I am teaching someone about statements, the very basis of mathematical thinking, but now I am a bit confused myself. For
$ \forall x \in \mathbb{R}, x > 2 \Rightarrow x > 3 $
to be a statement, does it need the $ \in \mathbb{R} $ part? And the quantifier?
So is $ x > 2 \Rightarrow x > 3 $ a statement or not?
On
Better to specify $\forall x \in \mathbb R$, because it could be possibly that your statement is about $\forall x \in \mathbb Q$, or $\forall x \in \mathbb Z$, etc.
On
If you don't specify it, there is a possibility of ambiguity. For example, consider the example $\forall x, x>3 \Rightarrow x>2$. If $x$ is real, it's obviously true. If however we take it over the lattice of divisibility, $x=9$ contradicts the statement, as $9$ is an odd number.
On
the very basis of mathematical thinking
is about communication as much as it is about formal statements.
Whether or not there's ambiguity in your example depends on the context. If it's part of a discussion about the real numbers you don't need the extra specificity. If it's a standalone example about quantifiers, you do.
If it's understood that the variable $x$ refers to a real number, then it can be omitted; otherwise, it cannot.
This practice is known as bounded quantification. In general, when you write $\forall x$, the variable $x$ is taken to range over the entire universe of discourse, whatever that may be. If the universe of discourse is not specified, then it is typically understood by context (e.g. the von Neumann universe in set theory).
If $p(x)$ is some statement with a free variable $x$, then the expression $\forall x \in X,\, p(x)$ is shorthand for $\forall x,\, (x \in X \Rightarrow p(x))$. It then doesn't matter what the universe of discourse is, because in order for the hypothesis $x \in X$ to be specified, you've instantly restricted yourself to elements of $X$.
Thus the statement $\forall x \in \mathbb{R},\, x > 2 \Rightarrow x > 3$ is shorthand for $$\forall x,\, (x \in \mathbb{R} \Rightarrow (x > 2 \Rightarrow x > 3))$$ If it were understood from context that the variable $x$ refers to a real number, then you could omit the "$\in \mathbb{R}$" part so that the statement becomes $\forall x,\, (x > 2 \Rightarrow x > 3)$; in fact, in this case, you could shorten this even further to become just $\forall x > 2,\, x > 3$.