Redundancy of the weak transitivity axiom in Zolin’s arithmetic decidability axiomatization

Determine whether the weak transitivity axiom in Zolin’s axiomatization of arithmetic decidability is redundant, equivalently, whether it can be derived from the remaining axioms.

Background

The paper discusses non-contingency logic under its provability-logical interpretation, where the non-contingency operator expresses arithmetic decidability: either a formula or its negation is provable in Peano Arithmetic. Zolin previously provided sound and complete axiomatizations and cut-based sequent systems for arithmetic decidability.

Zolin conjectured that the weak transitivity axiom in his axiomatization is redundant. The paper states that ongoing work derives this axiom from the remaining axioms, indicating that the issue is treated as an unresolved conjecture in the present discussion rather than as a result developed in the paper itself.

References

Zolin laid the foundation for this direction by providing sound and complete axiomatizations and cut-based sequent systems for arithmetic decidability. He conjectured that the weak transitivity axiom in his axiomatization is redundant.

— Proof Theory for Non-Contingency Logic  (2609.38975 - Wang et al., 30 Sep 2026) in Section 4, “Further Results and Future Work”