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.
 Premise 1 ~(%x)~Jx 1 QN 2 (^x)~~Jx 3 4 5 2 DN 6 (^x)Jx ~~~~ 7 ~ Part II ~ Premise 8 (^x)Jx 8 DN 9 ~~(^x)Jx 10 11 12 9 QN 13 ~(%x)~Jx
Problem 2.
 Premise 1 ~(^x)(Ax&Bx) 1 QN 2 (%x)~(Ax&Bx) 2 DM 3 (%x)(~Ax v ~Bx) 4 5 6 (%x)(~Axv~Bx)??? ~~~~ 7 ~ Part II ~ Premise 8 (%x)(~Axv~Bx) 8 DM 9 (%x)~(Ax&Bx) 10 11 12 9 QN 13 ~(^x)(Ax&Bx)
Problem 3.
 Premise 1 ~(^x)~Kxa 1 QN 2 (%x)~~Kxa 2 DN 3 (%x)Kxa ~~~~ 4 ~ Part II ~ Premise 5 (%x)Kxa 5 DN 6 (%x)~~Kxa 6 QN 7 ~(^x)~Kxa
Problem 4.
 Premise 1 ~(%x)(Px&Qx) 1 CM 2 ~(%x)(Qx&Px) ~~~~ 3 ~ Part II ~ Premise 4 ~(%x)(Qx&Px) 4 CM 5 ~(%x)(Px&Qx)

8.4ex V

1.
 Premise 1 ~(^x)(Ax v Bx) Premise 2 (^x)Ax 2 ^E 3 Aa 3 vI 4 Aa v Ba 4 ^I 5 (^x)(Ax v Bx)
3.
 Premise 1 ~(%x)Lxa&Lja 1 &E 2 ~(%x)Lxa 2 QN 3 (^x)~Lxa 1 &E 4 Lja 3 ^E 5 ~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