Conservative Extensions in Guarded and Two-Variable Fragments

We investigate the decidability and computational complexity of (deductive) conservative extensions in fragments of first-order logic (FO), with a focus on the two-variable fragment FO^2 and the guarded fragment GF. We prove that conservative extensions are undecidable in any FO fragment that contains FO^2 or GF (even the three-variable fragment thereof), and that they are decidable and 2-complete in the intersection GF^2 of FO^2 and GF.