$\forall a \in \mathbb{Z} : a^2 \geq 0$

Question

Can we decide whether this statement would be True or False or would that be meaningless?

The mathematical statement translates to For all $a$ in the set of Integers such that the square is greater than or equal to zero.

Answer 1

It is true. A square of an integer is indeed non-negative.

Answer 2

I see two possible interpretations of what you have said: one is true and one is incomplete.

If you wanted to say this: "For every integer, its square is non-negative", i.e. $$(\forall a\in\mathbb{Z})a^2\ge 0$$ then this is true.

If you wanted to say this: "For every integer whose square is non-negative...", i.e. $$(\forall a\in\mathbb{Z}: a^2\ge 0)$$ this is an incomplete sentence. (For every integer whose square is non-negative - what?) You can complete it (e.g. by adding something that could be true about that integer), after which it will make sense, and depending on what you've written it may be true or false. For example:

$$(\forall a\in\mathbb{Z}: a^2\ge 0) a\ge 0$$

("Every integer whose square is non-negative, is itself non-negative") - this is a sentence which happens to be false.

Logically, the sentence $(\forall a\in\mathbb{Z}:a^2\ge 0)P(a)$ is equivalent to $(\forall a\in\mathbb{Z})(a^2\ge 0\implies P(a))$, whatever predicate $P$ you choose.