Term-generic logic

We introduce term-generic logic (TGL), a first-order logic parameterized with terms defined axiomatically (rather than constructively), by requiring terms to only provide free variable and substitution operators satisfying some reasonable axioms. TGL has a notion of model that generalizes both first-order models and Henkin models of the λ-calculus. The abstract notions of term syntax and model are shown to be sufficient for obtaining the completeness theorem of a Gentzen system generalizing that of first-order logic. Various systems featuring bindings and contextual reasoning, ranging from pure type systems to the π-calculus, are captured as theories inside TGL. For two particular, but rather typical instances—untyped λ-calculus and System F—the general-purpose TGL models are shown to be equivalent with standard ad hoc models.