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.