Answers to Copi's Translation and Derivation Exercises
Predicate Logic: Singly General Monadic
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

Singly-General Monadic Predicate Logic Translations at pp. 69-70
  1. Snakes are reptiles. (x)(Sx Rx)
  2. Snakes are not all poisonous. (x)(Sx · ~Px)
  3. Children are present. (x)(Cx · Px)
  4. Executives all have secretaries. (x)(Ex Sx)
  5. Only executives have secretaries. (x)(Sx Ex)
  6. Only property owners may vote in special municipal elections. (x)(Vx Px)
  7. Employees may use only the service elevator. (x)(Ux Sx)
  8. Only employees may use the service elevator. (x)(Ux Ex)
  9. All that glitters is not gold. (x)(Gx ~Ax)
  10. None but the brave deserve the fair. (x)(Dx Bx)
  11. Not every visitor stayed for dinner. (x)(Vx · ~Sx)
  12. Not any visitor stayed for dinner. (x)(Vx ~Sx), or ~(x)(Vx · Sx)
  13. Nothing in the house escaped destruction.
  14. Some students are both intelligent and hard workers. (x)(Sx · Ix · Hx)
  15. No coat is waterproof unless it has been specially treated. (x)[Cx (~Wx Sx)]
  16. Some medicines are dangerous only if taken in excessive amounts. (x)[Mx · (Dx Ex)]
  17. All fruits and vegetables are wholesome and delicious. (x)[(Fx Vx) (Wx · Dx)]
  18. Everything enjoyable is either immoral, illegal, or fattening. (x)[Ex (~Mx ~Lx Fx)]
  19. A professor is a good lecturer if and only if he is both well-informed and entertaining. (x)[Px [Gx (Wx · Ex)]]
  20. Only policemen and firemen are both indispensable and underpaid. (x)[(Ix · Ux) (Px Fx)]
  21. Not every actor is talented who is famous. (x)(Ax · Fx · ~Tx)
  22. Any girl is healthy if she is well nourished and exercises regularly. (x)[Gx [(Wx · Ex) Hx]
  23. It is not true that every watch will keep good time if and only if it is wound regularly and not abused. ~(x)[Wx [(Rx · ~Ax) Kx]] or (x)~[Wx [(Rx · ~Ax) Kx]]
  24. Not every person who talks a great deal has a great deal to say. (x)(Px · Tx · ~Hx)
  25. No automobile that is over ten years old will be repaired if it is severely damaged. (x)[(Ax · Ox) (Dx ~Rx)]

    Translations at p. 70

    For problems 26-46 we are to use this dictionary:


  26. Some horses are gentle and have been well trained. (x)(Hx · Gx · Tx)
  27. Some horses are gentle only if they have been well trained. (x)[Hx · (Gx Tx)]
  28. Some horses are gentle if they have been well trained. (x)[Hx · (Tx Gx)]
  29. Any horse is gentle that has been well trained. (x)[Hx (Tx Gx)]
  30. Any horse that is gentle has been well trained. (x)[Hx (Gx Tx)]
  31. No horse is gentle unless it has been well trained. (x)[Hx (Gx Tx)]
  32. Any horse is gentle if it has been well trained. (x)[Hx (Tx Gx)]
  33. Any horse has been well trained if it is gentle. (x)[Hx (Gx Tx)]
  34. Any horse is gentle if an only if it has been well trained. (x)[Hx (Gx Tx)]
  35. Gentle horses have all been well trained. (x)[Hx (Gx Tx)]
  36. Only well trained horses are gentle. (x)[Hx (Gx Tx)]
  37. Only gentle horses have been well trained. (x)[Hx (Tx Gx)]
  38. Only horses are gentle if they have been well trained.
  39. Some horses are gentle even though they have not been well trained. (x)(Hx · Gx · ~Tx)
  40. If something is a well trained horse, then it must be gentle. (x)[(Hx · Tx) Gx]
  41. Some horses that are well trained are not gentle. (x)(Hx · Tx · ~Gx)
  42. Some horses are neither gentle nor well trained. (x)(Hx · ~Gx · ~Tx)
  43. No horse that is well trained fails to be gentle. (x)(Hx (Tx Gx)]
  44. A horse is gentle only if it has been well trained. (x)[Hx (Gx Tx)]
  45. If anything is a gentle horse, then it has been well trained. (x)[(Hx · Gx) Tx]
  46. If any horse is well trained, then it is gentle. (x)[(Hx · Tx) Gx]

