Finite-variable logics do not have weak Beth definability property

TL;DR

This paper proves that finite-variable logics with more than two variables lack the weak Beth definability property, providing a new, simplified proof for all n greater than 2 and settling the case for n=4.

Contribution

It establishes the weak Beth definability property does not hold for all n>2, including the previously unresolved case of n=4, with a unified proof approach.

Findings

Weak Beth definability fails for n>2

New simplified proof applicable to all n>2

Case n=4 resolved, n=2 remains open

Abstract

We prove that n-variable logics do not have the weak Beth definability property, for all n greater than 2. This was known for n=3 (Ildik\'o Sain and Andr\'as Simon), and for n greater than 4 (Ian Hodkinson). Neither of the previous proofs works for n=4. In this paper we settle the case of n=4, and we give a uniform, simpler proof for all n greater than 2. The case for n=2 is still open.