Natural Deduction Proof Help?
Having trouble with these two
1.
1 ~Y ⊃ ~L
2 W v ~Y
3 W ⊃ ~B
/ ~B v ~L
2.
1 ~U • (G = ~T)
2 (~U v D) ⊃ (W • ~R)
/ W • ~U
Sure! Let's go through each proof step by step.
For the first one:
1. ~Y ⊃ ~L - Given
2. W v ~Y - Given
3. W ⊃ ~B - Given
/ ~B v ~L - To prove
To prove ~B v ~L, we can use proof by cases. We'll consider two cases: W and ~W.
Case 1: W
4. ~Y - Assume (for conditional elimination)
5. ~L - Modus ponens (1, 4)
6. ~B v ~L - Disjunction introduction (5)
Case 2: ~W
7. ~Y - Assume (for disjunction elimination)
8. ~B - Modus ponens (3, 7)
9. ~B v ~L - Disjunction introduction (8)
Since we were able to derive ~B v ~L in both cases, we can conclude ~B v ~L using disjunction elimination (∨E) on line 2.
For the second proof:
1. ~U • (G = ~T) - Given
2. (~U v D) ⊃ (W • ~R) - Given
/ W • ~U - To prove
To prove W • ~U, we can use a proof by contradiction. Assume ~ (W • ~U), and try to derive a contradiction.
3. ~(W • ~U) - Assume (for contradiction)
4. ~W v ~~U - De Morgan's law (3)
5. ~W v U - Double negation elimination (4)
6. W v ~U - Commutation (5)
7. W v ~U - Disjunctive syllogism (2, 6)
8. ~U v D - Modus ponens (2, 7)
9. W • ~R - Modus ponens (8, 2)
10. ~W - Simplification (9)
11. W • ~U - Conjunction (9, 10)
Since we were able to derive the contradiction W • ~U in line 11, we can conclude W • ~U using proof by contradiction (~I) on line 3.