This paper consists of two parts. Part I contains a precise model-theoretic reconstruction of Quine's criterion for the ontological presuppositions of a theory. Two versions (K1), (K2) of the criterion are elaborated, (K2) being the more adequate one which is shown through a number of theorems for each version. Part II contains a critical discussion of (K2), in particular of the question wether (K2) is a criterion for ontological presuppositions, i.e. for entities existing independently of the theory. Its answer depends (...) on the meaning of the quantifiers used in (K2). It is shown that this meaning, contrary to Quine's opinion, does not commit one to the existence of entities. Quine's criterion is therefore not a criterion for ontological presuppositions of theories. If theories at all presuppose independently existing entities, then it is not by using the standard quantifiers of classical logic. One of the consequences of this result is that there is no need for a free logic, which is shown by discussing a system of Lambert and Meyer. Another consequence is that there is no problem about negative existentials. (shrink)