Selected Answers

      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)

Review Exercise answers

Group 1:

Problem One.  
Premise 1 P  
Premise 2 ~P  
Assumption 3 ....what if ~Q
1 R 4 ....then... P
2 R 5 ....then... ~P
3-4 ~E 6 Q  
Problem Five.  
Premise 1 A&B  
Assumption 2 ....what if A
1 &E 3 ....then... B
2-3 >I 4 A>B