5.7ex I

1.
 Premise 1 (L&J)vT Premise 2 ~T Premise 3 L>F 1,2 DS 4 L&J 4 &E 5 L 3,5 >E 6 F
5.
 Premise 1 L>(T=Y) Premise 2 (T=Y)>~M Premise 3 ~~M 2,3 MT 4 ~(T=Y) 1,4 MT 5 ~L 5,3 &I 6 ~L&~~M

5.7ex III

3.
 Premise 1 ~(~Av~B) 1 DM 2 ~~A&~~B 2 DN 3 A&~~B 3 DN 4 A&B
4.
 Premise 1 ~Av~B Premise 2 (A&B)v(~LvS) 1 DM 3 ~(A&B) 2,3 DS 4 ~LvS 4 IM 5 L>S

5.7ex IV

1.
 Premise 1 ~S>~T Premise 2 ~(T>S)vL 1 TR 3 T>S 3 DN 4 ~~(T>S) 2,4 DS 5 L
4.
 Premise 1 (~S&G)v(~S&K) 1 DI 2 ~S&(GvK) 2 CM 3 (GvK)&~S 3 CM 4 (KvG)&~S

5.7ex V

3.
 Assumption 1 ....what if A>C 1 IM 2 ....then... ~AvC 1-2 >I 3 (A>C)>(~AvC) Assumption 4 ....what if ~AvC 4 IM 5 ....then... A>C 4-5 >I 6 (~AvC)>(A>C) 3,6 &I 7 ((A>C)>(~AvC))&((~AvC)>(A>C)) 7 EQ 8 (A>C)=(~AvC)
Can you see how you could have saved in step in problem 3 and finished at line 7? Hint: don't forget the old I-rules!
5.
 Premise 1 ~(A>C) 1 IM 2 ~(~AvC) 2 DM 3 ~~A&~C 3 DN 4 A&~C ~~~~ 5 ~ part II ~ Premise 6 A&~C 6 DN 7 ~~A&~C 7 DM 8 ~(~AvC) 8 IM 9 ~(A>C)

5.8ex I

1.
 Premise 1 ~(A>B) 1 IM 2 ~(~AvB) 2 DM 3 ~~A&~B 3 &E 4 ~B
5.
 Assumption 1 What if.. ~(Rv~R) 1 DM 2 then...... ~R&~~R 2 &E 3 then...... ~R 2 &E 4 then...... ~~R 1-4 ~E 5 Rv~R

5.8ex II

3.
 Premise 1 ~L>(J=~K) Assumption 2 What if.. ~L&F 2 &E 3 then...... ~L 1,3 >E 4 then...... J=~K 4 EQ 5 then...... (J>~K)&(~K>J) 5 TR 6 then...... (~~K>~J)&(~K>J) 6 TR 7 then...... (~~K>~J)&(~J>~~K) 7 DN 8 then...... (K>~J)&(~J>~~K) 8 DN 9 then...... (K>~J)&(~J>K) 9 CM 10 then...... (~J>K)&(K>~J) 10 EQ 11 then...... ~J=K 2-11 12 (~L&F)>(~J=K)
4.
 Premise 1 [(~L>K)v(L>K)]>A Premise 2 ~(Av(~J>T)) Assumption 3 L R 4 L 3-4 >I 5 L>L 5 IM 6 ~LvL 6 vI 7 (~LvL)v(KvK) 7 AS 8 [(~LvL)vK]vK 8 AS 9 [(~Lv(LvK)]vK 9 AS 10 ~Lv[(LvK)vK] 10 CM 11 ~Lv[Kv(LvK)] 11 AS 12 (~LvK)v(LvK) 12 IM 13 (L>K)v(LvK) 13 DN 14 (L>K)v(~~LvK) 14 IM 15 (L>K)v(~L>K) 15, 1 >E 16 A 2 DM 17 ~A&~(~J>T) 17 &E 18 ~A
Notice: lines 16 and 18 provide the contradiction.

5.8ex III Logical Truths:

