 The two parts “there exists \(x\)” and the test “\(A\)” are glued together by sequential composition: “\(\exists x\); \(A\)”. Focusing on the part “\(\exists x\)”, what would be its natural meaning? An instruction to replace the old value of \(x\) by some arbitrary new value. This is again a relation between input states and output states, but the difference with definite assignments like \(x := y\) is that now the relation is not a function. In fact, this relational meaning of quantifiers shows up in the well known Tarski-style truth definition for first order logic (compare the entry on  Tarski’s truth definitions):