5.1a

Problem One.
 Premise 1 AvB Premise 2 A>I Premise 3 ~I 2,3 MT 4 ~A 1,4 DS 5 B 6 7 8 9 B???
Problem Two.
 Premise 1 K>~M Premise 2 LvM Premise 3 K 1,3 MP 4 ~M 2,4 DS 5 L 6 7 8 9 L???
Problem Three.
 Premise 1 J>L Premise 2 ~L Premise 3 JvR 1,2 MT 4 ~J 3,4 DS 5 R 6 7 8 9 R???
Problem Four.
 Premise 1 (F&K)>(DvS) Premise 2 F&K Premise 3 ~D 1,2 MP 4 DvS 3,4 DS 5 S 6 7 8 9 S???

5.1b

Problem One.
 Premise 1 AvB Premise 2 A>I Premise 3 ~I 2,3 MT 4 ~A 1,4 DS 5 B 6 7 8 9 B???
Problem Two.
 Premise 1 K>~M Premise 2 LvM Premise 3 K 1,3 MP 4 ~M 2,4 DS 5 L 6 7 8 9 L???
Problem Three.
 Premise 1 J>L Premise 2 ~L Premise 3 JvR 1,2 MT 4 ~J 3,4 DS 5 R 6 7 8 9 R???
Problem Four.
 Premise 1 (F&K)>(DvS) Premise 2 F&K Premise 3 ~D 1,2 MP 4 DvS 3,4 DS 5 S 6 7 8 9 S???
Problem Five.
 Premise 1 Jv(DvS) Premise 2 ~J Premise 3 ~D 2,1 DS 4 DvS 3,4 DS 5 S 6 7 8 9 S???

5.1c

Problem Two.
 Premise 1 (JvS)>(LvT) Premise 2 ~(LvT) 1,2 MT 3 ~(JvS) 4 5 6 7 8 9 ~(JvS)???
Problem Three.
 Premise 1 ~K>(~L>J) Premise 2 Lv~K Premise 3 ~L 2,3 DS 4 ~K 1,4 MP 5 ~L>J 3,5 MP 6 J 7 8 9 J???
Problem Five.
 Premise 1 T>(J>L) Premise 2 (R=L)vT Premise 3 ~(J>L) 1,3 MT 4 ~T 2,4 DS 5 R=L 6 7 8 9 R=L???

5.1g

Problem Two.
 Premise 1 LvM Premise 2 ~L Premise 3 M>(P&Q) 1,2 DS 4 M 3,4 MP 5 P&Q 6 7 8 9 P&Q???
Problem Four.
 Premise 1 A>B Premise 2 ~B Premise 3 ~A>K 1,2 MT 4 ~A 3,4 MP 5 K 6 7 8 9 K???

5.2a

Problem 1.
 Premise 1 (L&N)>P Premise 2 L Premise 3 N 2,3 &I 4 L&N 1,4 MP 5 P
Problem 2.
 Premise 1 (A&B)&(B>C) 1 &E 2 A&B 2 &E 3 B 1 &E 4 B>C 3,4 MP 5 C
Problem 4.
 Premise 1 (J&H)>(L&S) Premise 2 H&J 2 &E 3 H 2 &E 4 J 3,4 &I 5 J&H 1,5 MP 6 L&S 6 &E 7 S 7,4 &I 8 S&J

5.2b

Problem 1.
 Premise 1 A&B 1 &E 2 A 1 &E 3 B 4 2,3 &I 5 B&A
Problem 2.
 Premise 1 A&~B Premise 2 LvB 1 &E 3 ~B 2,3 DS 4 L 1 &E 5 A 6 4,5 &I 7 L&A
Problem 4.
 Premise 1 ~(~RvS)&J Premise 2 M>(~RvS) 1 &E 3 ~(~RvS) 2,3 MT 4 ~M