Translations at p. 71
  1. Blessed is he that considereth the poor. (x)(Cx Bx)
  2. He that hath knowledge spareth his words. (x)(Hx Sx)
  3. Whoso findeth a wife findeth a good thing. (x)(Wx Gx)
  4. He that maketh haste to be rich shall not be innocent. (x)(Rx ~Ix)
  5. They shall sit every man under his vine and under his fig-tree. (x)[Mx (Vx · Fx)]
  6. He that increaseth knowledge increaseth sorrow. (x)(Kx Sx)
  7. Nothing is secret which shall not be made manifest. (x)(Sx Mx)
  8. Whom The Lord loveth He chasteneth. (x)(Lx Cx)
  9. If a man desire the office of a bishop, he desireth a good work. (x)(Ox Wx)
  10. He that hateth dissembleth with his lips, and layeth up deceit within him. (x)[Hx (Dx · Lx)]

Derivations at p. 76

Since CP and IP were introduced before any of these exercises were assigned, I will feel free to use them.

Part I, exercise 2 (p. 76)

1. (x)(Cx Dx)
2. (x)(Ex ~Dx)
3. Cy Dy
4. Ey ~Dy
5. Dy ~Ey
6. Cy ~Ey
7. Ey ~Cy
8. (x)(Ex ~Cx)

/ (x)(Ex ~Cx)
1 UI
2 UI
4 trans
3,5 HS
6 trans
7 UG


Part I, exercise 3 (p. 76)
1. (x)(Fx ~Gx)
2. (x)(Hx · Gx)
3. Ha · Ga
4. Fa ~Ga
5. Ga
6. ~Fa
7. Ha
8. Ha · ~Fa
9. (x)(Hx · ~Fx)

/ (x)(Hx · ~Fx)
2 EI
1 UI
3 simp
4,5 MT
3 simp
7,6 conj
8 EG


Part I, exercise 4 (p. 76)
1. (x)(Ix Jx)
2. (x)(Ix · ~Jx)
3. Ia · ~Ja
4. Ia
5. Ia Ja
6. Ja
7. ~Ja
8. Ja (x)(Jx Ix)
9. (x)(Jx Ix)

/ (x)(Jx Ix)
2 EI
3 simp
1 UI
4,5 MP
3 simp
6 add
7,8 DS
I hope you noticed that you must instantiate 2 to an ordinary constant (like 'a'), but cannot universally generalize from an ordinary constant (like 'a'). So if you instantiate 2, you won't be able to generalize in the way needed to get the conclusion. The trick here is to notice that the premises are inconsistent. We get the conclusion because we can derive anything from a contradiction.

Part I, exercise 6 (p. 76)
1. (x)(Nx Ox)
2. (x)(Px Ox)
3. Ny Oy
4. Py Oy
5. (Ny Oy) · (Py Oy)
6. Ny Py
7. Oy Oy
8. Oy
9. (Ny Py) Oy
10. (x)[(Nx Px) Ox]

/ (x)[(Nx Px) Ox]
1 UI
2 UI
3,4 conj
ass, CP
5,6 CD
7 taut
6-8 CP
9 UG


Part I, exercise 7 (p. 76)
1. (x)(Qx Rx)
2. (x)(Qx Rx)
3. Qa Ra
4. Qa Ra
5. ~Ra
6. Qa
7. ~Qa
8. Qa · ~Qa
9. Ra
10. (x)Rx

