Chapter 7              Selected Answers                  

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Q7.1b

Problem Two.  
Premise1Db&Lac
1 &E2Db
 3 
2 %I4(%x)Dx
Problem Three.  
Premise1(^x)(Bx&Lx)
1 ^E2Ba&La
2 &E3La
3 %I4(%x)Lx

 

Q7.1d

Problem 2.  
Premise1Tjv(%x)~Kx
1 CM2(%x)~Kx v Tj
 3 
 4 
2 QN5~(^x)Kx v Tj
Problem 3.  
Premise1Lst>(^x)~Jx
Premise2Lst
1,2 MP3(^x)~Jx
 4 
3 QN5~(%x)Jx
Problem 4.  
Premise1~((%x)Txv(^x)Kx)
1 DM2~(%x)Tx&~(^x)Kx
2 &E3~(%x)Tx
 4 
3 QN5(^x)~Tx

 

7.1a

Problem One.  
Premise1(^x)Ax
1 ^E2Aa
1 ^E3Ab
2,3 &I4Aa&Ab
Problem Two.  
Premise 1 (^x)Ax
Premise 2 (^x)Bx
Premise 3 Ca
1 ^E 4 Ab
2 ^E 5 Bb
4,5 &I 6 Ab&Bb
6 ^I 7 (^x)(Ax&Bx)
Problem Five.  
Premise1(^x)(Tx>Ux)
Premise2Ta
1 ^E3Ta>Ua
2,3 MP4Ua
2,4 &I5Ua&Ta
7.1b

Problem One.  
Premise1~(^x)Gx
Premise2(%x)~Gx>Pj
1 QN3(%x)~Gx
2,3 MP4Pj
Problem Two.  
Premise1~(%x)Ax
1 QN2(^x)~Ax
2 ^E3~Aj
Problem Three.  
Premise1~(^x)Lx&~(%x)Lx
1 QN2(%x)~Lx&~(%x)Lx
2 QN3(%x)~Lx&(^x)~Lx
Problem Four.  
Premise1~(^x)~Jx
1 QN2(%x)~~Jx
2 DN3(%x)Jx

7.2a

Problem One.  
Premise1~(%x)(Tx&~Cx)
1 QN2(^x)~(Tx&~Cx)
2 DM3(^x)(~Tx v ~~Cx)
3 DN4(^x)(~Tx v Cx)
 5 
 6 
 7 
 8 
4 IM9(^x)(Tx>Cx)

 

7.2b

Problem Two.  
Premise1(^x)(Rx>~Mx)
1 IM2(^x)(~Rx v ~Mx)
2 DM3(^x)~(Rx&Mx)
3 CM4(^x)~(Mx&Rx)
 5 
 6 
 7 
 8 
4 QN9~(%x)(Mx&Rx)

 

7.2c

Problem One.  
Premise1(^x)(Ax>Bx)
1 IM2(^x)(~Ax v Bx)
2 DN3(^x)(~Ax v ~~Bx)
3 DM4(^x)~(Ax&~Bx)
4 CM5(^x)~(~Bx&Ax)
 6 
 7 
 8 
5 QN9~(%x)(~Bx&Ax)

7.3a

Problem One.  
Premise1(^x)(Ax&Bx)
1 ^E2Aa&Ba
2 &E3Ba
3 ^I4(^x)Bx
Problem Four.  
Premise 1 (%x)Px  
Assumption 2 ....what if ~(%y)Py
2 QN 3 ....then... (^y)~Py
3 ^E 4 ....then... ~Pa
4 ^I 5 ....then... (^x)~Px
5 QN 6 ....then... ~(%x)Px
1 R 7 ....then... (%x)Px
2-7RD 8 (%y)Py  

 

7.3b

Problem One.  
Premise1(^x)(Bx=Cx)
1 ^E2Ba=Ca
2 EQ3(Ba>Ca)&(Ca>Ba)
3 &E4Ba>Ca
4 ^I 5(^x)(Bx>Cx)
Problem Two.  
Premise1(^x)Bx
Premise2(^x)Cx
1 ^E3Ba
2 ^E4Ca
3,4 &I5Ba&Ca
5 DN6~~Ba&Ca
6 DN7~~Ba&~~Ca
7 DM8~(~Ba v ~Ca)
8 ^I9(^x)~(~Bx v ~Cx)
9 QN10~(%x)(~Bx v ~Cx)
Problem Three.  
Premise1~(%x)(Px&Mx)
1 QN2(^x)~(Px&Mx)
2 DM3(^x)(~Px v ~Mx)
3 CM4(^x)(~Mx v ~Px)
4 IM5(^x)(Mx>~Px)
Problem Four.  
Premise 1 Ca&~(%x)Px
1 &E 2 ~(%x)Px
2 QN 3 (^x)~Px
3 ^E 4 ~Pt
4 vI 5 ~Pt v Ct
5 IM 6 Pt>Ct
6 ^I 7 (^x)(Px>Cx)

