1. No cat is a dog (Cx; Dx)
(x) (Cx-> -Dx)  or  -(Ex)(Cx & Dx)
2. Some dogs are not nice (Dx; Nx)
(Ex) (Dx & -Nx)
3. Dogs are not cats (Dx; Cx)
(x) (Dx -> -Cx)
4. Dogs and cats are mammals (Dx; Cx; Mx)
(x) [(Dx v Cx) -> Mx]
5. Only members can enter (Mx; Cx)
(x) (Cx -> Mx)  This says that being a member is a necessary condition for entering
6. All and only members can enter (Mx; Cx)
(x) (Cx <-> Mx)  This says that being a member is a necessary and sufficient condition for entering
7. Peppers are good but not cheap (Px; Gx; Cx)
(x) [Px -> (Gx & -Cx)]
8. Peppers are neither good nor cheap (Px; Gx; Cx)
(x) [Px -> (-Gx & -Cx)]
9. Some peppers are good but not cheap (Px; Gx; Cx)
(Ex) (Px & Gx & -Cx)
10. Nobody but members can enter (Mx; Cx)
-(Ex) (-Mx & Cx)  or  (x) (Cx -> Mx).  This says that being a member is a necessary condition for entering
11. Apples aren't both cheap and good (Ax; Cx; Gx)
(x) [Ax -> -(Cx & Gx)]
note that (x) [Ax -> (-Cx & -Gx)] is wrong.  WHY?
12. Only members who pay may attend (Mx; Px; Ax)
(x) [Ax -> (Mx & Px)]  Being a paying member is a necessary condition for attending.
13. If only members attend,  Joe will not go (Mx; Ax; Joe=j; Gx)
(x) [(Ax -> Mx) -> -Gj]
14. Joe is a member but Mary isn't (Joe=j; Mx; Mary=m)
Mj & -Mm
15. All is lost (Lx)
(x) Lx.