Predicate logic universal quantifier not generally valid

Question

Prove that $(\forall x P(x) \to \forall x R(x)) \to \forall x (P(x) \to R(x))$ is not generally valid.

Therefore I want to find formulas for which there exists and assignment I s.t.
$I(\forall x P(x) \to \forall x R(x))=1$, but $I(\forall x (P(x) \to R(x)))=0$

The only thing I can of is $P(x)= \lnot x$ and $R(x) = x$, with the domain $D=\{a\}$, but I'm not sure if I can evaluate P(x) and R(x), before I use the values of my domain?

Answer 1

Hint

If $a$ does not have property $P$, and is the only object of the domain, then $\forall x P(x)$ is false, so $\forall x P(x) \rightarrow \forall x R(x)$ is true (as desired), but $\forall x (P(x) \rightarrow R(x))$ will also be true. Indeed, with a domain containing 1 object, the two statements will say the exact same thing. So, you will need to consider a domain with at least two objects!

Answer 2

If all human beings are left-handed then all human beings are Chinese. That's true, because both the hypothesis and the conclusion are false. But it's not the case that all left-handed human beings are Chinese.