Answers to Copi's Translation and Derivation Exercises
Propositional Logic
Peter Suber, Philosophy Department, Earlham College

References to Irving Copi, Symbolic Logic, are to the fifth edition, Macmillan, 1979.

I include the translation exercises, but not the derivation exercises, that Copi answers himself in the back of the book.

Remember that there are many ways to prove a valid argument valid with a derivation. If my way differs from your way, yours may still be perfectly good.

I welcome corrections.

If the steps of a proof are too long for their box and wrap to a second line, then the justifications to their right may become out of step with the lines they justify. To fix this, widen your browser window with your mouse.

Propositional Logic Derivations at p. 45

Copi assigns these exercises before he introduces CP and IP, and again after he introduces them. I do them here without CP and IP to show that it can be done. Almost all these problems are considerably easier with CP and/or IP.

Part III, exercise 2 (p. 45)

 1. C 2. C ~D 3. ~D C 4. D C / D C 1 add 2 com 3 imp

Part III, exercise 3 (p. 45)
 1. E (F G) 2. (E · F) G 3. (F · E) G 4. F (E G) / F (E G) 1 exp 2 com 3 exp

Part III, exercise 4 (p. 45)
 1. H (I · J) 2. ~H (I · J) 3. (~H I) · (~H J) 4. ~H I 5. H I / H I 1 imp 2 dist 3 simp 4 imp

Part III, exercise 6 (p. 45)
 1. N O 2. ~N O 3. (~N O) ~P 4. ~N (O ~P) 5. ~N (~P O) 6. (~N ~P) O 7. ~(N · P) O 8. (N · P) O / (N · P) O 1 imp 2 add 3 ass 4 com 5 ass 6 DeM 7 imp

Part III, exercise 7 (p. 45)
 1. (Q R) S 2. ~(Q R) S 3. (~Q · ~R) S 4. S (~Q · ~R) 5. (S ~Q) · (S ~R) 6. S ~Q 7. ~Q S 8. Q S / Q S 1 imp 2 DeM 3 com 4 dist 5 simp 6 com 7 imp

Part III, exercise 8 (p. 45)
 1. T ~(U V) 2. T ~(~U V) 3. T (U · ~V) 4. ~T (U · ~V) 5. (~T U) · (~T ~V) 6. ~T U 7. T U / T U 1 imp 2 DeM 3 imp 4 dist 5 simp 6 imp

Part III, exercise 9 (p. 45)
 1. W (X · ~Y) 2. ~W (X · ~Y) 3. (~W X) · (~W ~Y) 4. ~W ~Y 5. (~W ~Y) X 6. ~(W · Y) X 7. (W · Y) X 8. (W (Y X) / W (Y X) 1 imp 2 dist 3 com, simp 4 add 5 DeM 6 imp 7 exp

Part III, exercise 11 (p. 45)
 1. E F 2. E G 3. ~E F 4. ~E G 5. (~E F) · (~E G) 6. ~E (F · G) 7. E (F · G) / E (F · G) 1 imp 2 imp 3,4 conj 5 dist 6 imp

Part III, exercise 12 (p. 45)
 1. H (I J) 2. ~I 3. ~H (I J) 4. ~H (J I) 5. (~H J) I 6. ~H J 7. H J / H J 1 imp 3 com 4 ass 2,5 DS 6 imp

Part III, exercise 13 (p. 45)
 1. (K L) ~(M · N) 2. (~M ~N) (O P) 3. (O P) (Q · R) 4. (K L) (~M ~N) 5. (K L) (O P) 6. (K L) (Q · R) 7. (L K) (R · Q) / (L K) (R · Q) 1 DeM 4,2 HS 5,3 HS 6 com, com

Part III, exercise 14 (p. 45)
 1. S T 2. S T 3. ~T ~S 4. ~S T 5. ~T T 6. T T 7. T / T 1 trans 2 imp 3,4 HS 5 imp 6 taut

Part III, exercise 16 (p. 45)
 1. A (B C) 2. C (D · E) 3. ~C (D · E) 4. (~C D) · (~C E) 5. ~C D 6. C D 7. (A · B) C 8. (A · B) D 9. A (B D) / A (B D) 2 imp 3 dist 4 simp 5 imp 1 exp 7,6 HS 8 exp

Part III, exercise 17 (p. 45)
 1. E F 2. G F 3. ~E F 4. ~G F 5. (~E F) · (~G F) 6. (F ~E) · (F ~G) 7. F (~E · ~G) 8. (~E · ~G) F 9. ~(E G) F 10. (E G) F / (E G) F 1 imp 2 imp 3,4 conj 5 com, com 6 dist 7 com 8 DeM 9 imp

