For example, in FOL if we have $\neg \forall Cube(x)$, I know that this is equivalent to $\exists xCube(x)$.
My question is, if I translate the above into English sentence then:
$\neg \forall xCube(x)$ means "It's not the case that all objects x are Cubes"
$\exists xCube(x)$ means "There exists some object x such that x is a Cube"
Although by definition the above two sentences are equivalent. However, if we create a world where there are no Cubes at all, then the sentence "It's not the case that all objects x are Cubes" sounds reasonable, because if it's not that case all objects are cubes, then we probably could assume no objects are cubes, on the other hand, "There exists some object x such that x is a Cube" tells us that there must exist at least one object x that must be a cube.
Also, consider the following(example taken from textbook):
$\forall xCube(x) \lor \lnot \forall xCube(x) $ is a tautology. If we interpret this statement as "Either everything is cube or it's not the case that everything is cube", I can understand why it's a tautology based on the above argument. But, if we translate the sentence to:
$\forall xCube(x) \lor \exists xCube(x) $ Then it means either everything is cube or there exists some cube(s). But if we have a world that don't have any cubes at all, then this statement is false, thus, not a tautology.
I will appreciate it very much if anyone can point out the flaws in my reasoning.
Hint:
The correct negation is:
$\neg \forall x Cube(x)$ is equivalent to $\exists x(\neg Cube(x))$.
So the two sentences:
"It's not the case that all objects x are Cubes"
"There exists some object x such that x is a Cube"
are not equivalent.