1. Natural Deduction
Hurley, Logic 7.3

2. Our Replacement Rules De Morgan’s rule (DM): ~(p ● q) :: (~p v ~q)
~(p v q) :: (~p ● ~q)
Commutativity (Com):
(p v q) :: (q v p)
(p ● q) :: (q ● p)
Associativity (Assoc):
[p v (q v r)] :: [(p v q) v r]
[(p ● (q ● r)] :: [p ● (q ● r)]
Distribution (Dist):
[p ● (q v r)] :: [(p ● q) v (p ● r)]
[(p v (q ● r)] :: [(p v q) ● (p v r)]
Double Negation (DN):
p :: ~~p
Remember that p, q, r, and s stand for any well-formed formula, no matter how complex. For instance, below is an example of DN:
~~(K v J) ● ~B
__________
(K v J) ● ~B

3. Practice Finding Proof Steps
A → ~(B ● C)
A ● C / ~B
A , Simp
~(B ● C) ,3, MP
~B v ~C , DM
C ● A , Com
C , Simp
~~C , DN
~C v ~B , Com
~B ,8, DS

4. Practice Finding Proof Steps
D ● (E v F) (The long way…)
~D v ~F / D ● E
D , Simp
(E v F) ● D , Com
E v F , Simp
~~D , DN
~F ,6, DS
F v E , Com
E ,8, DS
D ● E ,9, Conj

5. Practice Finding Proof Steps
D ● (E v F) (The short way…)
~D v ~F / D ● E
(D ● E) v (D ● F) , Dist
(D ● F) v (D ● E) , Com
~(D ● F) , DM
D ● E ,5, DS

6. Practice Finding Proof Steps
(G ● H) v (G ● J)
(G v K) > L / L
G ● (H v J) , Dist
G , Simp
G v K , Add
L ,5, MP

7. Practice Finding Proof Steps
M v (N v O)
~O / M v N
(M v N) v O , Assoc
O v (M v N) , Com
M v N ,4, DS

8. Practice Finding Proof Steps
K > (F v B)
G ● K / (F ● G) v (B ● G)
K ● G , Com
K , Simp
F v B ,4, MP
G , Simp
G ● (F v B) ,6, Conj
(G ● F) v (G ● B) 7, Dist
(F ● G) v (G ● B) 8, Com
(F ● G) v (B ● G) 9, Com
(F ● G) v (B ● G)
(G ● F) v (B ● G) Com
(G ● F) v (G ● B) Com
G ● (F v B) Dist
If you have a complex conclusion, and you’re stuck, try altering it using replacement rules to see if it suggests a solution

9. Practice Finding Proof Steps
~S / ~(F ● S)
~S v ~F , Add
~F v ~S , Com
~(F ● S) , DM

10. Practice Finding Proof Steps
J v (K ● L)
~K / J
(J v K) ● (J v L) , Dist
J v K , Simp
K v J , Com
J ,5, DS

11. Practice Finding Proof Steps
H > ~A
A / ~(H v ~A)
~~A , DN
~H ,3, MT
~H ● ~~A ,4, Conj
~(H v ~A) , DM
~(H v ~A)
(~H ● ~~A) DM

12. Practice Finding Proof Steps
R > ~B
D v R
B / D
~~B , DN
~R ,4, MT
R v D , Com
D ,5, DS

13. Practice Finding Proof Steps
(O v M) > S
~S / ~M
~(O v M) ,2, MT
~O ● ~M , DM
~M ● ~O , Com
~M , Simp

14. Practice Finding Proof Steps
Q v (L v C)
~C / L v Q
(Q v L) v C , Assoc
C v (Q v L) , Com
Q v L ,4, DS
L v Q , Com

15. Practice Finding Proof Steps
~(~E ● ~N) > T
G > (N v E) / G > T
~~(E v N) > T , DM
(E v N) > T , DN
(N v E) > T , Com
G > T ,5, HS

16. Practice Finding Proof Steps
H ● (C ● T)
~(~F ● T) / F
~~F v ~T , DM
(H ● C) ● T , Assoc
T ● (H ● C) , Com
T , Simp
~T v ~~F , Com
~~T , DN
~~F ,8, DS
F , DN

17. Practice Finding Proof Steps
~(J v K)
B > K
S > B / ~S ● ~J
~J ● ~K , DM
~J , Simp
~K ● ~J , Com
~K , Simp
~B ,7, MT
~S ,8, MT
~S ● ~J ,9, Conj

18. Practice Finding Proof Steps
(G ● H) v (M ● G)
G > (T ● A) / A
(G ● H) v (G ● M) , Com
G ● (H v M) , Dist
G , Simp
T ● A ,5, MP
A ● T , Com
A , Simp

19. Practice Finding Proof Steps
~(U v R)
(~R v N) > (P ● H)
Q > ~H / ~Q
~U ● ~R , DM
~R ● ~U , Com
~R , Simp
~R v N , Add
P ● H ,7, MP
H ● P , Com
H , Simp
~~H , DN
~Q ,11, MT

