bound variable

A variable x is bound in a formula if it is within the scope of a quantifier (in first-order logic, (∀x) or (∃x)). Intuitively this means that as the formula is evaluated and x in this occurrence is assigned to an object, the quantified expression in which it occurs is evaluated with respect to that object. If a variable is not bound it is free. In (∀x)(Fx → Gx) all the variables are bound. In (∀x)(Fx → Gx) & Gx the final occurrence of the variable x is free, so the expression is an open sentence or predicate. To turn it into a closed sentence one must either replace the variable with a constant or closed term referring to a thing, or extend the scope of the initial quantifier, or introduce another quantifier: (∀x)(Fx → Gx) & (∃x)(Gx), for example.