Ontologies of paradigms θi (Quine would say theories, but I'm taking a broader view) are their domains of discourse, also referred to as universes of discourse Λn. They have admissible and inadmissible objects with respect to their epistemic scope. Paradigms can be compartmentalized, in which case we would be justified in speaking of universes (in the plural) of discourse Λn with respect to a singular paradigm.
∀x [x ∉ Λ1] or its logical equivalent ~(∃x) [x ∈ Λ1] is an assertion that x = ∅ with respect to Λi (it is suggestive that x would thus be a subset of Λ1, for all x). It is also suggestive that if there is even one Λi (say, Λ1) in which x = ∅, then
∀x [Λi ⊂ θ1 .⇒. x ∉ ∩i(i = 1, 2, …)Λi]
regardless of the fact that even if the statement that x ∈ Λi in which i ≠ 1 holds true for all Λi, there will always be reason to doubt the veracity of the assertion, if θ1 were the paradigm with universes of discourse Λi (i = 1, 2, ... ), that (∃x) [x ∈ paradigm θ1].
Note that this is not the same as making a general statement that "there is no x," i.e., that x does not exist. x is consigned to the null set relative to or with respect to lambda, so x does not exist in lambda, the paradigmatic universe of discourse. Nothing else is being stated.
This is merely simple fact and indisputable, as it is almost tautologically obvious. What a marvelous fact it is, friends; reflect for a moment. It means that if you want to argue a point ethically, scientifically, and correctly, you must remain true to your paradigm's inherent delimitations and horizons of epistemological demarcation. And as such, the correct attitude, perhaps to employ an E-Prime sort of idea and the logic of belief, would be to preface or append: 'according to my paradigm' to all ontologically based assertions.
This is the problem with fundamentalist thinking with respect to paradigms. A person A who embraces paradigm θ1 and person B who embraces paradigm θ2 of which ∃x [x ∉ θ1∩ θ2] cannot absolutely assert the existence or non-existence of x in terms of universal quantification; x exists or does not exist only relative to a Λn ⊂ [θ1 ∨ θ2]. In short, what is bullshit in one paradigm could be truth in another, and the twain can only meet in metalinguistic space that transcends both paradigms.