material de aula

1. Disciplina de Estruturas L´ogico-Dedutivas
Solu¸c˜oes da Lista de Exerc´ıcios 4
M´etodo dedutivo da l´ogica sentencial:
estrat´egias de prova, provas de teoremas
1. P ≡ Q ¬P ≡ ¬Q
1 P ≡ Q [P]
2 P → Q [≡E 1]
3 Q → P [≡E 1]
4 ¬P [H p/PC]
5 Q [H p/RAA]
6 P [MP 5,3]
7 ¬P ∧ P [∧I 4,6]
8 ¬Q [RAA 5-7]
9 ¬P → ¬Q [PC 4-8]
10 ¬Q [H p/PC]
11 P [H p/RAA]
12 Q [MP 11,2]
13 ¬Q ∧ Q [∧I 10,12]
14 ¬P [RAA 11-13]
15 ¬Q → ¬P [PC 10-14]
16 ¬P ≡ ¬Q [≡I 9,15]
2. ¬P ∨ Q ¬(P ∧ ¬Q)
1 ¬P ∨ Q [P]
2 ¬P [H p/PC]
3 P ∧ ¬Q [H p/RAA]
4 P [∧E 3]
5 ¬P ∧ P [∧I 2,4]
6 ¬(P ∧ ¬Q) [RAA 3-5]
1

2. 7 P → ¬(P ∧ ¬Q) [PC 2-6]
8 Q [H p/PC]
9 P ∧ ¬Q [H p/RAA]
10 ¬Q [∧E 9]
11 Q ∧ ¬Q [∧I 8,10]
12 ¬(P ∧ ¬Q) [RAA 9-11]
13 Q → ¬(P ∧ ¬Q) [PC 8-12]
14 ¬(P ∧ ¬Q) [∨E 1,7,13]
3. P → Q (P ∧ R) → (Q ∧ R)
1 P → Q [P]
2 P ∧ R [H p/PC]
3 P [∧E 2]
4 Q [MP 3,1]
5 R [∧E 2]
6 Q ∧ R [∧I 4,5]
7 (P ∧ R) → (Q ∧ R) [PC 2-6]
4. P → Q (P ∨ R) → (Q ∨ R)
1 P → Q [P]
2 P ∨ R [H p/PC]
3 P [H p/PC]
4 Q [MP 3,1]
5 Q ∨ R [∨I 4]
6 P → Q ∨ R [PC 3-5]
7 R [H p/PC]
8 Q ∨ R [∨I 7]
9 R → Q ∨ R [PC 7-8]
10 Q ∨ R [∨E 2,6,9]
11 (P ∨ R) → (Q ∨ R) [PC 2-10]
5. P → P
1 P [H p/PC]
2 P → P [PC 1-1]
2

3. 6. P → (Q → (P ∧ Q))
1 P [H p/PC]
2 Q [H p/PC]
3 P ∧ Q [∧I 1,2]
4 Q → (P ∧ Q) [PC 2-3]
5 P → (Q → (P ∧ Q)) [PC 1-4]
7. ¬(P ≡ ¬P)
1 P ≡ ¬P [H p/RAA]
2 P → ¬P [≡E 1]
3 ¬P → P [≡E 1]
4 P [H p/RAA]
5 ¬P [MP 4,2]
6 P ∧ ¬P [∧I 4,5]
7 ¬P [RAA 4-6]
8 ¬P [H p/RAA]
9 P [MP 8,3]
10 ¬P ∧ P [∧I 8,9]
11 ¬¬P [RAA 8-10]
12 P [¬E 11]
13 ¬P ∧ P [∧I 7,12]
14 ¬(P ≡ ¬P) [RAA 1-13]
8. (P → Q) → (¬Q → ¬P)
1 P → Q [H p/PC]
2 ¬Q [H p/PC]
3 P [H p/RAA]
4 Q [MP 3,1]
5 ¬Q ∧ Q [∧I 2,4]
6 ¬P [RAA 3-5]
7 ¬Q → ¬P [PC 2-6]
8 (P → Q) → (¬Q → ¬P) [PC 1-7]
3

4. 9. Q → (P ∨ ¬P)
1 Q [H p/PC]
2 ¬(P ∨ ¬P) [H p/RAA]
3 P [H p/RAA]
4 P ∨ ¬P [∨I 3]
5 ¬(P ∨ ¬P) ∧ (P ∨ ¬P) [∧I 2,4]
6 ¬P [RAA 3-5]
7 P ∨ ¬P [∨I 6]
8 ¬(P ∨ ¬P) ∧ (P ∨ ¬P) [∧I 2,7]
9 ¬¬(P ∨ ¬P) [RAA 2-8]
10 P ∨ ¬P [¬E 9]
11 Q → (P ∨ ¬P) [PC 1-10]
10. P ∨ (P → Q)
1 ¬(P ∨ (P → Q)) [H p/RAA]
2 P [H p/PC]
3 ¬Q [H p/RAA]
4 P ∨ (P → Q) [∨I 2]
5 ¬(P ∨ (P → Q)) ∧ (P ∨ (P → Q)) [∧I 1,4]
6 ¬¬Q [RAA 3-5]
7 Q [¬E 6]
8 P → Q [PC 2-7]
9 P ∨ (P → Q) [∨I 8]
10 ¬(P ∨ (P → Q)) ∧ (P ∨ (P → Q)) [∧I 1,9]
11 ¬¬(P ∨ (P → Q)) [RAA 1-10]
12 P ∨ (P → Q) [¬E 11]
11. ¬P ∨ (Q → P)
1 ¬(¬P ∨ (Q → P)) [H p/RAA]
2 Q [H p/PC]
3 ¬P [H p/RAA]
4 ¬P ∨ (Q → P) [∨I 3]
5 ¬(¬P ∨ (Q → P)) ∧ (¬P ∨ (Q → P)) [∧I 1,4]
6 ¬¬P [RAA 3-5]
7 P [¬E 6]
4

5. 8 Q → P [PC 2-7]
9 ¬P ∨ (Q → P) [∨I 8]
10 ¬(¬P ∨ (Q → P)) ∧ (¬P ∨ (Q → P)) [∧I 1,9]
11 ¬¬(¬P ∨ (Q → P)) [RAA 1-10]
12 ¬P ∨ (Q → P) [¬E 11]
5