Logic is metaphysics, only when one smuggles in one's intuitions as definitionally true; otherwise, it is not so: "If we look more closely how it comes about th

Logic is metaphysics, only when one smuggles in one's intuitions as definitionally true; otherwise, it is not so: "If we look more closely how it comes about that these existential statements are logical truths in these logical systems we see that it is only so because, by definition, a model for (standard) first order logic has to have a non-empty domain. It is possible to allow for models with an empty domain as well (where nothing exists), but models with an empty domain are excluded, again, by definition from the (standard) semantics in first order logic. Thus (standard) first order logic is sometimes called the logic of first order models with a non-empty domain. If we allow an empty domain as well we will need different axioms or rules of inference to have a sound proof system, but this can be done." (From Hofweber, Thomas, "Logic and Ontology" https://plato.stanford.edu/archives/sum2018/entries/logic-ontology/ https://plato.stanford.edu/archives/sum2018/entries/logic-on...)