Is it impossible for a quantifier-free formula to contain free variables?

Question

A first-order language must specify its signature to fix its alphabet of non-logical symbols. A signature $\Sigma$ contains the set $\Sigma^F$ of function symbols and the set $\Sigma^P$ of predicate symbols. Function symbols of arity 0 are called constants and predicate symbols of arity 0 are called propositional variables.

However, there exists another concept 'variable'. Some people say that variables are the technical tool for schematization. I don't understand where is the difference between constant and variable and discussions of variables are often involved with quantifiers. Can we say it is impossible for a quantifier-free formula to contain free variables?

Thanks for answer!

Answer 1

First-order formulae con contain free variables, like :

$x < 1$

or not.

Formulae without free variables are called sentences (or closed formulae) and they can have no occurrences of variables at all, like :

$0 < 1$

or all occurrences of variables are bounded by quantifiers, like :

$\exists x (x < 1)$.

The first two formuale above are both quantifier-free; the first one : $x < 1$, has a free occurrence of the variable $x$, while the second one : $0 < 1$, has no occurrences of variables at all.

Thus, it is possible for a quantifier-free formula to contain free variables.