Symbolic Logic II
Proofs Involving Quantifiers
Solutions
Use the Main Method to show the validity of each of the following arguments.
1.
xy(Fxy  Fyx)
1.
xy(Fxy  Fyx)
2.
-xy(Fxy  Fyx)
3.
xy-(FxyFyx)
4.
y-(FxyFyx)
5.
-(FxyFyx)
6.
y(FyxFxy)
7.
FxyFyx
xy(Fxy  Fyx)
negation of conclusion
2, prenex, Q.N.
3 EI x/x
4 EI y/y
1 UI x/x
6 UI y/y 5 and 7 are inconsistent
2.
x(Fx.y(Gy  Hy)); x(Fx  (-Lx  -z(Kz.Hz))); x(Kx.Gx)  xLx
1.
x(Fx.y(Gy  Hy))
2.
x(Fx  (-Lx  -z(Kz.Hz)))
3.
-(x(Kx.Gx)  yLy)
negation of conclusion, reletter
4.
xy(Kx.Gx.-Ly)
3, prenex, QN
5.
xy(Fx.GyHy)
1, prenex
6.
xz(Fx  . -Lx  -(Kz.Hz)) 2, prenex, QN
7.
y(Kx.Gx.-Ly)
4 EI x/x
8.
y(Fw.GyHy)
5 EI w/x
9.
Kx,Gx,-Lw
7 UI w/y
10.
Fw.GxHx
8 UI x/y
11.
z(Fw . –Lw  -(Kz.Hz))6 UI w/x
12.
Fw . –Lw  -(Kx.Hx)
11 UI x/z 8,9 and 12 are inconsistent
3.
xyz(Fxy.Fyz.  Fxz); x-Fxx
xy(Fxy -Fyx)
1.
xyz(Fxy.Fyz.  Fxz)
2.
x-Fxx
3.
-xy(Fxy -Fyx) negation of conclusion
4.
xy(Fxy.Fyx)
3 prenex, QN
5.
y(Fxy.Fyx)
4 EI x/x
6.
Fxy.Fyx
5 EI y/y
7.
-Fxx
2 UI x/x
8.
yz(Fxy.Fyz.Fxz) 1 UI x/x
9.
z(Fxy.Fyz.Fxz) 8 UI y/y
10.
Fxy.Fyx.Fxx
9 UI x/z 6,7, and 10 are inconsistent
4.
x(Fx  y(Gy  Hxy)); x(Gx  y(Hxy  Cy)) x(Fx.Gx)  y(Gy  Cy)
1.
x(Fx  y(Gy  Hxy))
2.
x(Gx  y(Hxy  Cy))
3.
4.
5.
6.
7.
8.
9.
-(x(Fx.Gx)  y(Gy  Cy))
xy(Fx.Gx.Gy.-Cy)
xy(Fx  .Gy  Hxy)
xy(Gx  .Hxy  Cy)
Fx.Gx.Gy.-Cy
Fx  Gy  Hxy
Gx . Hxy  Cy
negation of conclusion
3 prenex, QN
1 prenex
2 prenex
4 EI twice, x/x/, y/y
5 UI twice, x/x, y/y
6 UI twice, x/x, y/y 7-9 are inconsistent
5.
x(Fx  Gx) xFx  xGx
1.
x(Fx  Gx)
2.
-(xFx  xGx)
negation of conclusion
3.
-(xFxyGy.zGzwFw)
elimination biconditional, reletter
4.
ywzx(Fx.-Gy..Gz.-Fw)
3 prenex
5.
wzx(Fx.-Gy..Gz.-Fw)
4 EI y/y
6.
zx(Fx.-Gy..Gz.-Fw)
5 EI w/w
7.
x(Fx.-Gy..Gw.-Fw)
6 UI w/z
8.
(Fy.-Gy..Gw.-Fw)
7 UI y/x
9.
FyGy
1 UI y/x
10.
FwGw
1 UI w/x
8-10 are inconsistent
6.
x(Fx  Gx)
xFx  xGx
1.
x(Fx  Gx)
2.
-(xFx  xGx)
negation of conclusion
3.
xy(Fx.-Gy)
2 prenex
4.
Fx  Gx
1 EI x/x
5.
y(Fx.-Gy)
3 UI x/x
6.
Fx.-Gx
5 UI x/y
4 and 5 are inconsistent
Here is a different route to showing validity:
1. x(Fx  Gx)
2. x(-Fx  Gx)
1 mi
3. x-Fx ExGx 2 purified
4. -xFx  xGx 3 QN
5. xFx  xGx 4 mi
7.
xyz(Fxy.Fyz.  -Fxz) x-Fxx
1.
xyz(Fxy.Fyz.  -Fxz)
2.
-x-Fxx
negation of conclusion
3.
xFxx
2 prenex
4.
Fxx
3 EI x/x
5.
yz(Fxy.Fyz.  -Fxz)
1 UI x/x
6.
z(Fxx.Fxz.  -Fxz)
5 UI x/y
7.
Fxx.Fxx.-Fxx
6 UI x/z
4 and 7 are inconsistent
8.
x(Fx.y(Ty  Gy)); x(Fx  (z(Az.Gz))  Bxx)); y(Ay.Ty) xBxx
1.
x(Fx.y(Ty  Gy))
2.
x(Fx  (z(Az.Gz))  Bxx))
3.
y(Ay.Ty)
4.
-xBxx
negation of conclusion
5.
x-Bxx
4 prenex
6.
xy(Fx. Ty  Gy) 1 prenex
7.
xz(Fx : Az.Gz.  Bxx)
2 prenex
8.
y(Fx,Ty  Gy)
6 EI x/x
9.
Ay.Ty
3 EI y/y
10
Fx.Ty Gy
8 UI y/y
11.
z(Fx : Az.Tz.  Bxx) 7 UI x/x
12.
Fx :Ay.Ty.  Bxx
11 YUI y/z
13.
-Bxx
5 UI x/x
9,10, 12, and 13 are inconsistent