# A computational glimpse at the Leibniz and Frege hierarchies

## TL;DR

This paper investigates the computational complexity of classifying logical systems within Leibniz and Frege hierarchies, demonstrating that such classification is generally undecidable for finite Hilbert calculi.

## Contribution

It proves the undecidability of classifying logics in the Leibniz and Frege hierarchies from a computational perspective, highlighting fundamental limits.

## Key findings

- Classification problem is undecidable for finite Hilbert calculi
- No algorithm exists to determine logic placement in hierarchies
- Results apply to abstract algebraic logic frameworks

## Abstract

In this paper we consider, from a computational point of view, the problem of classifying logics within the Leibniz and Frege hierarchies typical of abstract algebraic logic. The main result states that, for logics presented syntactically, this problem is in general undecidable. More precisely, we show that there is no algorithm that classifies the logic of a finite consistent Hilbert calculus in the Leibniz and in the Frege hierarchies.