7.3c

Problem One.  
Premise 1 (^x)(Sx>Wx)
Premise 2 (^x)(Wx>~Hx)
1 ^E 3 Sa>Wa
2 ^E 4 Wa>~Ha
3,4 HS 5 Sa>~Ha
5 ^I 6 (^x)(Sx>~Hx)
Problem Two.  
Premise 1 (^x)(Ax&Bx)
Premise 2 Ca
1 ^E 3 At&Bt
3 &E 4 At
4 ^I 5 (^x)Ax
Problem Four.  
Premise 1 (^x)Px
1 ^E 2 Pc
2 ^I 3 (^y)Py

7.3d
Problem Two.  
Premise 1 (^x)(Sx>Wx)
Premise 2 Sa
1 ^E 3 Sa>Wa
2,3 MP 4 Wa
4 %I 5 (%x)Wx
Problem Three.  
Premise 1 ~(^x)Px
Premise 2 (%x)~Px>(^x)Cx
1 QN 3 (%x)~Px
2,3 MP 4 (^x)Cx
4 ^E 5 Ca

7.3e

Problem Two.  
Premise 1 (^x)(Sx=Wx)
Premise 2 Sa
1 ^E 3 Sa=Wa
2,3 =E 4 Wa
4 %I 5 (%x)Wx
Problem Three.  
Premise 1 ~(^x)Px
Premise 2 (%x)~Px>(QavQb)
1 QN 3 (%x)~Px
2,3 MP 4 QavQb
4 DN 5 ~~Qa v Qb
5 IM 6 ~Qa>Qb
Problem Four.  
Premise 1 (^x)Ax
Premise 2 (^x)(Ax=Bx)
1 ^E 3 Aa
2 ^E 4 Aa=Ba
3,4 =E 5 Ba
5 ^I 6 (^x)Bx

7.3f

Problem One.  
Premise 1 ~(%x)(Ax&~Bx)
1 QN 2 (^x)~(Ax&~Bx)
2 DM 3 (^x)(~Ax v ~~Bx)
3 DN 4 (^x)(~Ax v Bx)
4 IM 5 (^x)(Ax>Bx)
Problem Two.  
Premise 1 (^x)(SxvWx)
Premise 2 ~Sa
1 ^E 3 Sa v Wa
HINT: 4 Use DS
HINT: 5 USE %I
Problem Three.  
Premise 1 (^x)Ax&Aa
Premise 2 (^x)(Ax=Bx)
1 &E 3 (^x)Ax
3 ^E 4 Ab
2 ^E 5 Ab=Bb
4,5 =E 6 Bb
4,6 &I 7 Ab&Bb
7 ^I 8 (^x)(Ax&Bx)

7.3g

Problem One.  
Premise 1 (^x)(Wx&Tx)
1 ^E 2 Wa&Ta
2 &E 3 Ta
3 ^I 4 (^x)Tx
Problem Two.  
Premise 1 (^x)(Sx>Wx)
Premise 2 (^x)(Wx>~Hx)
1 ^E 3 Sa>Wa
2 ^E 4 Wa>~Ha
3,4 HS 5 Sa>~Ha
5 ^I 6 (^x)(Sx>~Hx)
Problem Three.  
Premise 1 (^x)(SxvWx)
Premise 2 (^x)~Sx
1 ^E 3 Sa v Wa
2 ^E 4 ~Sa
3,4 DS 5 Wa
5 ^I 6 (^x)Wx

7.3h

Problem One.  
Premise 1 (^x)(Wx>Mx)
Premise 2 ~Mu
1 ^E 3 Wu>Mu
2,3 MT 4 ~Wu
  5  
  6  
  7  
  8  
  9 ~Wu???
Problem Two.  
Premise 1 ~(%x)(Wx&Fx)  
Premise 2 Ws  
Assumption 3 ....what if Fs
2,3 &I 4 ....then... Ws&Fs
4 %I 5 ....then... (%x)(Wx&Fx)
1 R 6 ....then... ~(%x)(Wx&Fx)
3-6RD 7 ~Fs  
  8    
  9 ~Fs???  
Problem Three.  
Premise 1 (^x)(Wx>Mx)
Premise 2 Ws&~Fs
1 ^E 3 Ws>Ms
2 &E 4 Ws
3,4 MP 5 Ms
2 &E 6 ~Fs
5,6 &I 7 Ms&~Fs
  8  
7 %I 9 (%x)(Mx&~Fx)
Problem Six.