Part III, exercise 18 (p. 45)
 1. [(H · I) J] · [~K (I · ~J)] 2. (H · I) J 3. ~K (I · ~J) 4. H (I J) 5. ~(I · ~J) K 6. (~I J) K 7. (I J) K 8. H K / H K 1 simp 1 simp 2 exp 3 trans 5 DeM 6 imp 4,7 HS

Part III, exercise 19 (p. 45)
 1. [L · (M N)] (M · N) 2. L [(M N) (M · N)] 3. L [~(M N) (M · N)] 4. L [[~(M N) M] · [~(M N) N]] 5. L [[(~M · ~N) M] · [(~M · ~N) N] 6. L [(M ~M) · (M N) · (N ~M) · (N ~N)] 7. ~L [(M ~M) · (M N) · (N ~M) · (N ~N)] 8. [~L (M ~M)] · [~L (M N)] · [~L (N ~M)] · [~L (N ~N)] 9. ~L (N ~M) 10. ~L (~M N) 11. ~L (M N) 12. L (M N) / L (M N) 1 exp 2 imp 3 dist 4 DeM, DeM 5 dist, dist 6 imp 7 dist8 simp 9 con 10 imp 11 imp
There must be an easier way!

Part III, exercise 21 (p. 45)
 1. S (T · U) 2. (T U) V 3. ~S (T · U) 4. (~S T) · (~S U) 5. ~S T 6. S T 7. ~(T U) V 8. (~T · ~U) V 9. V (~T · ~U) 10. (V ~T) · (V ~U) 11. V ~T 12. ~T V 13. T V 14. S V / S V 3 dist 4 simp 5 imp 2 imp 7 DeM 8 com 9 dist 10 simp 11 com 12 imp 6,13 HS

Part III, exercise 22 (p. 45)
 1. ~W [(X Y) · (Z Y)] 2. W · (X Z) 3. W 4. X Z 5. (X Y) · (Z Y) 6. Y Y 7. Y / Y 2 simp 2 simp 1,3 DS 4,5 CD 6 taut

Part III, exercise 23 (p. 45)
 1. (A B) (C · D) 2. ~A (E ~E) 3. ~C 4. ~(A B) (C · D) 5. [~(A B) C] · [~(A B) D] 6. ~(A B) C 7. ~(A B) 8. ~A · ~B 9. ~A 10. E ~E 11. ~E ~E 12. ~E / ~E 1 imp 4 dist 5 simp 3,6 DS 7 DeM 8 simp 2,9 MP 10 imp 11 taut

Part III, exercise 24 (p. 45)
 1. (F G) · (H I) 2. F H 3. (F ~I) · (H ~G) 4. G I 5. ~I ~G 6. ~G ~I 7. G ~I 8. I G 9. ~I G 10. (G ~I) · (~I G) 11. G ~I / G ~I 1,2 CD 2,3 CD 5 com 6 imp 4 com 8 imp 7,9 conj 10 equiv

Part III, exercise 26 (p. 45)
 1. Q (R · S) 2. (Q T) · (T S) 3. (Q R) · (Q S) 4. Q S 5. Q T 6. T S 7. Q S 8. ~Q S 9. ~S Q 10. ~S S 11. S S 12. S / S 1 dist 3 simp 2 simp 2 com, simp 5,6 HS 4 imp 8 trans 9,7 HS 10 imp 11 taut

Part III, exercise 27 (p. 45)
 1. (U V) · (W X) 2. (U V) 3. ~U V 4. (~U V) X 5. ~U (V X) 6. (V X) ~U 7. W X 8. ~W X 9. (~W X) V 10. ~W (X V) 11. ~W (V X) 12. (V X) ~W 13. [(V X) ~U] · [(V X) ~W] 14. (V X) (~U · ~W) 15. (~U · ~W) (V X) 16. ~(U W) (V X) 17. (U W) (V X) / (U W) (V X) 1 simp 2 imp 3 add 4 ass 5 com 1 simp 7 imp 8 add 9 ass 10 com 11 com 6,12 conj 13 dist 14 com 15 DeM 16 imp
This is a very short proof with CP. See how much labor CP saves us!

