Free and bound variables in "if" statements

Question

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 ?

Answer 1

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 $x > 2 \models x > 3$.

Answer 2

Formulas with free variables are said to be TRUE if they hold for all values of the variable (in the domain of the structure at hand). So the unquantified implication IS true in that sense of the word true.

However, in this same technical sense, the formula "If x < 3 then x < 1" is NOT FALSE

since it does hold for some values of x (e.g. x = 5) even though in informal mathematical English, we would all say that it is false as a shorthand for "it is not true." Technically, it is neither true nor false, just like the formula x > 2. But with implications, we do make the informal assumption that the Handbook of Math. Discourse mentions because there is no purpose, outside of formal logic and model theory, for saying that the formula "If x < 3 then x < 1" is neither true nor false.