# Solutions to those ten proofs

Proofs using all 18 rules, from Hurley 9th ed. pp. 376-378

1.

1. Q ⊃ (F ⊃ A)

2. R ⊃ (A ⊃ F)

3. Q ∙ R / F ≡ A

4. Q 3      sm

5. R 3       cm, sm

6. F ⊃ A        mp 4,1

7. A ⊃ F         mp 5,2

8. (F ⊃ A) ∙ (A ⊃ F)       CN 6,7

9. F ≡ A                 EQ 8

2.

1. (J ∙ R) ⊃ H

2. (R ⊃ H) ⊃ M

3. ~(P v ~J) / M ∙ ~P

4. ~P ∙ ~~J            DM 3

5. ~P ∙ J                    DN 4

6. J ⊃ (R ⊃ H)                EXP 1

7. J                               COMM, SM 5

8. R ⊃ H                        MP 6,7

9. M                               mp 2,8

10. ~P                            SM 5

11. M ∙ ~P                    CN 9, 10

3.

1. F ⊃ (A ∙ K)

2. G ⊃ (~A ∙ ~K)

3. F v G / A ≡ K

4. [F ⊃ ( A ∙ K)] ∙ [G ⊃ (~A ∙ ~K)]             CN 1,2

5. (A ∙ K) v (~A ∙ ~K)                               CD 3,4

6. A ≡ K                                                         EQ 5

.4

1. T ⊃ G

2. S ⊃ G / (T v S) ⊃ G

3.~ T v G                          IMP 1

4. ~S v G                             IMP 2

5. (~T v G) ∙ (~S v G)                  CN 3,4

6. (G v ~T) . (G v ~S)                     cm, cm 5

7. G v (~T . ~S)                           Dist 6

8. (~T ∙ ~S) v G                             CM 7

9. ~(T vS) v G                                DM 6

10. (T v S) ⊃ G                             IMP 7

5.

1. S v ~N

2. ~S v Q / N ⊃ Q

3. S ⊃ Q                                IMP 2

4. ~S ⊃ ~N                           IMP 1

5. N ⊃ S                               Tran 4

6. N ⊃ Q                                HS 5,3

6.

1. (E ⊃ A) . (F ⊃ A)

2. E v G

3. F v ~G / A

4. ~E ⊃ G                           IMP 2

5. ~F ⊃ ~G                          IMP 3

6. G ⊃ F                              TRAN 5

7. ~E ⊃ F                              HS 4,6

8. E v F                                 IMP 7

9. A v A                                  CD 1,8

10. A                                     TAUT 9

7.

1. (F ∙ H) ⊃ N

2. F v S

3. H / N v S

4. ~(F ∙ H) v N                                    IMP 1

5. (~F v ~H) v N                                DM 4

6 . (~F v N) v ~H                                   AS, CM 5

7. ~~H                                                  DN 3

8. ~F v N                                            DS, CM 3,6

9. ~F ⊃ S                                           IMP 2

10. F ⊃ N                                              IMP 8

11. ~N ⊃ ~F                                       TRAN 10

12. ~N ⊃ S                                              HS 9,11

13. N v S                                               imp 12

8.

1. C ⊃ (~L ⊃ Q)

2. L ⊃ ~C

3. ~Q / ~C

4. ~C v (~~L v Q)                               IMP, IMP 1

5. (~C v L) v Q                                    AS, DN 4

6. ~C v L                                                 DS 3,5

7. C ⊃ L                                                   IMP 6

8. C ⊃ ~C                                               HS 7,2

9. ~ C v ~ C                                              IMP 8

10. ~C                                                       Taut 9

9.

1. K ≡ R

2. K ⊃ ( R ⊃ P)

3. ~P / ~R

4. (K ⊃ R) ∙ (R ⊃ K)                                   EQ 1

5. (K ∙ R) ⊃ P                                         exp 4

6. ~(K ∙ R)                                               mt 5,3

7. ~K v ~R                                                 dm 6

8. (K ∙ R) v (~ K . ~R)                                 EQ 1

9.~ K ∙ ~R                                                 DS 6,8

10. ~R                                                         CM, SM 9

In this version, line 4 turned out to be useless, but it is not illegally derived, so it’s fine. Also, line 7 never got put to use; again, that is not a mistake, just an unnecessary step.

10.

1. A ≡ W

2. ~A v ~W

3. R ⊃ A / ~(W v R)

4. ~(A ∙ W)                                  DM 2

5. (A ∙ W) v (~A ∙ ~W)               EQ 1

6. ~A ∙ ~W                                  DS 4,5

7. ~A                                              sm 6

8. ~R                                           MT 7, 3

9. ~W                                           CM, sm 6

10. ~W ∙ ~R                                     cn 9,8

11. ~(W v R)                                 DM 10

In this one, it is a huge help if you note that the conclusion is “neither nor,” so that it has a DM equivalence; that means that conjunction will be the second to last line, and that’s the goal to shoot for.