 It is well known that for \(\mathbf{FO}\) there is a sound and complete proof procedure. Because a first-order sentence \(\phi\) is inconsistent (non-satisfiable) iff its negation \({\sim}\phi\) is valid (true in all models), trivially \(\mathbf{FO}\) also has a sound and complete disproof  procedure.[40]  The latter property extends to \(\mathbf{IFL}\) (while the former does not, see Subsect. 4.3):