Part III, exercise 28 (p. 45)
 1. (Y Z) · (A B) 2. (Y Z) 3. ~Y Z 4. (~Y Z) ~A 5. ~Y (Z ~A) 6. ~Y (~A Z) 7. A B 8. ~A B 9. (~A B) ~Y 10. ~Y (~A B) 11. [~Y (~A Z)] · [~Y (~A B)] 12. ~Y [(~A Z) · (~A B)] 13. ~Y [~A (Z · B)] 14. ~Y [A (Z · B)] 15. Y [A (Z · B)] 16. (Y · A) (Z · B) / (Y · A) (Z · B) 1 simp 2 imp 3 add 4 ass 5 com 1 simp 7 imp 8 add 9 com 6,10 conj 11 dist 12 dist 13 imp 14 imp 15 exp
This is a very short proof with CP.

Part III, exercise 29 (p. 45)
 1. (C D) · (E F) 2. G (C E) 3. ~G (C E) 4. (~G C) E 5. (C ~G) E 6. C (~G E) 7. ~C (~G E) 8. C D 9. ~D ~C 10. ~D (~G E) 11. D (~G E) 12. D (E ~G) 13. (D E) ~G 14. (E D) ~G 15. E (D ~G) 16. ~E (D ~G) 17. E F 18. ~F ~E 19. ~F (D ~G) 20. F (D ~G) 21. (D ~G) F 22. (~G D) F 23. ~G (D F) 24. G (D F) / G (D F) 2 imp 3 ass 4 com 5 ass 6 imp 1 simp 8 trans 9,7 HS 10 imp 11 com 12 ass 13 com 14 ass 15 imp 1 simp 17 trans 18,16 HS 19 imp 20 com 21 com 22 ass 23 imp

Part III, exercise 30 (p. 45)
 1. (H I) · (J K) 2. H J 3. (H ~K) · (J ~I) 4. (I · ~K) L 5. K (I M) 6. I K 7. ~K ~I 8. ~I K 9. K ~I 10. ~I ~I 11. I ~I 12. (~K · I) L 13. ~K (I L) 14. I ~K 15. I (I L) 16. (I · I) L 17. I L 18. ~I (I M) 19. I (I M) 20. (I I) M 21. I M 22. ~I M 23. (I L) · (~I M) 24. L M / L M 1,2 CD 2,3 CD 6 imp 7 imp 8,9 HS 10 imp 4 com 12 exp 9 trans 14,13 HS 15 exp 16 taut 8,5 HS 18 imp 19 ass 20 taut 21 imp 17,22 conj 23,11 CD

Propositional Logic Translations + Derivations at pp. 45-48

Part IV, exercise 2 (p. 46)

 1. (S · U) P 2. (U P) W 3. S 4. S (U P) 5. U P 6. W / W 1 exp 3,4 MP 2,5 MP

Part IV, exercise 3 (p. 46)
 1. A (B · C) 2. (A B) D 3. (A B) · (A C) 4. A B 5. D 6. D C 7. C D / C D 1 dist 3 simp 2,4 MP 5 add 6 com

Part IV, exercise 4 (p. 46)
 1. G (B C) 2. G · ~C 3. G 4. ~C 5. B C 6. ~B / ~B 2 com, simp 2 simp 1,3 MP 4,5 MT

Part IV, exercise 6 (p. 46)
 1. P S 2. ~P S 3. (~P S) ~H 4. ~P (S ~H) 5. ~P (~H S) 6. P (~H S) 7. P (H S) / P (H S) 1 imp 2 add 3 ass 4 com 5 imp 6 imp

Part IV, exercise 7 (p. 46)
 1. (F R) · (L U) 2. (U · R) C 3. ~C 4. ~(U · R) 5. ~U ~R 6. ~R ~U 7. ~F ~L / ~F ~L 2, 3 MT 4 DeM 5 com 1, 6 DD

Part IV, exercise 8 (p. 46)
 1. (~A B) · (A C) 2. B (C E) 3. ~A B 4. A B 5. A (C E) 6. (A · C) E 7. (C · A) E 8. C (A E) 9. A C 10. A (A E) 11. (A · A) E 12. A E / A E 1 simp 3 imp 4, 2 HS 5 exp 6 com 7 exp 1 com, simp 9, 8 HS 10 exp 11 taut

Part IV, exercise 9 (p. 46)
 1. A (B · C) 2. (B ) I 3. ~A (B · C) 4. (~A B) · (~A C) 5. ~A B 6. A B 7. ~I ~(B C) 8. I ~(B C) 9. I (~B · ~C) 10. (I ~B) · (I ~C) 11. I ~B 12. ~B I 13. B I 14. A I / A I 1 imp 3 dist 4 simp 5 imp 2 trans 7 imp 8 DeM 9 dist 10 simp 11 com 12 imp 6, 13 HS

