E. Answers
(1) V(A) = F, V(B) = T, V(C) = F V(A) = F, V(B) = F, V(C) = F
(2) V(A) = T, V(B) = T, V(C) = F, V(D) = F V(A) = T, V(B) = F, V(C) = T, V(D) = F
(3) V(A) = T, V(B) = F, V(C) = T
(4) V(A) = F, V(B) = F, V(C) = F, V(D) = F, V(E) = T
F. Short Truth Table Test for Consistency Use the short truth table method to show that the following sets of premises are consistent:
(1) 1. (A B) ∙ ~ C (3) 1. A ∙ (B E) 2. C D 2. (C D) ~ A 3. D (A B) / B 3. B (C ∙ D) / E
(2) 1. D (A C) (4) 1. ~ B C 2. ~ [B (D E)] 2. (D C) ∙ (A ~ C) 3. ~ B ~ C / ~ D 3. (C ∙ ~ C) B 4. B (D A) / B
F. Answers
(1) V(A) = T, V(B) = T, V(C) = F, V(D) = F
(2) V(A) = T, V(B) = T, V(C) = F, V(D) = T, V(E) = F
(3) V(A) = T, V(B) = F, V(C) = F, V(D) = F, V(E) = F
(4) V(A) = F, V(B) = T, V(C) = T, V(D) = F
IV. CHAPTER FOUR: PROOFS WITHOUT CP OR IP Prove valid, using the eighteen valid argument forms (but not CP or IP):
(l) 1. A ~ A /~ A
(2) 1. ~ A B 2. C A 3. ~ B /~ C
11 (3) 1. A / (~ A C)
(4) 1. (A B) C (7) 1. (A B) (C D) / B C 2. ~ A ~ C /~ B
(5) 1. ~ [A ~ (~ A ~ B)] (8) 1. (A B) C 2. ~ (A B) (C D) / D 2. (B C) D 3. ~ (E ~ B) / ~ (~ D A) (6) 1. (A B) C 2. ~ C 3. ~ D B (9) 1. A B 4. ~ A ~ E /~ (~ E D) 2. C D / (A C) (B D)
IV. ANSWERS (l) 1. A ~ A / ~ A p 2. ~ A ~ A 1 Impl 3. ~ A 2 Taut
(2) 1. ~ A B p 2. C A p 3. ~ B / ~ C p 4. ~ A 1, 3 DS 5. ~ C 2, 4 MT
(3) 1. A/ (~ A C) p 2. A (B C) 1 Add 3. ~ ~ A (B C) 2 DN 4. ~ A (B C) 3 Impl 5. (~ A B) C 4 Exp 6. (B ~ A) C 5 Comm 7. B (~ A C) 6 Exp
(4) 1. (A B) C p 2. ~ A ~ C /~ B p 3. ~ C 2 Simp 4. ~ (A B) 1, 3 MT 5. ~ A ~ B 4 DeM 6. ~ B 5 Simp
12 (5) 1. ~ [A ~ (~ A ~ B)] p 2. ~ (A B) (C D) / D p ~ [A (~ ~ A ~ ~ B)] 1 DeM 4. ~ [A (A B)] 3 DN (2x) 5. ~ [(A A) B] 4 Assoc 6. ~ (A B) 5 Taut 7. C D 2,6 MP 8. D 7 Simp
(6) 1. (A B) C p 2. ~ C p 3. ~ D B p 4. ~ A ~ E /~ (~ E D) p 5. ~ (A B) 1,2 MT 6. ~ A ~ B 5 DeM 7. ~ A 6 Simp 8. ~ E 4,7 MP 9. ~ B 6 Simp 10. ~ D 3,9 DS 11. ~ E ~ D 8,10 Conj 12. ~ (E D) 11 DeM 13. ~ (~ ~ E D) 12 DN 14. ~ (~ E D) 13 Impl
(7) 1. (A B) (C D) / B C p 2. ~ (A B) (C D) 1 Impl 3. (~ A ~ B) (C D) 2 DeM 4. [(~ A ~ B) C] [(~ A ~ B) D] 3 Dist 5. (~ A ~ B) C 4 Simp 6. C (~ A ~ B) 5 Comm 7. (C ~ A) (C ~ B) 6 Dist 8. C ~ B 7 Simp 9. ~ B C 8 Comm 10. B C 9 Impl