5.2d

Problem 1.
 Premise 1 (L&M)>S Premise 2 L&(L>M) 2 &E 3 L 2 &E 4 L>M 3,4 MP 5 M 3,5 &I 6 L&M 1,6 MP 7 S 8 9 S???
Problem 2.
 Premise 1 J>~K Premise 2 J&(R>K) 2 &E 3 J 1,3 MP 4 ~K 2 &E 5 R>K 4,5 MT 6 ~R 4,6 &I 7 ~R&~K 8 9 ~R&~K???
Problem 3.
 Premise 1 T&~U Premise 2 Uv(T>S) 1 &E 3 ~U 2,3 DS 4 T>S 1 &E 5 T 4,5 MP 6 S 7 8 9 S???
Problem 4.
 Premise 1 MvN Premise 2 ~N Premise 3 (~N&M)>(O&P) 1,2 DS 4 M 2,4 &I 5 ~N&M 3,5 MP 6 O&P 6 &E 7 P 8 9 P???
Problem 5.
 Premise 1 J&(KvR) Premise 2 L>J Premise 3 ~K 1 &E 4 KvR 3,4 DS 5 R 6 7 8 9 R???

5.3a

Problem 1.
 Premise 1 ~~A&B 1 DN 2 A&B 2 DN 3 A&~~B 4 ??? 5 A&~~B???
Problem 3.
 Premise 1 A>~B Premise 2 B 1 IM 3 ~Av~B 2 DN 4 ~~B 3,4 DS 5 ~A 6 7 ??? 8 ~A???
Problem 4.
 Premise 1 ~K>T Premise 2 ~T 1,2 MT 3 ~~K 3 DN 4 K 5 6 7 ??? 8 K???

5.3b

Problem 1.
 Premise 1 [(J&K)>S]&[S>(J&K)] 2 3 4 1 EQ 5 (J&K)=S
Problem 2.
 Premise 1 L=M 1 EQ 2 (L>M)&(M>L) 2 &E 3 L>M 2 &E 4 M>L 5 6 7 8 3,4 &I 9 (M>L)&(L>M)
Problem 3.
 Premise 1 ~(J&K)vT 2 3 4 5 6 7 1 IM 8 (J&K)>T
Problem 4.
 Premise 1 ~(J>K) 1 IM 2 ~(~JvK) 2 DM 3 ~~J&~K 3 &E 4 ~K
Problem 5.
 Premise 1 (L>S)&(~SvL) 1 IM 2 (L>S)&(S>L) 2 EQ 3 L=S

5.4a

Problem 1.
 Premise 1 A&~B Assumption 2 ....what if A>B 1 &E 3 ....then... A 2,3 MP 4 ....then... B 1 &E 5 ....then... ~B 2-5 RD 6 ~(A>B) 7 8 ??? 9 ~(A>B)???
Problem 2.
 Premise 1 AvC Assumption 2 ....what if ~(CvA) 2 DM 3 ....then... ~C&~A 3 &E 4 ....then... ~C 1,4 DS 5 ....then... A 3 &E 6 ....then... ~A 2-6 RD 7 CvA 8 ??? 9 CvA???
Problem 3.
 Premise 1 ~(A>B) Assumption 2 ....what if A=B 2 EQ 3 ....then... (A>B)&(B>A) 3 &E 4 ....then... A>B 1 R 5 ....then... ~(A>B) 2-5 RD 6 ~(A=B) 7 ??? 8 ~(A=B)???
Problem 4.
 Premise 1 ~A&~B Assumption 2 ....what if ~(B>A) 2 IM 3 ....then... ~(~BvA) 3 DM 4 ....then... ~~B&~A 1 &E 5 ....then... ~B 4 &E 6 ....then... ~~B 2-6 RD 7 B>A

5.4b

Problem 1.
 Premise 1 Kv~I Assumption 2 ....what if ~K&I 2 &E 3 ....then... ~K 1,3 DS 4 ....then... ~I 2 &E 5 ....then... I 2-5 RD 6 ~(~K&I) 7 8 ??? 9 ~(~K&I)???
Problem 3.
 Premise 1 K>~K Premise 2 ~K>L Assumption 3 ....what if K 1,3 MP 4 ....then... ~K 3-4 RD 5 ~K 2,5 MP 6 L 7 ??? 8 L???
Problem 4.
 Premise 1 R>T Assumption 2 ....what if ~(R>(T&R)) 2 IM 3 ....then... ~(~Rv(T&R)) 3 DM 4 ....then... ~~R&~(T&R) 4 &E 5 ....then... ~~R 5 DN 6 ....then... R 1,6 MP 7 ....then... T 4 &E 8 ....then... ~(T&R) 8 DM 9 ....then... ~Tv~R 7 DN 10 ....then... ~~T 9,10 DS 11 ....then... ~R 2-11 RD 12 R>(T&R)
Problem 5.
 Premise 1 F>~K Premise 2 ~(~KvR) Assumption 3 ....what if ~(F>T) 3 IM 4 ....then... ~(~FvT) 4 DM 5 ....then... ~~F&~T 5 &E 6 ....then... ~~F 6 DN 7 ....then... F 1,7 MP 8 ....then... ~K 2 DM 9 ....then... ~~K&~R 9 &E 10 ....then... ~~K 3-10 RD 11 F>T ??? 12 F>T???

