Phil,
For physical things in the universe, existence means that somewhere at some point in time since the Big Bang, there was something. But this raises the question of what does it mean to make a true statement about somehting that no longer exists at the present moment.
Or what does it mean to say that if I do operation X on something Y that a predictable change on Y will occur? For example, I can predict: " If I drink this cup of coffee, the cup will be empty." Then somebody else says: "prove it." So I drink it, and turn the cup upside down and say: "See. Nothing spills out." .
Those are examples of statements about non-existent states. And they show that people can make such statements, and they are capable of causing them to become true.
Bit those are examples about physical things. Numbers are nonphysical things that people specify by axioms. Various people have invented physical methods for drawing symbols that have a one-to-one correspondence with those non-physical things. Since different people can consistently make exactly equivalent statements about those non-physical things, we use the same word 'exist' that we use for physical things that may or may not exist at the current moment in time and place.
That example can be generalized for all the kinds of non-physical entities for which we use phrases like "There exist", "Let there be,,, such that ,,,", and all the similar statements used in logic, mathematics, and every branch of mathematics, science, philosophy, linguistics, etc.
Since we can use language consistently to talk about all such things, it is meaningful for us to generalize words like "exist" to apply to such non-physical things that we call "things in our ontology".
Summary: This terminology is true by a convention that supports meaningful discussions, literature, philosophy and science. In short, it works.
Re ontology: That's another non-physical thing about which we can make consistent statements that other people can verify by doing similar reasoning in their own preferred language. In short, it works.
Does that mean those theoretical things really exist? That all depends on what language game you choose for defining the word 'really'. As Clark Gable said, "Frankly, my dear, I don't give a damn."
John,
Thanks for these comments. A natural question is whether an ontology is itself a thing that can exist. If existence includes ideas, thoughts, and propositions, then presumably an ontology can exist, as an idea that is a collection of propositions. And if we can have multiple different ontologies, e.g., with each ontology for a different domain of things that can exist, then we could have an ontology of these ontologies, i.e., a meta-ontology.
This seems to make sense, at least as something that may be designed, defined, imagined, proposed, suggested, etc. Of course, people are free to choose whether or not to spend their time creating a meta-ontology, or different meta-ontologies.
Phil