13 (8) 1. (A B) C p 2. (B C) D p 3. ~ (E ~ B) / ~ (~ D A) p 4. ~ E ∙ ~ ~ B 3 DeM 5. ~ E ∙ B 4 DN 6. B 5 Simp 7. B (C D) 2 Exp 8. C D 6, 7 MP 9. (A B) D 1, 8 HS 10. (B A) D 9 Comm 11. B (A D) 10 Exp 12. A D 6, 11 MP 13. ~ A D 12 Impl 14. ~ A ~ ~ D 13 DN 15. ~ (A ~ D) 14 DeM 16. ~ (~ D A) 15 Comm
(9) 1. A B p 2. C D / (A C) (B D) p 3. ~ A B 1 Impl 4. (~ A B) ~ C 3 Add 5. ~ A (B ~ C) 4 Assoc 6. ~ A (~ C B) 5 Comm 7. (~ A ~ C) B 6 Assoc 8. ~ C D 2 Impl 9. (~ C D) ~ A 8 Add 10 ~ A (~ C D) 9 Comm 11. (~ A ~ C) D 10 Assoc 12. [(~ A ~ C) B] [(~ A C) D] 7,11 Conj 13. (~ A ~ C) (B D) 12 Dist 14. ~ (A C) (B D) 13 DeM 15. (A C) (B D) 14 Impl
V. CHAPTER FIVE: PROOFS WITH CP OR IP
A. General Theory
1. Suppose you know that a particular twopremise argument is invalid. Now suppose we add the negation of the conclusion of the two premises to form a threesentence set of premises. Can a contradiction be derived from this threesentence set of premises? (Defend your answer.)
14 2. a. Use IP to prove that the following argument is valid.
A B A ~ B / ~ A
b. To illustrate how indirect proofs are a kind of shortened conditional proof, cross out the last line in the above proof and complete it as a conditional proof. (Hint: as an intermediate step prove A ~ A.)
A. Answers
1. No, because derivation of a contradiction would constitute an indirect proof of validity for the argument, but by the hypothesis of the problem, the argument in question is invalid.
2. a. 1. A B 2. A ~ B 3. A AP / ~A 4. B 1,4 MP 5. ~ B 2,4 MP 6. B ~ B 5,6 Conj 7. ~ A 37 IP
b. 7. B ~ A 4 Add 8. ~ A 5,7 DS 9. A ~ A 3-8 CP 10. ~ A ~ A 9 Impl 11. ~ A 10 Taut
B. Proofs with CP or IP Prove valid, using the eighteen valid argument forms and CP or IP:
(1) 1. A B 2. C D C ~ A / (A C) (B D) 3. D E 4. ~ D C (2) 1. (A B) C 5. E ~ A / B 2. (A ~ B) ~ C / C (4) 1. A (B C) 2. ~ C (A B) / C
15 (5) 1. (A B) (6) 1. ~ (A ~ B) [ (C D) E] 2. ~ [~ C (~ A ~ D)] / A 3. ~ [A (B ~ D)] [ ~ E ~ (C D)] / D C
B. Answers
(1) 1. A B p 2. C D / (A C) (B D) p 3. A C AP / B D 4. A 3 Simp 5. B 1, 4 MP 6. C 3 Simp 7. D 2, 6 MP 8. B C 5, 7 Conj 9. (A C) (B D) 3-8 CP
(2) 1. (A B) C p 2. (A ~ B) ~ C p / A (B C) 3. A AP / B C 4. A (B C) 1 Exp 5. B C 3, 4 MP 6. A (~ B ~ C) 2 Exp 7. ~ B ~ C 3, 6 MP 8. C B 7 Contra 9. (B C) (C B) 5, 8 Conj 10. B C 9 Equiv 11. A (B C) 3 -10 CP