Part IV, exercise 11 (p. 46)
 1. [(E · S) P] · [(E · ~S) ~P] 2. (E · S) P 3. (E · ~S) ~P 4. E (S P) 5. E (~S ~P) 6. ~E (S P) 7. ~E (~S ~P) 8. [~E (S P)] · [~E (~S ~P)] 9. ~E [(S P) · (~S ~P)] 10. ~E [(S P) · (P S)] 11. ~E (S P) 12. E (S P) 13. E [(S · P) (~S · ~P)] / E [(S · P) (~S · ~P)] 1 simp 1 com, simp 2 exp 3 exp 4 imp 5 imp 6,7 conj 8 dist 9 trans 10 equiv 11 imp 12 equiv

Part IV, exercise 12 (p. 46)
 1. A (B C) 2. B C 3. ~A (B C) 4. ~A (C B) 5. (~A C) B 6. ~(~A C) B 7. ~(~A C) C 8. (~A C) C 9. ~A (C C) 10. ~A C 11. A C / A C 1 imp 3 com 4 ass 5 imp 6,2 HS 7 imp 8 ass 9 taut 10 imp

Part IV, exercise 13 (p. 46)
 1. (A B) (C · E) 2. ~(A B) (C · E) 3. [~(A B) C] · [~(A B) E] 4. ~(A B) C 5. (~A · ~B) C 6. C (~A · ~B) 7. (C ~A) · (C ~B) 8. C ~A 9. ~C ~A 10. A C / A C 1 imp 2 dist 3 simp 4 DeM 5 com 6 dist 7 simp 8 imp 9 trans

Part IV, exercise 14 (pp. 46-47)
 1. (P W) (R · A) 2. (R J) · ~J 3. R J 4. ~J 5. ~R 6. ~R ~A 7. ~(R · A) 8. ~(P W) 9. ~P · ~W 10. ~P / ~P 2 simp 2 simp 3,4 MT 5 add 6 DeM 1,7 MT 8 DeM 9 simp

Part IV, exercise 16 (p. 47)
 1. P (C N) 2. (N · R) E 3. T (R · ~E) 4. (P · C) N 5. N (R E) 6. ~(R · ~E) ~T 7. (~R E) ~T 8. (R E) ~T 9. N ~T 10. (P · C) ~T 11. P (C ~T) / P (C ~T) 1 exp 2 exp 3 trans 6 DeM 7 imp 5,8 HS 4,9 HS 10 exp
The argument is also valid if step 3 is translated, "(T R) · ~E," but I won't provide the proof here.

Part IV, exercise 17 (p. 47)
 1. T (I · O) 2. T O 3. (T I) · (T O) 4. T O 5. O T 6. ~T O 7. ~O T 8. T O 9. ~O O 10. O O 11. O / O 1 dist 3 simp 4 com 2 imp 5 imp 6 imp 7,8 HS 9 imp 10 taut
The argument is also valid if step 1 is translated, "(T I) · O," but I won't provide the proof here.

Part IV, exercise 18 (p. 47)
 1. R N 2. R G 3. G ~S 4. (N B) · (B ~S) 5. S C 6. N B 7. B ~S 8. N ~S 9. R ~S 10. (R ~S) · (N ~S) 11. ~S ~S 12. ~S 13. C / C 4 simp 4 com, simp 6,7 HS 2,3 HS 9,8 conj 1,10 CD 11 taut 5,12 DS

Part IV, exercise 19 (p. 47)
 1. P S 2. S ~(B · D) 3. (~B T) · ~T 4. D 5. ~B T 6. ~T 7. B 8. B · D 9. ~S 10. ~P / ~P 3 simp 3 simp 5,6 MT 7,4 conj 8,2 MT 9,1 MT

Part IV, exercise 20 (p. 47)
 1. (S E) · (H L) 2. (E L) C 3. ~C 4. ~(E L) 5. ~E · ~L 6. ~E 7. ~L 8. S E 9. H L 10. ~S 11. ~H 12. ~S · ~H 13. ~(S H) / ~(S H) 2,3 MT 4 DeM 5 simp 5 simp 1 simp 1 simp 6,8 MT 7,9 MT 10,11 conj 12 DeM

Part IV, exercise 22 (p. 47)
 1. (T E) · (A L) 2. (T E) 3. ~T E 4. (~T E) ~A 5. ~T (E ~A) 6. ~T (~A E) 7. A L 8. ~A L 9. (~A L) ~T 10. ~T (~A L) 11. [~T (~A E)] · [~T (~A L)] 12. ~T [(~A E) · (~A L)] 13. ~T [~A (E · L)] 14. ~T [A (E · L)] 15. T [A (E · L)] 16. (T · A) (E · L) / (T · A) (E · L) 1 simp 2 imp 3 add 4 ass 5 com 1 simp 7 imp 8 add 9 com 6,10 conj 11 dist 12 dist 13 imp 14 imp 15 exp

