Translating statement into General Predicate Logic

Question

I have a question from Kleene's Mathematical Logic that I am struggling to answer.

'Every liberal advocates changes. Some conservatives favour no one who advocates changes. Therefore, some conservatives favour no liberal.' (Kleene, 1967, 147).

My solution is as follows. Every $x$ is a liberal, and every $y$ is a conservative. Every x advocates change $(Cy)$ and some y favour no-one who advocates change $\neg (Cx)$.

$\forall x (Cx) , \exists y(\neg Cy) \therefore \exists y(\neg Cx)$

I think this is right?

EDIT: Okay, so I know that the first premise is $\forall x(Lx \to Ax)$, or, for all $x$, such that $x$ is a (L)iberal, $x$ (A)dvocates change.

Answer 1

Hint

You have to translate every statement individually.

1st premise) Every liberal advocates changes.

"for every $x$, if $x$ is Liberal, then $x$ Advocate changes".

2nd premise) Some conservatives favour no one who advocates changes.

"there is some $y$ that is a Conservative and for every $z$, if $z$ Advocate changes, then $y$ does not Favour $z$".

Conclusion) Therefore, some conservatives favour no liberal.

"there is some $w$ that is Conservative and for every $u$, if $u$ is a Liberal, then $w$ does not Favour $u$".

Answer 2

You can't just stipulate that $x$'s will be used for liberals, and $y$'s for conservatives. All variables are assumed to be objects of one and the same domain.

So, what you need to do is use predicates $L(x)$ for '$x$ is a liberal' and $C(x)$ for '$x$ is a conservative. Of course, you just proposed to use $C(x)$ for '$x$ advocates change', but for that you could use $A(x)$ instead.

Also, for the favouring, you want to use a $2$-place predicate $F(x,y)$ that means '$x$ favours $y$'