First-order number theory : Usage of 'and' symbol?

Question

Whats the difference between the statements

∀x ∀y F

&

∀x ∧ ∀y F

Also, are these two statements equivalent?

∀x ∧ x>0 F

∀x x>0 F

Thanks.

Answer 1

We write $\forall x \varphi(x)$ where $\varphi$ is a proposition. $\forall x$ is one quantifier, and it is not a statement in and of itself. Therefore, $\forall x \land \forall y F$ makes no sense.

Answer 2

Intuitively, both pairs of statements are equivalent. If you look at your rules for constructing sentences, I suspect that the official way is $\forall x (\forall y F)$ or something like that to indicate that the range of the quantifiers is only $F$. This would come in if you wrote $\forall x \forall y F(x,y)\wedge G(x,y)$ The $x$ and $y$ in $G(x,y)$ might be bound or they might be free. The versions that I have seen without parentheses would only apply the quantifiers to $F(x,y)$ so you should read this as $\left(\forall x \forall y F(x,y)\right)\wedge G(x,y)$ and this is an expression with $x$ and $y$ free in $G(x,y)$ but bound in $F(x,y)$ (which I think is terrible practice because it makes your reader sort out what is going on. Much better to write $\left(\forall w \forall z F(w,z)\right)\wedge G(x,y)$. This is logically equivalent, but the range of the quantifiers if obvious.) For your second, what you really mean is $\forall x (x \gt 0 \implies F)$. I have seen this abbreviated to $\forall x\gt 0\ F$