What does Quine’s phrase “To be is to be a value of a bound variable” mean?

W.V.O. Quine’s famous criterion, “To be is to be a value of a bound variable,” serves as a method for determining the ontological commitment of a theory—that is, identifying what entities a theory asserts to exist.12

In this context, Quine argued that to determine what a theory is committed to, one must first express its propositions using predicate logic. Within this formal framework, a theory is committed to the existence of those entities that must serve as the values of its bound variables (the x in universal or existential quantifiers like $(\forall x)$ or $(\exists x)$) for the theory's sentences to be true.12

Essentially, if a theory requires that its variables range over a specific domain of objects to maintain its truth, the theory is ontologically committed to the existence of those objects. For example, if a theory uses bound variables to quantify over sets, it is committed to the existence of those sets. Quine used this criterion to advocate for a "sparse" ontology, often limiting his own commitments to physical objects and mathematical sets.12