# Self-referentiality in Justification Logic

## TL;DR

This paper investigates the role of self-referential justifications in justification logic, showing that prehistoric cycles are necessary but not sufficient for self-referential theorems unless the definition is expanded.

## Contribution

It clarifies the conditions under which prehistoric cycles lead to self-referential theorems in justification logic, resolving an open problem.

## Key findings

- Prehistoric cycles are not sufficient under the standard definition.
- An expanded definition makes prehistoric cycles sufficient.
- Self-referentiality is crucial for realizing S4 in LP.

## Abstract

The Logic of Proofs, LP, and other justification logics can have self-referential justifications of the form t:A. Such self-referential justifications are necessary for the realization of S4 in LP. Yu discovered prehistoric cycles in a particular Gentzen system as a necessary condition for S4 theorems that can only be realized using self-referentiality. It was an open problem whether prehistoric cycles also are a sufficient condition. The main results of this paper are: First, with the standard definition of self-referential theorems, prehistoric cycles are not a sufficient condition. Second, with an expansion on that definition, prehistoric cycles become sufficient for self-referential theorems.