1) If any marbles are round, they all are
My Answer: (Ǝx)(Mx and Lx) ⊃ (∀x)(Mx and Lx)
Is this correct?
No, not quite.
You are claiming that "If something a marble and round, then everything will be a marble and round."
You want to claim "If something is a marble and round, then all marbles will be round."
2) If a checker is jumped or crowned, then if all marbles are round it's red
My answer: (Ǝx)(Wx and Yx) v (Wx and Bx) ⊃ (∀x)(Mx and Lx) ⊃ (Ǝx)(Wx and Sx)
Is this correct?
No, you want "it is red" to refer back to any "checker that is jumped or crowned," so the predicates need to be in the scope of the same quantified term.
The statement may be read as, "For any checker of this type, if all marbles are such-and-such, then the checker will be of that type."
Much like the next question, this will take the form of:
(∀x) (____ ⊃ ((∀y) (____) ⊃ ____))
(Also, remember, you want to say "*all marbles are round" not "all things are marble and round".)
3) If a crowned checker is on the board but not jumped, then if some round things are marbles it's red if it's been moved.
(∀x)(Wx and Bx and Ix and -Yx) ⊃ (Ǝx)(Lx and Mx) ⊃ (Cx ⊃ Sx)
Is this correct?
Almost. A few brackets need to be added to make the scope of the quantifiers legal. Also, while it is technically correct to use the same terms, it is clearer to use distinct terms when nesting quantifiers.
$(∀x)\Big((Wx \land Bx \land Ix \land \lnot Yx) ⊃ \big((Ǝy)(Ly \land My) ⊃ (Cx ⊃ Sx)\big)\Big)$
