How to represent the below proposition ?

Question

Gold and silver ornaments are precious .

$G(x): x$ is a gold ornament.
$S(x): x$ is a silver ornament.
$P(x): x$ is precious.

Now why is this statement true? $$\forall x \Biggl ( \Bigl (G(x) \lor S(x) \Bigr ) \to P(x) \Biggr )$$
Why not the following? $$\forall x \Bigl (G(x) \Bigr ) \lor \forall x \Bigl (S(x) \Bigr ) \to P(x)$$
Although universal quantifier is not distributive over disjunction but in this case the ornament will belong to only 1 category, so is there any with this?

Answer 1

Here $\forall x \Biggl ( \Bigl (G(x) \lor S(x) \Bigr ) \to P(x) \Biggr )$: It implies that if an ornament is made of either gold or silver then it is precious.

However if you write like as below,entire meaning will change. $$\forall x \Bigl ( G(x) \Bigr ) \lor \forall x \Bigl ( S(x) \Bigr ) \to P(x)$$

First of all,you have made $x$ of $P(x)$ as free variable, so you should have out quantifier before $P(x)$ also.

Now coming to your confusion,here it implies all ornaments  made of gold or all ornaments made of silver are precious. So here there are  possibilities that some ornaments which are both gold and silver in your universe of discourse which is not true from given context of question.

I hope it is clear.