On Preservation Theorems for Two‐Variable Logic
Erich Grädel, Eric Rosen · Mathematical logic quarterly · 1999
Abstract We show that the existential preservation theorem fails for two‐variable first‐order logic FO 2 . It is known that for all k ≥ 3, FO k does not have an existential preservation theorem, so this settles the last open case, answering a question of Andreka, van Benthem, and Németi. In contrast, we prove that the homomorphism preservation theorem holds for FO 2 .