20. Practice Finding Proof Steps
~(F ● A)
~(L v ~A)
D > (F v L) / ~D
~F v ~A , DM
~L ● ~~A , DM
~~A ● ~L , Com
~~A , Simp
~A v ~F , Com
~F ,8, DS
~L , Simp
~F ● ~L ,10, Conj
~(F v L) , DM
~D ,12, MT

21. Practice Finding Proof Steps
E > ~B
U > ~C
~(~E ● ~U) / ~(B ● C)
~~E v ~~U , DM
E v ~~U , DN
E v U , DN
(E > ~B) ● (U > ~C) ,2, Conj
~B v ~C ,7, CD
~(B ● C) , DM
~ (B ● C)
~B v ~C DM

22. Practice Finding Proof Steps
(J v F) v M
(J v M) > ~P
~F / ~(F v P)
(F v J) v M , Assoc
F v (J v M) , Assoc
J v M ,5, DS
~P ,6, MP
~F ● ~P ,7, Conj
~(F v P) , DM
~(F v P)
~F ● ~P DM

23. Practice Finding Proof Steps
~(K v F)
~F > (K v C)
(G v C) > ~H / ~(K v H)
~K ● ~F , DM
~F ● ~K , Com
~F , Simp
K v C ,6, MP
~K , Simp
C ,8, DS
C v G , Add
G v C , Com
~H ,11, MP
~K ● ~H ,12, Conj
~(K v H) , DM
~ (K v H)
~K ● ~H DM

24. Practice Finding Proof Steps
(K ● P) v (K ● Q)
P > ~K / Q v T
K ● (P v Q)
K , Simp
~~K , DN
~P ,5, MT
(P v Q) ● K , Com
P v Q , Simp
Q ,8, DS
Q v T , Add

25. Practice Finding Proof Steps
(T ● R) > P
(~P ● R) ● G
(~T v N) > H / H
~P ● (R ● G) , Assoc
~P , Simp
~(T ● R) ,5, MT
~T v ~R , DM
(R ● G) ● ~P , Assoc
R ● G , Simp
R , Simp
~~R , DN
~R v ~T , Com
~T ,12, DS
~T v N , Add
H ,14, MP

26. Practice Finding Proof Steps
B v (S ● N)
B > ~S
S > ~N / B v W
(B v S) ● (B v N) , Dist
(B > ~S) ● (S > ~N) ,3, Conj
B v S , Simp
~S v ~N ,6, CD
~(S ● N) , DM
(S ● N) v B 1, Comm
B ,9 DS
B v W , Add

27. Practice Finding Proof Steps
(~M v E) > (S > U)
(~Q v E) > (U > H)
~(M v Q) / S > H
Line 3 can give you ~M and ~Q … from each, with addition (adding E), you can get the antecedents of both 1 and 2. Modus Ponens will give you what you need to attain the conclusion by Hypothetical Syllogism

28. Practice Finding Proof Steps
~Q > (C ● B)
~T > (B ● H)
~(Q ● T) / B
~Q v ~T , DM
[~Q > (C ● B)] ● [~T > (B ● H)] ,2, Conj
(C ● B) v (B ● H) ,5, CD
(B ● C) v (B ● H) , Com
B ● (C v H) , Dist
B , Simp

29. Practice Finding Proof Steps
~(A ● G)
~(A ● E)
G v E / ~(A ● F)
~A v ~G , DM
~A v ~E , DM
(~A v ~G) ● (~A v ~E) 4,5, Conj
~A ● (~G v ~E) , Dist
~A , Simp
~A v ~F , Add
~(A ● F) , DM
~(A ● F)
~A v ~F DM

30. Practice Finding Proof Steps
(M ● N) v (O ● P)
(N v O) > ~P / N
[(M ● N) v O] ● [(M ● N) v P] 1, Dist
(M ● N) v O , Simp
O v (M ● N) , Com
(O v M) ● (O v N) , Dist
(O v N) ● (O v M) , Com
O v N , Simp
N v O , Com
~P ,9, MP
[(M ● N) v P] ● [(M ● N) v O] 3, Com
(M ● N) v P , Simp
P v (M ● N) , Com
(P v M) ● (P v N) , Dist
(P v N) ● (P v M) , Com
P v N , Simp
N ,16, DS

31. Practice Finding Proof Steps
(T ● K) v (C ● E)
K > ~E
E > ~C / T ● K
[(T ● K) v C] ● [(T ● K) v E] , Dist
(T ● K) v C , Simp
C v (T ● K) , Com
(C v T) ● (C v K) , Dist
[(T ● K) v E] ● [(T ● K) v C] , Com
(T ● K) v E , Simp
E v (T ● K) , Com
(E v T) ● (E v K) , Dist
(K > ~E) ● (E > ~C) ,3, Conj
(E v K) ● (E v T) , Com
E v K , Simp
~E v ~C ,14 CD
~(E ● C) , DM
~(C ● E) , Com
(C ● E) v (T ● K) , Com
T ● K ,18, DS