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.