How Much Propositional Logic Suffices for Rosser's Essential Undecidability Theorem?

TL;DR

This paper investigates the minimal propositional logic requirements for proving Rosser's essential undecidability, demonstrating that very weak substructural logics suffice for a version of Robinson's R theory.

Contribution

It introduces a proof of essential undecidability within a weaker substructural logic and for a less expressive arithmetic theory, expanding the scope beyond classical and fuzzy logics.

Findings

Undecidability proven in a weaker substructural logic

Arithmetic operations interpreted as mere relations

Results extend beyond Boolean, intuitionistic, or fuzzy logic

Abstract

In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much weaker arithmetic theory, a version of Robinson's R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic.