Chapter 5             Selected Answers                  

Skip to: 5.1 5.2 5.3 5.4 5.5 5.6

5.1a

Problem One.  
Premise1AvB
Premise2A>I
Premise3~I
2,3 MT4~A
1,4 DS5B
 6 
 7 
 8 
 9B???
Problem Two.  
Premise1K>~M
Premise2LvM
Premise3K
1,3 MP4~M
2,4 DS5L
 6 
 7 
 8 
 9L???
Problem Three.  
Premise1J>L
Premise2~L
Premise3JvR
1,2 MT4~J
3,4 DS5R
 6 
 7 
 8 
 9R???
Problem Four.  
Premise1(F&K)>(DvS)
Premise2F&K
Premise3~D
1,2 MP4DvS
3,4 DS5S
 6 
 7 
 8 
 9S???

5.1b

Problem One.  
Premise1AvB
Premise2A>I
Premise3~I
2,3 MT4~A
1,4 DS5B
 6 
 7 
 8 
 9B???
Problem Two.  
Premise1K>~M
Premise2LvM
Premise3K
1,3 MP4~M
2,4 DS5L
 6 
 7 
 8 
 9L???
Problem Three.  
Premise1J>L
Premise2~L
Premise3JvR
1,2 MT4~J
3,4 DS5R
 6 
 7 
 8 
 9R???
Problem Four.  
Premise1(F&K)>(DvS)
Premise2F&K
Premise3~D
1,2 MP4DvS
3,4 DS5S
 6 
 7 
 8 
 9S???
Problem Five.  
Premise1Jv(DvS)
Premise2~J
Premise3~D
2,1 DS4DvS
3,4 DS5S
 6 
 7 
 8 
 9S???

5.1c

Problem Two.  
Premise1(JvS)>(LvT)
Premise2~(LvT)
1,2 MT3~(JvS)
 4 
 5 
 6 
 7 
 8 
 9~(JvS)???
Problem Three.  
Premise1~K>(~L>J)
Premise2Lv~K
Premise3~L
2,3 DS4~K
1,4 MP5~L>J
3,5 MP6J
 7 
 8 
 9J???
Problem Five.  
Premise1T>(J>L)
Premise2(R=L)vT
Premise3~(J>L)
1,3 MT4~T
2,4 DS5R=L
 6 
 7 
 8 
 9R=L???

5.1g

Problem Two.  
Premise1LvM
Premise2~L
Premise3M>(P&Q)
1,2 DS4M
3,4 MP5P&Q
 6 
 7 
 8 
 9P&Q???
Problem Four.  
Premise1A>B
Premise2~B
Premise3~A>K
1,2 MT4~A
3,4 MP5K
 6 
 7 
 8 
 9K???

 

5.2a

Problem 1.  
Premise1A&B
1 &E2A
1 &E3B
 4 
2,3 &I5B&A
Problem 4.  
Premise1~(~RvS)&J
Premise2M>(~RvS)
1 &E3~(~RvS)
2,3 MT4~M
 5 
 6 
 7 
 8~M???

5.2b

Problem 1.  
Premise1A&B
1 &E2A
1 &E3B
 4 
2,3 &I5B&A
Problem 2.  
Premise1A&~B
Premise2LvB
1 &E3~B
2,3 DS4L
1 &E5A
 6 
4,5 &I7L&A
Problem 4.  
Premise1~(~RvS)&J
Premise2M>(~RvS)
1 &E3~(~RvS)
2,3 MT4~M

5.2c

Problem 1.  
Premise1(L&M)>S
Premise2L&(L>M)
2 &E3L
2 &E4L>M
3,4 MP5M
3,5 &I6L&M
1,6 MP7S
 8 
 9S???
Problem 2.  
Premise1J>~K
Premise2J&(R>K)
2 &E3J
1,3 MP4~K
2 &E5R>K
4,5 MT6~R
4,6 &I7~R&~K
 8 
 9~R&~K???
Problem 3.  
Premise1T&~U
Premise2Uv(T>S)
1 &E3~U
2,3 DS4T>S
1 &E5T
4,5 MP6S
 7 
 8 
 9S???
Problem 4.  
Premise1MvN
Premise2~N
Premise3(~N&M)>(O&P)
1,2 DS4M
2,4 &I5~N&M
3,5 MP6O&P
6 &E7P
 8 
 9P???
Problem 5.  
Premise1J&(KvR)
Premise2L>J
Premise3~K
1 &E4KvR
3,4 DS5R
 6 
 7 
 8 
 9R???

5.3a

Problem 1.  
Premise1~~A&B
1 DN2A&B
2 DN3A&~~B
 4 
???5A&~~B???
Problem 3.  
Premise1A>~B
Premise2B
1 IM3~Av~B
2 DN4~~B
3,4 DS5~A
 6 
 7 
???8~A???
Problem 4.  
Premise1~K>T
Premise2~T
1,2 MT3~~K
3 DN4K
 5 
 6 
 7 
???8K???

5.3b

Problem 1.  
Premise1[(J&K)>S]&[S>(J&K)]
 2 
 3 
 4 
1 EQ5(J&K)=S
Problem 2.  
Premise1L=M
1 EQ2(L>M)&(M>L)
2 &E3L>M
2 &E4M>L
 5 
 6 
 7 
 8 
3,4 &I9(M>L)&(L>M)
Problem 3.  
Premise1~(J&K)vT
 2 
 3 
 4 
 5 
 6 
 7 
1 IM8(J&K)>T
Problem 4.  
Premise1~(J>K)
1 IM2~(~JvK)
2 DM3~~J&~K
3 &E4~K
Problem 5.  
Premise1(L>S)&(~SvL)
1 IM2(L>S)&(S>L)
2 EQ3L=S

5.4a

Problem 1.  
Premise1A&~B 
Assumption2....what ifA>B
1 &E3....then...A
2,3 MP4....then...B
1 &E5....then...~B
2-5 RD6~(A>B) 
 7  
 8  
???9~(A>B)??? 
Problem 2.  
Premise1AvC 
Assumption2....what if~(CvA)
2 DM3....then...~C&~A
3 &E4....then...~C
1,4 DS5....then...A
3 &E6....then...~A
2-6 RD7CvA 
 8  
???9CvA??? 
Problem 3.  
Premise1~(A>B) 
Assumption2....what ifA=B
2 EQ3....then...(A>B)&(B>A)
3 &E4....then...A>B
1 R5....then...~(A>B)
2-5 RD6~(A=B) 
 7  
???8~(A=B)??? 
Problem 4.  
Premise1~A&~B 
Assumption2....what if~(B>A)
2 IM3....then...~(~BvA)
3 DM4....then...~~B&~A
1 &E5....then...~B
4 &E6....then...~~B
2-6 RD7B>A 

5.4b

Problem 1.  
Premise1Kv~I 
Assumption2....what if~K&I
2 &E3....then...~K
1,3 DS4....then...~I
2 &E5....then...I
2-5 RD6~(~K&I) 
 7  
 8  
???9~(~K&I)??? 
Problem 3.  
Premise1K>~K 
Premise2~K>L 
Assumption3....what ifK
1,3 MP4....then...~K
3-4 RD5~K 
2,5 MP6L 
 7  
???8L??? 
Problem 4.  
Premise1R>T 
Assumption2....what if~(R>(T&R))
2 IM3....then...~(~Rv(T&R))
3 DM4....then...~~R&~(T&R)
4 &E5....then...~~R
5 DN6....then...R
1,6 MP7....then...T
4 &E8....then...~(T&R)
8 DM9....then...~Tv~R
7 DN10....then...~~T
9,10 DS11....then...~R
2-11 RD12R>(T&R) 
Problem 5.  
Premise1F>~K 
Premise2~(~KvR) 
Assumption3....what if~(F>T)
3 IM4....then...~(~FvT)
4 DM5....then...~~F&~T
5 &E6....then...~~F
6 DN7....then...F
1,7 MP8....then...~K
2 DM9....then...~~K&~R
9 &E10....then...~~K
3-10 RD11F>T 
???12F>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)  

5.5

Problem 3.  
Premise1PvQ 
Assumption2....what if~(QvP)
2 DM3....then...~Q&~P
3 &E4....then...~Q
1,4 DS5....then...P
3 &E6....then...~P
2-6 RD 7QvP 
 8  
 9  
 10  
???11QvP??? 

5.5c

Problem 1.  
Premise1T=R
Premise2(R=T)>(J=K)
Premise3J
1 CM4R=T
2,4 MP5J=K
3,5 =E6K
 7 
???8K???
Problem 2.  
Premise1~(M&S)
Premise2~(S&M)>T
1 CM3~(S&M)
2,3 MP4T
4 vI5Tv(X>Y)
 6 
 7 
 8 
???9Tv(X>Y)???
Problem 3.  
Premise1(JvL)>(T=U)
Premise2J&T
2 &E3J
3 vI4JvL
1,4 MP5T=U
2 &E6T
5,6 =E7U
 8 
???9U???
Problem 4.  
Premise1Pv~T
Premise2P>R
Premise3~T>R
1,2,3 vE4R
 5 
 6 
 7 
 8 
 9 
 10 
???11R???
Problem 5.  
Premise1J>R
Premise2(Rv~J)>F
1 IM3~JvR
3 CM4Rv~J
2,4 MP5F
5 vI6Fv~X
 7 
 8 
???9Fv~X???
5.5d

Problem 1.  
Premise1P&Q
1 &E2P
1 &E3Q
2,3 &I4Q&P
Problem 2.  
Premise1P=Q
1 EQ2(P>Q)&(Q>P)
2 &E3P>Q
2 &E4Q>P
3,4 &I5(Q>P)&(P>Q)
5 EQ6Q=P
Problem 3.  
Premise1PvQ 
Assumption2....what if~(QvP)
2 DM3....then...~Q&~P
3 &E4....then...~P
1,4 DS5....then...Q
3 &E6....then...~Q
2-6 RD7QvP 

 

5.5d

Problem 2.  
Premise1A>B 
Premise2B>C 
Assumption3....what if~(A>C)
3 IM4....then...~(~AvC)
4 DM5....then...~~A&~C
5 DN6....then...A&~C
6 &E7....then...A
1,7 MP8....then...B
2,8 MP9....then...C
6 &E10....then...~C
3-10 RD11A>C 
Problem 3.  
Premise1A>B 
Assumption2....what if~(~B>~A)
2 IM3....then...~(~~Bv~A)
3 DN4....then...~(Bv~A)
4 DM5....then...~B&~~A
5 DN6....then...~B&A
6 &E7....then...A
1,7 MP8....then...B
6 &E9....then...~B
2-9 RD10~B>~A 
Problem 4.  
Premise1P>(Q>R) 
Assumption2....what if~((P&Q)>R)
2 IM3....then...~(~(P&Q)vR)
3 DM4....then...~~(P&Q)&~R
4 DN5....then...(P&Q)&~R
5 &E6....then...P&Q
6 &E7....then...P
1,7 MP8....then...Q>R
6 &E9....then...Q
8,9 MP10....then...R
5 &E11....then...~R
2-11 RD12(P&Q)>R


5.5e

Problem 1.
Premise1(~AvB)vC
Premise2(BvC)>(B&C)
1 AS3~Av(BvC)
3 IM4A>(BvC)
2,4 HS5A>(B&C)
Problem 2.  
Premise1~P
1 vI2~PvT
2 IM3P>T
Problem 3.  
Premise1J>(~R&~K)
Premise2((RvK)>~J)=T
1 TR3~(~R&~K)>~J
3 DM4(~~Rv~~K)>~J
4 DN5(Rv~~K)>~J
5 DN6(RvK)>~J
2,6 =E7T

 

5.6a

Problem 1.  
Assumption1....what ifA&~A
1 &E2....then...A
1 &E3....then...~A
1-3 RD4~(A&~A) 
Problem 2.  
Assumption1....what if~((T=U)v~(T=U))
1 DM2....then...~(T=U)&~~(T=U)
2 DN3....then...~(T=U)&(T=U)
3 &E4....then...T=U
3 &E5....then...~(T=U)
1-5 RD6(T=U)v~(T=U) 
Problem 3.  
Assumption1....what if~(~~A>A)
1 DN2....then...~(A>A)
2 IM3....then...~(~AvA)
3 DM4....then...~~A&~A
4 &E5....then...~A
4 &E6....then...~~A
1-6 RD7~~A>A 
Problem 4.  
Assumption1....what ifJ=~J
1 EQ2....then...(J>~J)&(~J>J)
2 IM3....then...(~Jv~J)&(~J>J)
3 IM4....then...(~Jv~J)&(~~JvJ)
4 ID5....then...~J&(~~JvJ)
5 &E6....then...~J
5 &E7....then...~~JvJ
7 DN8....then...JvJ
8 ID9....then...J
1-9 RD10~(J=~J) 

 

5.6b

Problem 1.  
Premise1(A&A)&C
1 &E2A&A
2 &E3A
1 &E4C
3,4 &I5A&C
 6 
 7 
 8 
 9 
 10A&C???
~~~~11 ~ other direction~~~~~
Premise12A&C
12 ID13(A&A)&C
 14 
 15 
 16 
 17 
 18 
 19 
 20(A&A)&C???
Problem 2.  
Premise1~((AvB)vC)
1 DM2~(AvB)&~C
2 DM3(~A&~B)&~C
 4 
 5 
 6 
 7 
 8 
 9 
3 AS10~A&(~B&~C)
~~~~11 ~ other direction~~~~~
Premise12~A&(~B&~C)
12 AS13(~A&~B)&~C
13 DM14~(AvB)&~C
 15 
 16 
 17 
 18 
 19 
 20 
14 DM21~((AvB)vC)

5.6c

Problem 1.  
Premise1L&~L 
Assumption2....what if~M
1 &E3....then...L
1 &E4....then...~L
2-4 RD5M 
Problem 2.  
Premise1~T=U
Premise2W>~T
Premise3UvW
1 EQ4(~T>U)&(U>~T)
4 &E5U>~T
3,2,5 vE6~T
Problem 3.  
Premise1(M>N)v(O>P)
Premise2~N
1 IM3(~MvN)v(O>P)
3 CM4(Nv~M)v(O>P)
4 AS5Nv(~Mv(O>P))
2,5 DS6~Mv(O>P)
6 IM7M>(O>P)
7 EX8(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