In mathematics, we might have a statement: 

For all x in S, we have P(x) 

where S is a set, the domain of quantification that may or may not be empty. 

Classical first-order logic is a little weird in that every domain of quantification is assumed to be non-empty. It simply isn't necessary. I guess this assumption makes it possible to introduce new free variables by universal specification when required-- something you cannot usually do in mathematics.