1. The document contains solutions to exercises involving proofs in sentential logic using deduction methods. It presents 12 proofs demonstrating strategies for proving theorems about logical equivalences and implications between propositions. The proofs use sentential logic rules and inference steps like modus ponens, disjunctive syllogism, proof by contradiction, and equivalence introduction/elimination.

1. Disciplina de Estruturas Lógico-Dedutivas
Soluções da Lista de Exercı́cios 4
Método dedutivo da lógica sentencial:
estratégias 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

3. ¬P [H p/PC]
5

7. Q [H p/RAA]
6

11. P [MP 5,3]
7

15. ¬P ∧ P [∧I 4,6]
8

17. ¬Q [RAA 5-7]
9 ¬P → ¬Q [PC 4-8]
10

19. ¬Q [H p/PC]
11

23. P [H p/RAA]
12

27. Q [MP 11,2]
13

31. ¬Q ∧ Q [∧I 10,12]
14

33. ¬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

35. ¬P [H p/PC]
3

39. P ∧ ¬Q [H p/RAA]
4

43. P [∧E 3]
5

47. ¬P ∧ P [∧I 2,4]
6

49. ¬(P ∧ ¬Q) [RAA 3-5]
1

50. 7 P → ¬(P ∧ ¬Q) [PC 2-6]
8

52. Q [H p/PC]
9

56. P ∧ ¬Q [H p/RAA]
10

60. ¬Q [∧E 9]
11

64. Q ∧ ¬Q [∧I 8,10]
12

66. ¬(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

68. P ∧ R [H p/PC]
3

70. P [∧E 2]
4

72. Q [MP 3,1]
5

74. R [∧E 2]
6

76. 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

78. P ∨ R [H p/PC]
3

82. P [H p/PC]
4

86. Q [MP 3,1]
5

90. Q ∨ R [∨I 4]
6

92. P → Q ∨ R [PC 3-5]
7

96. R [H p/PC]
8

100. Q ∨ R [∨I 7]
9

102. R → Q ∨ R [PC 7-8]
10

104. Q ∨ R [∨E 2,6,9]
11 (P ∨ R) → (Q ∨ R) [PC 2-10]
5. ` P → P
1

106. P [H p/PC]
2 P → P [PC 1-1]
2

107. 6. ` P → (Q → (P ∧ Q))
1

109. P [H p/PC]
2

113. Q [H p/PC]
3

117. P ∧ Q [∧I 1,2]
4

119. Q → (P ∧ Q) [PC 2-3]
5 P → (Q → (P ∧ Q)) [PC 1-4]
7. ` ¬(P ≡ ¬P)
1

121. P ≡ ¬P [H p/RAA]
2

123. P → ¬P [≡E 1]
3

125. ¬P → P [≡E 1]
4

129. P [H p/RAA]
5

133. ¬P [MP 4,2]
6

137. P ∧ ¬P [∧I 4,5]
7

139. ¬P [RAA 4-6]
8

143. ¬P [H p/RAA]
9

147. P [MP 8,3]
10

151. ¬P ∧ P [∧I 8,9]
11

153. ¬¬P [RAA 8-10]
12

155. P [¬E 11]
13

157. ¬P ∧ P [∧I 7,12]
14 ¬(P ≡ ¬P) [RAA 1-13]
8. ` (P → Q) → (¬Q → ¬P)
1

159. P → Q [H p/PC]
2

163. ¬Q [H p/PC]
3

169. P [H p/RAA]
4

175. Q [MP 3,1]
5

181. ¬Q ∧ Q [∧I 2,4]
6

185. ¬P [RAA 3-5]
7

187. ¬Q → ¬P [PC 2-6]
8 (P → Q) → (¬Q → ¬P) [PC 1-7]
3

188. 9. ` Q → (P ∨ ¬P)
1

190. Q [H p/PC]
2

194. ¬(P ∨ ¬P) [H p/RAA]
3

200. P [H p/RAA]
4

206. P ∨ ¬P [∨I 3]
5

212. ¬(P ∨ ¬P) ∧ (P ∨ ¬P) [∧I 2,4]
6

216. ¬P [RAA 3-5]
7

220. P ∨ ¬P [∨I 6]
8

224. ¬(P ∨ ¬P) ∧ (P ∨ ¬P) [∧I 2,7]
9

226. ¬¬(P ∨ ¬P) [RAA 2-8]
10

228. P ∨ ¬P [¬E 9]
11 Q → (P ∨ ¬P) [PC 1-10]
10. ` P ∨ (P → Q)
1

230. ¬(P ∨ (P → Q)) [H p/RAA]
2

234. P [H p/PC]
3

240. ¬Q [H p/RAA]
4

246. P ∨ (P → Q) [∨I 2]
5

252. ¬(P ∨ (P → Q)) ∧ (P ∨ (P → Q)) [∧I 1,4]
6

256. ¬¬Q [RAA 3-5]
7

260. Q [¬E 6]
8

262. P → Q [PC 2-7]
9

264. P ∨ (P → Q) [∨I 8]
10

266. ¬(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

268. ¬(¬P ∨ (Q → P)) [H p/RAA]
2

272. Q [H p/PC]
3

278. ¬P [H p/RAA]
4

284. ¬P ∨ (Q → P) [∨I 3]
5

290. ¬(¬P ∨ (Q → P)) ∧ (¬P ∨ (Q → P)) [∧I 1,4]
6

294. ¬¬P [RAA 3-5]
7

298. P [¬E 6]
4

299. 8

301. Q → P [PC 2-7]
9

303. ¬P ∨ (Q → P) [∨I 8]
10

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