Having trouble with these two
1.
1 ~J
2 (B ⊃ M) ⊃ (~J ⊃ B)
3 ~J ⊃ (B ⊃ M)
/ B
2.
1 ~L
2 V v (A v L)
3 (V v A) ⊃ ~V
/ A
To solve these two proofs, you can use the method of natural deduction. This method involves applying logical rules and making deductions to reach the desired conclusion.
Let's start with the first proof:
1. 1 ~J (Given)
2. (B ⊃ M) ⊃ (~J ⊃ B) (Given)
3. ~J ⊃ (B ⊃ M) (Given)
/ B (To be proven)
To prove that B is true, we need to use the rules of implication elimination, modus ponens, and proof by contradiction. Here's a step-by-step breakdown:
4. Assume ~B. (Proof by Contradiction)
5. Assume ~(B ⊃ M). (Proof by Contradiction)
6. Using the rule of implication elimination (from line 2), we derive ~J ⊃ B.
7. Using the rule of modus ponens (from lines 1 and 6), we derive ~J.
8. Using the rule of modus ponens (from lines 3 and 7), we derive B ⊃ M.
9. Using the rule of modus ponens (from lines 5 and 8), we derive a contradiction: ~(B ⊃ M) and B ⊃ M cannot both be true.
10. Using the rule of proof by contradiction, we negate the assumption made in line 5 to derive B.
11. Using the rule of proof by contradiction, we negate the assumption made in line 4 to derive ~~B.
12. Applying double negation elimination (~~B → B), we derive B.
Therefore, B is proven to be true.
Now let's move to the second proof:
1. 1 ~L (Given)
2. V v (A v L) (Given)
3. (V v A) ⊃ ~V (Given)
/ A (To be proven)
To prove that A is true, we can use the rules of disjunction elimination and modus ponens. Here are the steps:
4. Assume ~(A). (Proof by Contradiction)
5. Using the rule of disjunction elimination (from line 2), we derive V v L.
6. There are two cases to consider:
- Case 1: V
7. Using the rule of disjunction elimination (from line 6), we can derive A from ~A (reaching a contradiction).
- Case 2: L
8. Using the rule of modus ponens (from lines 3 and 1), we can derive ~V.
9. In both cases, we reach a contradiction. Therefore, ~(A) is false, which means A is true.
Therefore, A is proven to be true.