(3) 1. A B p 2. C ~ A p 3. D E p 4. ~ D C p 5. E ~ A p / B 6. ~ B AP / B 7. A 1,6 DS 8. ~ ~ A 7 DN 9. ~ C 2, 8 MT 10. ~ ~ D 4,9 MT 11. D 10 DN 12. E 3, 11 MP 13. ~ E 5, 8 MT 14. E ~ E 12,13 Conj 15. B 6-14 IP
16 (4) 1. A (B C) p 2. ~ C (A B) p / C 3. ~ C AP / C 4. A B 2, 3 MP 5. (A B) C 1 Exp 6. C 4,5 MP 7. C ~ C 3, 6 Conj 8. C 3-7 IP
(5) 1. (A B) [ (C D) E] p / A [~ E ~ (C D)] 2. A AP / ~ E ~ (C D) 3. C D AP / E 4. A B 2 Add 5. (C D) E 1,4 MP 6. C 3 Simp 7. C D 6 Add 8. E 5, 7 MP 9. (C D) E 3-8 CP 10. ~ E ~ (C D) 9 Contra 11. A [~ E ~ (C D)] 2-10 CP
(6) 1. ~ (A ∙ ~ B) p 2. ~ [~ C ∙ (~ A ∙ ~ D)] p 3. ~ [A ∙ (B ∙ ~ D)] p / D C 4. ~ (D C) AP / D C 5. ~ D ~ C 4 DeM 6. ~ ~ C ~ (~ A ~ D) 2 DeM 7. C ~ (~ A ~ D) 6 DN 8. ~ C 5 Simp 9. ~ (~ A ~ D) 7,8 DS 10. ~ ~ A ~ ~ D 9 DeM 11. ~ A ~ (B ~ D) 3 DeM 12. ~ A ~ ~B 1 DeM 13. ~ ~ A D 10 DN 14. ~ D 5 Simp 15. ~ ~ A 13,14 DS 16. ~ ~ B 12,15 DS 17. ~ (B ~ D) 11,15 DS 18. ~ B ~ ~ D 17 DeM 19. ~ ~ D 16,18 DS 20. ~ D ~ ~ D 14,19 Conj 21. D C 4-20 IP
17 C. Show that premises in the following arguments are inconsistent:
(1) 1. A (B C) (3) 1. ~ (~ T ~ R) 2. C (A B) 2. ~ S T 3. (B ~ A) (D B) 3. R S / T R 4. B ~ C / ~ A
(2) 1. ~ (A ~ B) (4) 1. A (~ B ~ A) 2. ~ C A 2. B (~ C ~ B) 3. ~ C ~ B / C 3. C (~ A ~ B) / A (B C)
C. Answers
(1) 1. A (B C) p 2. C (A B) p 3. (B ~ A) (D B) p 4. B ~ C p 5. B 4 Simp 6. B ~ A 3 Simp 7. ~ A 5, 6 MP 8. B C 1, 7 DS 9. C 8 Simp 10. ~ C 4 Simp 11. C ~ C 9, 10 Conj / Premises are inconsistent.
(2) 1. ~ (A ~ B) p 2. ~ C A p 3. ~ C ~ B p 4. ~ A ~ ~ B 1 DeM 5. ~ A 4 Simp 6. ~ C 2, 5 DS 7. ~ B 3, 6 MP 8. ~ ~ B 4 Simp 9. B 8 DN 10. B ~ B 7, 9 Conj / Premises are inconsistent.
18 (3) 1. ~ (~ T ~ R) p 2. ~ S T p 3. R S p 4. (R S) (S R) 3 Equiv 5. R S 4 Simp 6. S T 2 Impl 7. R T 5,6 HS 8. ~ T ~ R 7 Contra 9 (~ T ~ R) ~ (~ T ~ R) 1,8 Conj / Premises are inconsistent.
(4) 1. A (~ B ~ A) p 2. B (~ C ~ B) p 3. C (~ A ~ B) p 4. [A (~ B ~ A)] [(~ B ~ A) A] 1 Equiv 5. A (~ B ~ A) 4 Simp 6. (~ B ~ A) A 4 Simp 7. [B (~ C ~ B)] [( ~ C ~ B) B] 2 Equiv 8. (~ C ~ B) B 7 Simp 9. ~ (~ C ~ B) B 8 Impl 10. (C B) B 9 DeM, DN (twice) 11. B (C B) 10 Comm 12. (B C) (B B) 11 Dist 13. B B 12 Simp 14. B 13 Taut 15. ~ (~ B ~ A) A 6 Impl 16. (B A) A 15 DeM, DN (twice) 17. A (B A) 16 Comm 18 (A B) (A A) 17 Dist 19. A A 18 Simp 20. A 19 Taut 21. ~ B ~ A 5,20 MP 22. ~ ~ A 20 DN 23. ~ B 21,22 DS 24. B ~ B 14,23 Conj / Premises are inconsistent.