/ (x)Rx
2 EI
1 UI
ass, IP
3,5 DS
4,5 MT
6,7 conj
5-8 IP
9 EG


Part I, exercise 8 (p. 76)
1. (x)[Sx (Tx Ux)]
2. (x)[Ux (Vx · Wx)]
3. Sy (Ty Uy)
4. Uy (Vy · Wy)
5. Sy
6. Ty Uy
7. Ty
8. Uy
9. Vy · Wy
10. Vy
11. Ty Vy
12. Sy (Ty Vy)
13. (x)[Sx (Tx Vx)]

/ (x)[Sx (Tx Vx)]
1 UI
2 UI
ass, CP
3,5 MP
ass, CP
6,7 MP
4,8 MP
9 simp
7-10 CP
5-11 CP
12 UG


Part I, exercise 9 (p. 76)
1. (x)[(Xx Yx) (Zx · Ax)]
2. (x)[(Zx Ax) (Xx · Yx)]
3. (Xy Yy) (Zy · Ay)
4. (Zy Ay) (Xy · Yy)
5. Xy
6. Xy Yy
7. Zy · Ay
8. Zy
9. Xy Zy
10. Zy
11. Zy Ay
12. Xy · Yy
13. Xy
14. Zy Xy
15. (Xy Zy) · (Zy Xy)
16. (Xy Zy)
17. (x)(Xx Zx)

/ (x)(Xx Zx)
1 UI
2 UI
ass, CP
5 add
3,6 MP
7 simp
5-8 CP
ass, CP
10 add
4,11 MP
12 simp
10-13 CP
9,14 conj
15 equiv
16 UG


Part I, exercise 10 (p. 76)
1. (x)[(Bx Cx) · (Dx Ex)]
2. (x)[(Cx Ex) {[Fx (Gx Fx)] (Bx · Dx)}]
3. (By Cy) · (Dy Ey)
4. (Cy Ey) [[Fy (Gy Fy)] (By · Dy)]
5. By Cy
6. Dy Ey
7. ~[Fy (Gy Fy)]
8. ~[~Fy (Gy Fy)]
9. Fy · ~(Gy Fy)
10. Fy
11. ~(Gy Fy)
12. ~(~Gy Fy)
13. Gy · ~Fy
14. ~Fy
15. Fy · ~Fy
16. Fy (Gy Fy)
17. By
18. Cy
19. Cy Ey
20. [Fy (Gy Fy)] (By · Dy)
21. By · Dy
22. Dy
23. By Dy
24. Dy
25. Ey
26. Ey Cy
27. Cy Ey
28. [Fy (Gy Fy)] (By · Dy)
29. By · Dy
30. By
31. Dy By
32. (By Dy) · (Dy By)
33. By Dy
34. (x)(Bx Dx)

/ (x)(Bx Dx)
1 UI
2 UI
3 simp
3 simp
ass, IP
7 imp
8 DeM
9 simp
9 simp
11 imp
12 DeM
13 simp
10,14 conj
7-15 IP
ass, CP
5,17 MP
18 add
4,19 MP
16,20 MP
21 simp
17-22 CP
ass, CP
6,24 MP
25 add
26 con
4,27 MP
16,28 MP
21 simp
24-30 CP
23,31 conj
32 equiv
33 UG

Translations + Derivations at pp. 76-78

Part II, exercise 2 (p. 76)
1. (x)(Cx ~Dx)
2. (x)(Cx · Ex)
3. Ca · Ea
4. Ca ~Da
5. Ca
6. ~Da
7. Ea
8. Ea · ~Da
9. (x)(Ex · ~Dx)

