Selected Answers from 8.2 and 8.3

8.2 ex I

1.	Premise 1 (^x)(Ax&Bx) 1 ^E 2 Aa&Ba 2 &E 3 Ba 3 ^I 4 (^y)By
4.	Premise 1 Aa>(^x)~Bx   Assumption 2 ....what if (^x)Ax 2 ^E 3 ....then... Aa 1,3 >E 4 ....then... (^x)~Bx 4 ^E 5 ....then... ~Bb 5 ^I 6 ....then... (^y)~By 2-6 >I 7 (^x)Ax>(^y)~By   Notice that you need to choose 'b' at line 5 to instantiate. If you choose 'a', it would not be arbitrary because 'a' occurs in the Premise of line 1.

8.2 ex II

2.	Premise 1 (%x)Ax>Ba   Assumption 2 ....what if Ab 2 %I 3 ....then... (%x)Ax 1,3 >E 4 ....then... Ba 2-4 >I 5 Ab>Ba   5 ^I 6 (^x)(Ax>Ba)
4.	Premise 1 (^x)(Tx&Sx) 1 ^E 2 Ta&Sa 2 &E 3 Ta 3 ^I 4 (^y)Ty 2 &E 5 Sa 5 ^I 6 (^z)Sz 4,6 &I 7 (^y)Ty&(^z)Sz

8.2 ex III

1.	Premise 1 (%x)(Tx&Lx)   Assumption 2 ....what if Tt&Lt 2 &E 3 ....then... Tt 3 %I 4 ....then... (%y)Ty 1,2-4 %E 5 (%y)Ty   The assumption at line 2 is a substitution instance of line 1. 't' is picked as the instantiating name; it's the temporary name to illustrate an object of the type said to exist by line 1.
3.	Premise 1 (%x)Mxc   Premise 2 (^x)[(%y)Myx>Tx]   Assumption 3 ....what if Mtc 3 %I 4 ....then... (%y)Myc 1,3-4 %E 5 (%y)Myc   2 ^E 6 (%y)Myc>Tc   5,6 >E 7 Tc

8.2ex IV

1.	Premise 1 (%z)Tz   Assumption 2 ....what if Ta 2 vI 3 ....then... Ta v Laa 3 %I 4 ....then... (%z)(Tz v Lzz) 1,2-4 %E 5 (%z)(Tz v Lzz)
2.	Premise 1 (%x)(Bx&Tt)   Assumption 2 ....what if Ba&Tt 2 &E 3 ....then... Ba 3 %I 4 ....then... (%x)Bx 2 &E 5 ....then... Tt 4,5 &I 6 ....then... (%x)Bx&Tt 1,2-6 %E 7 (%x)Bx&Tt

8.3 ex I

2.	Assumption 1 ....what if Tba 1 R 2 ....then... Tba 1-2 >I 3 Tba>Tba   3 ^I 4 (^x)(Txa>Txa)
5.	Premise 1 (%x)Bxx>Baa   Assumption 2 ....what if Baa 2 %I 3 ....then... (%x)Bxx 2-3 >I 4 Baa>(%x)Bxx   1,4 =I 5 (%x)Bxx=Baa

8.3ex II

1.	Premise 1 (%x)~Px     Assumption 2 ....what if ~Pt   Assumption 3 ....then... ....what if (^x)Px 3 ^E 4 ....then... ....then... Pt 2 R 5 ....then... ....then... ~Pt 3-5 ~I 6 ....then... ~(^x)Px   1,2-6 %E 7 ~(^x)Px
3.	Premise 1 (%x)(%y)Gxy     Premise 2 (^x)(^y)(Gxy>~Gyx)     Assumption 3 ....what if (%y)Gty   Assumption 4 ....then... ....what if Gtu 2 ^E 5 ....then... ....then... (^y)(Gty>~Gyt) 5 ^E 6 ....then... ....then... Gtu>~Gut 4,6 >E 7 ....then... ....then... ~Gut 7 %I 8 ....then... ....then... (%y)~Guy 8 %I 9 ....then... ....then... (%x)(%y)~Gxy 3,4-9 %E 10 ....then... (%x)(%y)~Gxy   1,3-10 %E 11 (%x)(%y)~Gxy
5.	Premise 1 (%x)Lxx>J   Assumption 2 ....what if Laa 2 %I 3 ....then... (%x)Lxx 1,3 >E 4 ....then... J 2-4 >I 5 Laa>J   5 ^I 6 (^y)(Lyy>J)

