Axiomatizations of free-logic are given by Theodore Hailperin (1957),[5] Jaakko Hintikka (1959),[6] Karel Lambert (1967),[7] and Richard L. Mendelsohn (1989).
[8] Karel Lambert wrote in 1967:[7] "In fact, one may regard free logic... literally as a theory about singular existence, in the sense that it lays down certain minimum conditions for that concept."
The question that concerned the rest of his paper was then a description of the theory, and to inquire whether it gives a necessary and sufficient condition for existence statements.
Lambert notes the irony in that Willard Van Orman Quine so vigorously defended a form of logic that only accommodates his famous dictum, "To be is to be the value of a variable," when the logic is supplemented with Russellian assumptions of description theory.
He criticizes this approach because it puts too much ideology into a logic, which is supposed to be philosophically neutral.
This amounts to the contribution that free logic makes to ontology.
An advantage of this is that formalizing theories of singular existence in free logic brings out their implications for easy analysis.
Lambert takes the example of the theory proposed by Wesley C. Salmon and George Nahknikian,[9] which is that to exist is to be self-identical.