19 VI. CHAPTER SIX: SENTENTIAL LOGIC TRUTH TREES Use the truth-tree method to determine the validity/invalidity of the following argument forms:
(1) 1. p ~ q (5) 1. (p q) r 2. ~ (~ q ~ p) / p 2. ~ (r q) 3. ~ (r ~ p)/ p q (2) 1. (p q) r 2. ~ q ~ r / p (6) 1. p q 2. ~ [p (~ r q)] (3) 1. p q 3. ~ p r / r 2. p r 3. ~ q r / ~ p (7) 1. p (q r) 2. p s (4) 1. ~ (p q) 3. ~ q ~ p / r 2. q ~ p / q ~ p (8) 1. (~ p q) (~ r ~ q) 2. s (r p) / s r
20 VI. ANSWERS (1) 1. p ~ q p 2. ~ (~ q ~ p) p 3. ~ p Negation of conclusion
4. ~ ~ q ~ ~ p From line 2 5. q From line 4
6. ~ p ~ q From line 1 * Invalid. Open branches. In the branch marked with *, let V(p) = F V(q) = T
(2) 1. (p q) r p 2. ~ q ~ r p 3. ~ p Negation of conclusion
4. ≁≁ q ~ r From line 2
5. ~ (p q) r ~ (p q) r From line 1 * Invalid. Open branches. In the branch marked with *, let V(p) = F V(q) = T V(r) = T (3) 1. p q p 2. p r p 3. ~ q r p 4. ≁≁ p Negation of conclusion 5. ~ q From line 3 r 6. ~ p q From line 1
All paths closed. Valid!
(4) 1. ~ (p q) p 2. q ~ p p 3. ~ (q ~ p) Negation of conclusion 4. ~ q From line 3 ≁≁p
5. ~ p ~ q From line 1
6. ~ q ~ p From line 2 * Invalid. Open branches. In the branch marked with *, let V(p) = T 21 V(q) = F
(5) 1. (p q) r p 2. ~ (r q) p 3. ~ (r ~ p) p 4. ~ (p q) Negation of conclusion 5. p From line 4 ~ q 6. ~ r From line 2 ~ q
7. ~ (p q) r From line 1
8. ~ p From line 7 ~ q
All paths closed. Valid!
(6) 1. p q p 2. ~ [p (~ r q)] p 3. ~ p r p 4. ~ r Negation of conclusion 5. ~ p From line 2 ~ (~ r q) 6. ≁≁p r From line 3
All paths closed. Valid!
(7) 1. p (q r) p 2. p s p 3. ~ q ~ p p 4. ~ r Negation of conclusion 5. p From line 2 s
6. ~ p q r From line 1
7. ≁≁q ~ p From line 3
8. ~ q r From line 6
All paths closed. Valid!
22 (8) 1. (~ p q) (~ r ~ q) p 2. s (r p) p 3. ~ (s r) Negation of conclusion 4. s From line 2 r p 5. ~ p q From line 1 ~ r ~ q
6. ~ s ~ r From line 3
7. r p From line 4
8. ~ p q From line 5
9. ≁≁r ~ q From line 5
All paths closed. Valid.
VII. CHAPTER SEVEN: PREDICATE LOGIC SYMBOLIZATION A. General theory:
1. Which of the following are sentences? a. (x)Fx (y)Gxy d. ~ (y) Fy (x)Gx b. (x)Fx Ga e. None of these c. (x)(Fx Ga)
2. Which of the following are sentences? a. (x)(Fx Gx) d. (x)Fy (y)Gx b. (x)Fx (x)Gx e. None of these c. (x)Fx Gx
3. Symbolize “Some mammals are not four-legged” a) when the domain is mammals and b) when the domain is unrestricted (using obvious abbreviations).
4. Symbolize “No whales are fish” a) when the domain is whales, and b) when the domain is unrestricted (using obvious abbreviations).
5. What sentence contradicts “No just acts are acts that cause pain”? a. All just acts are acts that cause pain. b. Some just acts are acts that cause pain. c. Some just acts are acts that do not cause pain. d. No acts that cause pain are just acts. e. None of these.
23