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.

Table of Contents

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 dist
8 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.

[Blue Ribbon] Peter Suber, Department of Philosophy, Earlham College, Richmond, Indiana, 47374, U. S. A.
peters@earlham. edu. Copyright © 1997, Peter Suber.