cleaner if I was allowed to use double negation, but in my mind, it shouldn't matter since both lines are a contradiction that essentially says: A & ~A.
Is this correct?
From there I assumed the premise of both sides to use indirect proofs, so:
1. ~(A > ~~A) AP
2. ~(~A or ~~A) 1 Implication
3. ~~A & ~~~A 2 DeMorgan's
4. A > ~~A 13 Indirect Proof
5. ~(~~A > A) AP
6. ~(~~~A or A) 5 Implication
7. ~~~~A & ~A 6 DeMorgan's
8. ~~A > A 57 Indirect Proof
9. (A >~~A) & (~~A > A) 4,8 Conjunction
10. A iff ~~A 9 Equivalence
To me, this looks like it would be correct. Obviously, lines 3 and 7 would look a lot cleaner if I was allowed to use double negation, but in my mind, it shouldn't matter since both lines are a contradiction that essentially says: A & ~A.
Is this correct?