Two-variable logic has weak, but not strong, Beth definability

TL;DR

This paper proves that the two-variable fragment of first-order logic possesses the weak Beth definability property but not the strong one, highlighting its unique position in the hierarchy of logical definability properties.

Contribution

It establishes that the two-variable fragment has the weak Beth property while lacking the strong Beth property, clarifying its role in the landscape of logical definability.

Findings

Two-variable logic has the weak Beth definability property.

Two-variable logic does not have the strong Beth definability property.

The two-variable fragment separates the weak and strong Beth properties.

Abstract

We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.