 Now define:   \[ \alpha[x]\beta := \forall v \in \textsf{VAR}\setminus\{x\}\ (\alpha(v) = \beta(v)). \]   So [\(x\)] is the binary relation “assignment \(\beta\) is a result of (at most) resetting the value of the variable \(x\) in assignment \(\alpha\)”. As already mentioned, this is an equivalence relation over variable assignments. Now the meaning \(G\) of \(\exists x P(x)\), will be:  \[ G := \{\alpha \in \textsf{ASSIGN } \mid \exists \beta \in F \alpha[x]\beta \}. \]   Thus, \(G\) is the set of assignments that can be successfully reset with respect to \(x\) and obtain an assignment in \(F\) as a result of this resetting. Viewed differently, \(G\) is the domain of the relation \(R\) given by  \[\alpha R\beta := \alpha[x]\beta \textrm{ and } \beta \in F. \]   