Selected Answers from 8.4

Q8.4

Problem 2.  
Premise 1 Tj v (%y)~Kmy
1 QN 2 Tj v ~(^y)Kmy
2 CM 3 ~(^y)Kmy v Tj

8.4 ex I

1.  
Premise 1 ~(^x)Gx
Premise 2 (%x)~Gx>(^y)Py
1 QN 3 (%x)~Gx
2,3 >E 4 (^y)Py
2.  
Premise 1 ~(%x)Ax
1 QN 2 (^x)~Ax
2 ^E 3 ~Aj
3.  
Premise 1 ~(^x)(%y)Lxy
1 QN 2 (%x)~(%y)Lxy
2 QN 3 (%x)(^y)~Lxy
4.  
Premise 1 ~(^x)~Jx
1 QN 2 (%x)~~Jx
2 DN 3 (%x)Jx

8.4 ex II

1.  
Premise 1 (^x)(Bx=Cx)
1 ^E 2 Ba=Ca
2 EQ 3 (Ba>Ca)&(Ca>Ba)
3 &E 4 Ba>Ca
4 ^I 5 (^z)(Bz>Cz)
3.  
Premise 1 ~(%x)(^y)(Bxy=Tyx)
1 QN 2 (^x)~(^y)(Bxy=Tyx)
2 QN 3 (^x)(%y)~(Bxy=Tyx)
3 CM 4 (^x)(%y)~(Tyx=Bxy)
4.
Assumption 1 ....what if ~(%x)(Px&Mxa)
1 QN 2 ....then... (^x)~(Px&Mxa)
2 DM 3 ....then... (^x)(~Px v ~Mxa)
3 CM 4 ....then... (^x)(~Mxa v ~Px)
4 IM 5 ....then... (^x)(Mxa>~Px)
1-5 >I 6 ~(%x)(Px&Mxa)>(^x)(Mxa>~Px)  
5.
Premise 1 ~(^x)(%y)~Fxy  
1 QN 2 (%x)~(%y)~Fxy  
2 QN 3 (%x)(^y)~~Fxy  
3 DN 4 (%x)(^y)Fxy  
Assumption 5 ....what if (^y)Fty
5 ^E 6 ....then... Fta
6 %I 7 ....then... (%x)Fxa
4,5-7 %E 8 (%x)Fxa  
8 ^I 9 (^y)(%x)Fxy  

8.4ex III

1.  
Assumption 1 ....what if ~(^y)Gyy
1 QN 2 ....then... (%y)~Gyy
2 DN 3 ....then... (%y)~~~Gyy
1-3 >I 4 ~(^y)Gyy>(%y)~~~Gyy  
2.  
Assumption 1 ....what if (^y)~Py      
Assumption 2 ....then... ....what if (%z)Pz    
Assumption 3 ....then... ....then... ....what if Pt  
Assumption 4 ....then... ....then... ....then... ....what if ~(Q&~Q)
3 R 5 ....then... ....then... ....then... ....then... Pt
1 ^E 6 ....then... ....then... ....then... ....then... ~Pt
4-6 ~E 7 ....then... ....then... ....then... Q&~Q  
2,3-7 %E 8 ....then... ....then... Q&~Q    
8 &E 9 ....then... ....then... Q    
8 &E 10 ....then... ....then... ~Q    
2-10 ~I 11 ....then... ~(%z)Pz      
1-11 >I 12 (^y)~Py>~(%z)Pz        
3.  
Assumption 1 ....what if ~[(^x)Ax v (%x)~Ax]
1 DM 2 ....then... ~(^x)Ax&~(%x)~Ax
2 &E 3 ....then... ~(^x)Ax
2 &E 4 ....then... ~(%x)~Ax
3 QN 5 ....then... (%x)~Ax
1-5 ~E 6 (^x)Axv(%x)~Ax  
4.  
Assumption 1 ....what if ~(^x)Ax  
1 QN 2 ....then... (%x)~Ax  
Assumption 3 ....then... ....what if ~At
3 vI 4 ....then... ....then... ~At v Bt
4 IM 5 ....then... ....then... At>Bt
5 %I 6 ....then... ....then... (%x)(Ax>Bx)
2,3-6 %E 7 ....then... (%x)(Ax>Bx)  
1-7 >I 8 ~(^x)Ax>(%x)(Ax>Bx)    

 

8.4ex IV

Problem 1.  
Premise1~(%x)~Jx
1 QN2(^x)~~Jx
 3 
 4 
 5 
2 DN6(^x)Jx
~~~~7~ Part II ~
Premise8(^x)Jx
8 DN9~~(^x)Jx
 10 
 11 
 12 
9 QN13~(%x)~Jx
Problem 2.  
Premise1~(^x)(Ax&Bx)
1 QN2(%x)~(Ax&Bx)
2 DM3(%x)(~Ax v ~Bx)
 4 
 5 
 6(%x)(~Axv~Bx)???
~~~~7~ Part II ~
Premise8(%x)(~Axv~Bx)
8 DM9(%x)~(Ax&Bx)
 10 
 11 
 12 
9 QN13~(^x)(Ax&Bx)
Problem 3.  
Premise1~(^x)~Kxa
1 QN2(%x)~~Kxa
2 DN3(%x)Kxa
~~~~4~ Part II ~
Premise5(%x)Kxa
5 DN6(%x)~~Kxa
6 QN7~(^x)~Kxa
Problem 4.  
Premise1~(%x)(Px&Qx)
1 CM2~(%x)(Qx&Px)
~~~~3~ Part II ~
Premise4~(%x)(Qx&Px)
4 CM5~(%x)(Px&Qx)

 

8.4ex V

1.  
Premise1 ~(^x)(Ax v Bx)
Premise2(^x)Ax
2 ^E3Aa
3 vI4Aa v Ba
4 ^I5 (^x)(Ax v Bx)
3.  
Premise1~(%x)Lxa&Lja
1 &E2~(%x)Lxa
2 QN3(^x)~Lxa
1 &E4Lja
3 ^E5~Lja
4.  
Premise 1 (%x)Px=(^x)~Px  
Assumption 2 ....what if ~(%x)Px
2 QN 3 ....then... (^x)~Px
1,3 =E 4 ....then... (%x)Px
2-4 ~E 5 (%x)Px  
1,5 =E 6 (^x)~Px  
6 QN 7 ~(%x)Px