/ (x)(Ex · ~Dx)
2 EI
1 UI
3 simp
4,5 MP
3 simp
7,6 conj
8 EG


Part II, exercise 3 (p. 76)
1. (x)(Fx Gx)
2. (x)(Hx · ~Gx)
3. Ha · ~Ga
4. Ha
5. ~Ga
6. Fa Ga
7. ~Fa
8. Ha · ~Fa
9. (x)(Hx · ~Fx)

/ (x)(Hx · ~Fx)
2 EI
3 simp
3 simp
1 UI
5,6 MT
4,7 conj
8 EG


Part II, exercise 4 (p. 77)
1. (x)(Jx ~Ix)
2. Ik
3. Jk ~Ik
4. ~Jk

/ ~Jk
1 UI
2,3 MT


Part II, exercise 6 (p. 77)
1. (x)(Ox ~Px)
2. (x)(Qx · Px)
3. Qa · Pa
4. Qa
5. Pa
6. Oa ~Pa
7. ~Oa
8. Qa · ~Oa
9. (x)(Qx · ~Ox)

/ (x)(Qx · ~Ox)
2 EI
3 simp
3 simp
1 UI
5,6 MT
4,7 conj
8 EG


Part II, exercise 7 (p. 77)
1. (x)(Rx Sx)
2. (x)(Rx · ~Tx)
3. Ra · ~Ta
4. Ra Sa
5. Ra
6. Sa
7. ~Ta
8. Sa · ~Ta
9. (x)(Sx · ~Tx)

/ (x)(Sx · ~Tx)
2 EI
1 UI
3 simp
4,5 MP
3 simp
6,7 conj
8 EG


Part II, exercise 8 (p. 77)
1. (x)(Ux Wx)
2. (x)(Wx ~Vx)
3. Uy Wy
4. Wy ~Vy
5. Uy ~Vy
6. (x)(Ux ~Vx)

/ (x)(Ux ~Vx)
1 UI
2 UI
3,4 HS
5 UG


Part II, exercise 9 (p. 77)
1. (x)(Bx Ax)
2. (x)(Ax Cx)
3. By Ay
4. Ay Cy
5. By Cy
6. (x)(Bx Cx)

/ (x)(Bx Cx)
1 UI
2 UI
3,4 HS
5 UG


Part II, exercise 11 (p. 77)
1. (x)(Dx Gx)
2. Sm
3. Dm
4. Dm Gm
5. Gm
6. Sm · Gm
7. (x)(Sx · Gx)


/ (x)(Sx · Gx)
1 UI
3,4 MP
2,5 conj
6 EG


Part II, exercise 12 (p. 77)
1. (x)[Tx (Fx · Dx)]
2. (x)(Tx · Bx)
3. Ta · Ba
4. Ta (Fa · Da)
5. Ta
6. Fa · Da
7. Ba
8. Da
9. Da · Ba
10. (x)(Dx · Bx)

/ (x)(Dx · Bx)
2 EI
1 UI
3 simp
4,5 MP
3 simp
6 simp
8,7 conj
9 EG


Part II, exercise 13 (p. 77)
1. (x)[(Bx Gx) Fx]
2. (x)[(Fx Vx) Nx]
3. (By Gy) Fy
4. (Fy Vy) Ny
5. By
6. By Gy
7. Fy
8. Fy Vy
9. Ny
10. By Ny
11. (x)(Bx Nx)

/ (x)(Bx Nx)
1 UI
2 UI
ass, CP
5 add
3,6 MP
7 add
4,8 MP
5-9 CP
10 UG


Part II, exercise 14 (p. 77)
1. (x)[Cx (Fx Kx)]
2. (x)(Fx Nx)
3. (x)(Cx · ~Nx)
4. Ca · ~Na
5. Ca (Fa Ka)
6. Fa Na
7. Ca
8. ~Na
9. Fa Ka
10. ~Fa
11. Ka
12. Ca · Ka
13. (x)(Cx · Kx)


/ (x)(Cx · Kx)
3 EI
1 UI
2 UI
4 simp
4 simp
5,7 MP
6,8 MT
9,10 DS
7,11 conj
12 EG