5.4c

Problem 4.
 Premise 1 K Premise 2 ~K Assumption 3 ....what if ~X 1 R 4 ....then... K 2 R 5 ....then... ~K 3-5 RD 6 X 7 ??? 8 X???

5.4d

Problem 3.
 Assumption 1 ....what if ~(P>P) 1 IM 2 ....then... ~(~PvP) 2 DM 3 ....then... ~~P&~P 3 &E 4 ....then... ~P 3 &E 5 ....then... ~~P 1-5 RD 6 P>P 7 8 ??? 9 P>P???

5.4e

Problem 2.

 Premise 1 ~(P=Q) Premise 2 P Assumption 3 ....what if Q 1 EQ 4 ....then... ~((P&Q)v(~P&~Q)) 4 DM 5 ....then... ~(P&Q)&~(~P&~Q) 5 &E 6 ....then... ~(P&Q) 2,3 &I 7 ....then... P&Q 3-7 RD 8 ~Q
Problem 3.
 Assumption 1 ....what if ~(Av~A) 1 DM 2 ....then... ~A&~~A 2 &E 3 ....then... ~A 2 &E 4 ....then... ~~A 1-4 RD 5 Av~A
Problem 4.
 Premise 1 ~(K>K) Assumption 2 ....what if ~X 1 IM 3 ....then... ~(~KvK) 3 DM 4 ....then... ~~K&~K 4 &E 5 ....then... ~K 4 &E 6 ....then... ~~K 2-6 RD 7 X

5.4f
Problem 2.
 Premise 1 P=Q Premise 2 Q 1 EQ 3 (P>Q)&(Q>P) 3 &E 4 Q>P 2,4 MP 5 P
Problem 5.
 Premise 1 P&(QvR) Assumption 2 ....what if ~((P&Q)v(P&R)) 2 DM 3 ....then... ~(P&Q)&~(P&R) 3 &E 4 ....then... ~(P&Q) 4 DM 5 ....then... ~Pv~Q 1 &E 6 ....then... P 6 DN 7 ....then... ~~P 5,7 DS 8 ....then... ~Q 3 &E 9 ....then... ~(P&R) 9 DM 10 ....then... ~Pv~R 7,10 DS 11 ....then... ~R 1 &E 12 ....then... QvR 11,12 DS 13 ....then... Q 2-13 RD 14 (P&Q)v(P&R)

5.5a

Problem 3.
 Premise 1 PvQ Assumption 2 ....what if ~(QvP) 2 DM 3 ....then... ~Q&~P 3 &E 4 ....then... ~Q 1,4 DS 5 ....then... P 3 &E 6 ....then... ~P 2-6 RD 7 QvP 8 9 10 ??? 11 QvP???

5.5c

Problem 1.
 Premise 1 T=R Premise 2 (R=T)>(J=K) Premise 3 J 1 CM 4 R=T 2,4 MP 5 J=K 3,5 =E 6 K 7 ??? 8 K???
Problem 2.
 Premise 1 ~(M&S) Premise 2 ~(S&M)>T 1 CM 3 ~(S&M) 2,3 MP 4 T 4 vI 5 Tv(X>Y) 6 7 8 ??? 9 Tv(X>Y)???
Problem 3.
 Premise 1 (JvL)>(T=U) Premise 2 J&T 2 &E 3 J 3 vI 4 JvL 1,4 MP 5 T=U 2 &E 6 T 5,6 =E 7 U 8 ??? 9 U???
Problem 4.
 Premise 1 Pv~T Premise 2 P>R Premise 3 ~T>R 1,2,3 vE 4 R 5 6 7 8 9 10 ??? 11 R???
Problem 5.
 Premise 1 J>R Premise 2 (Rv~J)>F 1 IM 3 ~JvR 3 CM 4 Rv~J 2,4 MP 5 F 5 vI 6 Fv~X 7 8 ??? 9 Fv~X???

