Necessary and sufficient conditions for extending the F–G equivalence to broader logical systems

Determine necessary and sufficient conditions under which the categorical equivalence between the Henkin-construction functor F: Th → Mod (assigning to each theory T its Henkin term model F(T) = Term(T*)/∼) and the compactness/saturation-based functor G: Th → Mod (assigning to each theory T a model G(T) built via compactness, ultraproducts, or saturation), as realized by the canonical natural isomorphism η: F ⇒ G with its 2-categorical rigidity, extends from classical first-order logic to broader classes of logical systems.

Background

The paper defines two functors from the category of first-order theories to the category of models: F, obtained via the Henkin construction yielding a term model, and G, obtained via compactness/saturation methods yielding a semantic model. A canonical natural transformation η: F ⇒ G is constructed and shown to be an isomorphism componentwise, with additional 2-categorical rigidity.

In the Future Directions, the authors explicitly highlight open problems and conjectures. A concrete task they single out is determining necessary and sufficient conditions governing when this categorical equivalence between syntactic (Henkin) and semantic (compactness/saturation) constructions extends beyond classical first-order logic to broader logical systems.