My question pertains to the semantics of classical predicate logic. If I'm working with a model M = $\langle D, v \rangle$ in which D is a non-empty set consisting of all existing obejects and $v$ is an assignment-function. The assignment-function is then determined by the condition:
(i) $v$: $\mathcal{C} \cup \mathcal{V} \rightarrow D$
$\mathcal{C}$ represents the set of all individual constants and $\mathcal{V}$ the set of all individual variables. As I understand this condition, the assignment function maps every constant and every variable onto an object in the domain. I can see, for example, that an individual constant 'a' gets mapped onto the individual 'John' in our domain. However, what does it mean to say that an individual variable 'x' gets mapped onto the domain? Isn't a variable undefined? I don't understand what exactly is assigned to it in the domain. Does an assignment function randomly map the 'x' onto an object in the domain?