$$∀x, p(x) ∨ ∀x, q(x)$$ let's say that p(x) means "is blue" and q(x) "is red".
if the "or" is an inclusive or, that means that, we can say "everything is blue or everything is red or everything is blue and red"
what does that "blue and red" mean? that every single object has 2 colors blue and red, or between all the objects I have, there are some red objects and blue objects?
this is getting me crazy I've been on this "everything this or that" and "everything this or everything that" all day.
if it's the latter, that is "between all the objects I have, there are some red objects and blue object" how is it different from
$$∀x, p(x) ∨ q(x)$$
which means "everything is blue or red or both blue and red"?
thank you!
Yes, in mathematics all "or"s are inclusive unless stated otherwise.
As to how they're different, consider a box with one red-but-not-blue ball and one blue-but-not-red ball (or perhaps more intuitively, "Every natural number is odd or even" vs. "Either every natural number is odd or every natural number is even" - noting that no natural number is both odd and even).
Note that the inclusive/exclusive issue plays no role here: the point is that there is a "situation" in which one of the sentences is true but the other is false, so they're not equivalent. On the other hand we do have $$[\forall x(P(x))\vee\forall x(Q(x))]\implies \forall x(P(x)\vee Q(x)),$$ it's just that the converse fails.