Q7.1b

Problem Two.
 Premise 1 Db&Lac 1 &E 2 Db 3 2 %I 4 (%x)Dx
Problem Three.
 Premise 1 (^x)(Bx&Lx) 1 ^E 2 Ba&La 2 &E 3 La 3 %I 4 (%x)Lx

Q7.1d

Problem 2.
 Premise 1 Tjv(%x)~Kx 1 CM 2 (%x)~Kx v Tj 3 4 2 QN 5 ~(^x)Kx v Tj
Problem 3.
 Premise 1 Lst>(^x)~Jx Premise 2 Lst 1,2 MP 3 (^x)~Jx 4 3 QN 5 ~(%x)Jx
Problem 4.
 Premise 1 ~((%x)Txv(^x)Kx) 1 DM 2 ~(%x)Tx&~(^x)Kx 2 &E 3 ~(%x)Tx 4 3 QN 5 (^x)~Tx

7.1a

Problem One.
 Premise 1 (^x)Ax 1 ^E 2 Aa 1 ^E 3 Ab 2,3 &I 4 Aa&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.
 Premise 1 (^x)(Tx>Ux) Premise 2 Ta 1 ^E 3 Ta>Ua 2,3 MP 4 Ua 2,4 &I 5 Ua&Ta
7.1b

Problem One.
 Premise 1 ~(^x)Gx Premise 2 (%x)~Gx>Pj 1 QN 3 (%x)~Gx 2,3 MP 4 Pj
Problem Two.
 Premise 1 ~(%x)Ax 1 QN 2 (^x)~Ax 2 ^E 3 ~Aj
Problem Three.
 Premise 1 ~(^x)Lx&~(%x)Lx 1 QN 2 (%x)~Lx&~(%x)Lx 2 QN 3 (%x)~Lx&(^x)~Lx
Problem Four.
 Premise 1 ~(^x)~Jx 1 QN 2 (%x)~~Jx 2 DN 3 (%x)Jx

7.1f With and without RD

7.2a

Problem One.
 Premise 1 ~(%x)(Tx&~Cx) 1 QN 2 (^x)~(Tx&~Cx) 2 DM 3 (^x)(~Tx v ~~Cx) 3 DN 4 (^x)(~Tx v Cx) 5 6 7 8 4 IM 9 (^x)(Tx>Cx)

7.2b

Problem Two.
 Premise 1 (^x)(Rx>~Mx) 1 IM 2 (^x)(~Rx v ~Mx) 2 DM 3 (^x)~(Rx&Mx) 3 CM 4 (^x)~(Mx&Rx) 5 6 7 8 4 QN 9 ~(%x)(Mx&Rx)

7.2c

Problem One.
 Premise 1 (^x)(Ax>Bx) 1 IM 2 (^x)(~Ax v Bx) 2 DN 3 (^x)(~Ax v ~~Bx) 3 DM 4 (^x)~(Ax&~Bx) 4 CM 5 (^x)~(~Bx&Ax) 6 7 8 5 QN 9 ~(%x)(~Bx&Ax)

7.3a

Problem One.
 Premise 1 (^x)(Ax&Bx) 1 ^E 2 Aa&Ba 2 &E 3 Ba 3 ^I 4 (^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.
 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 (^x)(Bx>Cx)
Problem Two.
 Premise 1 (^x)Bx Premise 2 (^x)Cx 1 ^E 3 Ba 2 ^E 4 Ca 3,4 &I 5 Ba&Ca 5 DN 6 ~~Ba&Ca 6 DN 7 ~~Ba&~~Ca 7 DM 8 ~(~Ba v ~Ca) 8 ^I 9 (^x)~(~Bx v ~Cx) 9 QN 10 ~(%x)(~Bx v ~Cx)
Problem Three.
 Premise 1 ~(%x)(Px&Mx) 1 QN 2 (^x)~(Px&Mx) 2 DM 3 (^x)(~Px v ~Mx) 3 CM 4 (^x)(~Mx v ~Px) 4 IM 5 (^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.