Part II, exercise 16 (p. 77)
1. (x)[(Hx · Bx) (Wx · Cx)]
2. (x)[(Hx · Ex) Bx]
3. (Hy · By) (Wy · Cy)
4. (Hy · Ey) By
5. Hy · Ey
6. By
7. Hy
8. Hy · By
9. Wy · Cy
10. Wy
11. (Hy · Ey) Wy
12. (x)[(Hx · Ex) Wx]

/ (x)[(Hx · Ex) Wx]
1 UI
2 UI
ass, CP
4,5 MP
5 simp
7,8 conj
3,8 MP
9 simp
5-10 CP
11 UG


Part II, exercise 17 (p. 77)
1. (x)(Px Lx)
2. (x)[(Px · Lx) Sx]
3. Py Ly
4. (Py · Ly) Sy
5. Py
6. Ly
7. Py · Ly
8. Sy
9. Ly · Sy
10. Py (Ly · Sy)
11. (x)[Px (Lx · Sx)]

/ (x)[Px (Lx · Sx)]
1 UI
1 UI
ass, CP
3,5 MP
5,6 conj
4,7 MP
6,8 conj
5-9 CP
10 UG


Part II, exercise 18 (p. 77)
1. (x)(Dx Px)
2. (x)(Dx · Ex)
3. (x)[(Px · Ex) Ox]
4. Da · Ea
5. Da Pa
6. (Pa · Ea) Oa
7. Da
8. Ea
9. Pa
10. Pa · Ea
11. Oa
12. Da. Oa
13. (x)(Dx · Ox)


/ (x)(Dx · Ox)
2 EI
1 UI
3 UI
4 simp
4 simp
5,7 MP
9,8 conj
6,10 MP
7,11 conj
12 EG


Part II, exercise 19 (p. 77)
1. (x)[(Dx Lx) Cx]
2. (x)(Ax Ix)
3. (x)(Lx · ~Ix)
4. (x)(Dx · Ax)
5. Da · Aa
6. (Da La) Ca
7. Aa Ia
8. Aa
9. Ia
10. Da
11. Da La
12. Ca
13. Ca · Ia
14. (x)(Cx · Ix)



/ (x)(Cx · Ix)
4 EI
1 UI
2 UI
5 simp
7,8 MP
5 simp
10 add
6,11 MP
12,9 conj
13 EG
I hope you noticed that you can't instantiate 3 and 4 to the same constant (such as 'a'). The trick here is to recognize that you don't need to instantiate 3 at all; it's not needed in the proof.

Part II, exercise 21 (p. 77)
1. (x)[Ax (Sx Wx)]
2. (x)(Ax Ix)
3. (x)(Ax · Sx · ~Wx)
4. Aa · Sa · ~Wa
5. Aa (Sa Wa)
6. Aa Ia
7. Aa
8. Sa Wa
9. (Sa Wa) · (Wa Sa)
10. Sa Wa
11. Sa
12. Wa
13. ~Wa
14. Wa (x)(Ix Ax)
15. (x)(Ix Ax)


/ (x)(Ix Ax)
3 EI
1 UI
2 UI
4 simp
5,7 MP
8 equiv
9 simp
4,11 simp
10,11 MP
4 simp
12 add
13,14 DS
Yes, the premises are inconsistent.
The argument is also valid (and the premises are still inconsistent) if step 1 is translated (x)[(Ax Sx) Wx].

