 In this entry IF first-order logic and extended IF first-order logic have been surveyed. Their metalogical properties have been explained and the philosophical relevance of these properties has been discussed. The suggested consequences of these logics for philosophical issues such as the existence of a self-applied truth-predicate, the logicist program, the philosophical relevance of axiomatic set theory, and informational independence in natural languages have been covered as well. Slash logic and dependence logic – both closely related to \(\mathbf{IFL}\) and inspired by it – were also briefly considered.