What is a 'criterion of existence'?--Hm ... hard to say ...--Why not start by making a theory of these criteria first, and then think about how that term might be defined? ... :-))
Natural Ontology
Ontology deals with existence questions. 'Does ... exist?'
The natural way of answering existence questions is to apply Criteria of Existence (ECC).
Natural Ontology, in the sense under consideration here, is the pursuit of existence questions with reference to ECC.
(The question how do answer/decide existence questions is a classical question of metapysics, in the sense of meta-ontology. Natural ontology assumes the basic metaphysical position that ECC are suitable means for answering/deciding existence questions.)
What is a Criterion of Existence?
A criterion of existence is a criterion which tells in favour of or against the existence of entities. Such a criterion may take the form '... exist(s) only if ___; or '... exist(s) if ___'; or '... exists(s) if and only if ___'; or 'To exist is to ___' (and there are numerous other forms which ECC can take).
What a Criterion of Existence does not need to be/do
ECC need not be general, or tout court. A criterion of the existence of material objects (in particular), for example, would be 'Material objects exist only if they are spatially located.' A more general criterion of existence would be: 'Objects exist only if they have causal effects on other entities.'
ECC need not be good; it is natural to speak of a "bad existence criterion". Such a bad criterion of existence (in our view) would be, 'Entities exist if people assume their existence.'
ECC need not be 'substantial' (whatever this may be supposed precisely to mean--A. Thomasson argues against "substantial" ECC ...).
ECC may (or may not) perhaps be 'purely formal' (whatever the opposite would be supposed to be--A. Thomasson seems to aim at an argument against ECC by proposing a "purely formal" criterion). In any case, they need not. (Why should they have to?)
Alternatives?
Is it not plausible to say, for example, that whatever our (best) theories assume to exist exists?
No, surely not. First of all, to exist is not to be assumed by someone(s), or by the mental product of someone(s) (in terms of existence criteria: this would violate the largely accepted Mind-Independence criterion). Secondly, it might be argued that existence is too high a claim to be plausibly claimed. Yet this, though perhaps possible, is not clear from the start; it would have to be shown in advance. But has it? (xxxxxxxx)
Is it not plausible to argue that ontology is 'easy' in that no thorough thinking, no systematic study and no criteria are required for the decision of existence questions? (xxxxxxxx)