1. (x)[Dx (y)(Py By)]  or (x)Dx (y)(Py By)
    • These variations seem acceptable but are not: (x)[Dx (y)(Py By)], and (x)(y)[Dx (Py By)]. The reason is that they are existentially quantified conditionals.  Existentially quantified conditionals are trivially true, they are true if there is but one thing that does not  satisfy the antecedent.
  2. (x)(Dx Cx)
  3. ~(x)Dx (y)(Py ~By) or (x)(y)[~Dx (Py ~By)]
  4. (x)[[Dx (y)(Py ~By)] ~Cx]  or (x)(y)[[Dx (Py ~By)] ~Cx]
    • Here the "if something" initially pushes us to use an existentially quantified conditional; but since such expressions don't mean what we want them to, we retreat to a universal quantifier. Hence the paraphrase underlying our translation is: All things that are damaged when nobody is blamed will not be charged to the tenant. Or: Nothing which is damaged when nobody is blamed will be charged to the tenant.
  5. (x)[(Bx Yx) Rx]
  6. (x)(Bx Yx) (y)(By Ry) or (x)(y)[(Bx Yx) (By Ry)]
    • Note this is not an existentially quantified conditional. It is an conditional with an existentially quantified antecedent and consequent.
  7. (x)[(Bx Yx) [(y)[(By Yy) Ry] Rx]]
    • See comments to exercises 4, above. The x quantifier cannot be existential, for then we'd have an existentially quantified conditional.
  8.  (x)[(Bx Rx) Yx] (y)(Yy Ry) or (x)(y){[(Bx Rx) Yx] (Yy Ry)}
    • See comments to exercises 4, above. The x quantifier cannot be existential, for then we'd have an existentially quantified conditional.
  9. (x)[(Ox Px) (Cx v Mx)] [(y)(Cy Py) v (z)(Mz Pz)]
  10. (x)[(Ox Px) [(y)(My ~Py) v Mx]]
    • See comments to exercises 4, above. The x quantifier cannot be existential, for then we'd have an existentially quantified conditional.
  11. (∃x)(Ox Px) [(y)((Oy Py) Cy) (z)(Cz Pz)]
  12. (x)[(Ox Px) [(y)[(Oy Py) Cy] Cx]]
  13. (x)[(Sx Fx) (Sx Wx)] [(y)(Sy) (z)(Wz Fz)]
  14. (x)[(Sx Wx) [(y)(Wy Fy) Fx]]
  15. (x)[[(Sx) (y)(Sy Wy)] Wx]
  16. [(x)(Px Fx) (y)(Ey ~Ly)] (z)(Ez Sz)
  17. (x)[(Ex Lx) [(y)(Py ~Fy) ~Sx]]
  18. [(x)(Ex Lx) (y)(Py ~Fy)] (z)(Ez ~Sz)
  19. (x)(Hx ~Sx) [(y)(Wy Ay) (z)(Wz Dz)]
  20. (x)[(Hx ~Sx) [(y)(Wy Ay) Ux]]

Prenexed and Purified forms of these answers