1.
 Assumption 1 ....what if ~A>G 1 TR 2 ....then... ~G>~~A 2 DN 3 ....then... ~G>A 1-3 >I 4 (~A>G)>(~G>A) Assumption 5 ....what if ~G>A 5 TR 6 ....then... ~A>~~G 6 DN 7 ....then... ~A>G 5-7 >I 8 (~G>A)>(~A>G) 4,8 =I 9 (~A>G)=(~G>A)
2.
 Assumption 1 ....what if J=~K 1 EQ 2 ....then... (J>~K)&(~K>J) 2 CM 3 ....then... (~K>J)&(J>~K) 3 EQ 4 ....then... ~K=J 1-4 >I 5 (J=~K)>(~K=J)
4.
 Assumption 1 ....what if ~R&~R) 1 DN 2 ....then... ~~(R&~R) 2 DN 3 ....then... R&~R 3 &E 4 ....then... R 3 &E 5 ....then... ~R 1-5 ~I 6 ~(R&~R)
5.
 Assumption 1 ....what if ~[A>(B>L)] 1 IM 2 ....then... ~[~Av(B>L)] 2 DM 3 ....then... ~~A&~(B>L) 3 DN 4 ....then... A&~(B>L) 4 IM 5 ....then... A&~(~BvL) 5 DM 6 ....then... A&(~~B&~L) 6 DN 7 ....then... A&(B&~L) 7 AS 8 ....then... (A&B)&~L 1-8 >I 9 ~[A>(B>L)]>[(A&B)&~L] 9 IM 10 ~~[A>(B>L)]v[(A&B)&~L] 10 DN 11 [A>(B>L)]v[(A&B)&~L]

5.8ex IV Logical Falsehood and Inconsistency

1.   Here's the way to do this one with DI. (You might instead have assumed 'T' at line 2. This leads to a contradiction within the subderivation and the conclusion of '~T' outside the subderivation. The rest is easy.)
 Premise 1 [(T>L)&(~T>L)]&~L 1 &E 2 (T>L)&(~T>L) 2 IM 3 (~TvL)&(~T>L) 3 IM 4 (~TvL)&(~~TvL) 4 DN 5 (~TvL)&(TvL) 5 CM 6 (Lv~T)&(TvL) 6 CM 7 (Lv~T)&(LvT) 7 DI 8 Lv(~T&T) 1 &E 9 ~L 8,9 DS 10 ~T&T 10 &E 11 T 10 &E 12 ~T
2.
 Premise 1 [A>(C&D)]&~(A>C) 1 &E 2 A>(C&D) 2 IM 3 ~Av(C&D) 3 DI 4 (~AvC)&(~AvD) 4 &E 5 ~AvC 5 IM 6 A>C 1 &E 7 ~(A>C)
3.
 Premise 1 A>(G>L) Premise 2 ~(G>L)vK Premise 3 ~(~K>~A) 2 IM 4 (G>L)>K 1,4 HS 5 A>K 3 TR 6 ~(A>K)
4.
 Premise 1 ~(A>~B)&T Premise 2 (A=~B)v(S&~T) Assumption 3 ....what if A=~B 3 EQ 4 ....then... (A>~B)&(~B>A) 4 &E 5 ....then... A>~B 1 &E 6 ....then... ~(A>~B) 3-6 ~I 7 ~(A=~B) 2,7 DS 8 S&~T 1 &E 9 T 8 &E 10 ~T

5.8ex V Logical Equivalence

2.
 Premise 1 (~A>A)>B 1 IM 2 (~~AvA)>B 2 DN 3 (AvA)>B 3 ID 4 A>B 4 TR 5 ~B>~A ~~~~ 6 ~ Part II ~ Premise 7 ~B>~A 7 TR 8 A>B 8 ID 9 (AvA)>B 9 DN 10 (~~AvA)>B 10 IM 11 (~A>A)>B
3.
 Premise 1 ~(A>(B>~C)) 1 IM 2 ~(~Av(B>~C)) 2 DM 3 ~~A&~(B>~C) 3 DN 4 A&~(B>~C) 4 IM 5 A&~(~Bv~C) 5 DM 6 A&(~~B&~~C) 6 DN 7 A&(B&~~C) 7 DN 8 A&(B&C) 9 8 AS 10 (A&B)&C ~~~~ 11 ~ Part II ~ Premise 12 (A&B)&C 12 AS 13 A&(B&C) 13 DN 14 A&(B&~~C) 14 DN 15 A&(~~B&~~C) 15 DM 16 A&~(~Bv~C) 16 IM 17 A&~(B>~C) 17 DN 18 ~~A&~(B>~C) 18 DM 19 ~(~Av(B>~C)) 19 IM 20 ~(A>(B>~C))
4.
 Premise 1 ~A&(BvC) 1 DI 2 (~A&B)v(~A&C) 2 DN 3 ~~(~A&B)v(~A&C) 3 IM 4 ~(~A&B)>(~A&C) 4 DM 5 (~~Av~B)>(~A&C) 5 DN 6 (Av~B)>(~A&C) 6 CM 7 (~BvA)>(~A&C) 7 IM 8 (B>A)>(~A&C) ~~~~ 9 ~ Part II ~~~ Premise 10 (B>A)>(~A&C) 10 IM 11 (~BvA)>(~A&C) 11 CM 12 (Av~B)>(~A&C) 12 DN 13 (~~Av~B)>(~A&C) 13 DM 14 ~(~A&B)>(~A&C) 14 IM 15 ~~(~A&B)v(~A&C) 15 DN 16 (~A&B)v(~A&C) 16 DI 17 ~A&(BvC)