8.3ex III

1.	Assumption 1 ....what if Ma   Assumption 2 ....then... ....what if Ja 1 R 3 ....then... ....then... Ma 2-3 >I 4 ....then... Ja>Ma   1-4 >I 5 Ma>(Ja>Ma)     5 ^I 6 (^x)[Mx>(Jx>Mx)]
2.	Assumption 1 ....what if Gab 1-1 >I 2 Gab>Gab   2 ^I 3 (^y)(Gay>Gay)   3 ^I 4 (^x)(^y)(Gxy>Gxy)
3.	Assumption 1 ....what if (^z)(Nz>Tz)   Assumption 2 ....then... ....what if (^z)Nz 1 ^E 3 ....then... ....then... Na>Ta 2 ^E 4 ....then... ....then... Na 3,4 >E 5 ....then... ....then... Ta 5 ^I 6 ....then... ....then... (^z)Tz 2-6 >I 7 ....then... (^z)Nz>(^z)Tz   1-7 >I 8 (^z)(Nz>Tz)>[(^z)Nz>(^z)Tz]
4.	Assumption 1 ....what if ~(Ga v ~Ga)   Assumption 2 ....then... ....what if Ga 2 vI 3 ....then... ....then... Ga v ~Ga 1 R 4 ....then... ....then... ~(Ga v ~Ga) 2-4 ~I 5 ....then... ~Ga   5 vI 6 ....then... Ga v ~Ga   1 R 7 ....then... ~(Ga v ~Ga)   1-7 ~E 8 Ga v ~Ga     8 ^I 9 (^x)(Gx v ~Gx)

8.3ex IV

1.	Premise 1 (^w)(Bw&Cw) 1 ^E 2 Ba&Ca 2 &E 3 Ba 3 ^I 4 (^w)Bw 2 &E 5 Ca 5 ^I 6 (^y)Cy   7   4,6 &I 8 (^w)Bw&(^y)Cy ~~~~ 9 ~Part II~~~~ Premise 10 (^w)Bw&(^y)Cy 10 &E 11 (^w)Bw 11 ^E 12 Ba 10 &E 13 (^y)Cy 13 ^E 14 Ca 12,14 &I 15 Ba&Ca   16   15 ^I 17 (^w)(Bw&Cw)
2.	Premise 1 Na>(^x)Tx   Assumption 2 ....what if Na 1,2 >E 3 ....then... (^x)Tx 3 ^E 4 ....then... Tb   5       6     2-4 >I 7 Na>Tb   7 ^I 8 (^x)(Na>Tx)   ~~~~ 9 ~ Part II ~ ~~~~ Premise 10 (^x)(Na>Tx)   Assumption 11 ....what if Na 10 ^E 12 ....then... Na>Tb 11,12 >E 13 ....then... Tb 13 ^I 14 ....then... (^x)Tx   15       16     11-14 >I 17 Na>(^x)Tx
4.	Premise 1 (%y)My>Ma     Assumption 2 ....what if Mb   2 %I 3 ....then... (%y)My   1,3 >E 4 ....then... Ma     5         6       2-4 >I 7 Mb>Ma     7 ^I 8 (^y)(My>Ma)     ~~~~ 9 ~ Part II ~ ~~~~ ~~~~ Premise 10 (^y)(My>Ma)     Assumption 11 ....what if (%y)My   Assumption 12 ....then... ....what if Mt 10 ^E 13 ....then... ....then... Mt>Ma 12,13 >E 14 ....then... ....then... Ma 11,12-14 %E 15 ....then... Ma     16       11-15 >I 17 (%y)My>Ma

