- #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 1-3 Indirect Proof

5. ~(~~A -> A) AP

6. ~(~~~A or A) 5 Implication

7. ~~~~A & ~A 6 DeMorgan's

8. ~~A -> A 5-7 Indirect Proof

9. (A ->~~A) & (~~A -> A) 4,8 Conjunction

10. A iff ~~A 9 EquivalenceTo 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 1-3 Indirect Proof

5. ~(~~A -> A) AP

6. ~(~~~A or A) 5 Implication

7. ~~~~A & ~A 6 DeMorgan's

8. ~~A -> A 5-7 Indirect Proof

9. (A ->~~A) & (~~A -> A) 4,8 Conjunction

10. A iff ~~A 9 EquivalenceTo 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?