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 ?