5.5d
Problem 1.
 Premise 1 P&Q 1 &E 2 P 1 &E 3 Q 2,3 &I 4 Q&P
Problem 2.
 Premise 1 P=Q 1 EQ 2 (P>Q)&(Q>P) 2 &E 3 P>Q 2 &E 4 Q>P 3,4 &I 5 (Q>P)&(P>Q) 5 EQ 6 Q=P
Problem 3.
 Premise 1 PvQ Assumption 2 ....what if ~(QvP) 2 DM 3 ....then... ~Q&~P 3 &E 4 ....then... ~P 1,4 DS 5 ....then... Q 3 &E 6 ....then... ~Q 2-6 RD 7 QvP

5.5d

Problem 2.
 Premise 1 A>B Premise 2 B>C Assumption 3 ....what if ~(A>C) 3 IM 4 ....then... ~(~AvC) 4 DM 5 ....then... ~~A&~C 5 DN 6 ....then... A&~C 6 &E 7 ....then... A 1,7 MP 8 ....then... B 2,8 MP 9 ....then... C 6 &E 10 ....then... ~C 3-10 RD 11 A>C
Problem 3.
 Premise 1 A>B Assumption 2 ....what if ~(~B>~A) 2 IM 3 ....then... ~(~~Bv~A) 3 DN 4 ....then... ~(Bv~A) 4 DM 5 ....then... ~B&~~A 5 DN 6 ....then... ~B&A 6 &E 7 ....then... A 1,7 MP 8 ....then... B 6 &E 9 ....then... ~B 2-9 RD 10 ~B>~A
Problem 4.
 Premise 1 P>(Q>R) Assumption 2 ....what if ~((P&Q)>R) 2 IM 3 ....then... ~(~(P&Q)vR) 3 DM 4 ....then... ~~(P&Q)&~R 4 DN 5 ....then... (P&Q)&~R 5 &E 6 ....then... P&Q 6 &E 7 ....then... P 1,7 MP 8 ....then... Q>R 6 &E 9 ....then... Q 8,9 MP 10 ....then... R 5 &E 11 ....then... ~R 2-11 RD 12 (P&Q)>R

5.5e

Problem 1.
 Premise 1 (~AvB)vC Premise 2 (BvC)>(B&C) 1 AS 3 ~Av(BvC) 3 IM 4 A>(BvC) 2,4 HS 5 A>(B&C)
Problem 2.
 Premise 1 ~P 1 vI 2 ~PvT 2 IM 3 P>T
Problem 3.
 Premise 1 J>(~R&~K) Premise 2 ((RvK)>~J)=T 1 TR 3 ~(~R&~K)>~J 3 DM 4 (~~Rv~~K)>~J 4 DN 5 (Rv~~K)>~J 5 DN 6 (RvK)>~J 2,6 =E 7 T

5.6a

Problem 1.
 Assumption 1 ....what if A&~A 1 &E 2 ....then... A 1 &E 3 ....then... ~A 1-3 RD 4 ~(A&~A)
Problem 2.
 Assumption 1 ....what if ~((T=U)v~(T=U)) 1 DM 2 ....then... ~(T=U)&~~(T=U) 2 DN 3 ....then... ~(T=U)&(T=U) 3 &E 4 ....then... T=U 3 &E 5 ....then... ~(T=U) 1-5 RD 6 (T=U)v~(T=U)
Problem 3.
 Assumption 1 ....what if ~(~~A>A) 1 DN 2 ....then... ~(A>A) 2 IM 3 ....then... ~(~AvA) 3 DM 4 ....then... ~~A&~A 4 &E 5 ....then... ~A 4 &E 6 ....then... ~~A 1-6 RD 7 ~~A>A
Problem 4.
 Assumption 1 ....what if J=~J 1 EQ 2 ....then... (J>~J)&(~J>J) 2 IM 3 ....then... (~Jv~J)&(~J>J) 3 IM 4 ....then... (~Jv~J)&(~~JvJ) 4 ID 5 ....then... ~J&(~~JvJ) 5 &E 6 ....then... ~J 5 &E 7 ....then... ~~JvJ 7 DN 8 ....then... JvJ 8 ID 9 ....then... J 1-9 RD 10 ~(J=~J)

