A Semantics for Nabla

We give a semantics for a classical variant of Dale Miller and Alwen Tiu’s logic FOλ∇. No such semantics seems to have existed for the nabla operator, except for one given by U. Schöpp. Our semantics validates the rule that nabla x implies exists x, but is otherwise faithful to the authors’ original intentions. The semantics is based on category of so-called nabla-sets, which we define as presheaves over the poset of natural numbers, with additional generic elements at each level. The semantics is sound, complete for Henkin structures, and complete for standard structures in the case