Logical Implication And Conjuction

Question

Given the sentence :

Whoever isn't fond off capitalism , isn't liberal.

I would like to convert them to First Order Logic . I tried 2 ways , not sure if they are true:

a) $\forall x ( \lnot FondOff(X,Capitalism) \rightarrow \lnot Is(X,Liberal))$

b) $\lnot \exists x ( FondOff(X,Capitalism) \land Is(X,Liberal))$

Answer 1

Answer a) works, though I would write it as:

$\forall x (\neg FondOfCapitalism(x) \rightarrow \neg Liberal(x))$

Using the equivalence $P \rightarrow Q \Leftrightarrow \neg P \lor Q$ we can rewrite this as:

$\forall x (\neg \neg FondOfCapitalism(x) \lor \neg Liberal(x)) \Leftrightarrow$ (DeMorgan)

$\forall x (\neg (\neg FondOfCapitalism(x) \land Liberal(x))) \Leftrightarrow$ (Quantifier Negation)

$\neg \exists x (\neg FondOfCapitalism(x) \land Liberal(x))$

And now you see that you were missing a $\neg$ in b)

Answer 2

First note that $\forall x(P)$ is equivalent to $\lnot \exists x(\lnot P)$ and $a \rightarrow b$ is equivalent to $\lnot a \lor b$. Now your first statement is correct and if you transform it into there exist statement then it will be $\lnot \exists x ( \lnot FondOff(X,Capitalism) \land Is(X,Liberal))$.