you will find the statement that classical logic presupposes equivalence between denotation and existence. This comes from the influence of Russellian description theory and the analysis of negative existential statements.
This presuppostion is embraced in the axioms for first-order logic given by
Ax(P(x)) -> P(t)
P(t) -> Ex(P(x))
The link on free logic will explain how it differs from classical first-order logic in relation to names and existential import.