Part II, exercise 22 (p. 77)
1. (x)[Px (Fx Tx)]
2. (x)[Px (Tx ~Wx)]
3. (x)(Px · Wx)
4. (x)(Px · ~Wx)
5. Pa · ~Wa
6. Pa (Ta ~Wa)
7. Pa
8. Ta ~Wa
9. (Ta ~Wa) · (~Wa Ta)
10. ~Wa Ta
11. ~Wa
12. Ta
13. Pa. Ta
14. (x)(Px · Tx)



/ (x)(Px · Tx)
4 EI
2 UI
5 simp
6,7 MP
8 equiv
9 simp
5 simp
10,11 MP
7,12 conj
13 EG
See comment to Part II, exercise 19, above. In this case we needn't instantiate step 3. (For that matter, we needn't instantiate step 1 either. )

Part II, exercise 23 (p. 77)
1. (x)[Mx (Ox · Gx)]
2. (x)(Ox Fx)
3. (x)[(Gx ~Fx) Px]
4. (x)[Px (Fx ~Gx)]
5. (x)[Mx · (Fx Ox)]
6. Ma · (Fa Oa)
7. Ma (Oa · Ga)
8. Oa Fa
9. (Ga ~Fa) Pa
10. Pa (Fa ~Ga)
11. Ma
12. Oa · Ga
13. Oa
14. Fa
15. Ga
16. Ga ~Fa
17. Pa
18. Fa ~Ga
19. ~Ga
20. ~Fa
21. Ma · ~Fa
22. (x)(Mx · ~Fx)




/ (x)(Mx · ~Fx)
5 EI
1 UI
2 UI
3 UI
4 UI
6 simp
7,11 MP
12 simp
8,13 MP
12 simp
15 add
9,16 MP
10,17 MP
14,18 MP
16,19 DS
11,20 conj
21 EG
Yes, the premises are inconsistent.

Part II, exercise 24 (p. 78)
1. (x)[(Wx Tx) Hx]
2. (x)[(Hx Lx) Dx]
3. (x)[Dx (Gx · Ux)]
4. (x)(Wx · ~Gx · ~Sx)
5. Wa · ~Ga · ~Sa
6. (Wa Ta) Ha
7. (Ha La) Da
8. Da (Ga · Ua)
9. ~Ga
10. ~Ga ~Ua
11. ~(Ga · Ua)
12. ~Da
13. ~(Ha La)
14. ~Ha · ~La
15. ~Ha
16. ~(Wa Ta)
17. ~Wa · ~Ta
18. ~Wa
19. Wa
20. Wa (x)(Tx · Sx · ~Gx)
21. (x)(Tx · Sx · ~Gx)



/ (x)(Tx · Sx · ~Gx)
4 EI
1 UI
2 UI
3 UI
5 simp
9 add
10 DeM
8,11 MT
7,12 MT
13 DeM
14 simp
6,15 MT
16 DeM
17 simp
5 simp
19 add
18,20 DS
Yes, the premises are inconsistent.

Part II, exercise 25 (p. 78)
1. (x)[(Mx · Cx) (Gx ~Dx)]
2. (x)[Mx (Rx Cx)]
3. (x)[Cx (~Dx Ex)]
4. (x)[(Mx · Ex) Gx]
5. (My · Cy) (Gy ~Dy)
6. My (Ry Cy)
7. Cy (~Dy Ey)
8. (My · Ey) Gy
9. My · Ry
10. (My · Ry) Cy
11. Cy
12. ~Dy Ey
13. My
14. My · Cy
15. Gy ~Dy
16. Ey
17. My · Ey
18. Gy
19. Ey Gy
20. Gy
21. ~Dy
22. Ey
23. Gy Ey
24. (Ey Gy) · (Gy Ey)
25. Ey Gy
26. (My · Ry) (Ey Gy)
27. (x)[(Mx · Rx) (Ex Gx)]



/ (x)[(Mx · Rx) (Ex Gx)]
1 UI
2 UI
3 UI
4 UI
ass, CP
6 exp
9,10 MP
7,11 MP
9 simp
13,11 conj
5,14 MP
ass, CP
13,16 conj
8,17 MP
16-18 CP
ass, CP
15,20 MP
12,21 MP
20-22 CP
19,23 conj
24 equiv
9-25 CP
26 UG



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.