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Exercise 8.4, I
(25) 1. (x)Ax (x)(Bx Cx)
2. (x)Dx (x)(Ex Bx)
14. (Cx • Ex) 13, UI
Part II
(1) 1. (x)(Ax Wx)
2. (x)(Rx Cx) / (x)[(Rx • Ax) (Cx • Wx)]
Exercise 8.4, II
(2) 1. (x)(Sx Wx)
(3) 1. (x)(Jx Wx) (x)(Ax • Rx)
2. (x)(Jx • Wx) (x)(Ax • Rx) / (x)(Ax • Rx)
8. Jm • Wm 7, DM
(4) 1. (x)[(Sx Ux) (Ix • Cx)]
Exercise 8.4, II
2. (x)[(Cx Vx) (Rx • Ax)] / (x)(Sx Ax)
(5) 1. (x)(Ax Dx)
2. (x)[(Ex • Ax) Cx] • (x)[(Cx • Dx) Fx] / (x)[(Ex • Ax) Fx]
(6) 1. (x)Sx (x)(Ex • Wx)
3. (x)Vx (x)Sx / (x)Lx
(7) 1. (x)Cx (x)[Ax (Sx • Dx)]
(8) 1. (x)Vx (x)(Px Ax)
Exercise 8.4, II
(9) 1. (x)(Sx Px) (x)(Rx Px)
2. (x)(Sx • Px) (x)(Wx Px) / (x)[(Rx • Px) • Wx]
9. (Rm Pm) 8, Impl
(10) 1. (x)(Gx • Px) (x)(Ax • Px)
13. (x)Px 12, QN
Exercise 8.4, II
Exercise 8.5
Part I
1. All cats are animals.
2. Some animals are fish.
Exercise 8.5, I
Part II
(1) 1. (x)(Ax Bx)
(2) 1. (x)(Ax Bx)
(3) 1. (x)Ax (x)Bx
(4) 1. (x)(Ax Bx)
(5) 1. (x)[Ax (Bx Cx)]
Exercise 8.5, II
(6) 1. (x)Ax
(7) 1. (x)(Ax Bx)
(8) 1. (x)(Ax • Bx) (x)Cx
(9) 1. (x)(Ax • Bx)
(10) 1. (x)(Ax • Bx)
Exercise 8.5, III
Part III
(1) 1. (x)[Vx • Px) (Ax • Mx)
(2) 1. (x)[(Px Hx) Mx]
(3) 1. (x)Ox (x)Bx
(4) 1. (x)(Tx Hx)
(5) 1. (x)[(Cx Vx) Mx]
Exercise 8.5, III
Exercise 8.6
Part I
1. Rcp
7. (x)[Px (y)Sxy]
Exercise 8.6, I
26. (x){Lx • (y)[(Py • Ryy) Rxy]}
Part II
(1) 1. (x)[Ax (y)Bxy]
(2) 1. (x)[Ax (y)(By Cxy)]
(3) 1. (x)[Ax • (y)(By Cxy)]
(4) 1. (x)(y)(Ax By) / (x)Ax (y)By
(5) 1. (x)Ax (y)By / (y)(x)(Ax By)
(6) 1. (x)(y)(Ax By)
2. (x)(y)(Ay Cx) / (x)(y)[Ax (By • Cy)]
(7) 1. (x)[Ax • (y)(Ay Bxy)] / (x)Bxx
Exercise 8.6, II
(8) 1. (x)[Ax • (y)(By Cxy)]
(9) 1. (x)(y)(Axy Bxy)
(10) 1. (x)(y)Axy (x)(y)Bxy
(11) 1. (x){Ax • [(y)By Cx]}
(12) 1. (x)(y)[(Ay • By) Cxy]
(13) 1. (x){Ax • (y)[(By Cy) Dxy]}
(14) 1. (x){Ax [(y)(By • Cy) Dx]}
(15) 1. (x)(y)(Ayx Axy) / (x)Axx
Exercise 8.6, II
(16) 1. (x)(Ax Bx)
(17) 1. (x)(y)(Axy Byx)
(18) 1. (x)[Ax (y)(By • Cxy)]
Exercise 8.6, II
(19) 1. (x)(y)Ayx (x)(y)Bxy
(20) 1. (x)(y)[Axy (Bx • Cy)]
Part III
(1) 1. (x)[Px (y)(Ay Oxy)]
(2) 1. (x)[(Fxm Fxp) Rx]
(3) 1. (x)(Hx Ax) / (x)[(y)(Hy • Oxy) (y)(Ay • Oxy)]
(4) 1. Po
2. (x)[(Px • Cx) Sox]
(5) 1. (x){(Hx • Px) [(y)(By Cy) Rx]}
6. (y)(By Cy) Rm 4, 5, MP
(6) 1. (x)[(Px • Cxx) Crx]
(7) 1. (x){Px • (y)[(Py • Kxy) Fxy]}
(8) 1. (x)[Px (y)(Ry Axy)]
