In this note I present a Lindström theorem characterizing the hybrid logic \(\mathcal {H}(\exists )\) as the most expressive logic having compactness, the Tarski union property, and invariance under quasi-generated substructures. The logic \(\mathcal {H}(\exists )\) is rather interesting, as it mixes the expressive power brought by the availability of world variables with an “almost local” quantification, which gives it a counting ability. However, \(\mathcal {H}(\exists )\) did not receive the same attention as the other logics in the hybrid family, and only quite recently bisimulation-related invariance results were obtained for it. The Lindström theorem presented here helps clarify further its characteristics and situate better its place among extensions. The result is based on a characterization of first-order logic obtained by Lindström and employs a strategy different from the one used in recent Lindström theorems for modal and intuitionistic logics, which require a characterization of the logic at issue with respect to some invariance notion derived from bisimulation. For \(\mathcal {H}(\exists )\) the strategy used in this note arguably provides a better Lindström theorem, as the invariance notion used gives a clearer perspective on the distinguishing capacities of the competing logics.
