Generality of the Sequent Calculus Notion

Determine how general a notion of sequent the infinitary and cyclic sequent calculi for first-order logic with non-monotone inductive definitions support.

Background

The paper restricts sequents to contain a single inductive definition, although earlier work considered a more general form in which the antecedent and succedent may contain sets of formulas involving multiple definitions. The authors note that definitions can sometimes be merged in a model-preserving way, but do not establish the full scope of the calculi under this broader formulation.

References

It is currently unknown how general of a notion of -sequent our calculi support.

— An Infinitary and a Cyclic Sequent Calculus for Non-Monotone Inductive Definitions  (2609.26337 - Eede, 22 Sep 2026) in Section 3, footnote accompanying the definition of SCFO(ID)-sequents

Proving totality of definitions is currently impossible, however, a first obstacle being that we cannot express totality in the syntax of .

— An Infinitary and a Cyclic Sequent Calculus for Non-Monotone Inductive Definitions  (2609.26337 - Eede, 22 Sep 2026) in Section 8, Conclusion, third item in the list of avenues for future research

Since the trace condition does not involve justifications, the question remains whether there exist notions of infinitary and cyclic calculus that are explicitly based on justification theory.

— An Infinitary and a Cyclic Sequent Calculus for Non-Monotone Inductive Definitions  (2609.26337 - Eede, 22 Sep 2026) in Section 8, Conclusion, fourth item in the list of avenues for future research