Calculi for neighboring intermediate and substructural logics

Develop block- or sequent-based calculi for neighboring logics by varying the restriction on which signed formulae may cross a transition, including calculi for intermediate logics such as Gödel–Dummett and Jankov logics and an assessment of the boundary imposed by substructural restrictions on natural-deduction discharge.

Background

The paper identifies the purge in the signed block calculus as a restriction on which formulae may be transported across a transition. It observes that modifying this restriction should yield neighboring logical systems. In particular, retaining selected F-signed formulae under shape-dependent conditions is proposed as a route toward intermediate logics, while restricting discharge on the natural-deduction side is associated with substructural logics.

The authors explicitly identify two directions as open: weakening the transition restriction to obtain calculi for intermediate logics, with Gödel–Dummett and Jankov logics suggested as initial cases, and restricting natural-deduction discharge in the style of Ilić. They also state that the block calculus cannot directly follow the latter route because structural contraction has been absorbed into its format, marking a boundary for the present approach.

References

Two directions of variation are open, and by Section~\ref{sec:comp} both are variations on a single parameter, the restriction governing what may cross a transition. Weakening the restriction --- allowing some $F$-signed formulae to be retained, under conditions on their shape --- yields the intermediate logics; the calculus for the logic of G\"odel and Dummett and that for Jankov's logic are the natural first cases, and they can be read off the corresponding multiple-succedent calculi through Theorem~\ref{thm:corr}. Restricting the discharge on the natural deduction side, in the manner of Ili\'c's initial rules , yields the substructural neighbours; the block calculus cannot follow it there, for the reason given in Section~\ref{sec:comp}, and this marks the boundary of the present approach rather than a defect of it.

— Sequent-style tableaux for intuitionistic propositional logic  (2608.21143 - Cuconato, 21 Aug 2026) in Remark 2.14, Section 6 (Concluding remarks), subsection “Neighbouring logics”