1. Natural Deduction 1
Hurley, Logic 7.4

2. Our Replacement Rules 2 Transportation (Trans): Exportation (Exp):
(p → q) :: (~q → ~p)
Material Implication (Impl):
(p → q) :: (~p v q)
Material Equivalence (Equiv):
(p Ξ q):: [(p → q) ● (q → p)]
(p Ξ p):: [(p ● q) v (~p ● ~q)]
Exportation (Exp):
[(p ● q) → r)] :: [(p → (q → r)]
Tautology (Taut):
p :: (p v p)
p :: (p ● p)

3. Practice Finding Proof Steps3
Practice Finding Proof Steps
~A / A → B
~A v B , Add
A → B , Impl

4. Practice Finding Proof Steps4
Practice Finding Proof Steps
F → G
F v G / G
~~F v G , DN
~F → G , Impl
~F → ~~G , DN
~G → F , Trans
~G → G ,6, HS
~~G v G , Impl
G v G , DN
G , Taut

5. Practice Finding Proof Steps5
Practice Finding Proof Steps
J → (K → L) / K → (J → L)
(J ● K) → L , Exp
(K ● J) → L , Com
K → (J → L) , Exp

6. Practice Finding Proof Steps6
Practice Finding Proof Steps
M → N
M → O / M → (N ● O)
~M v N , Impl
~M v O , Impl
( ~M v N) ● (~M v O) 4, Conj
~M v (N ● O) , Dist
M → (N ● O) , Impl

7. Practice Finding Proof Steps7
Practice Finding Proof Steps
P → Q
R → (S ● T)
~R → ~Q
S → (T → P) / P Ξ R
Q → R , Impl
P → R ,5, HS
(S ● T) → P , Exp
R → P ,7, HS
(P → R) ● (R → P) 6,8, Conj
P Ξ R , Equiv

8. Practice Finding Proof Steps8
Practice Finding Proof Steps
~S → K
S → (R v M) / ~R → (~M → K)
~K → ~~S , Trans
~K → S , DN
~K → (R v M) ,4, HS
~(R v M) → ~~K , Trans
~(R v M) → K , DN
(~R ● ~M) → K , DM
~R → (~M → K) , Exp

9. Practice Finding Proof Steps9
Practice Finding Proof Steps
K → M
L → M / (K v L) → M
(K → M) ● (L → M) ,2, Conj
(~K v M) ● (L → M) , Impl
(~K v M) ● (~L v M) , Impl
(M v ~K) ● (~L v M) , Com
(M v ~K) ● (M v ~L) , Com
M v (~K ● ~L) , Dist
M v ~(K v L) , DM
~M → ~(K v L) , Impl
~~(K v L) → ~~M , Trans
(K v L) → M , DN

10. Practice Finding Proof Steps10
Practice Finding Proof Steps
(S ● K) → R
K / S → R
(K ● S) → R , Com
K → (S → R) , Exp
S → R ,4, MP

11. Practice Finding Proof Steps11
Practice Finding Proof Steps
T → (F v F)
~(F ● F) / ~T
~F , Taut
T → F , Taut
~T ,4, MT

12. Practice Finding Proof Steps12
Practice Finding Proof Steps
G → E
H → ~E / G → ~H
~~E → ~H , Trans
E → ~H , DN
G → ~H ,4, HS

13. Practice Finding Proof Steps13
Practice Finding Proof Steps
S Ξ Q
~S / ~Q
(S → Q) ● (Q → S) , Equiv
(Q → S) ● (S → Q) , Com
Q → S , Simp
~Q ,5, MT

14. Practice Finding Proof Steps14
Practice Finding Proof Steps
~N v P
(N → P) → T / T
N → P , Impl
T ,3, MP

15. Practice Finding Proof Steps15
Practice Finding Proof Steps
F → B
B → (B → J) / F → J
(B ● B) → J , Exp
B → J , Taut
F → J ,4, HS

16. Practice Finding Proof Steps16
Practice Finding Proof Steps
(B → M) ● (D → M)
B v D / M
M v M ,2, CD
M , Taut

17. Practice Finding Proof Steps17
Practice Finding Proof Steps
Q → (F → A)
R → (A → F)
Q ● R / F Ξ A
Q , Simp
R ● Q , Com
R , Simp
F → A ,4, MP
A → F ,6, MP
(F → A) ● (A → F) ,8, Conj
F Ξ A , Equiv

18. Practice Finding Proof Steps18
Practice Finding Proof Steps
T → (~T v G)
~G / ~T
T → (T → G) , Impl
(T ● T) → G , Exp
T → G , Taut
~T ,5, MT

19. Practice Finding Proof Steps19
Practice Finding Proof Steps
(B → G) ● (F → N)
~(G ● N) / ~(B ● F)
(~G → ~B) ● (F → N) , Trans
(~G → ~B) ● (~N → ~F) 1, Trans
~G v ~N , DM
~B v ~F ,5, CD
~(B ● F) , DM

20. Practice Finding Proof Steps20
Practice Finding Proof Steps
(J ● R) → H)
(R → H) → M
~(P v ~J) / M ● ~P
J → (R → H) , Exp
J → M ,4, HS
~P ● ~~J , DM
~~J ● ~P , Com
J ● ~P , DN
J , Simp
M ,9, MP
~P , Simp
M ● ~P ,11, Conj

21. Practice Finding Proof Steps21
Practice Finding Proof Steps
T / S > T
T v ~S , Add
~S v T , Com
S > T , Impl

22. Practice Finding Proof Steps22
Practice Finding Proof Steps
(I > E) > C
C > ~C / I
~C v ~C , Impl
~C , Taut
~(I > E) ,4, MT
~(~I v E) , Impl
~~I & ~E , DM
~~I , Simp
I , DN