How to write the following sentence using propositional logic:
Only monkeys properly appreciate the value of art.
Is this good:
Let $M(x)$ be '$x$ is a monkey' and $A(x)$ be '$x$ properly appreciates the value of art', where the domain of $x$ is the set of all animals.
In this case, $\forall x\ A(x) \rightarrow M(x)$ or $\forall x\ A(x) \iff M(x)$?
You seem right, but you may want to formulate it as follows to reflect the nature of the question: If $x$ properly appreciates art, then $x$ is a monkey AND if $x$ is not a monkey, then $x$ does not appreciate art, but this latter statement is just the contrapositive of the previous statement which takes the form $(\forall x)(A(x)\to M(x))$. Thus, I do not see any issue with your formulation.