8.3ex V

1.	Premise 1 (^x)(Ax&~Ax) 1 ^E 2 Aa&~Aa 2 &E 3 Aa 2 &E 4 ~Aa
2.	Premise 1 (%x)(Ax&~Ax)     Assumption 2 ....what if At&~At   Assumption 3 ....then... ....what if ~(X&~X) 2 &E 4 ....then... ....then... At 2 &E 5 ....then... ....then... ~At 3-5 ~E 6 ....then... X&~X   1,2-6 %E 7 X&~X     7 &E 8 X     7 &E 9 ~X
3.	Premise 1 (%x)Axv(%x)Bx     Premise 2 ~(%y)(AyvBy)     Assumption 3 ....what if (%x)Ax   Assumption 4 ....then... ....what if At 4 vI 5 ....then... ....then... At v Bt 5 %I 6 ....then... ....then... (%y)(Ay v By) 3,4-6 %E 7 ....then... (%y)(Ay v By)   3-7 >I 8 (%x)Ax>(%y)(AyvBy)     Assumption 9 ....what if (%x)Bx   Assumption 10 ....then... ....what if Bt 10 vI 11 ....then... ....then... At v Bt 11 %I 12 ....then... ....then... (%y)(Ay v By) 9,10-12 %E 13 ....then... (%y)(Ay v By)   9-13 >I 14 (%x)Bx>(%y)(AyvBy)     1,8,14 vE 15 (%y)(AyvBy)
4.	Premise 1 (^x)(%y)(Txy>Bx)     Premise 2 (%y)(^x)(Tyx&~By)     Assumption 3 ....what if (^x)(Ttx&~Bt)   1 ^E 4 ....then... (%y)(Tty>Bt)   Assumption 5 ....then... ....what if Ttu>Bt 3 ^E 6 ....then... ....then... Ttu&~Bt 6 &E 7 ....then... ....then... Ttu 5,7 >E 8 ....then... ....then... Bt 6 &E 9 ....then... ....then... ~Bt 8,9 &I 10 ....then... ....then... Bt&~Bt 4,5-10 %E 11 ....then... Bt&~Bt   Assumption 12 ....then... ....what if ~(X&~X) 11 &E 13 ....then... ....then... Bt 11 &E 14 ....then... ....then... ~Bt 12-14 ~E 15 ....then... X&~X   2,3-15 %E 16 X&~X     16 &E 17 X     16 &E 18 ~X

8.3ex VI

1.	Premise 1 ~(^x)Ax     Assumption 2 ....what if ~(%x)~Ax   Assumption 3 ....then... ....what if ~Aa 3 %I 4 ....then... ....then... (%x)~Ax 2 R 5 ....then... ....then... ~(%x)~Ax 3-5 ~E 6 ....then... Aa   6 ^I 7 ....then... (^x)Ax   1 R 8 ....then... ~(^x)Ax   2-8 ~E 9 (%x)~Ax
2.	See 8.3ex II 1.
3.	Premise 1 ~(%x)(Ax&Bx)     Assumption 2 ....what if Aa   Assumption 3 ....then... ....what if Ba 2,3 &I 4 ....then... ....then... Aa&Ba 4 %I 5 ....then... ....then... (%x)(Ax&Bx) 1 R 6 ....then... ....then... ~(%x)(Ax&Bx) 3-6 ~I 7 ....then... ~Ba   2-7 >I 8 Aa>~Ba     8 ^I 9 (^x)(Ax>~Bx)
4.	Here's problem 4 completed using PD+ and the methods of 8.4. As you can see, this is far easier. After reading 8.4, you'll see that this is far clearer too! Premise 1 (^x)(Ax>~Bx) 1 IM 2 (^x)(~Ax v ~Bx) 2 DM 3 (^x)~(Ax&Bx) 3 QN 4 ~(%x)(Ax&Bx)