5.6b

Problem 1.
 Premise 1 (A&A)&C 1 &E 2 A&A 2 &E 3 A 1 &E 4 C 3,4 &I 5 A&C 6 7 8 9 10 A&C??? ~~~~ 11 ~ other direction~~~~~ Premise 12 A&C 12 ID 13 (A&A)&C 14 15 16 17 18 19 20 (A&A)&C???
Problem 2.
 Premise 1 ~((AvB)vC) 1 DM 2 ~(AvB)&~C 2 DM 3 (~A&~B)&~C 4 5 6 7 8 9 3 AS 10 ~A&(~B&~C) ~~~~ 11 ~ other direction~~~~~ Premise 12 ~A&(~B&~C) 12 AS 13 (~A&~B)&~C 13 DM 14 ~(AvB)&~C 15 16 17 18 19 20 14 DM 21 ~((AvB)vC)

5.6c

Problem 1.
 Premise 1 L&~L Assumption 2 ....what if ~M 1 &E 3 ....then... L 1 &E 4 ....then... ~L 2-4 RD 5 M
Problem 2.
 Premise 1 ~T=U Premise 2 W>~T Premise 3 UvW 1 EQ 4 (~T>U)&(U>~T) 4 &E 5 U>~T 3,2,5 vE 6 ~T
Problem 3.
 Premise 1 (M>N)v(O>P) Premise 2 ~N 1 IM 3 (~MvN)v(O>P) 3 CM 4 (Nv~M)v(O>P) 4 AS 5 Nv(~Mv(O>P)) 2,5 DS 6 ~Mv(O>P) 6 IM 7 M>(O>P) 7 EX 8 (M&O)>P

5.6e

Problem 1.
 Premise 1 (J=~K)>L Premise 2 J&~R Premise 3 (K=R)&S 3 &E 4 K=R 2 &E 5 ~R 4 EQ 6 (K>R)&(R>K) 6 &E 7 K>R 5,7 MT 8 ~K 2 &E 9 J 9,8 &I 10 J&~K 10 vI 11 (J&~K)v(~J&~~K) 11 EQ 12 J=~K 1,12 MP 13 L 3 &E 14 S 13,14 &I 15 L&S
Problem 2.
 Premise 1 A>(B&C) Premise 2 (KvR)>T Premise 3 RvK Assumption 4 ....what if ~(A>B) 4 IM 5 ....then... ~(~AvB) 5 DM 6 ....then... ~~A&~B 6 &E 7 ....then... ~~A 7 DN 8 ....then... A 1,8 MP 9 ....then... B&C 9 &E 10 ....then... B 6 &E 11 ....then... ~B 4-11 RD 12 A>B 3 CM 13 KvR 2,13 MP 14 T 12,14 &I 15 (A>B)&T
Problem 3.
 Premise 1 T>L Premise 2 (~L>~T)=K Assumption 3 ....what if ~(F>K) 3 IM 4 ....then... ~(~FvK) 4 DM 5 ....then... ~~F&~K 5 &E 6 ....then... ~K 1 TR 7 ....then... ~L>~T 2,7 =E 8 ....then... K 3-8 RD 9 F>K
Problem 4.
 Premise 1 Bv~A Premise 2 L=K 1 CM 3 ~AvB 3 IM 4 A>B 2 CM 5 K=L 4,5 &I 6 (A>B)&(K=L)
Problem 5.
 Premise 1 ~(K=R) Premise 2 R Assumption 3 ....what if K 2,3 &I 4 ....then... K&R 4 vI 5 ....then... (K&R)v(~K&~R) 5 EQ 6 ....then... K=R 1 R 7 ....then... ~(K=R) 3-7 RD 8 ~K 8 vI 9 ~KvX