Free and bound variables in “if” statements

The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion $\forall x(x>2)$ is false. In the first case $x$ is free while in the second case $x$ is a bound variable. Now for these two assertions:

If $x>2$ then $x>3$.($x$ is understood to be a real number).

For every $n$ if $n>2$ then $n>3$.

Is the $x$ in the first assertion free while the $n$ in the second assertion bound ? The Handbook of Mathematical Discourse states that in the first assertion $x$ is actually universally quantified like the second assertion. Can someone elaborate on that ?

Also, when proving them, their proofs are exactly the same except in the second one we add "let n be arbitrary"(How to Prove it). So do the two assertions differ in their Logical structure ?

In translating from English to formal language, one has to be careful to translate the meaning of a phrase rather than merely translating the individual words. Usually, one states "If $x > 2$ then $x > 3$" because I mean to assert its truth for all $x$. Thus, when formalizing the statement, I need to include $\forall x$.
An even better translation than $\forall x: x > 2 \implies x > 3$ would be $x > 2 \models x > 3$.
• In the entry "Open Sentence", The Handbook of Mathematical Discourse says "In many circumstances such an assertion is taken as being true for all instantiates of its variables." If I read the sentence, "If $x\gt 2$ then $x \gt 3$, I would assume that it is a (false) claim that it is universally true, so that $x$ is bound. But you have to be careful; there are places where that is not intended. That is why the book says "in many circumstances". But Ned is right: when the statement is an implication an open sentence like that is usually read as a claim that it is true for all $x$. – SixWingedSeraph Feb 25 '16 at 1:18