Natural Deduction 1 Hurley, Logic 7.4

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)

Practice Finding Proof Steps 3 Practice Finding Proof Steps ~A / A → B ~A v B 1, Add A → B 2, Impl

Practice Finding Proof Steps 4 Practice Finding Proof Steps F → G F v G / G ~~F v G 2, DN ~F → G 3, Impl ~F → ~~G 4, DN ~G → F 5, Trans ~G → G 1,6, HS ~~G v G 7, Impl G v G 8, DN G 9, Taut

Practice Finding Proof Steps 5 Practice Finding Proof Steps J → (K → L) / K → (J → L) (J ● K) → L 1, Exp (K ● J) → L 2, Com K → (J → L) 3, Exp

Practice Finding Proof Steps 6 Practice Finding Proof Steps M → N M → O / M → (N ● O) ~M v N 1, Impl ~M v O 2, Impl ( ~M v N) ● (~M v O) 4, Conj ~M v (N ● O) 5, Dist M → (N ● O) 6, Impl

Practice Finding Proof Steps 7 Practice Finding Proof Steps P → Q R → (S ● T) ~R → ~Q S → (T → P) / P Ξ R Q → R 3, Impl P → R 1,5, HS (S ● T) → P 4, Exp R → P 2,7, HS (P → R) ● (R → P) 6,8, Conj P Ξ R 9, Equiv

Practice Finding Proof Steps 8 Practice Finding Proof Steps ~S → K S → (R v M) / ~R → (~M → K) ~K → ~~S 1, Trans ~K → S 3, DN ~K → (R v M) 2,4, HS ~(R v M) → ~~K 5, Trans ~(R v M) → K 6, DN (~R ● ~M) → K 7, DM ~R → (~M → K) 8, Exp

Practice Finding Proof Steps 9 Practice Finding Proof Steps K → M L → M / (K v L) → M (K → M) ● (L → M) 1,2, Conj (~K v M) ● (L → M) 3, Impl (~K v M) ● (~L v M) 4, Impl (M v ~K) ● (~L v M) 5, Com (M v ~K) ● (M v ~L) 6, Com M v (~K ● ~L) 7, Dist M v ~(K v L) 8, DM ~M → ~(K v L) 9, Impl ~~(K v L) → ~~M 10, Trans (K v L) → M 11, DN

Practice Finding Proof Steps 10 Practice Finding Proof Steps (S ● K) → R K / S → R (K ● S) → R 1, Com K → (S → R) 3, Exp S → R 2,4, MP

Practice Finding Proof Steps 11 Practice Finding Proof Steps T → (F v F) ~(F ● F) / ~T ~F 2, Taut T → F 1, Taut ~T 3,4, MT

Practice Finding Proof Steps 12 Practice Finding Proof Steps G → E H → ~E / G → ~H ~~E → ~H 2, Trans E → ~H 3, DN G → ~H 1,4, HS

Practice Finding Proof Steps 13 Practice Finding Proof Steps S Ξ Q ~S / ~Q (S → Q) ● (Q → S) 1, Equiv (Q → S) ● (S → Q) 3, Com Q → S 4, Simp ~Q 2,5, MT

Practice Finding Proof Steps 14 Practice Finding Proof Steps ~N v P (N → P) → T / T N → P 1, Impl T 2,3, MP

Practice Finding Proof Steps 15 Practice Finding Proof Steps F → B B → (B → J) / F → J (B ● B) → J 2, Exp B → J 3, Taut F → J 1,4, HS

Practice Finding Proof Steps 16 Practice Finding Proof Steps (B → M) ● (D → M) B v D / M M v M 1,2, CD M 3, Taut

Practice Finding Proof Steps 17 Practice Finding Proof Steps Q → (F → A) R → (A → F) Q ● R / F Ξ A Q 3, Simp R ● Q 3, Com R 5, Simp F → A 1,4, MP A → F 2,6, MP (F → A) ● (A → F) 7,8, Conj F Ξ A 9, Equiv

Practice Finding Proof Steps 18 Practice Finding Proof Steps T → (~T v G) ~G / ~T T → (T → G) 1, Impl (T ● T) → G 3, Exp T → G 4, Taut ~T 2,5, MT

Practice Finding Proof Steps 19 Practice Finding Proof Steps (B → G) ● (F → N) ~(G ● N) / ~(B ● F) (~G → ~B) ● (F → N) 1, Trans (~G → ~B) ● (~N → ~F) 1, Trans ~G v ~N 2, DM ~B v ~F 4,5, CD ~(B ● F) 6, DM

Practice Finding Proof Steps 20 Practice Finding Proof Steps (J ● R) → H) (R → H) → M ~(P v ~J) / M ● ~P J → (R → H) 1, Exp J → M 2,4, HS ~P ● ~~J 3, DM ~~J ● ~P 6, Com J ● ~P 7, DN J 8, Simp M 5,9, MP ~P 6, Simp M ● ~P 10,11, Conj

Practice Finding Proof Steps 21 Practice Finding Proof Steps T / S > T T v ~S 1, Add ~S v T 2, Com S > T 3, Impl

Practice Finding Proof Steps 22 Practice Finding Proof Steps (I > E) > C C > ~C / I ~C v ~C 2, Impl ~C 4, Taut ~(I > E) 1,4, MT ~(~I v E) 5, Impl ~~I & ~E 6, DM ~~I 7, Simp I 8, DN