Difference between $\forall xp(x) \vee \forall xq(x)$ and $\forall x(p(x) \vee q(x))$ in english?

Question

I'm studying logic and I was reading about the distributive property of the universal quantifier, and this got me pretty confused.

Let $p(x)$ mean "eats cake" and $q(x)$ mean "eats ice cream"

Saying everyone eats cake or ice cream would be: $$\forall x( p(x) \vee q(x) )$$

Saying everyone eats cake or everyone eats ice cream would be: $$\forall xp(x) \vee \forall xq(x)$$

but what is the difference between these two sentences? I think I thought about it for too long and I'm making things over-complicated but I don't really get it.

If I go to a restaurant and they tell me "here everyone eats cake or everyone eats ice cream" this means that everyone in that restaurant, at the same time/day, eats ice cream or cake. If instead they told me "here everyone eats cake or ice cream" does it mean that everyone eats cake or ice cream and nothing else but maybe some people are having some cake and some people are eating some ice cream? Is that the difference?

Answer 1

Yes, that's the difference.

To illustrate, say we have a model with three people, Paul, Bob and Alice, such that:

• Paul eats cake, but no ice cream
• Bob eats cake and ice cream
• Alice eats ice cream, but no cake

Form this we can conclude:

• Paul eats cake or ice cream
• Bob eats cake or ice cream
• Alice eats cake or ice cream
• So everyone eats either cake or ice cream.

But:

• Paul doesn't eat ice cream
• So not everyone eats ice cream
• Alice doesn't eat cake
• So not everyone eats cake
• Since neither everyone eats cake nor everyone eats ice cream, it is not the case that everyone eats cake or everyone eats ice cream.

$\forall x (p(x) \lor q(x))$ is true, but $\forall x p(x) \lor \forall x q(x)$ is not, so the two can not be logically equivalent.