On the First-Order Prefix Hierarchy

Abstract

We investigate the expressive power of fragments of first-order logic that are defined in terms of prefixes. The main result establishes a strict hierarchy among these fragments over the signature consisting of a single binary relation. It implies that for each prefix p, there is a sentence $\phi_p$ in prenex normal form with prefix p, over a single binary relation, such that for all sentences θ in prenex normal form, if θ is equivalent to $\phi_p$, then p can be embedded in the prefix of θ. This strengthens a theorem of Walkoe.