Part IV, exercise 23 (p. 47)
 1. E S 2. E (S N) 3. S (N F) 4. (E · S) N 5. (S · E) N 6. S (E N) 7. E (E N) 8. (E · E) N 9. E N 10. (S · N) F 11. (N · S) F 12. N (S F) 13. E (S F) 14. (E · S) F 15. (S · E) F 16. S (E F) 17. E (E F) 18. (E · E) F 19. E F / E F 2 exp 4 com 5 exp 1,6 HS 7 exp 8 taut 3 exp 10 com 11 exp 9,12 HS 13 exp 14 com 15 exp 1,16 HS 17 exp 19 taut

Part IV, exercise 24 (p. 47)
 1. A (B C) 2. E (C P) 3. ~C 4. ~E (C P) 5. (C P) ~E 6. C (P ~E) 7. P ~E 8. ~E P 9. (~E P) B 10. ~E (P B) 11. ~E (B P) 12. (B P) ~E 13. ~A (B C) 14. (~A B) C 15. ~A B 16. (~A B) P 17. ~A (B P) 18. (B P) ~A 19. [(B P) ~E] · [(B P) ~A] 20. (B P) (~E · ~A) 21. (B P) ~(E A) 22. (B P) ~(A E) 23. ~(B P) ~(A E) / ~(B P) ~(A E) 2 imp 4 com 5 ass 3,6 DS 7 com 8 add 9 ass 10 com 11 com 1 imp 3 ass 3,14 DS 15 add 16 ass 17 com 12,18 conj 19 dist 20 DeM 21 com 22 imp

Part IV, exercise 25 (p. 47-48)
 1. (A B) [(C E) (~P · V)] 2. (~P ~N) U 3. ~(P · N) U 4. (P · N) U 5. U (P · N) 6. (U P) · (U N) 7. U P 8. ~U P 9. [(A B) · (C E)] (~P · V) 10. ~[(A B) · (C E)] (~P · V) 11. [~(A B) ~(C E)] (~P · V) 12. [(~A · ~B) (~C · ~E)] (~P · V) 13. [(~A · ~B) (~C · ~E)] ~P] · [(~A · ~B) (~C · ~E)] V] 14. [(~A · ~B) (~C · ~E)] ~P 15. (~A · ~B) [(~C · ~E) ~P] 16. (~A · ~B) [~P (~C · ~E)] 17. [(~A · ~B) ~P] (~C · ~E) 18. [(~A · ~B) ~P] ~C] · [(~A · ~B) ~P] ~E] 19. [(~A · ~B) ~P] ~C 20. [~P (~A · ~B)] ~C 21. ~P [(~A · ~B) ~C] 22. P [(~A · ~B) ~C] 23. ~U [(~A · ~B) ~C] 24. U [(~A · ~B) ~C] 25. U [~C (~A · ~B)] 26. U [(~C ~A) · (~C ~B)] 27. [U (~C ~A)] · [U (~C ~B)] 28. U (~C ~A) 29. U (~A ~C) 30. ~U (~A ~C) 31. ~U ~(A · C) 32. (A · C) U 33. A (C U) / A (C U) 2 DeM 3 imp 4 com 5 dist 6 simp 7 imp 1 exp 9 imp 10 DeM 11 DeM, DeM 12 dist 13 simp 14 ass 15 com 16 ass 17 dist 18 simp 19 com 20 ass 21 imp 7,22 HS 23 imp 24 com 25 dist 26 dist 27 simp 28 com 29 imp 30 DeM 31 trans 32 exp

Part IV, exercise 26 (p. 48)
 1. (S ~S) A 2. ~(S ~S) A 3. (~S · S) A 4. A (~S · S) 5. (A ~S) · (A S) 6. A ~S 7. A S 8. ~S A 9. S A 10. ~A S 11. ~A A 12. A A 13. A / A 1 imp 2 DeM 3 com 4 dist 5 simp 5 simp 6 com 8 imp 7 imp 10,9 HS 11 imp 12 taut

This file is an electronic hand-out for the course, Symbolic Logic.

Most of the logic symbols in this file are GIFs. See my Notes on Logic Notation on the Web.

Peter Suber, Department of Philosophy, Earlham College, Richmond, Indiana, 47374, U. S. A.